Mean field theories and dual variation [2 ed.] 9789462391536, 9789462391543

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Mean field theories and dual variation [2 ed.]
 9789462391536, 9789462391543

Table of contents :
Preface to the Second Edition......Page 6
Preface to the First Edition......Page 8
Contents......Page 12
1.1 Basic Notions......Page 15
1.2 Collapse Formation......Page 19
1.3 Mean Field Hierarchy......Page 24
1.4 Stationary State......Page 27
1.5 Localization and Symmetrization......Page 29
1.6 Weak Solutions......Page 32
1.7 Rescaling......Page 38
1.8 Collapse Mass Quantization......Page 41
1.9 Blowup in Infinite Time......Page 47
1.10 Simple Blowup Points......Page 55
1.11 Summary......Page 58
2.1 Full System of Chemotaxis......Page 60
2.2 Non-local Parabolic Equation......Page 61
2.3 Smoluchowski-ODE System......Page 64
2.4 Harmonic Heat Flow......Page 73
2.5 Normalized Ricci Flow......Page 78
2.6 Concentration of Probability Measures......Page 88
2.7 Summary......Page 91
3.1 Full System of Chemotaxis Revisited......Page 93
3.2 Lagrangian and Duality......Page 99
3.3 Gradient Systems with Duality......Page 110
3.4 Zygmund Spaces......Page 116
3.5 Summary......Page 122
4.1 Non-convex Evolution......Page 123
4.2 Gradient and Skew-Gradient Systems......Page 132
4.3 Semi-unfolding-Minimality......Page 138
4.4 Kuhn-Tucker Duality......Page 144
4.5 Summary......Page 151
5.1 Thermal Equilibrium......Page 152
5.2 Stefan Problem......Page 158
5.3 Phase Field Model......Page 161
5.4 Summary......Page 168
6.1 Non-equilibrium Thermodynamics......Page 169
6.2 Penrose-Fife Theory......Page 176
6.3 Penrose-Fife Equation......Page 180
6.4 Coupled Cahn-Hilliard Equation......Page 184
6.5 Shape Memory Alloys......Page 188
6.6 Summary......Page 211
7.1 Ideal Fluids......Page 213
7.2 Gas Dynamics......Page 220
7.3 Self-gravitating Fluids......Page 225
7.4 Plasma Confinements......Page 242
7.5 Related Models......Page 249
7.6 Summary......Page 255
8.1 Interface Vanishing......Page 256
8.2 Plasma Equilibrium......Page 268
8.3 MHD Fluids......Page 273
8.4 Summary......Page 277
9.1 Point Vortex......Page 278
9.2 Boltzmann Relation......Page 283
9.3 Ensemble......Page 285
9.4 Turbulence......Page 289
9.5 Summary......Page 304
10.1 Kramers-Moyal Expansion......Page 306
10.2 Kinetic Model......Page 313
10.3 Maximum Entropy Production......Page 320
10.4 Summary......Page 326
11.1 Semilinear Equations......Page 327
11.2 Degenerate Equations......Page 332
11.3 Blowup Threshold......Page 335
11.4 Structure of the Blowup Set......Page 338
11.5 Other Properties......Page 345
11.6 Summary......Page 354
12.1 Field Theory......Page 356
12.2 Exponential Nonlinearity Revisited......Page 365
12.3 Scaling Invariance......Page 370
12.4 Liouville-Bandle Theory......Page 372
12.5 Alexandroff-Bol's Inequality......Page 380
12.6 Pre-scaled Analysis......Page 393
12.7 Entire Solution......Page 400
12.8 Blowup Analysis......Page 405
12.9 Summary......Page 416
13.1 Method of Duality......Page 418
13.2 Higher-Dimensional Quantization......Page 423
13.3 Dimension Control of the Blowup Set......Page 426
13.4 Summary......Page 429
Bibliography......Page 430
Index......Page 447

Citation preview

Atlantis Studies in Mathematics for Engineering and Science Series Editor: Charles K. Chui

Takashi Suzuki

Mean Field Theories and Dual Variation Mathematical Structures of the Mesoscopic Model Second Edition

Atlantis Studies in Mathematics for Engineering and Science Volume 11

Series editor Prof. Charles K. Chui, Department of Statistics, Stanford University, Stanford, CA, USA

Aims and scope of the series The series ‘Atlantis Studies in Mathematics for Engineering and Science’ (AMES) publishes high quality monographs in applied mathematics, computational mathematics, and statistics that have the potential to make a significant impact on the advancement of engineering and science on the one hand, and economics and commerce on the other. We welcome submission of book proposals and manuscripts from mathematical scientists worldwide who share our vision of mathematics as the engine of progress in the disciplines mentioned above. For more information on this series and our other book series, please visit our website at: www.atlantis-press.com/publications/books. Atlantis Press 8, square des Bouleaux 75019 Paris, France

More information about this series at http://www.atlantis-press.com

Takashi Suzuki

Mean Field Theories and Dual Variation Mathematical Structures of the Mesoscopic Model Second Edition

Takashi Suzuki Graduate School of Engineering Science Osaka University Toyonaka, Osaka Japan

ISSN 1875-7642 ISSN 2467-9631 (electronic) Atlantis Studies in Mathematics for Engineering and Science ISBN 978-94-6239-153-6 ISBN 978-94-6239-154-3 (eBook) DOI 10.2991/978-94-6239-154-3 Library of Congress Control Number: 2015954602 © Atlantis Press and the author(s) 2008, 2015 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. Printed on acid-free paper

Preface to the Second Edition

Seven years have past since the first edition of this book is published. This new edition is composed of 13 chapters. Actually, we have broken two chapters of the first edition into several ones, providing with additional sections. We also added new chapters, following fundamental concepts of the first edition. Thus we deal with several models in physics, chemistry and biology, taking regarding two mathematical structures, that is the duality and scaling. In this new edition, we added several topics in physics, particularly the theory of perfect fluids and that of static magnetic fields. Some parts, particularly, the chapters of chemotaxis and time realization are also modified according to the study after the first edition. Below we record the descriptions at the beginning of the chapters of the first edition.

Duality-Sealed Variation Self-organization is a phenomenon widely observed in physics, chemistry, and biology. Usually, it is mentioned in the context of far-from-equilibrium of dissipative open systems describing the discharge of entropy, but the other aspect, formation of self-assembly, is sealed in the total set of stationary solutions of thermodynamically closed systems. This stationary problem is contained in a hierarchy of the mean field of many self-interacting particles, whereby the transmission of free energy is a leading factor. We can observe a unified structure of calculus of variation in these mathematical models, and obtain a guideline of mathematical analysis from it, especially in the total structure of stationary solutions and its roles in dynamics. Then we will see the features of self-organization achieved near from equilibrium. Equations describing self-assembly have been classified into models (A), (B), and (C). Although model (C) equations indicate models other than those of model (A) or (B) equations, see [113], some of them are formulated as the (skew-) gradient system with semi-duality. In this part, first, we take a simplified system of

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chemotaxis to confirm the above story, and then develop the abstract theory of Toland duality. Next, we examine the closed characters of model (A) and model (B) equations, and, finally, confirm the profiles of dual variation sealed in several model (C) equations.

Scaling–Revealing Hierarchy Details of the self-organization are hierarchical achievements between self-assembly and dissipative structures [425]. The latter is top-down and occurs in a far-from-equilibrium formulation of an open system that involves dissipation of entropy and is characterized by periodic structures, spiral waves, traveling waves, self-similar evolutions, and so forth [269]. The former is, as described in the previous part, bottom-up and controlled by the stationary state of the closed system which is involved by condensates, collapses, spikes, quantizations, free energy transmissions, variational structures, and so forth. The control of the total set of stationary states to the global dynamics, however, is not restricted to thermodynamics. This profile is observed widely in mathematical models involved by the mean field hierarchy and sometimes referred to as the nonlinear spectral mechanics [364]. In more precise terms, there is a unified mathematical principle in each mean field hierarchy provided with the underlying physical principle, such as the conservation laws, decrease of the free energy, and so forth. This part describes the quantized blow-up mechanism which is one of the leading principle of self-assembly. It arises in self-interacting fluid, turbulence in the context of the propagation of chaos, mean field hierarchy derived from the friction–fluctuation self-interaction in the molecular kinetics, and gauge field concerning condensate of microscopic states. Actually, this profile of quantization is revealed by a blow-up analysis which is one of the important products of the method of scaling and is valid even to the higher space dimension. Osaka, Japan June 2015

Takashi Suzuki

Preface to the First Edition

In general, systems of nonlinear partial differential equations are formulated and studied individually according to specific physical or biological models of interest. There are, however, certain classes of Lagrangian systems for which the asymptotic behavior of the nonstationary solutions, including the blow-up mechanism, is controlled by their total sets of stationary solutions. In this book, we will show that these equations have their origin in mean field theory and that the associated elliptic problems are provided with mass and energy quantizations because of their scaling properties. Mean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews, with some success, though still in progress, in many areas of science, such as the study of turbulence, gauge field, plasma physics, self-interacting fluids, kinetic theory, distribution function method in quantum chemistry, tumor growth modeling, phenomenology of critical phenomena. In this last scientific area, phase transition, phase separation, and shape memory alloys are included. However, in spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature. The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics. One of the key concepts to be used repeatedly in our discussion in this book is the notion of duality, which has been the origin of the extension of functions to distributions [322] that provides a useful tool for the formulation of weak solutions in the theory of partial differential equations. In addition, Riesz’ representation theorem [117] provides the formulation of duals for such important function spaces

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as CðKÞ0 ffi MðKÞ and L1 ðΩÞ0 ffi L1 ðΩÞ, where K and Ω are compact and open sets in Rn , respectively. Thus if a family of continuous approximate solutions is not compact but is provided with a priori estimates, say, in the L1 or L1 norm, then the weak solution is realized as an L1 function or a measure, respectively, whereby several fundamental profiles caused by the nonlinearity can be observed (e.g., interface, shock, localization, concentration, condensate, blowup, self-similar evolution, traveling and spiral waves, …). Another concept of duality, namely, that of Hardy–BMO [344] is reflexive. It is represented by L log LðΩÞ0 ffi ExpðΩÞ and ExpðΩÞ0 ffi L log LðΩÞ, where Ω  Rn is a bounded domain. The Hardy–BMO duality is associated with the mean field hierarchy through the entropy functional and the Gibbs measure. Thus this duality notion can be applied to the study of phenomena between particles and fields in the microscopic level as governed by the Boltzmann principle and the zeta function, respectively. Yet another duality concept that will be used in this book is operator theoretical. For example, for some given rectangular matrix A a column vector b, the duality of the linear model Ax ¼ b is the transposed operation that arises in the context of the Fredholm formulation concerning the existence and uniqueness of the solution. On the other hand, the linear inequality Ax  b in convex analysis has an analogous duality formulation. For example, this formulation results from integrating a linear functional equation where the Legendre transformation takes on the role of the transposed operator [35]. This involutive operation is often referred to as the Fenchel–Moreau duality, which results in the more complicated dualities of Kuhn–Tucker and Toland. Here, the Toland duality is concerned with the functional represented by the difference of two convex functionals, such as the Helmholtz free energy. The field functional then is obtained in its dual form, and these two principal functionals are regarded as unfoldings of the Lagrangian. This book consists of two main chapters. The first chapter is devoted to the study of variational structures, where the quantized blow-up mechanism observed in the chemotaxis system [364] is first described. Within this section, the free energy transmission, which results in the formation of self-assembly, is emphasized which means that the source of the emergence is the wedge of blow-up envelope, while entropy and mass are exchanged to create a clean self with the quantized mass. The leading principle derived from this study can be summarized as follows: dual variation sealed in the mean field hierarchy and scaling that reveals this hierarchy. In fact, the phenomenon of quantization is sealed in the total set of stationary solutions, referred to as the nonlinear spectral mechanics. The structure of dual variation controls, not only the local dynamics around the stationary state, but also the formation of self-assembly realized as a global dynamics including the blowup of the solution. This duality is associated with the convexity of the functional; and thus, our description involves convex analysis as mentioned above. We will examine the phenomena and logistics of both variational and dynamical structures of several problems including phase transition, phase separation, hysteresis in shape memory alloys which review the theory of nonequilibrium thermodynamics.

Preface to the First Edition

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Chapter 2 is devoted to the discussion of the method of scaling for revealing the mathematical principles hidden in the mean field hierarchy subject to the second law of thermodynamics. More precisely, we will describe the quantized blow-up mechanism observed in self-gravitating fluid, turbulence, and self-dual gauge field. In particular, blow-up analysis, one of the most important applications of the method of scaling, is used to clarify this mechanism. This analysis consists of four ingredients, namely, scaling invariance of the problem, classification of the solution of the limit problem defined on the whole space and time, control of the rescaled solution at infinity, and hierarchical arguments. Meanwhile we will also describe the method of statistical mechanics for the study of macroscopic phenomena caused by the microscopic principles. In particular, we will reformulate the simplified system of chemotaxis as a fundamental equation of the material transport, the Smoluchowski–Poisson equation. Unfortunately, a complete overview of the topics mentioned above is beyond the scope of this book, and the list of references is far from being sufficient. Furthermore, several important related topics are missing, among which are the complex Ginzburg–Landau free energy associated with superconductivity in low temperature [24, 269], Arnold’s variational principle for describing the MHD equilibrium [12], the density function theory of quantum chemistry [48], kinetic equations in fluid dynamics [32], and the duality between Aleksandrov’s problem in integral geometry and the optimal mass transport theory [281]. There are also several stationary problems in engineering associated with duality [131] which may be extended to the nonstationary problems of their own. We take this opportunity to clarify several terminologies used in this book that are the concern of thermal phenomena associated with dissipation. First, we follow the classical concepts of equilibrium thermodynamics and classify thermal systems into isolated, closed, and open systems. This classification is also associated with the theory of equilibrium statistical mechanics. More precisely, these systems are provided with the microscopic structures of microcanonical, canonical, and grand canonical ensembles, respectively. A closed system here indicates the lack of transport of materials between the outer systems, whereby the transport of heat and that of the energy are permitted. In the open system, on the other hand, the transport of material between the outer systems is also permitted. Next, in view of the theory of nonequilibrium thermodynamics, we add one more aspect to the openness, that is the dissipation of entropy to the outer system [266]. Under this agreement, closedness is reformulated as a system provided with decreasing total free energy or that provided with increasing total entropy. Thus we have two kinds of closedness: physical closedness in terms of kinetic and/or material, and thermodynamical closedness described by two typical models discussed in detail in this book. Free energy transmission occurs in the latter case and could be the origin of “self-assembly”. In this connection, we note that the dissipative system is defined by the presence of an attractor in the theory of dynamical systems. Gradient system with compact semi-orbits is a typical dissipative system in this case [151]. We believe, however, that this definition of dissipative system does not describe what were observed by

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[266]. More precisely, decrease of the free energy or increase of the entropy means thermodynamical closedness. It is formulated by the gradient system in the same two models as mentioned above, and, therefore, is never provided with the dissipation in the sense of nonequilibrium thermodynamics [266] in spite of the fact that these two models are typical dissipative systems in the theory of dynamical systems. To avoid this confusion, the above mentioned terminology of dissipative system in the theory of dynamical systems [151] is not used in this book. Actually, a recent paradigm asserts two aspects of self-organization, far-from-equilibrium (top-down self-organization) and self-assembly (bottom-up self-organization), emphasizing the role of their hierarchical developments, with the Hopf bifurcation casting the threshold [425]. Dissipation occurs in a state far-from-equilibrium, which results in a spiral wave, a traveling wave, a periodic structure, a self-similar development, and so forth; while self-assembly, we believe, is formed near-from-equilibrium of thermodynamically closed system induced by the stationary states, developing a condensate, a collapse, a blowup, a spike, and so forth. This book is concerned with the formation of self-assembly. Its profile is provided with the “triple seal,” of which basic concepts have already been described. First, several features of self-assembly, that is, one aspect of the self-organization are sealed in thermodynamically closed systems. Secondly, the dynamics of the closed system is sealed in the total set of stationary states. Finally, the stationary states themselves are sealed in the (skew-) Lagrangian, provided with the structure of dual variation. These features of the mean field hierarchy are certainly revealed by the method of scaling derived from the microscopic principle. In summary, formation of self-assembly arises in thermodynamically closed systems, and hence this process is subject to the calculus of variation. The closed system follows the microscopic principle. This principle controls the mean field hierarchy totally, and, therefore, fundamental profiles of the macroscopic mean field equation are revealed by the method of scaling. We would like to thank Prof. Tomohiko Yamaguchi for several stimulative discussions on nonequilibrium thermodynamics. Thanks are also due to Profs. Piotr Biler, Fumio Kikuchi, Futoshi Takahashi, and Gershon Wolansky for careful reading over the primary version of the manuscript. Osaka, Japan June 2008

Takashi Suzuki

Contents

1

Chemotaxis . . . . . . . . . . . . . . . . . . . 1.1 Basic Notions . . . . . . . . . . . . . 1.2 Collapse Formation . . . . . . . . . 1.3 Mean Field Hierarchy . . . . . . . . 1.4 Stationary State . . . . . . . . . . . . 1.5 Localization and Symmetrization 1.6 Weak Solutions . . . . . . . . . . . . 1.7 Rescaling . . . . . . . . . . . . . . . . 1.8 Collapse Mass Quantization. . . . 1.9 Blowup in Infinite Time . . . . . . 1.10 Simple Blowup Points . . . . . . . 1.11 Summary . . . . . . . . . . . . . . . .

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1 1 5 10 13 15 18 24 27 33 41 44

2

Time 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Relaxization . . . . . . . . . . . . . . . . . . Full System of Chemotaxis . . . . . . . . Non-local Parabolic Equation. . . . . . . Smoluchowski-ODE System . . . . . . . Harmonic Heat Flow . . . . . . . . . . . . Normalized Ricci Flow . . . . . . . . . . . Concentration of Probability Measures Summary . . . . . . . . . . . . . . . . . . . .

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Toland Duality. . . . . . . . . . . . . . . . . . . . 3.1 Full System of Chemotaxis Revisited 3.2 Lagrangian and Duality. . . . . . . . . . 3.3 Gradient Systems with Duality. . . . . 3.4 Zygmund Spaces . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . .

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Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.1 Non-convex Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2 Gradient and Skew-Gradient Systems . . . . . . . . . . . . . . . . . . . 120

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4.3 4.4 4.5

Semi-unfolding-Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Kuhn-Tucker Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

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Phase Transition . . . . . . . 5.1 Thermal Equilibrium. 5.2 Stefan Problem . . . . 5.3 Phase Field Model . . 5.4 Summary . . . . . . . .

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141 141 147 150 157

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Critical Phenomena of Isolated Systems 6.1 Non-equilibrium Thermodynamics . 6.2 Penrose-Fife Theory . . . . . . . . . . . 6.3 Penrose-Fife Equation . . . . . . . . . . 6.4 Coupled Cahn-Hilliard Equation. . . 6.5 Shape Memory Alloys . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . .

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159 159 166 170 174 178 201

7

Self-interacting Fluids. . . . . 7.1 Ideal Fluids . . . . . . . . 7.2 Gas Dynamics . . . . . . 7.3 Self-gravitating Fluids. 7.4 Plasma Confinements . 7.5 Related Models . . . . . 7.6 Summary . . . . . . . . .

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8

Magnetic Fields . . . . . . . 8.1 Interface Vanishing 8.2 Plasma Equilibrium 8.3 MHD Fluids . . . . . 8.4 Summary . . . . . . .

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9

Boltzmann-Poisson Equation. 9.1 Point Vortex . . . . . . . . 9.2 Boltzmann Relation . . . 9.3 Ensemble . . . . . . . . . . 9.4 Turbulence . . . . . . . . . 9.5 Summary . . . . . . . . . .

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269 269 274 276 280 295

10 Particle Kinetics. . . . . . . . . . . . . . . 10.1 Kramers-Moyal Expansion . . . 10.2 Kinetic Model . . . . . . . . . . . . 10.3 Maximum Entropy Production . 10.4 Summary . . . . . . . . . . . . . . .

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11 Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 11.1 Semilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 11.2 Degenerate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

Contents

11.3 11.4 11.5 11.6

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Blowup Threshold . . . . . . . Structure of the Blowup Set. Other Properties . . . . . . . . . Summary . . . . . . . . . . . . .

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327 330 337 346

12 Gauge Fields . . . . . . . . . . . . . . . . . . . . 12.1 Field Theory . . . . . . . . . . . . . . . . 12.2 Exponential Nonlinearity Revisited . 12.3 Scaling Invariance . . . . . . . . . . . . 12.4 Liouville-Bandle Theory . . . . . . . . 12.5 Alexandroff-Bol’s Inequality . . . . . 12.6 Pre-scaled Analysis . . . . . . . . . . . 12.7 Entire Solution. . . . . . . . . . . . . . . 12.8 Blowup Analysis . . . . . . . . . . . . . 12.9 Summary . . . . . . . . . . . . . . . . . .

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349 349 358 363 365 373 386 393 398 409

13 Higher-Dimensional Blowup . . . . . . . . . . . 13.1 Method of Duality . . . . . . . . . . . . . . 13.2 Higher-Dimensional Quantization . . . . 13.3 Dimension Control of the Blowup Set 13.4 Summary . . . . . . . . . . . . . . . . . . . .

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411 411 416 419 422

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

Chapter 1

Chemotaxis

Chemotaxis is a feature that living things are attracted by special chemical source, which results in, for example, the formation of spores in the case of cellular slime molds. In more details, under sufficient foods and water, the spores of cellular slime molds break and small amoebas are born. First, the dissipative structure makes them quite active. Then they are combined as spores again after the first twenty-four hours. At this occasion, the chemotactic feature to the secreted chemical substances takes a role. This process is made up of a mathematical model, and then formation of the delta-function singularity was conjectured. The effect of chemotaxis is formulated also in the material transport theory subject to the total mass conservation and the decrease of the free energy. The quantized blowup mechanism to such a system is actually proven in this context, using the weak formulation made by the duality and the scaling invariance of the problem.

1.1 Basic Notions If  ⊂ Rn , n = 2, 3, is a bounded domain with smooth boundary ∂ and v = v(x) is a C 1 vector field defined on , then the divergence formula of Gauss holds as   ∇ · v dx = ν · v dS, (1.1) 

∂

where · is the inner-product in Rn , ⎞ ⎛

⎛ ⎞ ∂ ν1 ⎜ ∂x1 ⎟ ∇ = ⎝ . . . ⎠ , ν = ⎝ . . . ⎠ , dS ∂ νn ∂x n

are the gradient operator, the outer normal vector, and the area (or line) element, respectively. © Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_1

1

2

1 Chemotaxis

Thus ⎛ 1⎞ v ∂v n ∂v 1 + ··· + , v = ⎝ . . . ⎠ , x = (x1 , . . . , xn ) ∈ Rn ∇ ·v = ∂x1 ∂xn vn is the divergence of the vector field v = v(x). We have ν=

x2 (s) , dS = ds −x1 (s)

if n = 2 and ∂ is parametrized as x = (x1 (s), x2 (s)) by the arc-length s satisfying x1 (s)2 + x2 (s)2 = 1, while ν=

xu × xv , dS = |xu × xv | dudv = EG − F 2 dudv |xu × xv |

if n = 3 and ∂ is parametrized as x = x(u, v) ∈ R3 by (u, v) ∈ R2 , where xu =

∂x ∂x , xv = , E = |xu |2 , F = xu · xv , G = |xv |2 ∂u ∂v

and × denotes the outer product in R3 . From the theory of ordinary differential equations, we can define a local flow {Tt } by x(t) = Tt x0 , where x = x(t) is a unique solution to dx = v(x), dt

x|t=0 = x0 ∈ Rn .

(1.2)

Then, if ω ⊂ Rn is a bounded domain, the set Tt (ω) = {Tt x | x ∈ ω} indicates the region occupied by the particles at t = t that have moved from those in ω at t = 0, subject to the velocity field v. Its volume (or area) is equal to  |Tt (ω)| =

Tt (ω)

 dξ =

ω

|Jt (x)| dx

using the transformation of variables ξ = Tt x, where Jt (x) = det denotes the Jacobian.

∂ξi ∂xj

1≤i,j≤n

1.1 Basic Notions

3

We can derive [375],

∂ξi

= δij , ∂xj t=0

d ∂ξi

∂v i = dt ∂xj t=0 ∂xj

or, equivalently, ∂ξi ∂v i = δij + t + o(t) ∂xj ∂xj from (1.2), and, therefore, Jt (x) = 1 + t

∂v 1 ∂v n + ··· + ∂x1 ∂xn

+ o(t).

This relation implies  |Tt (ω)| = |ω| + t or, equivalently,

ω

∇ · v dx + o(t),



d |Tt (ω)|

= ∇ · v dx. dt ω t=0

Thus the left-hand side of (1.1) indicates the total amount of the particles flowing into  per unit time from inside source, which is equal to the total integral of the normal component of the velocity v on the boundary indicated by the right-hand side. For the proof of (1.1), we approximate  by h using small rectangles, for example, and reduce it to   ∇ · v dx = ν · v dS. (1.3) h

∂h



 To prove (1.3), next, we take the rectangles ih composing h , and show 

 ih

∇ · v dx =

∂ih

ν · v dS

(1.4)

 for each i. Then, i of the left-hand side is equal to that of (1.3). The same is true for the right-hand side because the inner boundary integrals cancel. Thus (1.4) implies (1.3). Equality (1.4), on the other hand, is proven by changing the triple (double) integral of the left-hand side to the successive integral, and then using the fundamental theorem of differentiation and integration.

4

1 Chemotaxis

Divergence formula of Gauss (1.1) is equivalent to  

∂f dx = ∂xi

 ∂

νi f dS,

1 ≤ i ≤ n,

where f = f (x) is a C 1 scalar field. This formula implies the integration by parts,  

w

∂g dx = ∂xi





∂

wνi g dS −



∂w g dx, ∂xi

1 ≤ i ≤ n,

(1.5)

where w = w(x) and g = g(x) are C 1 scalar fields. Direction derivative of f toward ν, on the other hand, is defined by

d ∂f = f (· + sν)

. ∂ν ds s=0 Using ⎛

∂f ∂x1



⎜ ⎟ ∇f = ⎝ . . . ⎠ , ∂f ∂xn

we obtain

∂f = ν · ∇f by ∂ν

∂f d ∂f ∂f = f (x1 + sν1 , . . . , xn + sνn )

= ν1 + · · · νn ∂ν ds ∂x1 ∂xn s=0 which means that f (x + sν) = f (x) + sν · ∇f (x) + o(s) as s → 0 and, therefore, ∇f (x) has the direction where f (x) obtains the infinitesimally maximum increase, and the length of this vector is equal to the degree of the variation of f (x) in this direction [375]. Applying (1.1) for v = ∇f , we obtain 

 

f dx =

∂

∂f dS, ∂ν

using Laplacian, =∇ ·∇ =

∂2 ∂2 + · · · + , ∂xn2 ∂x12

1.1 Basic Notions

5

∂f provided that f is a C 2 scalar field. Similarly, putting w = ∂x and taking i we obtain    ∂f dS − ∇g · ∇f dx = g gf dx  ∂ ∂ν 

 i

in (1.5),

if g and f are C 1 and C 2 scalar fields, respectively. This equality implies Green’s formula   ∂g ∂f −f dS gf − f g dx = g ∂ν  ∂ ∂ν valid to C 2 scalar fields indicated by f = f (x) and g = g(x). Heat Equation If  is occupied with a heat conductor and ω is a sub-domain with C 1 boundary ∂ω, then  cρθt dx ω

denotes the infinitesimal change of the heat energy inside ω, where c, ρ, and θ denote the specific heat, the density, and the temperature, respectively. Then, from Newton-Fourier-Fick’s law, the heat energy lost per unit time is proportional to the temperature gradient, and thus we obtain   ∂θ dS (1.6) cρθt dx = κ ω ∂ω ∂ν with the conductivity κ. The right-hand side of (1.6) is equal to   ν · κ∇θ dS = ∇ · κ∇θ dx ∂ω

ω

by (1.1), and hence it holds that cρθt = ∇ · κ∇θ

(1.7)

called the heat equation because ω is arbitrary.

1.2 Collapse Formation Simplified system of chemotaxis [183, 249] has backgrounds in statistical mechanics, thermodynamics, and biology [364]. For this system, we have a striking profile of the solution, called the quantized blowup mechanism [364, 365].

6

1 Chemotaxis

In the context of biology, this model is associated with the chemotactic feature of cellular slime molds [199], typically formulated by 1 ut = ∇ · (∇u − u∇v) , −v = u − ||

 ∂u ∂v ∂v

−u , = 0, v = 0, ∂ν ∂ν ∂ν ∂ 

 

u in  × (0, T ) (1.8)

where  ⊂ Rn , n = 2, 3, is a bounded domain with smooth boundary ∂ and ν is the outer unit normal vector. Here, u = u(x, t) stands for the density of cellular slime molds, and the first equation of (1.8) is written as ut = −∇ · j with j = −∇u + u∇v

(1.9)

which means that d dt



 ω

u=−

∂ω

ν·j

for any sub-domain ω with C 1 boundary ∂ω by (1.1). Thus, the vector field j stands for the flux of u, and the first equation of (1.8) describes mass conservation. The null flux condition, ν · j = 0, therefore, is imposed on the boundary in (1.8), which guarantees the total mass conservation, u(·, t)1 = u0 1

(1.10)

for u0 = u(·, 0) ≥ 0 because u = u(x, t) ≥ 0 follows from the maximum principle [305]. Here and henceforth,  · p denotes the standard L p norm: f p =

 

|f|

 p 1/p

, 1≤p 0 is a constant. Under these reduction we reach a parabolic-parabolic system. Finally, the second form of (1.8) is derived [183], associated with an asymptotic expansion on v. The system (1.8) is called the Smoluchowski Poisson equation in non-equilibrium statistical mechanics. Later we shall see its derivation and the general form. System (1.8) is well-posed local-in-time [26, 364, 375, 424]. If the initial value u0 ≥ 0 of u is sufficiently regular, then there is a unique (classical) non-negative solution local-in-time. Therefore, if Tmax denotes the supremum of its existence time we obtain Tmax > 0. In case T = Tmax < +∞, this T is called the blowup time, and actually, there arises lim u(·, t)∞ = +∞. t↑T

For this system (1.8), we have the following theorems. Theorem 1.2.1 If n = 2 and T = Tmax < +∞, then it holds that u(x, t)dx 



m(x0 )δx0 (dx) + f (x)dx

(1.24)

x0 ∈S

as t ↑ T , where  S = x0 ∈  | ∃(xk , tk ) → (x0 , T ) such that u(xk , tk ) → +∞}

(1.25)

denotes the blowup set of u and 0 ≤ f = f (x) ∈ L 1 () ∩ C( \ S). Theorem 1.2.2 We obtain m(x0 ) ∈ m∗ (x0 )N in (1.24), where  m∗ (x0 ) ≡

8π, x0 ∈  4π, x0 ∈ ∂

(1.26)

and hence 2 · (S ∩ ) + (S ∩ ∂) ≤ u0 1 /(4π) by (1.10).

(1.27)

10

1 Chemotaxis

The convergence (1.24) is ∗-weakly in the set of measures on , denoted by M(), which means    lim u(·, t)ϕ = m(x0 )ϕ(x0 ) + fϕ t↑T



x0 ∈S



for any ϕ ∈ C(). This phenomenon of the appearance of the delta-function singularity in u(x, t)dx is called the formation of collapse, and the relation m(x0 ) ∈ m∗ (x0 )N is referred to as the mass quantization. Actually, we have the strict inequality in (1.27), which guarantees global-in-time existence of the solution for the critical mass λ = 4π. There arises blowup in infinite time to this solution, in case there is no stationary solution. The condition m(x0 ) = m∗ (x0 ) with (1.26) is called the simplicity of the blowup point. In 1973, Nanjundiah [258] conjectured the formation of collapse, and ChildressPercus [69] refined this conjecture in 1981 by heuristic arguments. More precisely, formation of collapse is supposed to hold only in the case of n = 2, and 8π is expected to be the threshold of λ = u0 1 for the existence of the solution global in time. The latter means that λ < 8π implies Tmax = +∞, while any λ > 8π admits u0 ≥ 0 such that u0 1 = λ and Tmax < +∞. In 1995, Nagai [249] showed that this threshold is affirmative for radially symmetric solutions, but later, 4π is proven to be the actual threshold in non-radially symmetric case [26, 126, 250, 251, 328]. Theorem 1.2.1 is regarded as a localization of this threshold, particularly, the estimate from below, combined with the formation of delta-function singularity [329]. Theorem 1.2.2 arises with a detailed blowup mechanism. Namely, formation of a collapse is a process of collision of sub-collapses provided with the normalized mass 8π (and 4π if they are on the boundary). Hence m(x0 ) ∈ m∗ (x0 )N indicates the vanishing of the residual part other than these sub-collapses. Also, the case m(x0 ) = mm∗ (x0 ) with m ≥ 2, m ∈ N, does not arise if the solution is radially symmetric. Collision of sub-collapse was first unaware [364], but was noticed later by a formal calculation [323]. For the rigorous proof of residual vanishing, we require several technical lemmas described below [373, 374]. Finally, the simplicity of the collapse in realized when the free energy is bounded. Concerning the other space dimensions, it is known that there is no blowup for n = 1 and no L 1 threshold blowup for n = 3, see [163, 164, 249, 325]. In the case of n = 2, however, the formation of collapse is valid even to perturbed systems [208].

1.3 Mean Field Hierarchy System (1.8) arises also in statistical mechanics, that is the Smoluchowski Poisson equation, whereby u = u(x, t) indicates the distribution function of self-gravitating particles, and v = v(x, t) is the gravitational field created by them. In more details, the second equation of (1.8) is equivalent to

1.3 Mean Field Hierarchy

11

 v(x, t) =



G(x, x  )u(x  , t)dx  ,

where G = G(x, x  ) denotes the Green’s function associated with the elliptic boundary value problem called the Poisson equation 1 − v = u − ||

 ∂v

= 0, v = 0. ∂ν ∂ 

 

u,

(1.28)

This Green’s function behaves like the gravitational potential, G(x, x  ) ≈ (x − x  ) for  (x) =

1 4π 1 2π

1 · |x| , n=3 1 log |x| , n = 2.

This equation thus describes the formation of gravitational field by the particle density. System (1.8) is contained in a hierarchy of the mean fields of many self-gravitating particles subject to the second law of thermodynamics obtained by the fluctuationfriction approach, see Sect. 10.1. More precisely, it is a macroscopic description of this mean field hierarchy in accordance with the microscopic Langevin equation, and the mezoscopic Fokker-Planck-Poisson equation [22, 417, 418]. Since it is associated with the canonical ensemble, this hierarchy of mean fields is governed by the decrease of Helmholtz’ free energy F. This free energy is defined by the inner energy minus entropy if the temperature is normalized to 1, that is, 

1 u(log u − 1) − F(u) = 2 

 ×

G(x, x  )u(x)u(x  )dxdx 

(1.29)

because μ(dx, t) = u(x, t)dx stands for the particle density, and G = G(x, x  ) casts the potential, see Sect. 5.1. Thus, the first and the second terms on the right-hand side of (1.29) indicates (−1) times entropy and the inner (kinetic) energy, respectively. The inner force is self-attractive in this case, and actually, − 21 factor appears in the second term because of Newton’s third law of action-reaction. This potential, therefore, is symmetric, G(x  , x) = G(x, x  ).

(1.30)

There are several microscopic and kinetic derivations of (1.8), see Sects. 10.1 and 10.3.

12

1 Chemotaxis

The first variation δF(u) of F(u) is defined by

d w, δF(u) = F(u + sw)

. ds s=0

(1.31)

Identifying this paring , with the L 2 inner product, we obtain  δF(u) = log u − v, v =



G(·, x  )u(x  )dx  .

Thus (1.8) reads; ut = ∇ · u∇δF(u),

∂ u δF(u)

= 0. ∂ν ∂

(1.32)

This problem is a form of the model B equation [113, 153, 171, 297] derived from the free energy F = F(u), and, consequently, it follows that   ∂ d u=− u δF(u) = 0 dt  ∂ν ∂ d F(u) = − u |∇δF(u)|2 ≤ 0, dt 

(1.33)

see Sect. 4.1. The second inequality of (1.33) means the decrease of the free energy, while the first equality of (1.33), combined with u = u(x, t) ≥ 0, assures the total mass conservation (1.10), λ ≡ u0 1 = u(·, t)1 , t ∈ [0, Tmax ).

(1.34)

Relation (1.34) leads to the selection n = 2 for the formation of collapse, using the dimension analysis [69]. In more details, if u is concentrated on a region with the radius δ > 0, then it is of order δ −n because of this property, (1.34). Then, we replace δ −1 by ∇ in (1.8), and take δ 2−n and δ n for v and t, respectively, which results in     δ −2n δ 0 , δ 0 , δ n−2 = 0, δ −n δ 0 , ·, δ 0 = 0 by 1 ut + ∇ · (u∇v) − u = 0, u − ||

 

u + v = 0.

Balance of these relations, thus, implies n = 2. This heuristic argument is the origin of a rigorous proof of the formation of collapse and with mass quantization, that is, the scaling.

1.3 Mean Field Hierarchy

13

Theorem 1.2.2 concerned with the mass quantization actually describes the “local” L 1 threshold for the post-blowup continuation. The reasons why 8π was conjectured first and why it was modified later to 4π lie in the structure of the total set of stationary states.

1.4 Stationary State As we described, the above mentioned quantized blowup mechanism of nonstationary solutions is sealed in the total set of stationary solutions, which are defined by the zero free energy consumption. More precisely, from (1.33), the stationary state of (1.8) is realized by  δF(u) = constant,



u=λ

or, equivalently, u1 = λ,

log u − v = constant,

(1.35)

where λ = u0 1 and 

v = (−N )−1 (u − u),



v=0

(1.36)

for u=

1 ||

 

u.

Here and henceforth, − N denotes the Laplacian − provided with the Neumann ∂ · = 0. Eliminating u in (1.35), now we obtain the relation boundary condition, ∂ν ∂

λev u =  v, e

(1.37)

and, therefore, the nonlinear eigenvalue problem with non-local term,

v e 1 in , − v = λ  v − || e

 ∂v

= 0, v=0 ∂ν ∂ 

(1.38)

arises from (1.36). It is worth noting that the quantized blowup mechanism at this level is already observed in the blowup family of solutions [277, 327].

14

1 Chemotaxis

More precisely, if {(λk , vk )}∞ k=1 is a family of solutions to (1.38) for λ = λk and v = vk , satisfying λk → λ0 ∈ [0, ∞) and vk ∞ → +∞, then the blowup set of {vk }, denoted by   S = x0 ∈  | ∃xk → x0 such that vk (xk ) → +∞ , is finite and furthermore, passing through a subsequence, it holds that uk (x)dx 



m∗ (x0 )δx0 (dx)

(1.39)

x0 ∈S

in M() as k → ∞, where λk evk uk =  v . k e This relation implies λ0 ∈ 4πN , and furthermore, it holds that ⎡ ∇x ⎣m∗ (x0 )K(x, x0 ) +





m∗ (x0 )G(x, x0 )⎦ = 0, x = x0

(1.40)

x0 ∈S \{x0 }

for each x0 ∈ S, where only tangential derivative is taken in (1.40) if x0 ∈ ∂, and  



K(x, x ) = G(x, x ) +



1 

2π log x − x ,

1 

π log x − x ,

x∈ x ∈ ∂

(1.41)

stands for the regular part of the Green’s function G(x, x  ). Corresponding to (1.39), we obtain  m∗ (x0 )G(x, x0 ) vk (x) → v0 (x) = x0 ∈S

locally uniformly in x ∈  \ S and this v0 = v0 (x) is called the singular limit of {vk }. The above mentioned blowup mechanism emerged from the total set of stationary solutions is associated with the structures of complex analysis and differential geometry [360], and also the scaling invariance [215]. For example, −v = ev is invariant under the transformation v μ (x) = v(μx) + 2 log μ for μ > 0, see Sect. 12.2. The singular limit of {vk }, λk → λ0 , arising at the first quantized value λ0 = 4π to (1.38), on the other hand, is given by 4πG(·, x0 ) with x0 ∈ ∂ satisfying

1.4 Stationary State

15

∂ K(x, x0 )

= 0, ∂τx x=x0 where τ is the unit tangential vector. This singular limit is “stable”, attracts the non-stationary solution in infinite time, and casts the underlying driving force to the self-assembly, see Sect. 1.9. The collision of collapses is prevented in (1.39). This mechanism is broken, if the boundary condition is not provided to vk , see [215]. In other words, the quantized blowup mechanism of the family of solutions to the elliptic equation, free from the boundary condition, is the origin of the quantized blowup mechanism of (1.8). In the latter case we shall actually see the collision of sub-collapses.

1.5 Localization and Symmetrization We turn to the proof of Theorems 1.2.1 and 1.2.2, where by n = 2 is always assumed. First, Theorem 1.2.1 is proven by localizing the following criterion concerning the existence of the solution global-in-time [26, 126, 251]. Theorem 1.5.1 If λ = u0 1 < 4π, then it holds that Tmax = +∞. The dual Trudinger-Moser inequality inf {F(u) | u ≥ 0, u1 = 4π} > −∞

(1.42)

is applicable for the proof of the above theorem [364], where F = F(u) denotes the free energy defined by (1.29). In fact, from (1.42) and the first equality of (1.33) we obtain  sup u(log u − 1)(·, t) ≤ C t∈[0,Tmax ) 

with a constant C > 0 in the case of λ < 4π, which guarantees Tmax = +∞ with sup u(·, t)∞ < +∞ t≥0

by the parabolic regularity. Later in Sect. 3.3, we shall show the duality between (1.42) and Chang-Yang’s inequality [53, 54],  inf{J4π (v) | v ∈ H 1 (),



v = 0} > −∞,

(1.43)

16

1 Chemotaxis

where 1 Jλ (v) = ∇v22 − λ log 2

 

ev + λ(log λ − 1).

Here and henceforth, H m () = W m,2 () with W m,p () standing for the standard Sobolev space, where m = 0, 1, . . . and 1 ≤ p ≤ ∞: W m,p () = {f ∈ L p () | Dα f ∈ L p (), |α| ≤ m}. We shall write W0 () for the closure of C0∞ () in W m,p () and H0m () = W0m,2 (). Chang-Yang’s inequality is relative to the Trudinger-Moser inequality [245, 402], m,p

inf{J8π (v) | v ∈ H01 ()} > −∞,

(1.44)

see also Sect. 9.4. The total mass λ = 4π in (1.43) is a half of that in (1.44) because of the concentration on the boundary of the minimizing sequence for the former. The same profile occurs to the solution to (1.8) blowing-up in finite time and this profile provides the reason why λ < 4π is sharp in Theorem 1.5.1. The proof of Theorem 1.2.1 is based on the methods of localization and symmetrization, and the details are the the following. 1. Using nice cut-off functions, we show the formation of collapse at each isolated blowup point. 2. The Gagliardo-Nirenberg inequality, see [1], guarantees ε-regularity. In more details, there is an absolute constant, denoted by ε0 > 0, such that lim lim sup u(·, t)L1 (∩B(x0 ,R)) < ε0 R↓0 t↑Tmax



x0 ∈ /S

which means that x0 ∈ S



lim sup u(·, t)L1 (∩B(x0 ,R)) ≥ ε0

(1.45)

t↑Tmax

for any R > 0. 3. If we can replace lim supt↑Tmax by lim inf t↑Tmax in (1.45), then we obtain S < +∞ because the total mass is conserved by (1.34). When it is done, any blowup point is shown to be isolated, and the formation of collapse, (1.24), is proven with m(x0 ) ≥ ε0 . 4. The above replacement is justified by the weak formulation of (1.8),

1.5 Localization and Symmetrization

d dt

17

 u(·, t)ϕ = u(·, t)ϕ    1 + ρϕ (x, x  )u(x, t)u(x  , t)dxdx  , 2 ×



(1.46)

obtained by the method of symmetrization where ϕ ∈ C 2 (),

∂ϕ

=0 ∂ν ∂

(1.47)

and ρϕ (x, x  ) ≡ ∇ϕ(x) · ∇x G(x, x  ) + ∇ϕ(x  ) · ∇x G(x, x  ) ∈ L ∞ ( × ).

(1.48)

This formulation is possible because of the symmetry of the Green’s function, indicated by (1.30). 5. In more details, we obtain the monotonicity formula



d

u(·, t)ϕ

≤ C∇ϕC 1 (λ + λ2 )

dt 

(1.49)

 from (1.46), and, therefore, lim t↑T



u(·, t)ϕ exists for ϕ = ϕ(x) in (1.47). Using

this property, we can replace (1.45) by x0 ∈ S



lim inf u(·, t)L1 (∩B(x0 ,R)) ≥ ε0 , ∀R > 0. t↑Tmax

6. The collapse mass estimate from below, m(x0 ) ≥ m∗ (x0 ), is proven by the localized free energy. This argument, however, may be replaced by the use of parabolic envelope described below, and the limit equation derived from the backward selfsimilar transformation. Thus, all the argument necessary for the proof of Theorem 1.2.1 and the first part of Theorem 1.2.2 are free from the free energy. This point of view is used in the study of chemotactic systems [97]. Mass quantization (1.26), on the other hand, is a consequence of the formation of sub-collapses with residual vanishing. We approach this phenomenon from the local blowup criterion. Here we note that the global blowup criterion also follows from the weak formulation (1.46), using the second moment. A plot-type argument is given by Biler-Hilhorst-Nadzieja [27], and we obtain the following theorem [328, 364].

18

1 Chemotaxis

Theorem 1.5.2 There is an absolute constant η > 0 such that if we have x0 ∈  and 0 < R  1 satisfying 1 R2



 ∩B(x0 ,2R)

|x − x0 |2 u0 < η,

∩B(x0 ,R)

u0 > m∗ (x0 ),

then Tmax < +∞. More precisely, it holds that Tmax = o(R2 ) as R ↓ 0. We note that the assumptions of the above theorem are extended to the case that u0 = u0 (x) is a measure, say, u0 (x) = m(x0 )δx0 (dx) with m(x0 ) > m∗ (x0 ) in which case we will not have any weak solution even local-in-time. This observation leads to the following considerations. 1. The argument employed in the proof of Theorem 1.5.2 is valid to the weak solution. In particular, formation of the over-quantized collapse in finite time, m(x0 ) > m∗ (x0 ) in (1.24), means the instant blowup, that is, the impossibility of the post-blowup continuation of the solution. Thus the weak (measure-valued) post-blowup continuation of the blowup solution u = u(x, t) assures mass quantization at the first level, m(x0 ) = m∗ (x0 ). 2. The Fokker-Planck-Poisson equation, on the other hand, admits a weak solution global-in-time for appropriate initial values [282, 405], and, therefore, we use the rescaled variables, regarding the hierarchy of the mean field of particles. This argument called blowup analysis is used in a variety of fields. This process is applicable because of the scaling invariance of the equation. 3. For the blowup analysis to complete, however, we require two more important factors. The first is the classification of the solution to the limit equation, that is, the Liouville property. The second is the control of the behavior of the rescaled solution at infinity in the space variable. This part of the solution is called the tail. For the Liouville property to establish, we use the scaling invariance again. Control of the tail is done by the parabolic envelope and the concentration profile of the sequence of probability measures, called the concentration compactness principle [221]. 4. Finally, the simplicity of the blowup point arises under the boundedness of the free energy, which is the case if the total mass is dis-quantized. There are still several technicalities in the proof of mass quantization. In the next section, we shall describe the notion of weak solution, its definition and fundamental properties.

1.6 Weak Solutions The weak form (1.46) implies the continuation of the classical solution u(x, t) dx up to the blowup time t = T as a ∗-weakly continuous measure on  denoted by M(). Since M() = C() and ρϕ (x, x  ) ∈ L ∞ ( × ) \ C( × ) in (1.48),

1.6 Weak Solutions

19

we take an extension of the product measure μ ⊗ μ = μ(dx, t) ⊗ μ(dx  , t) provided with the non-negativity and linking with μ ⊗ μ. This concept is due to [331]. First, let X be the closure in L ∞ ( × ) of the linear space

  ∂ϕ

2 = 0, ψ ∈ C( × ) . X0 = ρϕ + ψ | ϕ ∈ C (), ∂ν ∂ Then we say that 0 ≤ μ = μ(dx, t) ∈ C∗ ([0, T ], M()) is a weak solution to (1.8) if there is 0 ≤ N = N (·, t) ∈ L∗∞ ([0, T ], X  )

(1.50)

called multiplicate operator, satisfying the following properties. 1. For ϕ = ϕ(x) such that (1.47) the mapping t ∈ [0, T ] → ϕ, μ(dx, t) is absolutely continuous and it holds that 1 d ϕ, μ(dx, t) = ϕ, μ(dx, t) + ρϕ , N (·, t)X ,X  a.e. t. dt 2

(1.51)

2. We have N (·, t)|C(×) = μ(dx, t) ⊗ μ(dx  , t) a.e. t.

(1.52)

Equality (1.52) indicates the linking of N with μ(dx, t) ⊗ μ(dx  , t), while N ≥ 0 in (1.50) is due to the lattice structure of X and is equivalent to the property f , g ∈ X , |f | ≤ g a.e. in  × 





f , N X ,X  ≤ g, N X ,X  .

The monotonicity formula (1.49) is thus valid to the weak solution:



d

μ(dx, t), ϕ ≤ C∇ϕ 1 C () a.e. t,

dt provided that (1.47). Weak solution has two important properties, ∗-weak compactness of the bounded sequence and the Liouville property. The following fact holds by a diagonal argument because X is separable. Theorem 1.6.1 Let 0 ≤ μk (dx, t) ∈ C∗ ([0, T ], M()), k = 1, 2, . . . be a sequence of weak solutions to (1.8) associated with the multiplicate operator 0 ≤ Nk (·, t) ∈ L∗∞ ([0, T ], M()). Then, if μk (, 0) + sup Nk (·, t)X  ≤ C, k = 1, 2, . . . 0≤t≤T

(1.53)

20

1 Chemotaxis

we have a subsequence denoted by the same symbol such that μk (dx, t)  μ(dx, t) in C∗ ([0, T ], M()) with some μ = μ(dx, t). It is a weak solution to (1.8) with the multiplicate operator N (·, t) satisfying   (1.54) sup μ(, t) + N (·, t)X  ≤ C. 0≤t≤T

The classical solution 0 ≤ u = u(x, t) ∈ C 2,1 ( × [0, T ]) to (1.8) is regarded as a weak solution μ(dx, t) = u(x, t)dx with the multiplicate operator  f , N 

X ,X 

=

×

f (x, x  )u(x, t)u(x  , t)dxdx 

f = f (x, x  ) ∈ X ⊂ L ∞ ( × ). Under this notation if uk = uk (·, t) ∈ C 2,1 ( × [0, T ]), k = 1, 2, . . . is a sequence of classical solutions, then condition (1.53) is reduced to u0k 1 ≤ C, uk0 = uk |t=0 . The generated weak solution {μ(dx, t), N (·, t)} also satisfies (1.54), and hence we can construct weak solutions more, using this limit. Sometimes we omit to mention the uniform boundedness of the multiplicate operator in these processes. Regularity of the weak solution μ(dx, t) is also defined, using v(·, t) ∈ W 1,q (), 1 ≤ q < 2 such that − v = μ,

v|∂ = 0.

(1.55)

Let ω ⊂  be a sub-domain and μ(dx, t) take the density u = u(·, t) ∈ L p (ω), 1 < 2,p 1 (ω)2 . Hence we p < ∞, in ω. Then we have v ∈ Wloc (ω) which implies u∇v ∈ Lloc can require another relation on the linking of N (·, t), that is 1 ρϕ , N (·, t)X ,X  = ∇ϕ · ∇[(−)−1 μ(·, t)], μ(dx, t) a.e. t 2 for any ϕ ∈ C02 (ω), in which case we say that the weak solution μ(dx, t) is regular in ω. If μk (dx, t), k = 1, 2, . . ., is regular in ω ⊂  in Theorem 1.6.1 with the densities uk (x, t) on ω such that sup uk (·, t)Lp (ω) ≤ C, k = 1, 2, . . .

0≤t≤T

for some 1 < p < ∞, then the generated weak solution μ(dx, t) is regular in ω with a density u(x, t) satisfying u(·, t)Lp (ω) ≤ C.

1.6 Weak Solutions

21

The Liouville property, on the other hand, is concerned with the full orbit of the weak solution 0 ≤ a = a(dx, t) ∈ C∗ (−∞, +∞; M(R2 ))

(1.56)

to the Smoluchowski-Poisson equation on the entire space at = a − ∇ · a∇ ∗ a in R2 × (−∞, +∞),

(1.57)

recalling the fundamental solution (x) =

1 1 log 2π |x|

to −. Here, we use the one-point compactification of R2 , denoted by R2 ∪ {∞} which is identified with the sphere   S 2 = (x1 , x2 , x3 ) ∈ R3 | x12 + x22 + x32 = 1 by the stereographic projection yi = xi /(1 − x3 ) (i = 1, 2). Let M(R2 ) = C∞ (R2 ) , C∞ (R2 ) = {f ∈ C(R2 ∪ {∞}) | f (∞) = 0}. Next, put ρ0ϕ (x, x  ) = −

∇ϕ(x) − ∇ϕ(x  ) · (x − x  ), ϕ ∈ C02 (R2 ), 2π|x − x  |2

(1.58)

recalling C02 (R2 ) the set of C 2 functions on R2 with compact supports, and define E as the closure of E0 = {ψ + ρ0ϕ | ψ ∈ C0 (R2 × R2 ), ϕ ∈ C02 (R2 )} in L ∞ (R2 × R2 ) where C0 (R2 × R2 ) denotes the set of continuous functions on R2 × R2 with compact supports. Then we say that a = a(dx, t) in (1.56) with a(R2 , t) ≤ C, ∀t

(1.59)

is a weak solution to (1.57) if there is 0 ≤ κ(·, t) ∈ L∗∞ (−∞, +∞; E  ), called the multiplicate operator, satisfying the following properties:

22

1 Chemotaxis

1. The mapping t ∈ (−∞, +∞) → ϕ, a(dx, t) is locally absolutely continuous for each ϕ ∈ C02 (R2 ) and it holds that d 1 ϕ, a(dx, t) = ϕ, a(dx, t) + ρ0ϕ , κ(·, t)E ,E  a.e. t. dt 2

(1.60)

2. There arises the relation κ(·, t)|C0 (R2 ×R2 ) = a(dx, t) ⊗ a(dx  , t) a.e. t. Let 0 ≤ ϕ0,r = ϕ0,r (x) ≤ 1 be a smooth function with the support contained in B(0, r), taking the value 1 on B(0, r/2). Put ϕ(x) = ϕ0,1 (x/R), R > 0, in (1.60), integrate in t, and then R ↑ +∞, using (1.59) and the dominated convergence theorem. Differentiating in t, we see that a(R2 , t) is a constant in t, denoted by m ≥ 0: a(R2 , t) = m, −∞ < t < +∞. The following theorem may be indicated as the weak Liouville property. From the proof, (1.57) for t ≤ 0 implies either m ≥ 8π or m = 0, while (1.57) for t ≥ 0 implies m ≤ 8π. For the proof we use the method of [206], that is the local second moment and scaling invariance of (1.57), aβ (dx, t) = β 2 a(dx  , t  ), x  = βx, t  = β 2 x, β > 0.

(1.61)

Theorem 1.6.2 If a = a(dx, t) in (1.56) is a weak solution to (1.57) with the multiplicate operator κ(·, t) satisfying κ(·, t)K ≤ C, ∀t. Then it holds that m = 0 or m = 8π in (1.61). Proof We shall show that (1.57) for t ≤ 0 implies m = 0 or m ≥ 8π, see [373] for the reverse inequality, that is inequality (1.57) for t ≥ 0 implies m ≤ 8π. Let c = c(s), s ≥ 0, be a smooth function such that 0 ≤ c (s) ≤ 1, −1 ≤ c(s) ≤ 0, s ≥ 0  s − 1, 0 ≤ s ≤ 1/4 c(s) = 0, s ≥ 4. Applying (1.60) for ϕ(x) = c(|x|2 ), we obtain d c(|x|2 ) + 1, a(dx, t) dt 

= 4c (|x|2 )|x|2 + 4c (|x|2 ) −

 m  2 c (|x| ), a(dx, t) − J, κ(·, t) 2π

(1.62)

1.6 Weak Solutions

23

for J = J(x, x  ) =

(c (|x|2 ) − c (|x  |2 ))(|x|2 − |x  |2 ) . 4π|x − x  |2

Now we divide the xx  -space into four parts, A = {|x|, |x  | ≤ 2}, B = {|x| ≤ 1, |x  | ≥ 2}, C = {|x  | ≤ 1, |x| ≥ 2}, and D = {|x|, |x  | ≥ 2}. Since  |x| ≤ 1 |x  | ≥ 2 ⇒ |x − x  | ≥ 1 |x| ≥ 2 |x  | ≤ 1  |x|, |x  | ≤ 1/2 ⇒ c (|x|2 ) = c (|x  |2 ) |x|, |x  | ≥ 2 s ≥ 1/4 ⇒ c(s) + 1 ≥ 1/4 it holds that |J| ≤ C(ϕ0,8 (x) + ϕ0,8 (x  )){(c(|x|2 ) + 1) + (c(|x  |2 ) + 1)}. Since (1.62) implies |4c (s)s| ≤ C(c(s) + 1) we obtain



d

c(|x|2 ) + 1, a(dx, t) − (4 − m )c (|x|2 ), a(dx, t)

dt 2π ≤ Cc(|x|2 ) + 1, a(dx, t) a.e. t. We have, furthermore, δ > 0 satisfying c(s) + 1 + c (s) ≥ δ by (1.62). In case of 0 < m < 8π, therefore, it follows that d c(|x|2 ) + 1, a(dx, t) ≥ −Cc(|x|2 ) + 1, a(dx, t) dt m ) +δm(4 − 2π

(1.63)

from the non-negativity and linking property of κ(·, t). Inequality (1.63) implies c(|x|2 ) + 1, a(dx, t) < 0 for t  −1 in case c(|x|2 ) + 1, a(dx, 0) < η ≡ δm(4 −

m )/C, 2π

a contradiction. Hence it follows that 0 < m < 8π



c(|x|2 ) + 1, a(dx, 0) ≥ η > 0.

(1.64)

From the scaling invariance of (1.57), the measure aβ (dx, t) in (1.61) is a weak solution satisfying aβ (R2 , t) = m. We thus obtain (1.64) for a = aβ (dx, t), that is c(|x|2 ) + 1, aβ (dx, 0) = c(β −2 |x|2 ) + 1, a(dx, 0) ≥ η

24

1 Chemotaxis

for any β > 0. The dominated convergence with β ↑ +∞ now guarantees 0 ≥ η, a contradiction again. Hence it holds that either m = 0 or m ≥ 8π.  

1.7 Rescaling Fundamental ingredients of blowup analysis are illustrated in Sect. 12.2. Concerning (1.8), first, we assume T = Tmax < +∞ and x0 ∈ S and take the standard backward self-similar variables defined by y = (x − x0 )/(T − t)1/2 , s = − log(T − t)

(1.65)

for t < T . The formal blowup rate, on the other hand, is (T − t)−1/(p−1) if the nonlinearity is of degree p, and p = 2 in this system of chemotaxis (1.8), see Sect. 1.10 for details. Putting z(y, s) = (T − t)u(x, t), w(y, s) = v(x, t),

(1.66)

we obtain   zs = ∇ · ∇z − z∇(w + |y|2 /4) 0 = w + z − ∂z ∂w = =0 ∂ν ∂ν

e−s λ in ||

es/2 ( − {x0 }) × {s} s>− log T

es/2 (∂ − {x0 }) × {s}.

on

(1.67)

s>− log T

Parabolic Envelope Taking a nice cut-off function around x0 ∈ S with the support radius 2R > 0 denoted by ϕx0 ,R , we can derive from (1.49) that



d 2 −2

u(·, t)ϕ x0 ,R ≤ C(λ + λ )R

dt  with a constant C > 0 independent of 0 < R < 1. This cut-off function ϕ = ϕx0 ,R ∈ C 2 () was introduced by [329] and satisfies

∂ψ

= 0, ϕ = 1 on  ∩ B(x0 , R), ∂ν ∂

1.7 Rescaling

25

and |∇ϕ| ≤ CR−1 ϕ5/6 , |∇ 2 ϕ| ≤ CR−2 ϕ2/3 in , see also Chap. 11 of [364] for the construction. Then it follows that

!

ϕx

0 ,R

" ! " , μ(·, T ) − ϕx0 ,R , μ(·, t) ≤ C(λ + λ2 )R−2 (T − t)

(1.68)

because u(x, t)dx = μ(dx, t) is extended as ∗-weakly continuous in M() on [0, T ], that is μ = μ(dx, t) ∈ C∗ ([0, T ], M()). Since 0 < R < 1 is arbitrary in (1.68), we can put R = bR(t) for given b > 0, provided that 0 < R(t) ≡ (T − t)1/2 < b−1 . Then it holds that

!

ϕx

0 ,bR(t)

" ! " , μ(·, T ) − ϕx0 ,bR(t) , μ(·, t) ≤ Cb−2 , x0 ∈ S

! " which implies lim sup m(x0 ) − ϕx0 ,bR(t) , μ(·, t) ≤ Cb−2 by t↑T

μ(dx, T ) =



m(x0 )δx0 (dx) + f (x)dx,

x0 ∈S

and, therefore,

! " lim lim sup ϕx0 ,bR(t) , μ(·, t) − m(x0 ) = 0.

b↑+∞

(1.69)

t↑T

Relation (1.69) indicates that infinitely wide parabolic region concerning the backward self-similar variables, called parabolic envelope, contains the whole blowup mechanism. Similarly, we obtain



d

2 2

u(·, t)|X| ϕ x0 ,R ≤ C(λ + λ )

dt 

(1.70)

2 , 0 < R  1, a conformal where X = x − x0 for x0 ∈  and X :  ∩ B(x0 , R) → R+ diffeomorphism for x0 ∈ ∂.

Weak Scaling Limit Given sk ↑ +∞, we have {sk } ⊂ {sk } and ζ(dy, s) such that z(y, s + sk )dy  ζ(dy, s)

(1.71)

26

1 Chemotaxis

in C∗ (−∞, +∞; M(R2 )). Here, 0-extension to z(y, s) is taken where it is not defined. This ζ = ζ(dy, s) satisfies supp ζ(·, s) ⊂ L, and is a weak solution to    in L × (−∞, +∞) zs = ∇ · ∇z − z∇ w + |y|2 /4

 ∂z

= 0, ∇w(y, s) = ∇(y − y )z(y , s)dy , ∂ν ∂L L where L is R2 if x0 ∈ , and a half space with ∂L parallel to the tangent line of ∂ at x0 if x0 ∈ ∂, and 1 1 log . (y) = |y| 2π If x0 ∈ ∂, we take the even extension of ζ(dy, s) denoted by the same symbol without confusion. Then, all the above cases are reduced to L = R2 ,   in R2 × (−∞, +∞) zs = ∇ · ∇z − z∇(w + |y|2 /4)  ∇w(y, s) = ∇(y − y )z(y , s)dy . (1.72) R2

We thus obtain a full-orbit of the weak solution ζ = ζ(dy, s) ∈ C∗ (−∞, +∞; M(R2 )) to (1.72). This ζ(dy, s) is a finite measure on R2 satisfying  m(x0 ), x0 ∈  ε0 ≤ m ≡ ζ(R2 , s) = 2m(x0 ), x0 ∈ ∂

(1.73)

for each s ∈ (−∞, +∞) from the parabolic envelope (1.69). Similarly we obtain I(s) ≡ |y|2 , ζ(dy, s) ≤ C, ∀s

(1.74)

by (1.70). Since ζ = ζ(dy, s) is a weak solution to (1.72) the mapping s → I(s) is locally absolutely continuous and it holds that m2 dI = 4m − + I a.e. s ds 2π

(1.75)

by (1.73), (1.74), and the dominated convergence theorem. Then (1.74) and (1.75) imply m2 I(s) = c0 ≡ −4m + ≥ 0, ∀s, (1.76) 2π and, in particular, m ≥ 8π or m(x0 ) ≥ m∗ (x0 ).

1.7 Rescaling

27

We may use the scaling back A(dy , s ) = es ζ(dy, s), y = e−s/2 y, s = −e−s

(1.77)

to confirm m ≥ 8π. In fact, this A = A(dy, s) ∈ C∗ (−∞, 0; M(R2 )) is a weak solution to As = ∇ · (∇A − A∇ ∗ A) in R2 × (−∞, 0)

(1.78)

provided with A(R2 , s) = m. Then we obtain m ≥ 8π from the proof of Theorem 1.6.2. This argument does not require the second parabolic envelope, (1.74), but (1.74) implies |y|2 , A(dy, s) = −c0 s, s < 0

(1.79)

A(R2 , s) = m, s < 0

(1.80)

besides

1.8 Collapse Mass Quantization Since (1.78) is a half orbit defined for s < 0, we take s˜ ↑ +∞ to translate A(dy, s − s˜ ),  = 1, 2, . . .. Since A (dy) = A(dy, −˜s )/m

(1.81)

is a probability measure on R2 and concentration-compactness principle controlles its behavior as  → ∞. Passing to a subsequence, we have the following alternatives [221]. 1. compact

Any 0 < ε < 1 admits y ∈ R2 and R > 0 such that A (B(y , R)) > 1 − ε, ∀  1

2. vanishing

For any R > 0 it holds that lim sup A (B(x, R)) = 0.

→∞ x

3. dichotomy such that

(1.82)

(1.83)

There is 0 < λ < 1 such that any ε > 0 admits y ∈ R2 and R > 0

28

1 Chemotaxis

lim inf A (B(y , R)) ≥ λ − ε →∞

lim lim inf A (R2 \ B(y , R )) ≥ 1 − λ − ε

R ↑+∞ →∞

(1.84)

Using this property with (1.79)–(1.80), we obtain the following lemma [373]. Lemma 1.8.1 Given s˜ ↑ +∞, we have a subsequence denoted by the same symbol and σ ∈ N ∪ {0}, σ ≤ m/(8π), such that for {A (dy)} in (1.81) any ε > 0 admits j j j y ∈ R2 and bj > 0, 1 ≤ j ≤ σ, such that for B = B(y , bj ) it holds that j

j

lim |yi − y | = +∞, ∀i = j, lim sup |m · A (B ) − 8π| < ε, ∀j →∞ ⎛ ⎞

→∞

lim sup A ⎝B(y, R) \

→∞ y∈R2 j |y |



σ

B ⎠ = 0, ∀R > 0 j

j=1

1/2 C(1 + max bj )˜s , j

∀  1, ∀j.

(1.85)

Since (1.85) we have ⎛ lim inf m · A ⎝R2 \ →∞

σ

⎞ B ⎠ ≥ m − 8πσ − m · ε > 0 j

(1.86)

j=1

in case m ∈ / 8πN. Here are several remarks. First, the ball B(y , R) in (1.82) is not unique, but one may take R > 0 to be minimal. Then we can assign the minimum of maxj bj in (1.85). Under this agreement, the argument of contradiction guarantees b0 > 0, independent of s˜ ↑ +∞ and its subsequence, such that bj ≤ b0 . The argument of contradiction guarantees also that any 0 < ε  1 admits s1  1 such that for s˜ ≥ s1 there are yj (˜s) ∈ R2 , |yj (˜s)| ≤ C˜s1/2 , 0 < bj (˜s) ≤ b0 , 1 ≤ j ≤ σ(˜s) for σ(˜s) ∈ N ∪ {0}, σ(˜s) ≤ m/(8π) such that   ˜ s), −˜s = 0, ∀R  1 lim sup A B(y, R) \ E(˜

s˜ ↑+∞ y∈R2

lim |yi (˜s) − yj (˜s)| = +∞, i = j

max A(Bj (˜s), −˜s) − 8π < ε

s˜ ↑+∞

1≤j≤σ(˜s)

1.8 Collapse Mass Quantization

29

where ˜ s) = B˜ j (˜s) = B(yj (˜s), bj (˜s)), E(˜

σ(˜s)

B˜ j (˜s).

j=1

This property implies that any R > 0 admits s1  1 such that lim lim sup R(tk )2 u(·, T − s˜ R(tk )2 )L∞ (B(x0 ,bR(tk ))\F k (˜s)) ≤ CR−2



(1.87) lim sup max u(·, T − s˜ R(tk )2 )L1 (Bk (˜s)) − 8π < ε b↑+∞ k→∞

k→∞ 1≤j≤σ(˜s)

j

for s˜ ≥ s1 where xjk (˜s) − x0 = R(tk )yj (˜s), rjk (˜s) = R(tk )bj (˜s) σ(˜s)

Bjk (˜s) = B(xjk (˜s), rjk (˜s)), F k (˜s) =

Bjk (˜s). j=1

Here, the first inequality of (1.87) is a consequence of the following lemma, which may be called the scaling invariant ε regularity [373]. Lemma 1.8.2 Let u = u(x, t) be a classical solution to (1.8) on  × (−T , T ) and u0 = u(·, 0). Then we have ε0 , ξ0 , and C0 independent of x0 ∈  and 0 < R  1 such that u0 L1 (B(x0 ,R)) < ε0 implies sup t∈[−ξ0 R2 ,ξ0 R2 ]

u(·, t)L∞ (B(x0 ,R/2)) ≤ C0 R−2 .

Lemma 1.8.2 is a consequence of the smoothing effect of local norms on u, which is used in the study of blowup in infinite time of the solution [330]. Here, we note that property (1.87) arises to a subsequence of {tk } denoted by the same symbol. Since {tk } is arbitrary, again, argument of contradiction guarantees (1.87) without passing to a subsequence. Now we define tk ↑ T by R(tk ) = 2−k/2 , k  1, and take s1  1, σ(˜s), and Bjk (˜s) for 1 ≤ j ≤ σ(˜s) and s˜ ≥ s1 , so that (1.87) arises. We hereby use the first inequality of (1.87) on s1 ≤ s˜ ≤ 2s1 . In this case the intervals [T − 2s1 R(tk ), 1 − s1 R(tk )], k = 1, 2, . . . covers 0 < T − t  1 by T − s1 R(tk )2 = T − 2s1 R(tk+1 )2 , and, therefore, any ε > 0 admits xj (t) ∈ B(x0 , CR(t)), 0 < rj (t) ≤ CR(t), 0 < T − t  1, 1 ≤ j ≤ σ(t) with σ(t) ∈ N ∪ {0}, σ(t) ≤ m/(8π), such that

30

1 Chemotaxis

Bi (t) ∩ Bj (t) = ∅, i = j



lim lim sup max u(·, t)L1 (B(xj (t),rj (t))) − 8π < ε b↑+∞

t↑T

1≤j≤σ(t)

lim lim sup R(t)2 u(·, t)L∞ (B(x0 ,bR(t))\G(t)) ≤ C

b↑+∞

(1.88)

t↑T

where σ(t)

Bj (t) = B(xj (t), rj (t)), G(t) =

Bj (t). j=1

The relation (1.88) now provides with additional properties to ζ = ζ(dy, s) in (1.71). Hence, each 0 < ε  1 admits s1  1 such that for s˜ ≥ s1 there are σ(˜s) ∈ N ∪ {0}, σ(˜s) ≤ m/(8π), and yj (˜s) ∈ R2 , |yj (˜s)| ≤ C˜s1/2 , 0 < bj (˜s) ≤ b0 , 1 ≤ j ≤ σ(˜s) such that ζ = ζ(dy, − log s˜ ) is regular in

(R2 \ Es˜ ) × {− log s˜ } and it holds that s˜ ≥˜s1

Bi (˜s) ∩ Bj (˜s) = ∅, i = j, sup s˜ ≥s1 , 1≤j≤σ(˜s)

sup ζ(dy, − log s˜ )L∞ (R2 \Es˜ ) ≤ C

s˜ ≥s1



ζ(Bj (˜s), − log s˜ ) − 8π < ε

(1.89)

where σ(˜s)

Es˜ =

Bj (˜s), Bj (˜s) = B(˜s−1/2 yj (˜s), s˜ −1/2 bj (˜s)).

(1.90)

j=1

In case of m ∈ / 8πN, furthermore, we have always (1.85) in Lemma 1.8.1, which implies the existence of δ > 0 such that inf ζ(R2 \ Es˜ , − log s˜ ) ≥ δ.

s˜ ≥s1

(1.91)

Here we take s˜ ↑ +∞ and a subsequence denoted by the same symbol such that ˜ s) in C∗ (−∞, +∞; M(R2 )), ζ(dy, s − s˜ )  ζ(dy,

(1.92)

˜ where ζ˜ = ζ(dy, s) is a weak solution to (1.72). The family of measures, {ζ(dy, s − s˜ )}, on the other hand, is tight by (1.74), and hence it follows that ˜ ˜ 2 , s) = m, |y|2 , ζ(dy, s) ≤ C. ζ(R

(1.93)

1.8 Collapse Mass Quantization

31

We have also s˜ −1/2 |yj (˜s)| ≤ C,

lim s˜ −1/2 bj (˜s) = 0

s˜ ↑+∞

in (1.89) and (1.90). ˜ The singular part of ζ(dy, s), s ∈ R, denoted by ζ˜s (dy, s), is composed of a finite sum of delta functions. This property is confirmed by ε regularity of the classical solution, tracing back the limiting process. Next, Theorem 1.5.2 applied to the scaling ˜ ˜ back A(dy, s) of ζ(dy, s), defined as in (1.77), implies that the coefficient of each delta function of ζ˜s (dy, s) must be less than or equal to 8π. These properties, together with ˜ (1.89), guarantee that the singular support of ζ(dy, s), denoted by Ss , is composed of a finite number of collisionless accumulating points of the set S˜s˜ = {˜s−1/2 yj (˜s) | 1 ≤ j ≤ σ(˜s)}, − log s˜ = s + s˜ ,  → ∞, and the coefficient of each delta function of ζs (dy, s) is exactly 8π. We may assume also that Ss , s ∈ Q, is composed of the converging points of S˜s˜ by a diagonal argument. We thus end up with ζ˜s (dy, s) =

σ(s)  j=1

j

8πδyj

∞ (s)

(dy), |y∞ (s)| ≤ C, s ∈ R

(1.94)

j

with σ(s) ∈ N ∪ {0}, σ(s) ≤ m/(8π), and y∞ (s) ∈ R2 , 1 ≤ j ≤ σ(s). It holds also that ˜ ζ(dy, s) = ζ˜s (dy, s) + g(y, s)dy

(1.95)

with  0 ≤ g = g(y, s) ≤ C,

R2

|y|2 g(y, s)dy ≤ C, s ∈ R

(1.96)

and g(·, s)1 ≥ δ, s ∈ Q

(1.97)

by (1.89), (1.91), and (1.93). Inequalities (1.96) and (1.97), however, are impossible because the term |y|2 /4 in (1.72) attracts the density g = g(y, s) to y = ∞ as s ↑ +∞. This property was first noted in the form of the non-existence of non-trivial stationary classical solution to (1.72), see [256]. We have thus the following lemma [374] to conclude the proof of Theorem 1.2.2.

32

1 Chemotaxis

˜ Lemma 1.8.3 There is no weak solution ζ(dy, s) ∈ C∗ (−∞, +∞; M(R2 )) to (1.72) satisfying (1.94)–(1.97). ˜ Lemma 1.8.3 implies ζ(dy, s) = ζ˜s (dy, s). Concerning ζ = ζ(dy, s) in (1.71), we have, see [332], ζ(dy, s) =



8πδy0 (dy) + g(y, s)dy, s ∈ R

(1.98)

y0 ∈Bs

where Bs is a finite set and 0 ≤ g = g(·, s) ∈ L 1 (R2 ) ∩ C(R2 \ Bs ). The case 0 < g(·, s)1 ∈ m∗ (x0 )N in (1.98), however, may arise, that is the residual vanishing of ˜ ˜ ζ(dy, s), ζ(dy, s) = ζ˜s (dy, s), but not necessarily that of ζ(dy, s). Collision of Sub-collapses ˜ Once Theorem 1.2.2 is proven, we have σ(t) = σ ≡ m/(8π) and ζ(dy, s) = ζ˜s (dy, s) with (1.94), that is ˜ ζ(dy, s) =

σ 

8πδyj

j

∞ (s)

j=1

(dy), |y∞ (s)| ≤ C, s ∈ R.

(1.99)

˜ ˜ The scaling back A(dy, s) ∈ C∗ (−∞, 0; M(R2 )) of ζ˜ = ζ(dy, s) defined as in (1.77) is a weak solution to (1.78), taking the form ˜ A(dy, s ) =

σ 

8πδy˜ j

∞ (s

j=1

(dy), |˜y∞ (s )| ≤ C(−s )1/2 , s < 0. j

)

(1.100)

Equality (1.99) or (1.100) indicates the formation of sub-collapses with quantized mass and residual vanishing. Then the center of these sub-collapses move under the control of Hamiltonian [276]. j

Theorem 1.8.1 If σ ≥ 2 the mapping s → y˜ ∞ (s) is locally Lipschitz continuous and it holds that j

d y˜ ∞ 1 σ = 8π∇j Hσ0 (˜y∞ , . . . , y˜ ∞ ), 1 ≤ j ≤ σ ds  (yi − yj ). in (1.100), where Hσ0 (y1 , . . . , yσ ) = 1≤i 0. It is a weak solution to (1.8) on β × (−∞, +∞), β = β −1 , with the total mass and uniform estimate of the multiplicate operator, independent of β. By the diagonal argument, therefore, any βk ↓ 0 admits a subsequence denoted by the same symbol such that 

˜ t) in C∗ (−∞, +∞; M(R2 )) μβk (dx, t)  μ(dx,

(1.104)

where μ˜ = μ(dx, ˜ t) is a weak solution to (1.57) with uniformly bounded multiplicate operator. We also take the even extension of μ(dx, ˜ t) in the case x0 ∈ ∂. By (1.104) we have μ(R ˜ 2 , 0) ≥ μ({0}, 0) > 0, and, therefore, Theorem 1.6.2 implies μ(R ˜ 2 , 0) = 8π.

(1.105)

Given r > 0, take 0 < β  1, and put ϕ = ϕ0,r . Then it holds that ϕ, μβ (dx, 0) =

N(0) 

mi (0)ϕ((xi (0))/β)

i=1



+



ϕ(x/β)f (x, 0)dx

(1.106)

by (1.103). Now we put β = βk and make k → ∞ in (1.106). By the absolute continuity of f (·, 0)dx, we obtain ˜ 0) = μ({0}, 0), ∀r > 0. ϕ0,r , μ(dx,

(1.107)

Then it follows that μ(dx, ˜ 0) = 8πδ0 (dx) and hence μ({0}, 0) = m∗ (x0 ) from (1.107) and (1.105). Having (1.103), we put f (x, t)dx for the regular part of μ(dx, t): μ(dx, t) = μs (dx, t) + f (x, t)dx.

(1.108)

This density f = f (x, t) satisfies a parabolic equation with the Neumann boundary condition in the relatively open set ( \ St ) × {t}, St = {xi (t) | 1 ≤ i ≤ n(t)} t∈R

in  × R. We obtain, therefore, μ(dx, t) = μs (dx, t) for t < t0 if f (x0 , t0 ) = 0, ∃(x0 , t0 ) ∈ ( \ St0 ) × {t0 }

(1.109)

1.9 Blowup in Infinite Time

35

by the strong maximum principle, and then it holds that n(0) 

λ=

m∗ (xi (0)).

(1.110)

i=1

This case is called the residual vanishing. Lemma 1.9.2 If there arises (1.101) to (1.8), n = 2, with lim F(u(·, t)) > −∞,

t↑+∞

(1.111)

then it holds that (1.109), t0 = +∞. Proof We have seen that f ≡ 0 implies f (·, t) > 0 in  \ St , −∞ < ∀t < +∞. From N(0) ≥ 1, furthermore, it follows that S0 = ∅. Let K be a relatively compact set in  \ S0 . Since % f (x, 0)dx is absolutely continuous, there is a finite number of open balls {Bi }, K ⊂ i Bi , such that μ( ∩ Bˆ i , 0) < 2π where Bˆ i = B(xi , 2ri ) for Bi = B(xi , ri ). Since (1.102) we have 0 < ε  1 independent of i such that μ(Bi ∩ , t) < 4π for |t| ≤ ε. Then it holds that K ∩ St = ∅, |t| ≤ ε, by (1.103). In Lemma 1.9.1 we may assume tk + 2ε < tk+1 , passing to a subsequence. Then (1.111) implies   tk +ε  dt u|∇(log u − v)|2 < +∞, k

tk −ε



and, therefore, 



tk +ε

lim

k→∞ tk −ε

u|∇(log u − v)|2 = 0.

dt K

From the elliptic and parabolic regularity, there is a smooth g = g(x, t) defined on K × [−ε, ε] such that  ε  dt f |∇(log f − g)|2 = 0. (1.112) −ε

K

In accordance with (1.102) we have v(x, t + tk )  ∃v∗ (x, t) in C∗ (−∞, +∞, W 1,q ()), 1 ≤ q < 2.

36

1 Chemotaxis

Then (1.112) implies log f∗ − g∗ = constant in  \ S0 for f∗ = f (·, 0) > 0 and g∗ = v∗ (·, 0), that is f∗ = σeg∗ in  \ S0

(1.113)

where σ > 0 is a constant. In particular, we obtain 



σ

g∗

e

=

K

K

f∗ ≤ λ,

(1.114)

while (1.103) implies v∗ (x) ≥ m∗ (x0 )G(x, x0 ) for x0 = x1 (0) ∈ S0 . Then it holds that  eg∗ → +∞, K ↑  \ S0 K

by Fatou’s lemma, see [37]. We have σ = 0 by (1.114), and, then, (1.113) implies   f∗ = f (·, 0) = 0 in  \ S0 , a contradiction. We use the following lemma to complete the proof of Theorem 1.9.1. This lemma is the dual form of the improved Trudinger-Moser inequality described in Sect. 2.6. Lemma 1.9.3 Given n ∈ N, θ ∈ (0, 1), and δ > 0, we have c0 = c0 (n, θ) > 0 and C = C(n, θ, δ) > 0 such that if u = u(x) ≥ 0, x ∈ , admits a family of open sets denoted by i ⊂ , 1 ≤ i ≤ n, such that dist (i , j ) ≥ δ, 1 ≤ i < j ≤ n  u < mi θ, 1 ≤ i ≤ n i

(1.115) (1.116)

and  0

n

u < c0 , 0 =  \

i , i=1

then it holds that  F(u) ≡



u(log u − 1) −

1 2

 ×

G(x, x  )u ⊗ u ≥ −C,

(1.117)

1.9 Blowup in Infinite Time

37

where  mi =

8π, ∂ ∩ i = ∅ 4π, ∂ ∩ i = ∅

in (1.116). Proof Let ui = u · χi , 0 ≤ i ≤ n. Below, the constant C > 0 changes from line to line. First, we have  1 ui (log ui − 1) − (−)−1 ui , ui  ≥ −C, 1 ≤ i ≤ n 2θ i by (1.116), recalling (1.41). Similarly, any ε > 0 admits c0 > 0 such that 

 0

u < c0

We have 8





u0 (log u0 − 1) −

(1.118)



n   

i=0

1 (−)−1 u0 , u0  ≥ −C. 2ε

ui (log ui − 1) =



u(log u − 1),

and also (−)−1 ui , uj  ≤ C, 1 ≤ i < j ≤ n by (1.115). Thus we obtain (1.117), regarding (1.118) and (−)−1 u0 , ui  ≤ (−)−1 u0 , u0 1/2 · (−)−1 ui , ui 1/2 .   Proof of Theorem 1.9.1 We have only to show (1.111), assuming u(x, tk )dx 



m∗ (x0 )δx0 (dx) + f (x)dx in M()

x0 ∈S0

with tk ↑ +∞. First, we take m ∈ N satisfying  

f =λ−

 x0 ∈S

m∗ (x0 ) < 4π · m,

(1.119)

38

1 Chemotaxis

and then 0 < 1 − θ  1, n ∈ N, n ≤ m, and open sets 1 , . . . , n1 ⊂  \ S, n = n1 + S, such that i ∩ j = ∅, 1 ≤ i < j ≤ n1 , 

 i

f < mi θ, 1 ≤ i ≤ n1 ,

%n1

\

i=1 i

f
0 independent of k  1 such that dist (ki , kj ) ≥ δ for 1 ≤ i < j ≤ n. Furthermore, it holds that 

 ki

u(·, tk ) < m ˜ i θ, 1 ≤ i ≤ n,

%n

\

k i=1 i

u(·, tk ) < c0 (n, θ),

where  m ˜i =

1 ≤ i ≤ n1 mi , m∗ (x0 ), n1 + 1 ≤ i ≤ n, ki =  ∩ B(x0 , rk (x0 )), x0 ∈ S.

Then Lemma 1.9.3 implies F(u(·, tk )) ≥ −C, and hence (1.111).

 

Collapse Dynamics By Theorem 1.9.1, if λ ∈ / 4πN, the solution u = u(x, t) to (1.8) satisfies either T < +∞, or T = +∞ with compact orbit denoted by O. In the latter case the ω-limit set is defined by ω(u0 ) = {u∞ ∈ C() | there is tk ↑ +∞ such that u(·, tk ) → u∞ in C()}. Under the presense of the Lyapunov function F = F(u), this ω-limit set is non-void, compact, connected, and is contained in E, the total set of stationary solutions, see [161] and also Sect. 3.3.

1.9 Blowup in Infinite Time

39

In particular, if λ ∈ / 4πN and Eλ = ∅ then it holds that T < +∞, where Eλ denotes the set of stationary solutions defined by (1.38). Even if Eλ = ∅ we have F(u0 )  −1



T < +∞

(1.120)

in the case of λ ∈ / 4πN. Property (1.120) is valid even for n ≥ 3 if the solution is radially symmetric [415]. See also [276, 330, 372] for related results. Equality (1.101) means ∃tk ↑ +∞, u(·, tk )∞ → +∞.

(1.121)

By Theorem 1.9.1 and its proof, (1.121) arises with μ(dx, t) = μs (dx, t) for μ(dx, t) and μs (dx, t) defined by (1.102) and (1.103), respectively. We have also (1.111), which implies that μs (dx, t) is stationary [330]. Hence there are  ∈ N ∪ {0}, k ∈ N ∪ {0}, and x1 , . . . , x ∈ ∂, x+1 , . . . , xm ∈ , m =  + k, satisfying ∇τ i Hm (x1 , . . . , xm ) = 0, 1 ≤ i ≤  ∇i Hm (x1 , . . . , xm ) = 0,  + 1 ≤ i ≤ m where Hm (x1 , . . . , xm ) =

1 R(xi ) + 2 m



i=1

1≤i 0 for x ∈  \ S in (1.24), and in particular, 

u0 1 =

m∗ (x0 ) + f 1 >

x0 ∈S



m∗ (x0 ).

x0 ∈S

 

Hence (1.123) follows.

Concerning the radially symmetric solution u = u(|x|, t) on  = B ≡ B(0, 1) the inequality d dt

 |x| 8π, since there is neither stationary solution nor blowup in infinite time, proven similarly to Theorem 1.9.3, we have the blowup in finite time. If λ = 8π we have T = +∞ similarly to (1.8) with λ = 4π. Since there is no stationary solution, we have blowup in infinite time in this case. Finally, there is no blowup point on the boundary in (1.124) in the general case of T = Tmax < +∞, see [370]. Very similar result holds for (1.124) on the whole space R2 , although there are several function spaces for its well-posedness to be valid. Roughly speaking, if the second moment of the initial mass is finite, then the blowup mechanism is included in a bounded region. In particular, if λ = u0 1 = 8π then we obtain blowup in infinite time because the classical stationary solution indicated by the Louville family, see Sect. 12.4, does not take the finite second moment.

1.10 Simple Blowup Points ˜ If the solution is radially symmetric, so is A˜ = A(dy, s) in (1.100), and, therefore, it holds that σ = 1. Such a case of σ = 1 is called the simple blowup point. A direct proof of this property is given in [276], that is, any radially symmetric blowup solution u = u(|x|, t) to (1.8) satisfies m(x0 ) = 8π, x0 = 0. If the blowup point is simple, then we have m = 8π and hence I(s) ≡ 0 in (1.76). This property implies ζ(dy, s) = 8πδ0 (dy) which is independent of the choice of tk ↑ T . Theorem 1.8.1, therefore, is simplified for the case of σ = 1 as in the following theorem [256, 326]. This type of blowup mechanism has been noticed in [165, 333, 362, 363].

42

1 Chemotaxis

Theorem 1.10.1 If x0 ∈ S is simple, that is, m(x0 ) = m∗ (x0 ) in (1.24), then it holds that z(y, s + s )dy  m∗ (x0 )δ0 (dy)

(1.127)

in C∗ (−∞, +∞; M(R2 )) as s ↑ +∞, where R(t) = (T − t)1/2 and z = z(y, s) denotes the backward self-similar transformation of u = u(x, t) defined by (1.65)– (1.66). In particular, it holds that lim R(t)2 u(·, t)L∞ (B(x0 ,bR(t))∩) = +∞ t↑T

(1.128)

for any b > 0. ˜ Relation (1.127) implies A(dy, s) = 8πδ0 (dy) in (1.100). The ODE part of (1.8), ut = u − ∇u · ∇v + u2 , may be formulated by ˙ = M2 M which takes the solution M(t) = (T − t)−1 blowing-up at t = T . We say that the rate of the blowup solution to (1.8) is type (I) if lim sup(T − t)u(·, t)∞ < +∞ t↑T

and type (II) in the other case. The classification of the blowup point, on the other hand, is in accordance with the backward self-similar transformation (1.65)–(1.66). We thus say that x0 ∈ S is of type (I) if lim sup R(t)2 u(·, t)L∞ (B(x0 ,bR(t)∩) < +∞ t↑T

for any b > 0, where R(t) = (T − t)1/2 , and that x0 ∈ S is of type (II) if it is not of type (I), that is if there is tk ↑ T and b > 0 such that lim R(tk )2 u(·, tk )L∞ (B(x0 ,bR(tk ))∩) = +∞.

k→∞

(1.129)

Theorem 1.10.1 implies that any simple blowup point is of type (II), and equality (1.129) is refined as (1.128). The convergence (1.127), on the other hand, implies that the total blowup mechanism is enclosed in the infinitesimally small parabolic region, which we call the hyper-parabola.

1.10 Simple Blowup Points

43

Entropy Dissapation The hyper-parabola, (1.127), suggests the other scaling r(t)  R(t) = (T − t)1/2 to describe real blowup envelope for simple blowup points. There is a radially symmetric solution to (1.8) satisfying & ' √ () 1/2 e− 2|log(T −t)| u(x/r(t)) · 1x≥r(t) 1 + o(1) + O u(x, t) = r(t)2 |x|2

(1.130)

as t ↑ T = Tmax < +∞ uniformly in x ∈ B(0, bR(t)) for any b > 0, see [165], where √

r(t) = (T − t)1/2 e−

1

2|log(T −t)|1/2

|log(T − t)| 4 8 · (1 + o(1))  R(t), u(y) = # $2 . 1 + |y|2

#

|log(T −t)|−1/2 −1

$

Thus, r(t) and u(y) indicate the blowup rate and a stationary solution of (1.57), − log u = u

in R2 ,

respectively. The solution (1.130) enjoys the property lim Fx0 ,br(t) (u(·, t)) = +∞, ∀b > 0 t↑T

(1.131)

where  Fx0 ,R (u) =

u(log u − 1)ϕx0 ,R  1 − G(x, x  )(u · ϕx0 ,R )(x)(u · ϕx0 ,R )(x  )dxdx  2 × 

denotes the local free energy. This profile indicates the entropy dissapation in the context of the theory of self-organization, see [364], and is expected for a wide class of solutions with r(t) = u(x0 , t)−1/2 . We complete this chapter with the following theorem. Theorem 1.10.2 In Theorem 1.2.1, any x0 ∈ S is simple if and only if the free energy F(u(·, t)) is bounded. Proof The only if part is proven similarly to Theorem 1.9.1. For the counter part to prove, let lim F(u(·, t)) > −∞. (1.132) t↑T

44

1 Chemotaxis

Then it holds that 



T

dt 0



u|∇(log u − v)|2 < +∞.

(1.133)

Given ϕ = ϕ(x) with (1.47), we have





d





ϕu = u∇(log u − v) · ∇ϕ

dt   ≤ ∇ϕ∞ λ1/2 u1/2 ∇(log u − v)2 and, therefore,



d

ϕx0 ,R u

dt

dt  t  T  1/2 1/2 −1 1/2 2 ≤ Cλ R (T − t) dt u|∇(log u − v)|

ϕx

0 ,R , μ(dx, t) − ϕx0 ,R , μ(dx, t) ≤

t



T



for (1.68). Then it holds that ϕx0 ,b(T −t)1/2 , μ(dx, t) = m(x0 ) + o(1), t ↑ T by (1.133) where b > 0 is arbitrary. There arises thus the hyper-parabola, ζ(dy, s) = m(x0 )δ0 (dy) in (1.71), and, therefore, m(x0 ) = m∗ (x0 ) follows from the proof of the first part of this theorem.  

1.11 Summary We have studied the chemotaxis system and observed that the formation of selfassembly is realized in thermally and materially closed system, whereby the method of nonlinear spectral mechanics is applicable. 1. The simplified system of a chemotax is provided with the total mass conservation and decrease of Helmholtz’ free energy. Hence it is called the SmoluchowskiPoisson equation. 2. This variational structure of the Smoluchowski-Poisson equation fits the scaling invariance in the case of two space dimensions. Then the critical mass 8π arises. 3. This equation has the quantized blowup mechanism in the stationary state, the blowup in finite time, and the blowup in infinite time.

1.11 Summary

45

4. These profiles are obtained by a weak formulation accompanied with the symmetry of the Green’s function, that is bounded in time variation of the local L 1 -norm. Then, we apply a hierarchical weak scaling limits to prove the collapse mass quantization. 5. The Liouville property applied to the scaling limit assures the quantization of subcollapses. Then the residual vanishing achieves the quantization of collapses. 6. Dual form of the improved Trudinger-Moser inequality assures the quantization of the blowup in infinite time. Also any blowup point formed in finite time is simple if and only if the free energy is bounded. Then any blowup point is of type II.

Chapter 2

Time Relaxization

Different models can share common stationary states. In the previous chapter we described the quantized blowup mechanics of Smoluchowski-Poisson equation in two-space dimensions. This chapter is devoted to the full system of chemotaxis and its relatives. Among them are the tumor growth model and the geometric flow. The quantized blowup mechanism is realized in some models, but is violated in the other cases.

2.1 Full System of Chemotaxis A typical form of the full system of chemotaxis is 1 u t = ∇ · (∇u − u∇v), τ vt = v + u − ||    ∂u ∂v ∂v  −u , = 0, v=0 ∂ν ∂ν ∂ν ∂ 

 

u in  × (0, T ) (2.1)

where τ > 0 in the left-hand side of the second equation stands for the relaxization time which describes a chemical process of the formation of the field v. The fundamental structure is the existence of the Lagrangian, 

1 u(log u − 1) + ∇v22 − L(u, v) = 2 

 

vu

(2.2)

 defined for



v = 0. Then, (2.1) is equivalent to

u t = ∇ · u∇ L u (u, v), τ vt = −L v (u, v) in  × (0, T )    ∂ ∂v  u L u (u, v), = 0, v = 0. ∂ν ∂ν ∂  © Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_2

47

48

2 Time Relaxization

Thus it holds that d L(u, v) = − dt

 

u |∇ L u |2 + τ −1 vt2 d x ≤ 0

for the solution (u, v) = (u(·, t), v(·, t)). This formulation to (2.1) is summarized as dual variation, where the stationary state is described by both components equivalently. Provided with the variational structures individually, their linearly stable critical points are dynamically stable, see Chap. 3. Well-posedness local-in-time and the standard blowup criterion are valid also to this system. More precisely, we have always 0 < Tmax ≤ +∞. If T = Tmax < +∞, furthermore, then lim u(·, t)∞ = +∞ holds, and the blowup set S of u, defined t↑T

by (1.25), is not empty. Similarly to Theorem 1.5.1, furthermore, λ = u 0 1 < 4π implies Tmax = +∞ for the general case, while λ < 8π implies Tmax = +∞ in the radially symmetric case [26, 126, 251]. The local blowup criterion, or ε-regularity, also holds and we obtain lim sup u(·, t) L 1 (∩B(x0 ,R)) ≥ m ∗ (x0 ) t↑T

for each x0 ∈ S, where R > 0 is arbitrary [252]. The bounded variation in time of the local L 1 -norm, however, is not known, and, consequently, the finiteness of the blowup points, or the blowup threshold as in Theorem 1.5.2, is not valid.

2.2 Non-local Parabolic Equation Replacing u t by εu t in (2.1) and making ε ↓ 0, we obtain (1.37) which induces another simplified system of chemotaxis,   v e 1 in  × (0, T ) τ vt = v + λ  v − || e   ∂v  = 0, v=0 ∂ν ∂ 

(2.3)

formulated by Wolansky [420, 421]. This problem is provided with the Lyapunov function   1 2 v Jλ (v) = ∇v2 − λ log e + λ(log λ − 1), v = 0, 2   and it holds that

d Jλ (v) = −τ −1 vt 22 ≤ 0. dt

2.2 Non-local Parabolic Equation

49

Not so much is known to (2.3) also, but the following theorems [192, 420] show the dis-quantized blowup mechanism of the relative model, λev τ vt = v +  v in  × (0, T ), e

v|∂ = 0.

(2.4)

Theorem 2.2.1 If  = B(0, R), v0 = v0 (|x|), and λ ≥ 8π, then λev u ≡  v  λδ0 e

(2.5)

as t ↑ T = Tmax ∈ (0, +∞] in C(), where v = v(·, t) is the solution to (2.4) with v(·, 0) = v0 . Theorem 2.2.2 If λ > 8π in the previous theorem, it holds that T = Tmax < +∞, and, therefore, we have the formation of collapse (2.5) in finite time with the disquantized mass λ > 8π. Blowup in infinite time is expected at the threshold case λ = 8π. Non-local ODE The spatially homogeneous part of (2.3) or (2.4) is formulated by the non-local ODE. Adopting the normalization τ = 1, λ = ||, we obtain ||ev vt =  v − 1 e

in  × (0, T )

(2.6)

||ev vt =  v e

in  × (0, T ).

(2.7)

or

Equation (2.6) is reduced to (2.7) by v˜ = v + t. In (2.7) it holds that 1 d || dt

 

v=1

and hence 1 ||

 

v=t+

1 ||

 

v0 .

(2.8)

50

2 Time Relaxization

|| Then we use −(e−v )t = a(t) ≡  v to derive e e

−v0 (x)

−e

−v(x,t)



t

= A(t) ≡

a(t )dt ,

(2.9)

0

where v0 = v(·, 0). Equality (2.9) implies e−v0 (x1 ) − e−v(x1 ,t) = e−v0 (x2 ) − e−v(x2 ,t) , and, hence, the order preserving property v0 (x1 ) ≤ v0 (x2 )



v(x1 , t) ≤ v(x2 , t).

In particular, it holds that v0 ∞ = v0 (x∗ )



v(·, t)∞ = v(x∗ , t).

Since vt > 0, we have lim v(x, t) = v(x, T ) ∈ (−∞, +∞] for each x ∈ , and, t↑T

therefore, lim v(·, t)∞ = +∞



t↑T

v(x∗ , T ) = +∞.

It thus follows that e−v0 (x∗ ) = e−v0 ∞ = A(T ) = e−v0 (x) − e−v(x,T ) = A(T ) from (2.9), so that v(x, T ) = − log{e−v0 (x) − e−v0 ∞ }. Pluging this relation into (2.8), we obtain −

1 ||

 

log{e−v0 − e−v0 ∞ } = T +

1 ||

 

v0 ,

and hence the following theorem. Theorem 2.2.3 The solution v = v(x, t) to the non-local ODE (2.7) blows-up at the blowup time 1 T =− ||

 

log{e

−v0

−e

−v0 ∞

1 }− ||

 

v0 .

(2.10)

2.2 Non-local Parabolic Equation

51

The case T = +∞ is admitted in (2.10), while the right-hand side is always positive by   1 1 −v0 log{e } − v0 = 0. − ||  || 

2.3 Smoluchowski-ODE System Several mathematical models are proposed to describe the movement of living things attracted by non-diffusive chemical factors using re-inforced random walk which results in parabolic-ODE systems in the limit state, that is pt = ∇ · (D∇ p − pχ(w)∇w), wt = g( p, w),

(2.11)

see [291]. Here, p and w are due to the conditional probability of the decision of the walkers and the density of the control species, respectively, D > 0 the diffusion constant, χ the chemotactic sensitivity, and g the chemical growth rate. Angiogenesis is the formation of blood vessels from pre-existing vasculature. It is a process whereby capillary sprouts are formed in response to externally supplied stimuli and provides with a drastic stage to the tumor growth. A parabolic-ODE system modelling tumor-induced angiogenesis is proposed in this connection [9], using the endotherial-cell density per unit area n, the TAF (tumor angiogenic factors) concentration f , and the matrix macromolecule fibronectin concentration c, that is, n t = Dn − ∇ · (χ(c)n∇c) − ρ0 ∇ · (n∇ f ) f t = βn − μn f, ct = −γnc

in  × (0, T ),

(2.12)

where χ(c) =

χ0 1 + αc

denotes the chemotactic sensitivity and D, ρ0 , β, μ, γ, χ0 , α are positive constants. System (2.11) is formulated as an evolution equation with strong dissipation [205, 212, 430]. There is also an approach from the comparison theorem [122]. Here we use the calculus of variation. We formulate the problem as the parabolic-ODE system qt = ∇ · (∇q − q∇ϕ(v)), vt = q in  × (0, T )  ∂q  = 0, (q, v)|t=0 = (q0 (x), v0 (x)), ∂ν ∂

(2.13)

52

2 Time Relaxization

where  ⊂ Rn is a bounded domain with smooth boundary ∂, ν is the outer unit normal vector, q0 > 0 and v0 are smooth function on , and ϕ : R → R is a smooth function [378]. We impose the compatibility condition  ∂v0  = 0. ∂ν ∂ Then we obtain the null flux boundary condition  ∂q ∂ϕ(v)  =0 −q ∂ν ∂ν ∂ by a simple calculation using the ODE part. Although the form (2.13) is restrictive, several important cases of (2.11) are reduced to it. First, if g( p, w) = ( p − μ)w, w > 0, then we obtain (2.13) with v = log w, q = p − μ, ϕ(v) = A(ev ), A = χ. Next, if g( p, w) = p(μ − w), w < μ, then (2.13) follows from v = − log(μ − w), q = p, ϕ(v) = A(μ − e−v ),

A = χ.

Finally, if g( p, w) = − pw, w > 0, then (2.13) holds for v = − log w, q = p, ϕ(v) = A(e−v ),

A = χ.

System (2.12) is also transformed into a similar form, qt = ∇ · (∇q − q∇ϕ(v, w)), vt = q, wt = q in  × (0, T ).

(2.14)

In 1979, Rascle [312] studied (2.13) for ϕ(v) = −v. This system looks like the case of (2.1) where the diffusion of the second equation is neglected. Furthermore, the chmotaxis term on the right-hand side of the first equation has the negative sign, with the sensitivity ϕ(v) = −v regarded as the self-impulsive factor. In the actual interpretation, however, we replace −v by v. Thus u is attractive to the material v and this v is consumed by u itself. In [312], global-in-time solution is obtained using the Lyapunov function in the case of one-space dimension, and a related system of angiogenesis is studied by [119]. This method is applicable to (2.13) under the assumption ϕ ∈ C 3 (R), ϕ

≥ 0 ≥ ϕ ,   ∂ q0 , v0 ∈ C 2+α (), (q0 , v0 ) = 0 ∂ν ∂

(2.15)

2.3 Smoluchowski-ODE System

53

for 0 < α < 1. System (2.13) is equivalent to the one studied by [74], n t = ∇ · (∇n − nχ(c)∇c), ct = −cn in  × (0, T )  ∂n ∂c  − χ(c)  = 0, ∂ν ∂ν ∂ where c, n > 0, and χ = χ(c) is a C 1 —function satisfying χ(c) ≥ 0, cχ (c) + χ(c) ≥ 0. Global-in-time existence of the weak solution with the convergence q(·, t) → q 0 ≡

1 ||

 

q0

(2.16)

as t ↑ +∞ is derived formally, see also [75]. Similar to [312], this property arises in the classical sense when the space-dimension is one, using the continuous embedding, see J.L. Lions [219], L 2 (0, T ; H 1 ()) ∩ L ∞ (0, T ; L 2 ()) → L 4 (0, T ; L ∞ ()). This conclusion is a counter part of the result [430], that is if ϕ(v) = v, we have both global and blowup in finite time solutions depending on their initial data. We note that ϕ(v) = v does not satisfy ϕ (v) ≤ 0. If ϕ(v) = v, system (2.13) is a relative to (2.1) with the diffusion term −v in the second equation neglected: 1 u t = ∇ · (∇u − u∇v), τ vt = u − ||  ∂u ∂v  − u  = 0. ∂ν ∂ν ∂

 

u in  × (0, T )

The key structure of (2.13) is  dL 1 = − q −1 |∇q|2 + ϕ

(v)q|∇q|2 d x dt 2   1 L= q(log q − 1) + 1 − ϕ (v)|∇v|2 d x, 2 

(2.17)

and thus L can be a Lyapunov function. We can show the following theorems [378]. Theorem 2.3.1 If (2.15) holds, then there exists a unique solution to (2.13) such that q, v ∈ C 2+α,1+α/2 (Q T ) with q = q(x, t) > 0, provided that T is sufficiently small.

54

2 Time Relaxization

Theorem 2.3.2 If (2.15) holds and the space dimension n = 1, the solution in the previous theorem exists for all T > 0. Given tk ↑ +∞ and δ > 0, furthermore, we have tk ∈ (tk − δ, tk + δ) such that q(·, tk ) → q 0 for q 0 defined by (2.16) uniformly on . Similar results to Theorems 2.3.1 and 2.3.2 are valid to (2.14). Actually, we obtain the following theorem [369, 378]. Theorem 2.3.3 If 0 < n 0 , f 0 , c0 are C 2+α on , f0 >

β μ

  ∂ (n 0 , f 0 , c0 ) = 0, ∂ν ∂

(2.18)

and the space dimension n = 1, then there is a unique solution global-in-time to (2.12) with the initial boundary condition  ∂n ∂c ∂ f  D − χ(c)n − ρ0 n = 0, ∂ν ∂ν ∂ν ∂

(n, f, c)|t=0 = (n 0 (x), f 0 (x), c0 (x)).

Any tk ↑ +∞ and δ > 0, furthermore, admits tk ∈ (tk − δ, tk + δ) such that n(·, tk ) → n 0 ≡

1 ||

 

n0

uniformly on . In the other case of the first Assumption of (2.18), 0 < f0
0, and ϕ (v) ≤ 0 we obtain 

0 ≤

1 (q) − ϕ (v) |∇v|2 d x 2   t  1

+ dt · ϕ (v)q|∇v|2 + q −1 |∇q|2 d x = L(0), 0  2

(2.21)

and, then, it follows that ∇q 1/2  L 2 (0,t;L 2 ()) ≤ L(0)1/2 /2. Next, we have

d dt

 

q = 0 which implies q(t)1 = q0 1 .

(2.22)

We thus get  sup q(s)

1/2

t

2 + 0

s∈(0,t)

1/2 ∇q(s)1/2 22

ds

≤ C.

(2.23)

Now we show the following lemma. Lemma 2.3.1 Assume (2.15) and let 1 Q δ (x, t) = 2δ



t+δ

q(x, t )dt , δ > 0.

t−δ

Then it holds that lim Q δ (t) − q 0 ∞ = 0.

t↑+∞

(2.24)

Proof By (2.22) and (2.21) we have 



h(t)2 = h 0 2 , 0

∇h(t)22 dt < +∞

(2.25)

56

2 Time Relaxization 1/2

where h(t) = q(t)1/2 and h 0 = q0 . Then it holds that ∇ Q δ (·, t)1 ≤

1 2δ



t+δ

2h2 ∇h2 dt =

t−δ

2h 0 2 · 2δ



t+δ

∇h2 dt

t−δ

which implies ∇ Q δ (·, t)21 ≤ 4h 0 22 ·

1 2δ



t+δ

t−δ

∇h(·, t)22 dt → 0, t ↑ +∞

(2.26)

by (2.25). The equality 1 ||

 

Q δ (·, t) = q 0

(2.27)

implies (2.24) by n = 1.



The second relation of (2.25) implies Theorem 2.3.2. In fact, given tk ↑ +∞, we have 

tk +δ

lim

k→∞ tk −δ

∇h(t)22 dt = 0,

and, therefore, lim ∇h(tk )2 = 0, ∃tk ∈ (tk − δ, tk + δ),

k→∞

which implies the convergence of {h(·, tk )} to a constant. Then we obtain lim q(·, tk ) = q 0 by (2.25).

k→∞

Turning to Theorem 2.3.4, we use 1/2

1/2

 p L 4 (0,t;L ∞ ()) ≤ C3  p L ∞ (0,T ;L 2 ()) · ∇ p L 2 (0,T ;L 2 ())

(2.28)

 valid to n = 1 and



p = 0. Here, the constant C on the right-hand side independent

of T , see [210] p. 74 and p. 63. We apply (2.28) for p = q 1/2 − Since

  p22 = q1 −

1 ||

1 ||



q 1/2 .

2

 



q 1/2

≤ q1 , ∇ p = ∇q 1/2

2.3 Smoluchowski-ODE System

57

we obtain  0

T

4  1/2 1/2 q − 1 q dt ≤ C. ||  ∞

(2.29)

Let 1 q≡ ||

 

q = q 0.

(2.30)

Since  2  1 q 1/2 = q − q + h q 1/2 − ||  2    2q 1/2 1 1/2 1/2 h=q− q + q ||  || 

(2.31)

inequality (2.29) means 

T

q − q + h2∞ dt ≤ C.

0

(2.32)

Here we have ∇h2

 2 = q 1/2 · ∇q 1/2 2 ||    1/2 1 ≤2 q ∇q 1/2 2 = 2q 1/2 ∇q 1/2 2 || 

by (2.31), and, therefore, 

T

0

∇h22 dt ≤ C.

(2.33)

h − h2∞ ≤ C

(2.34)

By n = 1 inequality (2.33) implies  0

T

where 1 h= ||



 

h=q−

1 ||

2

 

q

1/2

,

58

2 Time Relaxization

and, therefore, 2   T  1 q(·, t) − a(t)2∞ dt ≤ C, a(t) = q(·, t)1/2 . ||  0

(2.35)

Now we show the following lemma. Lemma 2.3.2 Under the Assumptions of (2.15) and (2.19) it holds that d q22 + ∇q22 ≤ b(t), dt



T

b(t)dt ≤ C.

(2.36)

0

Proof Inequality q(x, t) ≥ 0 implies v(x, t) ≥ v0 (x)

in Q T =  × (0, T )

(2.37)

while ϕ (v(x, t)) is uniformly bounded by ϕ ≤ 0 ≤ (ϕ ) :   ϕ (v(x, t)) ≤ C. Here we use 1 d q22 = 2 dt



(2.38)



∇q · (∇q − q∇ϕ(v))   = −∇q22 + a(t) ϕ (v)∇q · ∇v + (q − a(t))ϕ (v)∇q · ∇v 

q · qt = −





to derive



     (q − a(t))ϕ (v)∇q · ∇v  ≤ q − a(t)∞ · C∇q2   



1 ∇q22 + C 2 q − a(t)2∞ 4

(2.39)

by (2.21) and (2.38). We have, on the other hand,          

1/2 1/2   = 2a(t) a(t) ϕ (v)∇q · ∇v ϕ (v)q ∇v · ∇q        ≤ 2C ϕ

(v)1/2 q 1/2 |∇v · ∇q 1/2 |   ≤C ϕ

(v)q |∇v|2 + |∇q 1/2 |2 d x 

(2.40)

by (2.19). Now we apply (2.35) and (2.23), (2.21) to (2.39) and (2.40), respectively, and then, obtain the result. 

2.3 Smoluchowski-ODE System

59

Inequality (2.36) implies 

T

sup q(t)2 +

t∈(0,T )

0

∇q22 dt ≤ C

(2.41)

and then it follows that  0

T

q − q2∞ dt ≤ C

(2.42)

from n = 1. Now we apply (2.28) to p = q − q which results in q − q L 4 (0,t;L ∞ ()) ≤ C.

(2.43)

Now we show the following lemma. Lemma 2.3.3 Under the Assumptions of (2.15) and (2.19) it holds that d q − q44 ≤ C(q − q4∞ + q − q2∞ ). dt Proof Letting A(q) = (q − q)4 and a(q) = A

(q) = 12(q − q)2 , we obtain d dt



 

A(q) =





A (q)qt = −



A

(q)∇q · (∇q − q∇ϕ(v))

which means    d 2 A(q) + a(q)|∇q| = a(q)q∇q · ∇ϕ(v) dt      = a(q)(q − q)∇q · ∇ϕ(v) + q a(q)∇q · ∇ϕ(v) = I + I I. 



Here, inequality (2.38) implies    

1/2

1/2  |I | =  a(q)(q − q)(−ϕ (v)) ∇q · (−ϕ (v)) ∇v    1/2 ≤ (2C)1/2 a(q)2 |q − q|2 (−ϕ (v))|∇q|2 

1/2



≤ C a(q) (q − q)∞ · a(q)|∇q|   1 ≤ a(q)|∇q|2 + (C )2 a(q)1/2 (q − q)2∞ 4  1/2

2

(2.44)

60

2 Time Relaxization

and 

|I I | ≤ q · a(q)(−ϕ (v))1/2 ∇q · (−ϕ (v))1/2 ∇v   1 ≤ a(q)|∇q|2 + Ca(q)1/2 2∞ . 4  Then it holds that  1 A(q) + a(q)|∇q|2 2  

≤ C a(q)1/2 (q − q)2∞ + a(q)1/2 2∞ ,

d dt



(2.45) 

and, hence, (2.44). Proof of Theorem 2.3.4: We have  0



q(t) − q4∞ + q(t) − q2∞ dt < +∞

(2.46)

by (2.42) and (2.43). Here we take tk ↑ +∞ in Lemma 2.3.1 to apply (2.44) and (2.46). It holds for t > tk that q(t) − q44 ≤ q(tk ) − q44 + C





tk

q(t ) − q4∞ + q(t ) − q2∞ dt .

We obtain (2.20) by making t ↑ +∞ and then k → ∞.



2.4 Harmonic Heat Flow Several geometric flows are known to have the quantized blowup mechanism [352]. In the harmonic heat flow case, the total energy acts as the Lyapunov function. Differently from (1.8), this Lyapunov function is non-negative, and then type (II) blowup rate arises from this non-negativity. In this connection, the solution constructed by [207] to (1.8) has the bounded total free energy. This property holds in the case of λ∈ / 4πN as is shown in the proof of Theorem 1.2.2. If we take the flat torus  = R2 /aZ × bZ, a, b > 0 and the (n − 1)-dimensional sphere S n−1 = {x ∈ Rn | |x| = 1} as the domain and the target, respectively, this flow is described by u = u(x, t) :  × [0, T ) → S n−1 ⊂ Rn

2.4 Harmonic Heat Flow

61

satisfying u t − u = u |∇u|2 , |u| = 1 in  × (0, T ).

(2.47)

In this case, it holds that u t 22

1 d ∇u22 = + 2 dt



1 u · u t |∇u| = 2 

 

2



 ∂ 2 |u| |∇u|2 = 0 ∂t

and hence dE = − u t 22 ≤ 0, dt

E=

1 ∇u22 . 2

(2.48)

Thus, the above E casts the Lyapunov function to (2.47). The stationary solution to (2.47) is called the harmonic map. In the general setting, we take the m-dimensional compact Riemannian manifold (, g) and the compact Riemannian manifold N without boundaries. By Nash’s theorem this N is isometrically imbedded in Rn for large n. We define the Sobolev space composed of a class of the mappings from  to N provided with the finite energy as in the previous section, that is

H 1 (, N ) = u ∈ H 1 (, Rn ) | u ∈ N , a.e. on   1 |∇u|2 dv , E(u) = 2  where dv is a volume element of (, g). We call a mapping u ∈ H 1 (, N ) (weakly) harmonic if   d E((u + εφ)) =0 dε ε=0 for any φ ∈ C0∞ (, Rn ), where  : U → N is a smooth nearest point projection from some tubular neighborhood U of N to N . This relation is equivalent to saying that u is a weak solution of the Euler-Lagrange equation − u = A(u)(∇u, ∇u)

on ,

(2.49)

sometimes called the harmonic map equation, where  and A(u)(·, ·) denote the Laplace-Beltrami operator on (M, g) and the second fundamental form of the imbedding N → Rn at y ∈ N , respectively. Harmonic heat flow is the mapping u = u(x, t) :  × [0, T ) → N ⊂ Rn satisfying u t = −δ E(u) and, therefore, the harmonic map is regarded as a stationary state of the harmonic heat flow.

62

2 Time Relaxization

First, we have the energy quantization for the sequence of harmonic maps [89, 184, 247, 307–309, 352, 408]. Theorem 2.4.1 Let {u k }k be a harmonic map sequence satisfying sup E(u k ) < +∞. k

Then, passing to a subsequence we assume u k  u in H 1 (, N ) weakly to some map u ∈ H 1 (, N ). This u is a harmonic map, and there exist p

• p-sequences of points {xk1 }, . . . , {xk } in  p • p-sequences of positive numbers {δk1 }, . . . , {δk } converging to 0 1 p • p-non-constant harmonic maps {ω }, . . . , {ω } : S 2 → N satisfying lim E(u k ) = E(u) +

k→∞

p

E 0 (ω j )

j=1

 ⎫ ⎧ ⎨ δ i δ j xki − xkj  ⎬ k , k, lim max = +∞ j k→∞ i= j ⎩ δ j δ i δki + δk ⎭ k k     p j · − x k j j lim u k − u − ω − ω (∞) j k→∞ δk j=1 where E 0 (ω) =

1 2

=0

(2.50)

H 1 (,N )

 S2

|∇ω|2 dv S 2 .

The first equality of (2.50) is an energy identity which says that there is no unaccounted energy loss during the iterated rescaling process near the point of singularity, sometimes referred to as the bubbling process, and that the only reason for failure of strong convergence to the weak limit is the formation of several bubbles due to the non-constant harmonic maps ω j : S 2 → N , j = 1, . . . , p. Differently from (1.38), j there is a possibility that some of xk , j = 1, . . . , p, converge to the same point, k and this process is classified into two cases—the separated bubbles and the bubbles on bubbles [376]. The ε-regularity and the monotonicity formula are known to the harmonic heat flow, similarly to the simplified system of chemotaxis (1.8). In the case of (2.47) for  = R2 /aZ × bZ, first, there is ε0 > 0 such that u = u(x, t) is smooth in B R/2 × [0, T ], provided that sup E(u(·, t), B R ) < ε0 , t∈[0,T ]

2.4 Harmonic Heat Flow

63

where E(u, B R ) =

1 2

 |∇u|2 ,

B R = B(0, R).

BR

Second, E(u(·, T ), B R ) ≤ E(u 0 , B2R ) + C E 0 T /R 2 ,

E0 =

1 ∇u 0 22 2

holds with C > 0. These analytic structures guarantee that there is a weak solution global-in-time with a finite number of sigular points in  × [0, +∞), and then we will obtain a homotopy between the initial state and the expected ultimate state, see [351, 352]. Similarly to the chemotaxis system, (1.8) with (1.132), the non-negativity of E implies 1 sup ∇u(·, t)22 + 2 t∈[0,T )



T 0

u t (·, s)22 ds ≤

1 ∇u 0 22 2

= E0

(2.51)

by (2.48). Then, there is tk ↑ T satisfying (T − tk ) u t (tk )22 → 0

(2.52)

because otherwise it holds that 

T

0

u t (·, s)22 ds = +∞,

a contradiction. For this u k = u(·, tk ), it is known, see [401], that the conclusion of Theorem 2.4.1 arises. The hyper-parabola also works, and thus there are similarities and differences between (1.8) and (2.47). More precisely, we have 

 

|u t |2 ϕ2 +





∇u · ∇(u t ϕ2 ) =

u t · u|∇u|2 ϕ2    ∂ 2 1 |u| |∇u|2 ϕ2 = 0 = 2  ∂t 

with 

 ∇u · ∇(u t ϕ ) = 2





(∇u · ∇u t )ϕ + [(u t · ∇)u] · ∇ϕ2   1 d 2 2 = |∇u| ϕ + [(u t · ∇)u] · ∇ϕ2 2 dt   2



64

2 Time Relaxization

from (2.47), and, therefore,    1 d |u t |2 ϕ2 + |∇u|2 ϕ2 + [(u t · ∇)u] · ∇ϕ2 = 0 2 dt    for each ϕ ∈ C 1 () which implies that   1/2   1 d  2 2 2 2 2 2 2  |∇u| ϕ  ≤ u t 2 ϕ∞ + u t 2 · |∇u| |∇ϕ |  2 dt   √ 1/2 ≤ u t 22 ϕ2∞ + 2u t 2 · E 0 ∇ϕ2 ∞ , (2.53) and then it follows that 

T

   d  2 2  |∇u| ϕ  dt < +∞.  dt 

0

Thus μ(d x, t) = |∇u(x, t)|2 d x ∈ C∗ ([0, T ], M()) follows again from (2.51). Using tk ↑ T satisfying (2.52), we obtain μ(d x, T ) =



m(x0 )δx0 (d x) + f (x)d x

(2.54)

x0 ∈S

with a finite set S and 0 ≤ f = f (x) ∈ L 1 (). The above described result [401] guarantees the energy quantization. More precisely, m(x0 ) > 0 is a finite sum of the energies of non-constant harmonic maps: S 2 → N . Inequality (2.53) implies also      2  |∇u(·, t)|2 ϕ2 − f (x)ϕx0 ,R − m(x0 ) x0 ,R     T 1/2  T √ 1/2 2 2 ≤ 2C u t (·, s)2 ds + 2 2E 0 u t (·, s)2 ds · C(T − t)1/2 /R t

t

for x0 ∈ S with C > 0 independent of 0 < R  1. Then it follows that  lim t↑T



|∇u(·, t)|2 ϕ2x0 ,b R(t) = m(x0 )

from (2.51) again, where b > 0 is arbitrary and R(t) = (T − t)1/2 . The hyperparabola thus arises and we obtain type (II) blowup rate at each x0 ∈ S similarly to (1.8) provided with (1.132).

2.5 Normalized Ricci Flow

65

2.5 Normalized Ricci Flow The normalized Ricci flow describes an evolution in time of the metric g = g(t) on a compact Riemannian manifold. If  is a compact Riemannian surface without boundary, this flow is given by ∂g = (r − R) g, ∂t

t >0

(2.55)

where R = R(·, t) stands for the scalar curvature of (, g(t)) and r = r (t) represents the average scalar curvature given by  r=



R(·, t) dμt   dμt

with the volume element μ = μt . R. Hamilton [154] introduced the above flow to approach the Poincaré conjecture, and this idea was realized later [298–300]. In the case that  is a compact Riemannian surface is described in [155], and it is shown that the solution to (2.55) exists globally in time, converges in C ∞ —topology as t ↑ +∞, and the scalar curvature of the limit metric is constant. Here, we describe an arguement using the analytic form of (2.55). First, we obtain ∂R = t R + R(R − r ) ∂t and, therefore, R(·, t) > 0 everywhere on  follows from R(·, 0) > 0 everywhere on , where t denotes the Laplace-Beltrami operator associated with g = gt . Henceforth, we deal with this case. Then, from Gauss-Bonnet’s theorem there follows  R(·, t) dμt = 4 πχ(). (2.56) 

Here, χ() = 2 − 2 k() stands for the Euler characteristic of , and hence k() is the genus of . Since R(·, t) > 0 in , this formula gives k() = 0, and then the uniformization theorem reduces the problem to the case  = S 2 , g(t) = ew(·,t) g0 , where S 2 is the two dimensional sphere, g0 is its standard metric, and w = w(·, t) is a smooth function. In this case, the scalar curvature R0 corresponding to the metric g0 is a constant, and it is related to R = R(·, t) through R = e−w (−w + R0 )

(2.57)

66

2 Time Relaxization

with  = g0 . Here, we obtain  S2

R(·, t) dμt = 8π

(2.58)

by (2.56) and hence r=

8π 8π = w , dμ 2 2 t S S e dx

(2.59)

setting d x = dμg0 . Finally, we have |S 2 |R0 = 8 π

(2.60)

because (2.58) holds and R0 is a constant. By pluging (2.57) into (2.55) and using (2.59)–(2.60), we end up with  w  e 1 ∂ew  = w + 8π in S 2 × (0, T ). − 2 w ∂t |S | S2 e

(2.61)

The result in [155] thus reads as follows. The solution to (2.61) exists globally in time and w(·, t) → w∞ as t ↑ +∞ in C ∞ —topology, with w∞ standing for a stationary solution:   w∞ e 1 in S 2 . (2.62) − w∞ = 8π  w − 2 ∞ |S | e 2 S From the viewpoint of dynamical systems, the proof of [155] consists of three ingredients: extension of the solution globally in time, compactness of the orbit, and uniqueness of the ω-limit set. All the steps are based on the geometric structure of (2.55), involving Harnack’s inequality for the scalar curvature, monotonicity of an awkward geometric quantity called “entropy”, soliton solutions of the Ricci flow, the modified Ricci flow, and so forth. There are, however, several complementary analytic arguments. First, the third step originally achieved by modifying (2.55) via a transformation group may be replaced by the uniqueness of the solution to  − w = 8π

ew 1  −  2 w  e S  S2



 in

In fact, since d dt

 S2

ew = 0

S2,

S2

w = 0.

(2.63)

2.5 Normalized Ricci Flow

67

follows from (2.61), the stationary solution w = w∞ to (2.61) constitutes of (2.62) with   w∞ e = ew0 , w|t=0 = w0 (x). S2

S2

This w∞ , on the other hand, must be a constant, provided that only the trivial solution w = 0 is admitted to (2.63). The uniqueness of the steady state of (2.61), thererfore, means that (2.63) implies w = 0. This property is actually the case because the metric on S 2 with constant Gaussian curvature is the standard one. Thus the steady state of (2.61) is unique:  1 ew0 . w∞ = || S 2 Next, a gradient estimate of the form |∇ S 2 w| ≤ C

(2.64)

is obtained using the symmetry of S 2 , that is the moving sphere method, with C depending only on w0 = w(·, 0), see [20]. This estimate, combined with the argument in [431] based on Harnack’s inequality, induces also global-in-time existence of the solution to (2.61) and also the compactness of the orbit. Then, what happens to  w  e 1 ∂ew = w + λ  w − ∂t || e

in  × (0, T ),

(2.65)

where λ > 0 is a constant and  is a compact Riemannian surface without boundary? Unless λ = 8π and  = S 2 , it is not the normalized Ricci flow (2.61) any more. An estimate like (2.64) may not be obtained because of the lack of the symmetry of . The arguments of [155] using the geometric structure such as the covariant and Lie derivatives, Bochner-Weitzenböck’s formula, and so forth, see [71], are also invalid, and even global in time existence of the solution is not obvious. Similar to (2.61), however, we have  d ew = 0, dt  and, therefore, λ w e

r=

68

2 Time Relaxization

is a constant. Under the change of variables u = r ew and t = r −1 τ , and writing t for τ , problem (2.65) is transformed into u t =  log u + u −

1 ||

 

u

in  × (0, T )

(2.66)

with  

u(·, t) = λ.

(2.67)

This form, (2.66), is a non-local perturbation of the logarithmic diffusion equation u t =  log u

in  × (0, T )

(2.68)

which also describes the evolution of surfaces by Ricci flow [155]. There are also some other physical problems which could be described by (2.68); the spread over a thin colloidal film at a flat surface [41, 82, 414], the modelling of the expansion of a thermalized cloud of electrons [225], and the central limit approximation to the Calerman’s model of the Boltzmann equation [209, 224]. Due to a variety of applications logarithmic diffusion equation (2.68) has attracted the interests, see for instance [403]. The spatially homogeneous part of (2.66) is formulated by the linear non-local ODE,  1 u in  × (0, T ) ut = u − ||  but the non-local term is removable by (2.67). The regions + (t) = {x ∈  | u(x, t) > λ/||} − (t) = {x ∈  | u(x, t) < λ/||} are invariant in t, and, therefore, the extinction limt↑T inf  u(·, t) = 0 arises before the blowup limt↑T sup u(·, t) = +∞ comes. This fact is also known to (2.68), and we obtain u(·, t)1 = u 0 1 − 4πt in the case of  = R2 , see [404] and the references therein. Equation (2.66), however, may be written as 1 u t = (log u − v), −v = u − ||  v=0 

 

u in  × (0, T ) (2.69)

2.5 Normalized Ricci Flow

69

which is to be compared with the Smoluchowski-Poisson Eq. (1.8) formulated by (1.32),  1 u t = ∇ · (u∇(log u − v)) , −v = u − u in  × (0, T ) ||   v = 0. (2.70) 

Actually, (2.66) is again a model (B) equation formulated by Helmholtz’ free energy 

1 u(log u − 1) − (−)−1 u, u 2    1 −1 v = (−) u ⇔ −v = u − u, v = 0. ||  

F(u) =

(2.71)

From this fact, then, it is easy to derive the total mass conservation and the decrease of the free energy, that is d dt

 

u = 0,

d F(u) = − dt

 

|∇(log u − v)|2 ≤ 0.

(2.72)

Concerning the Smoluchowski-Poisson Eq. (2.70), there is a fundamental property obtained by the method of symmetrization that is (1.49) which results in the formation of collapse with the quantized mass. This control in time of the local L 1 —norm of the solution u = u(·, t) to (2.69) is not available. The equivalent form (2.66), however, is provided with the comparison principle thanks to (2.67). An important consequence of this property is the monotonicity formula of the Benilan-Crandall type, ∂ ∂t



u et − 1

 ≤ 0,

(2.73)

which guarantees the point-wise convergence of u(x, T ) = lim u(x, t) t↑T

in the case of T = Tmax < +∞, where Tmax ∈ (0, +∞] denotes the existence time of the solution. The following theorem [193, 194] is obtained by this structure and Trudinger-Moser-Fontana’s inequality [118],    v = 0 > −∞ inf J8π (v) | v ∈ H 1 (),   1 2 Jλ (v) = ∇v2 − λ log ev + λ(log λ − 1). 2 

(2.74)

70

2 Time Relaxization

Theorem 2.5.1 If 0 < λ ≤ 8π, then the solution u = u(·, t) to (2.66) satisfies the uniform estimates 0 < u(x, t), u(x, t)−1 ≤ C

in  × (0, T ).

(2.75)

The solution w = w(·, t) to (2.65) with 0 < λ ≤ 8π, therefore, exists global-intime, and the orbit O = {w(·, t)}t≥0 is compact in C(). The ω-limit set of O is thus non-empty, connected, comapct, and contained in the set of stationary solutions, and in particular, any tk ↑ +∞ admits {tk } ⊂ {tk } and w∞ = w∞ (x) satsifying 

ew 1 − w = λ  w − || e with

 

ew =

 

 in 

(2.76)

ew0

such that w(·, tk ) → w∞ uniformly on . If only w = 0 satisfies (2.76) with  

w = 0,

therefore, w∞ is a constant and it holds that w(·, t) → w∞ uniformly on  as t ↑ +∞. The asymptotic behavior w = w(·, t) to λ ≤ 8π is thus controlled by the uniqueness of the stationary solution. This property holds for 0 < λ ≤ 8π if either b π  = S 2 or  = R2 /aZ × bZ, ≤ ≤ 1, see [55, 68, 217, 218]. 4 a The proof of Theorem 2.5.1 for the critical case λ = 8π relies on the following lemma [45, 274, 276] of which proof is given later. Here we recall (2.71). Lemma 2.5.1 Let  be a compact Riemannian surface without boundary isometrically embedded in R N , and suppose that {u k } is a family of positive measurable functions on  satisfying   u k 1 = 8π, F(u k ) ≤ C, lim (−)−1 u k , u k = +∞ k→∞  N lim xu k = 8πx∞ ∈ R .

k→∞ 

Then it holds that x∞ ∈  and u k (x)d x  8πδx∞ (d x) in M(). Now we show the following lemma. Lemma 2.5.2 In (2.65), 0 < λ ≤ 8π, it holds that lim inf t↑∞ w(t) > −∞.

2.5 Normalized Ricci Flow

71

Proof Rewriting w = log u in (2.66) and (2.67), we reach ∂ew λ = w + ew − , ∂t ||

 

ew = λ.

(2.77)

For the moment, we work with (2.77). First, p = wt satisfies pt = e−w p + p − p 2 in  × (0, T ) for 0 < t < T = Tmax which implies p = wt (·, t) ≤

et in , et − 1

(2.78)

equivalent to (2.73). Using Jλ (v) in (2.74) for v = w − w with w=

1 ||

 

w,

next, we obtain the field functional J˜λ (w) = Jλ (w − w) − λ(log λ − 1)     1 = ∇w22 − λ log ew − w 2 

(2.79)

which casts the Lyapunov function to (2.77). The solution w = w(·, t) to (2.65), thus, satisfies  w    e 1 d ˜ ∇w · ∇wt − λ  w − ew wt2 , (2.80) wt d x = − Jλ (w) = || dt   e where Fontana’s inequality (2.74) is applicable as J˜8π (w(·, t)) ≥ −C, 0 ≤ t < T. Since we obtain J˜λ (w) = (8π − λ) log

 

 ew−w + J˜8π (w)

= (8π − λ)(log λ − w) + J˜8π (w) in (2.79), the lemma is obvious for 0 < λ < 8π by (2.80) and (2.81).

(2.81)

72

2 Time Relaxization

Let λ = 8π. The elementary inequality ew w ≥ −e−1 , w ∈ R, now, implies  H (t) ≡



ew w ≥ −e−1 || .

By (2.65) and (2.81), next, we have   ∂ew d w e wt + e wwt d x = e + ·w dt   ∂t      8π w + ew − w = − ∇w22 + ew w − 8 πw = ||    ≤ ew w + 8πw + C = H + 8πw + C. (2.82)

dH = dt



w

w



If lim inf w(t) = −∞ t↑∞

is the case, inequality (2.78) implies the existence of tk ↑ +∞ in tk+1 > tk + δ with δ > 0 such that 8πw(t) ≤ −C − k, k = 1, 2, · · · , tk − δ < t < tk in (2.82), and, therefore, d  −t  e H ≤ −ke−t , tk − δ < t < tk . dt 

tk

Operating

(2.83)

· dt, t ∈ (tk − δ, tk − δ/2), to (2.83) implies

t

e−tk H (tk ) ≤ e−t H (t) + k(e−tk − e−t ). Then it holds that H (t) ≥ et−tk H (tk ) + k(1 − et−tk ) ≥ −e−δ−1 || + k(1 − e−δ/2 ), tk − δ < t < tk − δ/2, and, in particular,  lim

inf

k→∞ t∈(tk −δ,tk −δ/2) 

(ew w)(·, t) = +∞.

(2.84)

2.5 Normalized Ricci Flow

73

We have, on the other hand, (2.79) with λ = 8π and (2.81), which implies ∞ 



tk −δ/2

dt

k=1 tk −δ



(ew wt2 )(·, t) ≤







dt 0



(ew wt2 )(·, t) < +∞.

(2.85)

It holds also that ew wt 21 ≤



ew ·



 

ew wt2 = 8π

 

ew wt2 ,

and, therefore, 

tk −δ/2

lim

k→∞ tk −δ

(ew wt )(·, t)21 dt = 0

(2.86)

by (2.85). From (2.84) and (2.86) there is tk ∈ (tk − δ, tk − δ/2) such that  lim

k→∞ 

e

w

w ∂e

(·, tk ) = 0. = +∞, lim k→∞ ∂t 1

w(·, tk )

(2.87)

Using u = ew and u k = u(·, tk ), we rewite the first relation of (2.87) as  lim

k→∞ 

u k log u k = +∞,

(2.88)

while (2.72) implies F(u k ) ≤ F(u 0 ).

(2.89)

We thus obtain lim (−)−1 u k , u k  = +∞

k→∞

by (2.88)–(2.89). We have also u k 1 = 8π by (2.77) with λ = 8π, and, furthermore, may assume x∞ ∈ R N such that  lim

k→∞ 

xu k = 8πx∞ ∈ R N

up to a subsequence. By Lemma 2.5.1 we have x∞ ∈  and it holds that

ew(·,tk ) = u k  8 πδx∞ in M() by a subsequence.

(2.90)

74

2 Time Relaxization

Here we apply the second relation of (2.87) and (2.90) to both sides of (2.65) for t = tk . From the elliptic L 1 estiamte [38] it holds that w(·, tk ) → 8πG(·, x∞ )

in W 1,q (), 1 ≤ q < 2

(2.91)

where G = G(x, x ) stands for the Green’s function to v = (−)−1 u defined by (2.71). Using the asymptotics of G(x, x∞ ) as x → x∞ , we can derive  lim

k→∞ 

ew(·,tk ) = +∞

from (2.91), see [37], a contradiction by (2.77), where λ = 8π.



To complete the proof of Theorem 2.5.1, first, we show ∇w2 ≤ C, 0 ≤ t < T

(2.92)

which is obvioius for 0 < λ < 8π by (2.80), (2.81), and   λ 1 λ ˜ 1− ∇w22 + J˜λ (w) = J8π (w). 2 8π 8π Even in the case of λ = 8π we obtain   1 1 2 w−w ˜ ∇w = ∇w22 + 8πw − 8π log(8π). e J8π (w) = 2 − 8π log 2 2  Then, (2.92) follows from (2.80) and Lemma 2.5.2. Jensen’s inequality, next, assures  exp

1 ||



  1 λ , w ≤ ew = ||  || 

and, therefore, |w(t)| ≤ C by Lemma 2.5.2. Poincaré-Wirtinger’s inequality then implies  log u H 1 () = w H 1 () ≤ C, and hence

p log u − p log u , e ≤ C p, e 1

1

p>0

by Fontana’s inequality. In particular, it holds that u(·, t) p , u −1 (·, t) p ≤ C p ,

p ≥ 1,

(2.93)

and, then, Moser’s iteration scheme implies (2.75), see [193].



2.6 Concentration of Probability Measures

75

2.6 Concentration of Probability Measures This paragraph is devoted to the proof of Lemma 2.5.1. First, we show the following lemma [63]. Lemma 2.6.1 Let  → R N be a compact Riemannian surface without boundary, and δ > 0, γ0 ∈ (0, 1/2), and ε ∈ (0, 1) are given. Then, there is K = K (ε, δ0 , γ0 ) > 0 such that for v ∈ H01 () satisfying  S1 , S2 ⊂ : closed, dist (S1 , S2 ) ≥ δ0 ,

Si

ev

e

v

≥ γ0 , i = 1, 2

(2.94)

it holds that  log



ev ≤

1 + 2ε ∇v22 + K . 32π

(2.95)

Proof Letting g1 , g2 ∈ C0∞ (R2 ) be such that supp g1 ∩ supp g2 = ∅, gi = 1 on Si , i = 1, 2, we obtain  

ev ≤

1 γ0



ev ≤ Si

1 γ0

 

egi v , i = 1, 2.

Then, Fontana’s inequality implies 

1 log ||

 

e

gi v

 ≤

1 ∇(gi v)22 + C, i = 1, 2. 16π

(2.96)

Here we assume ∇(g1 v)2 ≤ ∇(g2 v)2 , g = g1 + g2 , without loss of generality, to get  1 1 ∇(g1 v)22 + ∇(g2 v)22 = ∇(g1 + g2 )v22 2 2 1 = g 2 |∇v|2 + 2∇g · ∇v + v 2 |∇g|2 d x 2  1+ε ≤ ∇v22 + Cv22 + Cε (2.97) 2

∇(g1 v)22 ≤

76

2 Time Relaxization

From (2.96) and (2.97) it follows that 

1 ||

log

 

ev

 ≤

1+ε ∇v22 + Cv22 + C 32π

(2.98)

in the case (2.94). Now we apply Rellich’s theorem to the second term on the righthand side on (2.98), to obtain (2.95). 

Let P() = ρ ∈ L 1 () | ρ ≥ 0, ρ1 = 1 be the space of absolutely continuous probability measures. We put 1 I(ρ) = 2



1 G(x, x )ρ ⊗ ρ − 8π ×

 

ρ log ρ, ρ ∈ P(),

recalling the Green’s function G = G(x, x ) to v = (−)−1 u defined by (2.71). For 0 ≤ u ∈ L 1 (), u1 = 8π, it holds that ρ = u/8π ∈ P() with the relation    1 1

u(log u − 1) − G(x, x )u ⊗ u I(ρ) = − 64π 2  2 × 1 1 {1 − log(8π)} = − F(u) + constant. − 64π 2 64π 2 The dual form of Fontana’s inequality arises as sup {I(ρ) | ρ ∈ P()} < +∞, see the next section, while we have I = K + E/(8π) for K(ρ) =

1 2

 ×

G(x, x )ρ ⊗ ρ, E(ρ) = −

 

ρ(log ρ − 1).

With this Iβ = K + E/β, β > 8π, Lemma 2.6.1 is stated as follows. Lemma 2.6.2 Each d > 0 admits C = C(d) > 0 provided with the following property. Namely, given m > 0, we have β = β(m) > 8π such that if ρ ∈ P() satisfies 



dist(A1 , A2 ) ≥ d,

ρ ≥ m, A1

ρ≥m A2

for some measurable sets A1 , A2 ⊂ , then it holds that Iβ (ρ) ≤ C(d). Using Lemma 2.6.2, now we show the following lemma, which is equaivalent to Lemma 2.5.1.

2.6 Concentration of Probability Measures

77

Lemma 2.6.3 If {ρk } ⊂ P() assumes  lim K(ρk ) = +∞, lim I(ρk ) = I∞ > −∞, lim

k→∞

k→∞

xρk = x∞ ∈ R N

k→∞ 

then it holds that x∞ ∈  and ρk (x)d x  δx∞ (d x) in M(). Proof We take the concentration function of ρk = ρk (x) ∈ P() denote by  Q k (r ) = sup

y∈ ∩B(y,r )

ρk ,

to show lim {1 − Q k (r )} = 0, 0 < r  1.

(2.99)

k→∞

For this purpose we take  xk ∈ ,

∩B(xk ,r/2)

ρk = Q k (r/2),

regarding the compactness of . Since  1 − Q k (r ) ≤ 1 −



∩B(xk ,r )

ρk =

\B(xk ,r )

ρk

it holds that  min {Q k (r/2), 1 − Q k (r )} ≤ min

 ∩B(xk ,r/2)

ρk ,

\B(xk ,r )

 ρk .

Applying Lemma 2.6.2 for d = r/2, we have C > 0. Hence each m > 0 admits β = β(m) > 8π such that m ≤ min {Q k (r/2), 1 − Q k (r )}



Iβ (ρk ) ≤ C,

while we have Iβ (ρk ) =

8π β



  β − 1 K(ρk ) + I(ρk ) → +∞ 8π

from the assumption. Therefore, it holds that lim min {Q k (r/2), 1 − Q k (r )} = 0.

k→∞

78

2 Time Relaxization

We obtain, on the other hand, c0 > 0 such that Q k (r ) ≥ c0 r 2 , k = 1, 2, . . . , 0 < r  1 from the covering theorem of Vitali, and, hence (2.99). k  1 such that 1 − For the proof of x∞ ∈ , first, each 0 < r  1 admits  Q k (r/2) ≤ r by (2.99). This property implies for xk =



xρk that

     |xk − xk | =  (x − xk )ρk  ≤ |x − xk | ρk  ∩B(xk ,r )   |x − xk | ρk ≤ r + diam  · + \B(xk ,r )

\B(xk ,r/2)

ρk

= r + diam  · (1 − Q k (r/2)) ≤ (1 + diam ) r and hence limk→∞ |xk − xk | = 0. Thus we obtain x∞ ∈ . Similarly, each ζ = ζ(x) ∈ C() admits k  1 such that        ≤ ζ(xk ) − ζρ k   

∩B(xk ,r )

|ζ(xk ) − ζ(x)| ρk



|ζ(xk ) − ζ(x)| ρk ≤ ζ − ζ(xk ) L ∞ (B(xk ,r ))  + 2 · diam  · ζ∞ ρk +

\B(xk ,r )

\B(xk ,r/2)

≤ ζ − ζ(xk ) L ∞ (B(xk ,r )) + 2 · diam  · ζ∞r. Since ζ ∈ C() is uniformly continuous, we get       ζρk  = 0 lim ζ(xk ) − k→∞  

with r ↓ 0, and, therefore, ρk (x)d x  δx∞ (d x).



2.7 Summary We studied several models related to the Smoluchowski-Poisson equation, in accordance with the stationary and ODE parts. 1. Some non-stationary problems sharing the same stationary problem arise in biology and geometry which, however, obey different features from those of the Smoluchowski-Poisson equation.

2.7 Summary

79

2. Non-local parabolic equation arises as the other limit of the full system of chemotaxis, which is subject to the dis-quantized blowup mechanism. 3. Type (II) blowup rate, formation of sub-collapses, and possible collision of collapses are observed when the free energies are bounded in the energy quantization, such as harmonic heat flow and semilinear parabolic equation with critical Sobolev exponent. 4. In the harmonic heat flow, sub-collapses suffer fast collisions inside the hyperparabola. 5. There is a global-in-time compact orbit for the normalized Ricci flow even at the critical level of the total mass.

Chapter 3

Toland Duality

The collapse mass quantization, (1.26) to (1.8), has its origin in a stationary state (1.38) which provides a variational structure of duality between the particle density u and the field distribution v. These variational problems split into each component, and, then, the infinitesimal stability of the stationary state implies the dynamical stability. First, we observe the above duality in the full system of chemotaxis. Next, we develop the abstract theory, and then examine the continuity of the entropy functional.

3.1 Full System of Chemotaxis Revisited A full system of chemotaxis takes the form u t = ∇ · (∇u − u∇v), τ vt − v = u    ∂u ∂v − u , v  = 0 ∂ν ∂ν ∂

in  × (0, T ) (3.1)

where τ > 0 is a constant. The positivity of the solution u = u(x, t) > 0 is kept, and it holds that  d u = 0. (3.2) dt  Hence there arises the total mass conservation u(·, t)1 = u 0 1 ≡ λ. System (3.1) is provided with the Lyapunov function, L = L(u, v) defined by  L(u, v) =



u(log u − 1) +

1 ∇v22 − v, u . 2

© Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_3

(3.3)

81

82

3 Toland Duality

With this Lyapunov function, (3.1) is written as u t = ∇ · (u∇ L u (u, v)) , τ vt = −L v (u, v) in  × (0, T )    ∂ u L u (u, v), v  = 0. ∂ν ∂

(3.4)

System (3.4) is a model (B)—model (A) equation, see the next section, which results in  d L(u, v) + τ vt 22 + u |∇ (log u − v)|2 = 0 (3.5) dt  besides (3.4), where (u, v) = (u(·, t), v(·, t)) is the solution to (3.1) and  ,  denotes the duality,  v, u = uv. 

By (3.4) and (3.5), the stationary state of (3.1) is formulated by log u − v = constant, u1 = λ, u > 0 v = (−)−1 u

(3.6)

where the second equation stands for the Poisson part v|∂ = 0.

− v = u,

(3.7)

Using λev u= v e derived from the first equation of (3.6), we eliminate u in the second equation to reach the Boltzmann-Poisson equation λev − v =  v , e

v|∂ = 0.

(3.8)

Equation (3.8) is associated with the field functional Jλ (v) =

1 ∇v22 − λ log 2

 

 ev + λ(log λ − 1)

(3.9)

3.1 Full System of Chemotaxis Revisited

83

defined for v ∈ H01 (), that is δ Jλ (v) = 0 where   d . w, δ Jλ (v) = Jλ (v + sw) ds s=0 If the Poisson part (3.7) is replaced by 1 − v = u − ||



 ∂v  =0 ∂ν ∂



 u,



v = 0,

(3.10)

as in (1.8), the stationary state is described by    v e 1 , − v = λ  v − v = 0, ||  e

 ∂v  =0 ∂ν ∂

(3.11)

which is reduced to the stationary state of (2.65) if  is a compact Riemann surface without boundary. Besides the chemotaxis model, we find several elliptic eigenvalue problems with non-local term whereby the two-dimensional Laplacian competes with an exponential nonlinearity, for example, gauged Shrödinger equations [393, 394, 416] and the mean field of stationary point vortices [284]. The eigenvalue problem (3.8) is involved in a complex analysis and a theory of surfaces [360] where the quantized blowup mechanism is observed [254]. More precisely, in this problem, non-compact solution sequence arises only at the quantized value of λ ∈ 8πN, and the location of the blowup points is controlled by the Green’s function. This quantized blowup mechanism admits several approaches to the solution set to (3.8), the singular perturbation [16], the topological degree calculation [60, 61, 214], and variational methods [85, 88, 98, 353]. The uniqueness of the solution, on the other hand, holds if 0 < λ < 8π, see [19, 52, 359, 376]. It is proven by the bifurcation analysis combined with the isoperimetric inequality on surfaces [15], see Sect. 12.2 and the recent work . Since (3.8) is the Euler-Lagrange equation to the variational functional (3.9) defined for v ∈ H01 (), and the Morse index of its critical point v = v, denoted by i(v), indicates the maximum dimension of the linear subspace where the associated quadratic form   1 d2 2 δ Jλ (v)[w.w] = J λ (v + sw) 2 2 ds s=0 defined for w ∈ H01 () is negative definite. Actually, the Morse index of the solution v = v to (3.8) is shown to be equal to the number of eigenvalues in μ < 1 minus one of the eigenvalue problem  − φ = μuφ,

φ|∂ = constant,

∂

∂φ = 0, ∂ν

(3.12)

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3 Toland Duality

where λev u =  v, e

(3.13)

see [359]. Here, we emphasize that μ = 0 is the first eigenvalue of (3.12) with the corresponding eigenfuction being a constant. In fact, since  2  1 − u w− uw d x, w ∈ H01 () λ   

δ Jλ (v)[w, w] = 2

∇w22

the Morse index i(v) is equal to the number of eigenvalues in μ < 1, including multiplicities, of the eigenvalue problem    1 uw , − w = μu w − λ 

w|∂ = 0.

(3.14)

Putting Hc1 () = {φ ∈ H 1 () | φ = constant on ∂} and φ=w−

1 λ

 

uw ∈ Hc1 (),

we reduce (3.14) to  − φ = μuφ,

φ|∂ = constant,



uφ = 0.

(3.15)

Here, any eigenvalue of (3.12) is in μ ≥ 0, and the eigenfunction corresponding to the minimal eigenvalue μ = 0 is constant. Since μ = 0 is is not the eigenvalue in (3.15), i(v) is the number of eigenvalues of (3.12) in 0 < μ < 1. Since the weak form of (3.12) is given by φ ∈ Hc1 (), (∇φ, ∇ψ) = μ(φ, ψ)u , ∀ψ ∈ Hc1 ()   (∇φ, ∇ψ) = ∇φ · ∇ψ, (φ, ψ)u = uφψ, 



the min-max principle guarantees i(v) = max{k | μk < 1} − 1 μk = where φ2u = (φ, φ)u .

max Y ⊂Hc1 (), co-dim(Y )=k

min

φ∈Y \{0}

∇φ22 φ2u

,

3.1 Full System of Chemotaxis Revisited

85

Eliminating v in (3.6), we can reduce it to (−)−1 u = log u + constant, u > 0 in , u1 = λ.

(3.16)

This stationary state is nothing but the Euler-Lagrange equation of the variational functional with constraint, that is,  1 u(log u − 1) − (−)−1 u, u F(u) = 2  u = u(x) > 0, u1 = λ. (3.17) If v is a critical point of Jλ (v) defined for v ∈ H01 () by (3.9), then u given by (3.13) satisfies (3.16). Hence this u is a critical point of F(u) in (3.17). Its Morse index i(u) indicates the maximal dimension of of the linear space where the quadratic form    1 d2  δ F(u)[ϕ, ϕ] = F(u + sϕ) = u −1 ϕ2 − (−)−1 ϕ, ϕ 2 ds 2  s=0 2

defined for ϕ ∈ L 20 () is negative definite, where  L 20 () = {ϕ ∈ L 2 () |



ϕ = 0}.

Then the min-max principle guarantees i(u) = max{k | σk < 1} σk =

max Z ⊂L 20 (), co-dim(Z )=k

min

ϕ∈Z \{0}

ϕ2 −1 u

(−)−1/2 ϕ22

(3.18)

for  ϕ2u −1

=



u −1 ϕ2 .

(3.19)

Here we replace the inner product of L 2 () for the one associated with the norm (3.19), to take the orthogonal projection P : L 2 () → L 20 (), that is u Pv = v − λ

 

v,

and furthermore, the bounded linear operator T : L 20 () → L 20 (): T ϕ = Pu(−)−1 ϕ.

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3 Toland Duality

Then, (3.18) implies that i(u) is the number of eigenvalues in 0 < μ < 1, including multiplicities, of the eigenvalue problem T ϕ = μϕ, ϕ ∈ L 20 ()

(3.20)

which is equivalent to (3.14). The problem (3.14) is transformed into (3.15) by φ=w−

1 λ

 

uw

while (3.15) implies (3.14) by w = φ − φ|∂ . We thus end up with the followig theorem. Theorem 3.1.1 Let (u, v) be the stationary solution to (3.1), u 0 1 = λ, and regard u and v as the critical points of free energy F and field functional Jλ defined by (3.17) and (3.9), respectively. Then their Morse indicies coincide as i(u) = i(v). Theorem 3.1.1 is valid even when the Poisson part is replaced by (3.10), that is the other full system of chemotaxis (2.1) appeared in Sect. 2.1, provided with the Lyapunov function (2.2), see [364]. Such a spectral equivalence of variational problems, however, is a consequence of the general theory of dual variation. Before turning to the theory, we define the semi-stationary states. In fact, if if v is stationary in (3.1), then v = (−)−1 u,

(3.21)

which results in the Smoluchowski-Poisson equation in the form of u t = ∇ · (∇u − u∇v) , −v = u in  × (0, T )    ∂u ∂v − u , v  = 0 ∂ν ∂ν ∂

(3.22)

where the free energy F(u) defined by (3.17) casts the Lyapunov function: d F(u(·, t)) = − dt

 

u|∇(log u − v)|2 ≤ 0.

Actually, the Lyapunov function L(u, v) to (3.1) is reduced to F(u) in the semistationary state (3.21) as L|v=(− D )−1 u = F.

(3.23)

In the other semi-stationary state of (3.1), the u component is stationary, which results in λev (3.24) u =  v. e

3.1 Full System of Chemotaxis Revisited

87

Then (3.1) is reduced to the parabolic equation with nonlocal term, (2.4). The field functional Jλ (v) of (3.9) acts as the Lyapunov function, and it holds that d Jλ (v(·, t)) = −vt 22 ≤ 0 dt while L(u, v) is reduced to this Jλ (v) similarly: L|u= λev = Jλ . e

(3.25)

v

These relations, (3.23) and (3.25), are called the unfolding of the Legendre transformations in the general theory. We have, on the other hand, the minimality indicated by L(u, v) ≥ max {F(u), Jλ (v)} ,

(3.26)

where u1 = λ. In fact, the first inequality is a direct consequence of Schwarz’ inequality, while the second inequality is proven by Jensen’s inequality. We may write (3.26) as inf

u≥0, u1 =λ, v∈H01 ()

L(u, v) =

inf

v∈H01 ()

Jλ (v) =

inf

u≥0, u1 =λ

F(u)

(3.27)

of which boundedness for λ = 8π is referred to as the Trudinger-Moser inequality and its dual form. This boundedness is applicable to ensure the global-in-time existence of the solution to (3.1) in the case of λ = u 0 1 < 8π, see [364] and also Sect. 3.3. These structures are valid also to (2.1). We take L = L(u, v) of (3.3) for v ∈  H 1 () with  v = 0. Then, it holds that 

1 u(log u − 1) − (−)−1 u, u 2    1 2 v = Jλ (v) ≡ ∇v2 − λ log e + λ(log λ − 1), 2 

L|v=(− J L )−1 u = F(u) ≡ L|u= λev e

v

and (3.26). This time v = (−)−1 u stand for (3.10), and the boundedness of (3.10) arises for λ = 4π.

3.2 Lagrangian and Duality We are in position to formulate the general theory of dual variation. Let X be a Banach space over R. Its dual space and the duality paring are denoted by X ∗ and  ,  =  ,  X,X ∗ , respectively.

88

3 Toland Duality

Given F : X → [−∞, +∞], we define its Legendre transformation F ∗ : X ∗ → [−∞, +∞] by F ∗ ( p) = sup {x, p − F(x)} ,

p ∈ X ∗.

x∈X

Then, Fenchel-Moreau’s theorem guarantees that if F : X → (−∞, +∞] is proper, convex, lower semi-continuous, then so is F ∗ : X ∗ → (−∞, +∞], and the second Legendre transformation defined by   F ∗∗ (x) = sup x, p − F ∗ ( p) , p∈X ∗

x∈X

is equal to F(x), see [35, 94]. Here and henceforth, we say that F : X → (−∞, +∞] is proper if its effective domain defined by D(F) = {x ∈ X | F(x) ∈ R} is not empty; convex if F(θx + (1 − θ)y) ≤ θF(x) + (1 − θ)F(y) for any x, y ∈ X and 0 < θ < 1; and lower semi-continuous if F(x) ≤ lim inf F(xk ), k

provided that xk → x in X . Let F, G : X → (−∞, +∞] be proper, convex, lower semi-continuous, with the effective domains D(F) and D(G), respectively. Let D(F) = {x ∈ X | F(x) < +∞} D(G) = {x ∈ X | G(x) < +∞} and ϕ(x, y) = F(x + y) − G(x) for x ∈ D(G). Then, y ∈ X → ϕ(x, y) ∈ (−∞, +∞]

(3.28)

3.2 Lagrangian and Duality

89

is proper, convex, lower semi-continuous, and its Legendre transformation is defined by L(x, p) = sup {y, p − ϕ(x, y)}

(3.29)

y∈X

for p ∈ X ∗ . Thus L(x, ·) : X ∗ → (−∞, +∞] is proper, convex, lower semi-continuous. Sometimes −L(x, p) is referred to as the Lagrangian, but in this book, L(x, p) is called the Lagrangian directly. If (x, p) ∈ D(G) × X ∗ , then it holds that L(x, p) = sup {y + x, p − F(x + y) + G(x) − x, p} y∈X ∗

= F ( p) + G(x) − x, p .

(3.30)

Putting L(x, p) = +∞ for x ∈ / D(G), on the other hand, we obtain (3.30) for any (x, p) ∈ X × X ∗ . Thus we define J ∗ ( p) =



F ∗ ( p) − G ∗ ( p), p ∈ D(F ∗ ) +∞, otherwise

(3.31)

for p ∈ X ∗ . Then, it holds that inf L(x, p) = F ∗ ( p) − sup {x, p − G(x)}

x∈X

x∈X

= F ∗ ( p) − G ∗ ( p) = J ∗ ( p) / D(F ∗ ) by (3.30) for p ∈ D(F ∗ ). Let us note that this relation is valid even to p ∈ and (3.31). Similarly, we define J (x) =

G(x) − F(x), x ∈ D(G) +∞, otherwise

for x ∈ X , and obtain   inf L(x, p) = G(x) − sup x, p − F ∗ ( p)

p∈X ∗

p∈X ∗ ∗∗

= G(x) − F (x) = J (x)

(3.32)

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3 Toland Duality

for x ∈ D(G), which is valid even to x ∈ / D(G) by (3.30) and (3.32). Thus, we have D(J ) = {x ∈ X | J (x) = ±∞} = D(G) ∩ D(F)   D(J ∗ ) = p ∈ X ∗ | J ∗ ( p) = ±∞ = D(G ∗ ) ∩ D(F ∗ ) and inf x∈X L(x, p) = J ∗ ( p), p ∈ X ∗ inf p∈X ∗ L(x, p) = J (x), x ∈ X.

(3.33)

Relation (3.33) implies inf

(x, p)∈X ×X ∗

L(x, p) = inf ∗ J ∗ ( p) = inf J (x), p∈X

x∈X

(3.34)

called the Toland duality [399, 400]. Here, we call v, u of (3.30) the hook term. The functional ϕ(x, y) = F(x + y) − G(x) : X × X → [−∞, +∞] of (3.28) is called the cost function, which is convex in y-component. Differently from the above properties of the Toland duality, the cost function ϕ = ϕ(x, y) : X × X → [−∞, +∞] of the Kuhn-Tucker duality has the property that x ∈ X → ϕ(x, y) and y → ϕ(x, y) are concave and convex, respectively. In this case, this cost function defines the skewLagrangian by (3.29), L(x, p) = sup {y, p − ϕ(x, y)} , y

see Sect. 4.4. Variational Equivalence Above mentioned global theory can be localized by sub-differentials. First, given F : X → [−∞, +∞], x ∈ X , and p ∈ X ∗ , we define p ∈ ∂ F(x) ∗

F(y) ≥ F(x) + y − x, p , ∀y ∈ X ⇔ F ∗ (q) ≥ F ∗ ( p) + x, q − p , ∀q ∈ X ∗ .



x ∈ ∂ F ( p)

It is obvious that ∂ F(x) = ∅ implies x ∈ D(F), but if F : X → (−∞, +∞] is proper, convex, lower semi-continuous, then x ∈ ∂ F ∗ ( p)



p ∈ ∂ F(x),

(3.35)

3.2 Lagrangian and Duality

91

and Fenchel-Moreau’s identity F(x) + F ∗ ( p) = x, p

(3.36)

follows, see [94]. If p ∈ ∂ F(x), then x ∈ ∂ F ∗ ( p), and, therefore, F ∗ ( p) − x, p = −F(x). This relation implies L| p∈∂ F(x) = J,

(3.37)

where L(x, p) = F ∗ ( p) + G(x) − x, p for J (x) = G(x) − F(x),

J ∗ ( p) = F ∗ ( p) − G ∗ ( p).

Similarly, we obtain L|x∈∂G ∗ ( p) = J ∗ ,

(3.38)

and, thus, unfolding Legendre transformation (3.37) and (3.38) comparable to (3.23), (3.25) arises in this abstract framework. The minimality such as (3.26), on the other hand, follows immediately from (3.34). More precisely, we have L(x, p) ≥ J (x),

L(x, p) ≥ J ∗ ( p),

and, therefore,   L(x, p) ≥ max J (x), J ∗ ( p)

(3.39)

for any (x, p) ∈ X × X ∗ , and in this way, properties (3.23), (3.25), and (3.26) are obtained in the context of convex analysis. The first part of our local theory is the following variational equivalence. Let F, G : X → (−∞, +∞] be proper, convex, lower semi-continuous, and L = L(x, p) be defined by (3.30). Given (x, p) ∈ X × X ∗ , the sets of minimizers of p ∈ X ∗ and x ∈ X of the variational problems J (x) = inf ∗ L(x, p), p∈X

J ∗ ( p) = inf L(x, p) x∈X

are denoted by A∗ (x) and A( p). We say that x ∈ X and p ∈ X ∗ are critical points of J and J ∗ if ∂G(x) ∩ ∂ F(x) = ∅ and ∂G ∗ ( p) ∩ ∂ F ∗ ( p) = ∅, respectively, and that (x, p) is a critical point of L if 0 ∈ ∂x L(x, p), 0 ∈ ∂ p L(x, p).

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3 Toland Duality

Theorem 3.2.1 Under the above notations, first, it holds that A∗ (x) = ∂ F(x), A( p) = ∂G ∗ ( p)

(3.40)

for any (x, p) ∈ X × X ∗ . Next, the following conditions are equivalent. 1. (x, p) ∈ X × X ∗ is a critical point of L. 2. x ∈ X is a critical point of J and it holds that p ∈ ∂G(x) ∩ ∂ F(x). 3. p ∈ X ∗ is a critical point of J ∗ and it holds that x ∈ ∂ F ∗ ( p) ∩ ∂G ∗ ( p). Finally, we have L(x, p) = J (x) = J ∗ ( p)

(3.41)

if one of the above conditions is satisfied. Proof By (3.30) and (3.35), it holds that 0 ∈ ∂x L(x, p) = 0 ⇔ p ∈ ∂G(x) ⇔ x ∈ ∂G ∗ ( p) 0 ∈ ∂ p L(x, p) = 0 ⇔ x ∈ ∂ F ∗ ( p) ⇔ p ∈ ∂ F(x)

(3.42)

for any (x, p) ∈ X × X ∗ . Given x ∈ X , we take p ∈ A∗ (x). This p attains J (x) = inf ∗ L(x, p), p∈X

and, therefore, 0 ∈ ∂ p L(x, p). Thus A∗ (x) = ∂ F(x) holds by (3.42). The relation A( p) = ∂G ∗ ( p) follows similarly and the first part, (3.40), is proven. The second part, the equivalence of the above mentioned three conditions, is obtained also by (3.42) because (x, p) ∈ X × X ∗ is a critical point of L(x, p) = F ∗ ( p) + G(x) − x, p if and only if p ∈ ∂G(x), x ∈ ∂ F ∗ ( p).

(3.43)

3.2 Lagrangian and Duality

93

Finally, (3.41) follows from (3.40) and (3.43). More precisely, L(x, p) = F ∗ ( p) + G(x) − x, p = F ∗ ( p) − G ∗ ( p) = G(x) − F(x). 

The proof is complete. Spectral Equivalence Under the assumptions of Theorem 3.2.1, we have p ∈ ∂G(x) ∩ ∂ F(x)



x ∈ ∂ F ∗ ( p) ∩ ∂G ∗ ( p),

and, therefore, each critical point of J produces that of J ∗ , and the converse is also true. This correspondence x



p

may be called the Legendre transformation of the critical point, and furthermore, (x, p) is a critical point of L in this case. This equivalence of critical points is valid up to their Morse indices, under reasonable assumptions as we are showing. This property is the second part of the local theory of the Toland duality, the spectral equivalence. Recall that X denotes a Banach space over R and F, G : X → (−∞, +∞] are proper, convex, lower-semicontinuous. We define L(x, p), J (x), and J ∗ ( p) by (3.30), (3.31), and (3.32), respectively, and let (x, p) ∈ X × X ∗ be a critical point of L = L(x, p). Let Y be a C 1 manifold or a closed convex set in X containing x, and let Y 0 be the tangent hyper-plane of Y ∩ D(J ) at x. First, if there is Yk ⊂ Y ∩ D(J ) containing x and locally homeomorphic to a closed subspace of Y 0 of codimension k, such that x is a minimizer (or strict minimizer) of J |Yk , then we write i Y (x) ≤ k

(or i Ys (x) ≤ k).

Next, we call the minimum of such k the (strict) local index of x relative to Y and write i Y (x)

(or i Ys (x)).

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3 Toland Duality

If there is Yˇk ⊂ Y ∩ D(J ) containing x and locally homeomorphic to a closed subspace of Y 0 of dimension k, such that x is a maximizer (or strict maximizer) of J |Yˇk , then we write ˇ ≥k i(x)

(or iˇs (x) ≥ k),

and the maximum of such k is called the (strict) anti-local index of xˆ relative to Y , and is written by ˇ i(x)

(or iˇs (x)).

If Y∗ is a C 1 manifold or a closed convex set in X ∗ containing p, then the (strict) local index of p relative to Y∗ , denoted by iˇY∗ ( p)

(or iˇYs ∗ ( p)),

is defined similarly, using the tangent hyper-plane Y∗0 of Y∗ ∩ D(J ∗ ) at p. If O∗ ⊂ Y∗ is an open set containing p, then the mapping T : O∗ → Y with T ( p) = x is said to be k-(Y, Y∗ ) faithful if T −1 (Yk ) is locally homeomorphic to a closed subspace of Y∗0 of codimension k containing p, provided that Yk is a closed subspace of Y of codimension k containing x. Similarly, it is anti k-(Y, Y∗ ) faithful if T −1 (Yˇk ) is locally homeomorphic to a closed subspace of Y∗0 of dimension k containing p, provided that Yˇk is a closed subspace of Y of dimension k containing x. If O ⊂ Y is an open set containing x, then the (anti) k-(Y∗ , Y ) faithfulness of the mapping T∗ : O → Y∗ with T∗ (x) = p is defined similarly. Theorem 3.2.2 Under the assumptions of Theorem 3.2.1, let (x, p) be a critical (s) (s) point of L = L(x, p), and let k = i Y (x) and k∗ = i Y∗ ( p) be the (strict) local indices of x and p relative to Y and Y∗ , respectively. Then, if we have k∗ -(Y∗ , Y ) faithful T∗ ⊂ ∂G and k-(Y, Y∗ ) faithful T ⊂ ∂ F ∗ , it follows that k = k∗ , that is (s)

(s)

i Y (x) = i Y ( p). A similar fact holds for the anti-local indices.

3.2 Lagrangian and Duality

95

Proof We describe the proof only for the strict indices because the other cases are similar. Since k = i Ys (x), there is Yk ⊂ Y ∩ D(J ) containing x and locally homeomorphic to a closed subspace of Y 0 of codimension k such that x is a strict minimizer of J |Yk : x ∈ Yk \ {x} ⇒ J (x) > J (x).

(3.44)

Since p ∈ A∗ (x), we have J (x) = inf ∗ L(x, p) = L(x, p) ≥ J ∗ ( p) = inf L(x, p) p∈X

and also

x∈X

(3.45)

p ∈ A∗ (x) ⇔ x ∈ ∂ F ∗ ( p).

If p ∈ T −1 (Yk ) \ { p}, therefore, then T p ∈ ∂ F ∗ ( p) by T ⊂ ∂ F ∗ , and hence p ∈ A∗ (T p) which implies J (T p) = inf ∗ L(T ( p), q) = L(T ( p), p) ≤ inf L(x, p) = J ∗ ( p). q∈X

x∈X

We have, on the other hand, J (T p) > J (x) ≥ J ∗ ( p) by T p ∈ Yk , (3.44), and (3.45), and, therefore, it holds that J ∗ ( p) > J ∗ ( p), ∀ p ∈ T −1 (Yk ) \ { p} which means that i Ys ∗ ( p) ≤ k = i Ys (x) because T is k-(Y, Y∗ ) faithful.



Stability The stability of the stationary state is derived only from the unfolding-minimality which comprises the third part of the local theory of the Toland duality. Here we introduce the notion of infinitesimal stablility, weaker than linear stability. Let X be a Banach space over R, and F : X → (−∞, +∞] be proper, convex, lower semi continuous. Let J : X → [−∞, +∞], satisfy L| p∈∂ F(x) = J,

L : X × X ∗ → [−∞, +∞]

L(x, p) ≥ J (x), ∀(x, p) ∈ X × X ∗ .

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3 Toland Duality

Let Y0 be a closed subset of a Banach space Y , which is continuously imbedded in X , and Y∗ be a Banach space continuously imbedded in X ∗ , and assume the existence of (x, p) ∈ D(L) = {(x, p) | L(x, p) = ±∞} ⊂ X × X ∗ satsifying p ∈ ∂ F(x) ∩ Y∗ and x ∈ Y0 . Suppose that x is a linearly stable local minimizer of J |Y0 in the sense that there is ε0 > 0 such that any ε ∈ (0, ε0 /4] admits δ > 0 satisfying x ∈ Y0 , x − xY < ε0 , J (x) − J (x) < δ



x − xY < ε,

(3.46)

and suppose also that L|Y0 ×Y∗ is continuous at (x, p). Then we obtain the following theorem. Theorem 3.2.3 The above (x, p) is dynamically stable in x component concerning the orbit {(x(t), p(t))}0≤t 0 admits δ > 0 such that x(0) − xY < δ,  p(0) − pY∗ < δ

(3.48)

sup x(t) − xY < ε.

(3.49)

implies t∈[0,T )

Proof Given ε ∈ (0, ε0 /4], we take δ of (3.46), denoted by δ1 > 0. Since L|Y0 ×Y∗ is continuous at (x, p), there is δ ∈ (0, ε0 /2] such that x(0) − xY < δ,  p(0) − pY∗ < δ

(3.50)

L (x(0), p(0)) − L(x, p) < δ1 .

(3.51)

imply

We have, on the other hand, W (x, p) ≥ J (x) ≥ J (x) = W (x, p) for any (x, p) ∈ Y0 × X ∗ with x − xY < ε0 from the assumption. Therefore, as far as x(t) − xY < ε0 (3.52)

3.2 Lagrangian and Duality

97

it holds that 0 ≤ J (x(t)) − J (x) ≤ W (x(t), p(t)) − J (x) ≤ L (x(0), p(0)) − L(x, p) < δ1 .

(3.53)

Now, we have x(0) − xY < δ ≤ ε0 /2, and if there is t0 ∈ (0, T ) such that x(t0 ) − xY = ε0 /2, then it holds that (3.52) and hence (3.53) for t = t0 which implies x(t0 ) − xY < ε ≤ ε0 /4

(3.54)

from (3.46) (with δ = δ1 ), a contradiction. Since t ∈ [0, T ) → x(t) ∈ Y0 ⊂ Y is continuous, the relation



x(t) − xˆ < ε0 /2 Y keeps for t ∈ [0, T ), and hence (3.52). Again this implies (3.53) and (3.54) for any t ∈ [0, T ), and the proof is complete.  Similarly, let G : X → (−∞, +∞] be proper, convex, lower semi continuous, let J ∗ : X ∗ → [−∞, +∞] satisfy L|x∈∂G ∗ ( p) = J ∗ ,

L(x, p) ≥ J ∗ (x), ∀(x, p) ∈ X × X ∗ ,

and let (x, p) ∈ D(L) be such that x ∈ ∂G ∗ ( p) ∩ Y and p ∈ Y0∗ , where Y0∗ is a closed subset of Y∗ . Assume, furthermore, that p is a linearly stable local minimizer of J ∗ |Y0∗ in the sense that any ε ∈ (0, ε0 /4] admits δ > 0 such that p ∈ Y0∗ ,  p − pY∗ < ε0 , J ∗ ( p) − J ∗ ( p) < δ



 p − pY∗ < ε.

Finally, let an orbit {(x(t), p(t))}0≤t 0 admits δ > 0 such that (3.48) implies sup  p(t) − pY∗ < ε.

t∈[0,T )

In the case that the variational functional is made from real-analytic functions, any local minimum becomes infinitesimally stable, even without linear stability. This structure is useful, particularly, in the study of thermodynamical models involved by hysteresis [294, 368].

98

3 Toland Duality

3.3 Gradient Systems with Duality This paragraph is devoted to several comments concerning the above abstract results. Let X , Y be Banach spaces with continuous inclusion Y ⊂ X , and Y∗ be another Banach space continuously imbedded in X ∗ . Furthermore, Y0 ⊂ Y and Y0∗ ⊂ Y∗ are closed subsets, and L(x, p) = F ∗ ( p) + G(x) − x, p be the Lagrangian defined by proper, convex, lower semi-continuous mappings F, G : X → (−∞, +∞]. Given a continuous orbit t ∈ [0, ∞) → (x(t), p(t)) ∈ Y0 × Y0∗ , we can define its ω-limit set by ω (x(0), p(0)) = {(x∞ , p∞ ) | there exists tk → +∞ such that (x(tk ), p(tk )) → (x∞ , p∞ ) in Y0 × Y0∗ .} In the case that t → L(x(t), p(t)) ∈ R is non-increasing and (x 1 , p 1 ) and (x 2 , p 2 ) are elements in ω(x(0), p(0)), we obtain tk1 → +∞ such that (x(tk1 ), p(tk1 )) → (x 1 , p 1 ) in Y0 × Y0∗ tk2 → +∞ such that (x(tk2 ), p(tk2 )) → (x 2 , p 2 ) in Y0 × Y0∗ 1 such that tk1 < tk2 < tk+1 (k = 1, 2, . . .) which implies 1 1 L(x(tk1 ), p(tk1 )) ≥ L(x(tk2 ), p(tk2 )) ≥ L(x(tk+1 ), p(tk+1 )),

and, therefore, L(x 1 , p 1 ) = L(x 2 , p 2 ), provided that L : Y0 × Y0∗ → R is continuous. Thus L is invariant on ω(x(0), p(0)). If there is a continuous local semi-flow {Tt } on Y0 × Y0∗ such that (x(t), p(t)) = Tt (x(0), p(0)), then we obtain Tt (x∞ , p∞ ) ∈ ω(x(0), p(0)) for any (x∞ , p∞ ) ∈ ω(x(0), p(0)), and, therefore, L(Tt (x∞ , p∞ )) = constant

(3.55)

3.3 Gradient Systems with Duality

99

for any t ≥ 0. This relation means ω(x(0), p(0)) ⊂ E, where E denotes the set of (x∞ , p∞ ) ∈ Y0 × Y0∗ satisfying (3.55), which may be called the stationary set. The orbit O = {(x(t), p(t)) | 0 ≤ t < +∞} is continuous in Y0 × Y0∗ , and, therefore, the ω-limit set ω(x(0), p(0)) is connected and compact, provided that O is compact [161]. Under the assumption of Theorem 3.2.3, on the other hand, the critical point (x, p) of L is isolated. Therefore, if the above semi-flow {Tt } is global in time around it, then it is asymptotically stable and it holds that lim x(t) − xY = lim  p(t) − pY∗ = 0. (3.56) t→+∞

The Lagrangian

t→+∞

L(x, p) = F ∗ ( p) + G(x) − x, p

induces the gradient system, say p˙ ∈ −L p (x, p), τ x˙ ∈ −L x (x, p),

(3.57)

using the combination of model (A)–model (A) equations, see Sect. 4.1, where τ > 0 is the relaxization time. Then, its simplified system is defined by p˙ ∈ −L p (x, p), 0 ∈ L x (x, p) or 0 ∈ L p (x, p), τ x˙ ∈ −L x (x, p). Equation (3.57) means ˙ q − p , ∀q ∈ X ∗ F ∗ (q) − F ∗ ( p) ≥ x − p, G(y) − G(x) ≥ y − x, p − τ x ˙ , ∀y ∈ X. More precisely, assuming a Hilbert space Y over R with the continuous inclusion X ⊂ Y ≈ Y ∗ ⊂ X ∗,

(3.58)

we define the solution ( p, x) = ( p(t), x(t)) to (3.57) by p ∈ C([0, T ), Y ), p˙ ∈ C([0, T ), X ) x ∈ C([0, T ), X ), x˙ ∈ C([0, T ), Y ).

(3.59)

100

3 Toland Duality

Then, it holds that lim sup h↓0

lim sup h↓0

 1 ∗ 2 F ( p(t)) − F ∗ ( p(t − h)) ≤ x(t), p(t) ˙ −  p(t) ˙ Y h 1 2 {G(x(t)) − G(x(t − h))} ≤ x(t), ˙ p(t) − τ x(t) ˙ Y h

for t ∈ (0, T ), and, therefore, p(t) ∈ D(F ∗ ), x(t) ∈ D(G), 0 ≤ t < T ⇒ p(T ) ∈ D(F ∗ ), x(T ) ∈ D(G), L(x(T ), p(T )) ≤ lim inf L(x(t), p(t)). t↑T

In particular, if (x, p) is a linearly stable critical point of L, and (3.57) is locally well-posed in a neighborhood of (x, p) in the sense of (3.59), then it is globally well-posed there, (x, p) is dynamically stable, and (3.56) holds. Entropy Functional Here we apply the above formulated abstract theory to Helmholtz’ free energy defined in Sect. 1.3, where  ⊂ Rn is a bounded domain with smooth boundary ∂. This paragraph is devoted to a convex analytic approach to what is written in Chap. 9 of [364]. Given a positive definite self-adjoint operator A in L 2 () with compact resolvent, we take the Gel’fand triple [189, 390] X → L 2 () ≈ L 2 ()∗ → X ∗ for X = D(A1/2 ), and define the dual entropy functional F : X → (−∞, +∞] by  F(v) = λ log



Ve

v

 + λ(log λ − 1),

where V = V (x) > 0 is a continuous function of x ∈ . This functional is proper, convex, lower semi-continuous, and it holds that  v D(F) = v ∈ X | V e < +∞ 

u ∈ ∂ F(v)



λV ev . u= v  Ve

3.3 Gradient Systems with Duality

101

The entropy functional F ∗ : X ∗ → (−∞, +∞] is defined by the second Legendre transformation of the dual entropy functional, F ∗ (u) = sup {v, u − F(v)} , v∈X

and it holds that F ∗ (u) = where



 u(log u

+∞,

− 1 − log V ), u ∈ D(F ∗ ) otherwise,

  D(F ∗ ) = u ∈ X ∗ | u ≥ 0, u ∈ (L log L)(), u1 = λ

with (L log L)() denoting the Zygmund space, see Sect. 3.4. Thus v ∈ ∂ F ∗ (u) if and only if u ∈ D(F ∗ ) and v = log u − log V + constant ∈ X. Putting G(v) =

1

1/2 2

A v , 2 2

next, we obtain a proper, convex, lower semi-continuous mapping G : X → (−∞, +∞). The self-adjoint operator A induces the isomorphism Aˆ : X → X ∗ , and we have G ∗ (u) =

1 ˆ −1 A u, u 2

for u ∈ X ∗ . Then, the Lagrangian is defined by L(u, v) = F ∗ (u) + G(v) − v, u , with the stationary state described by 0 ∈ L u (u, v), 0 ∈ L v (u, v) or, equivalently,

u = Aˆ −1 v, v ∈ ∂ F ∗ (u).

Here, we adopt the notation L(u, v), following (2.2).

(3.60)

102

3 Toland Duality

From Theorem 3.2.1, this relation splits into the conditions on u and v, that is, to be a critical point of J (v) = G(v) − F(v)  

2 1



V ev + λ(log λ − 1) = A1/2 v − λ log 2 2   defined for v ∈ X ,



V ev < +∞, and that of

J ∗ (u) = F ∗ (u) − G ∗ (u) 

1 ˆ −1 A u, u = u(log u − 1 − log V ) − 2  defined for u ∈ X ∗ ∩ L 1 (), u ≥ 0, u1 = λ. These variational problems are equivalent to  v ˆ = λV e ∈ X ∗ v ∈ X, V ev < +∞, Av (3.61) v   Ve and u ∈ X ∩ (L log L)(), u ≥ 0, u1 = λ Aˆ −1 u = log u − log V + constant ∈ X,

(3.62)

respectively. The condition  v ∈ L 2 (),



v = 0, A−1 v = constant



v=0

is satisfied in many cases. Assuming this property, we can define (Y, Y∗ )-faithful T ⊂ ∂ F ∗ for Y = D(J )/R or Y = D(J ) and Y∗ = D(J ∗ ), namely, v = T u if and only if v = log u − log V + constant ∈ X, or, equivalently,

λV ev . u= v  Ve

This situation is the same as that of Chap. 9 of [364] and details are omitted. The Fréchet derivative dG = Aˆ : X → X ∗ ,

3.3 Gradient Systems with Duality

103

next, is an isomorphism, and hence T = dG|Y : Y → Y∗ is also faithful by the duality E X P  () ≈ L log L(), see [1, 311] and Sect. 3.4 for these function spaces. Thus, we obtain the variational and spectral equivalences of these J and J ∗ , together with the unfolding-minimality using the Lagrangian L = L(u, v) defined by (3.60). If (u, v) is a linearly stable critical point of L, then it is dynamically stable for u t = ∇ · (u∇ L u (u, v)) ,

u

τ vt = −L v (u, v)

  ∂ L u (u, v) = 0 ∂ν ∂ (3.63)

in the sense that any ε > 0 admits δ > 0 such that v(·, 0) − v X < δ, u(·, 0) − u X ∗ ∩L log L < δ, u(·, 0)1 = λ = u1 implies sup v(·, t) − v X < ε,

t∈[0,T )

sup u(·, t) − u X ∗ ∩L log L < ε.

t∈[0,T )

This dynamical stability is actually the case if the solution is sufficiently regular, say, t → (u(·, t), v(·, t)) ∈ D(F ∗ ) × X is continuous, t → L(u(·, t), v(·, t)) is nonincreasing, and u(·, t)1 is invariant in t, see Theorem 3.2.3. If v is slightly regular, say, ev ∈ L p () for p > 1, then this critical point (u, v) of L = L(u, v) comprises the classical solutions to (3.61) and (3.62). In this case, which is always true if n = 2, the functionals 

1 −1 A u, u , u(log u − 1 − log V ) − u ≥ 0, u1 = 1 2   

2 1



V ev + λ log λ − λ J (v) = A1/2 v − λ log 2 2 

J ∗ (u) =

are twice differentiable at u = u, v = v which results in the derivation of Morse indices by linearized operators and which are equal to the strict local indices and the strict local anti-indices of the critical point (u, v) of 

1

1/2 2 u(log u − 1 − log V ) − v, u . L(u, v) = A v + 2 2 

104

3 Toland Duality

More precisely, the linearized operator derived from above J = J (v) and v = v is defined by    V ev ψ V ev ψ v  Lψ = Aψ − λ  −  2 · V e v V ev  Ve 

with the domain D(L) = D(A) in L 2 (), and then, the non-local term is eliminated by the transformation  V ev ψ . ϕ = ψ −  v  Ve Thus if v = Au means (1.28), then we obtain the eigenvalue problem ∂ϕ = 0 on ∂ ∂ν

− ϕ = μuϕ in , for

λV ev u= , v  Ve

(3.64)

(3.65)

and the (strict) local index of v and hence that of u are equal to the number of eigenvalues in 0 < μ ≤ 1 (0 < μ < 1) defined by (3.64). Similarly, if v = Au is defined by −v = u in ,

v = 0 on ∂,

then the (strict) local index of v and hence that of u are equal to the number of eigenvalues in 0 < μ ≤ 1 (0 < μ < 1) of (3.12), that is  − ϕ = μuϕ in , ϕ = constant on ∂,

∂

∂ϕ =0 ∂ν

(3.66)

for u = u(x) defined by (3.65), see also [364].

3.4 Zygmund Spaces The above mentioned dynamical stability to (3.63) requires the continuity of the entropy functional defined on the Zygmund space [361]. In fact, given an open set  ⊂ Rn , we define the Zygmund space by   (L log L)() = f :  →  measurable |  f  L log L < +∞ ,

3.4 Zygmund Spaces

where

105

   |f|  f  L log L = inf k > 0 | | f | log e + ≤k k 

denotes the Luxemburg norm, see [1]. Then, we obtain the following theorem [182]. Theorem 3.4.1 There is an order-preserving norm in (L log L)() equivalent to the Luxemburg norm defined by   | f (x)| | f (x)| log e + , =  f 1  

[ f ] L log L

under the agreement that [ f ] L log L = 0 if  f 1 = 0. The last paragraph of this section is devoted to the proof of Theorem 3.4.1 based on a series of lemmas. Lemma 3.4.1 It holds that    a g gb a f fb log ≥ g log e + − log f log e + − a a−b b a a−b b

(3.67)

for f ≥ g ≥ 0 and a > b > 0. Proof We have log x ≤ x − 1, x > 0, and, therefore, b log Then, putting

a ≤ a − b, b

a > b > 0.

 s sb a h(s) = s log e + − log , a a−b b

we obtain  1 s +s· h  (s) = log e + a e+ ≥ 1−1=0

s a

·

b a 1 − log a a−b b

and hence (3.67) follows from h( f ) ≥ h(g).



Lemma 3.4.2 If f, g ∈ L log L() satisfies |g(x)| ≤ | f (x)| a.e. then it holds that [g] L log L ≤ [ f ] L log L . Proof We may suppose that 0 < g1 <  f 1 . Then, we obtain     |g(x)| | f (x)| |g(x)| log e + ≤ | f (x)| log e + g1  f 1   log  f 1 − log g1 + |g(x)|  f 1 − | f (x)| g1 ·  f 1 − g1

106

3 Toland Duality

and, therefore, 

     |g(x)| | f (x)| |g(x)| log e + | f (x)| log e + ≤ . g1  f 1   

The proof is thus complete. Lemma 3.4.3 We have     x+y x y (x + y) log e + ≤ x log e + + y log e + , a+b a b

(3.68)

where x, y ≥ 0 and a, b > 0.

  Proof The function h(s) = log e + s −1 is convex in s > 0, and hence it holds that  h

a+b x+y

 ≤

a  x y h + h x+y x x+y

  b . y

This inequality implies (3.68).



Lemma 3.4.4 The triangle inequality [ f + g] L log L ≤ [ f ] L log L + [g] L log L holds true. Proof We have [ f + g] L log L ≤ [| f | + |g|] L log L by Lemma 3.4.2, and, therefore, may assume f, g ≥ 0. Then, Lemma 3.4.3 guarantees   f +g ( f + g) log e +  f + g1       f g f log e + + g log e + ≤  f 1 g1  = [ f ] L log L + [g] L log L , 

[ f + g] L log L =



and the proof is complete.

Lemma 3.4.5 The norm [ · ] L log L is equivalent to the Luxemburg norm, and it holds that  f 1 ≤  f  L log L ≤ [ f ] L log L ≤ 2  f  L log L . Proof From the definition of the Luxemburg norm, we obtain   |f| | f | log e + ≥  f 1 K = K  

3.4 Zygmund Spaces

107

for K =  f  L log L , which implies  K ≤

  |f| | f | log e + = [ f ] L log L .  f 1 

We have, on the other hand,       |f| |f| K | f | log e + | | ≤ f log e · +  f 1  f 1  f 1      |f| K K | f | log e + | f | log + = = K +  f 1 · log ,    f f 1 K   1 and here, the right-hand side is estimated from above by K+ because log s ≤

K ≤ 2K = 2 [ f | L log L e

s for s ≥ 1. The proof is complete. e



We turn to the following theorem proven by a series of lemmas [182]. Theorem 3.4.2 If  is bounded, the mapping  f ∈ (L log L)() →



f log | f | ∈ R

is well-defined and continuous. Lemma 3.4.6 If x, y are elements in a normed space and t ∈ R, then it holds that   x |x|ıt + y |y|ıt − (x + y) |x + y|ıt    |x| + |y| . ≤ 2 |t| |x + y| log e + |x + y|

(3.69)

  Proof Given a ≥ b > 0, we use eit − 1 ≤ |t| valid to t ∈ R. It thus holds that     a   ıt   ıt log a   ıt a  e − bıt  =  a  = e b − 1 ≤ |t| log ≤ |t| −1 . − 1  b  b b From the symmetry and the homogenity, we may assume |x| + |y| = 1,

0 < |x| ≤

1 ≤ |y| 2

108

3 Toland Duality

to prove (3.69). Then, it follows that |x + y| ≤ |x| + |y| = 1, |y| ≤ |x| + |y| = 1, and, therefore,   x |x|ıt + y |y|ıt − (x + y) |x + y|ıt         = (x + y) 1 − |x + y|ıt − (x + y) 1 − |y|ıt − x |y|ıt − |x|ıt    |y| 1 1 ≤ |t| |x + y| log + |x + y| log + |x| · −1 . |x + y| |y| |x| Here, from |y| − |x| ≤ |y + x| the right-hand side is equal to |t| {− |x + y| log |x + y| − |x + y| log |y| + |y| − |x|}   1 ≤ |t| |x + y| log + |x + y| |x + y| · |y| 2e e ≤ |t| |x + y| log = |t| |x + y| log |x + y| · |y| |x + y| 2    1 1 . ≤ |t| |x + y| log e + = 2 |t| |x + y| log e + |x + y| |x + y| 

The proof is complete. Lemma 3.4.7 If x, y are elements in a normed space, then it holds that |x log |x| + |y| log |y| − (x + y) log |x + y||   |x| + |y| ≤ 2 |x + y| log e + . |x + y|

Proof We obtain the result by dividing both sides of inequality in the previous lemma by |t| and making t → 0.  Lemma 3.4.8 It holds that      | f | + |g|  f 1 + g1 | f − g| log e + ≤  f − g1 log e + | f − g|  f − g1  for f, g ∈ L 1 (). Proof Both sides are zero if f = g a.e. by the definition, and, therefore, we can assume  f − g1 = 1. Thus μ(d x) = | f (x) − g(x)| d x

3.4 Zygmund Spaces

109

is a probability measure while h(s) = log(e + s) is a concave function of s > 0. Hence Jensen’s inequality guarantees      | f | + |g| | f | + |g| dμ ≤ log e + log e + dμ | f − g|   | f − g|      | f | + |g| d x = log e +  f 1 + g1 , = log e +





and the proof is complete.



Lemma 3.4.9 It holds that         f log | f | − g log |g| d x  ≤  f − g1 log  f − g1  + ||      f 1 + g1 +2  f − g1 log e + + [ f − g] L log L (3.70)  f − g1 for f, g ∈ L log L(). Proof We obtain | f log | f | − g log |g| − ( f − g) log | f − g||   | f | + |g| ≤ 2 | f − g| log e + | f − g|

(3.71)

by Lemma 3.4.7. Here we have          | f − g| log | f − g| ≤  | f − g| log | f − g|      f − g1     +  f − g1 log  f − g1  and           | f − g| log | f − g|  ≤ log | f − g|  | f − g|     f − g1  f − g1   | f −g|≤ f −g1    | f − g| + | f − g| log e +  f − g1 | f −g|≥ f −g1   f − g1 + [ f − g] L log L . ≤ | f − g| log | f − g| | f −g|≤ f −g1 Since

   f − g  f − g1 ≤ | f − g| − 1 ≤  f − g1 | f − g| log | f − g| | f − g|

we obtain (3.70).



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3 Toland Duality

3.5 Summary We described the fundamental structure of dual variation, particularly, the Toland duality. 1. In the gradient system associated with the Toland duality, stationary states are equivalently formulated variations, in terms of the field and the particle components. 2. These structures are packaged in the Lagrangian. Then, a linearly stable stationary solution is dynamically stable. This stability splits into both components, field distribution and particle density. 3. A typical example of the Toland duality is the full system of chemotaxis, associated with the entropy functional, which is continuous on the Zygmund space (L log L)().

Chapter 4

Phenomenology

An important feature of dual variation is the unfolding-minimality concerning the stationary state. There are, however, systems provided only with the semi-duality, that is semi-unfolding and minimality. Among them are the problems arising in phase transition, phase separation, and shape memory alloys. In the Kuhn-Tucker duality, on the other hand, the stable critical state is realized as a saddle. We have several such systems in mathematical biology, game theory, and linear programing. This chapter is devoted to these variants of the Toland duality.

4.1 Non-convex Evolution The Ginzburg-Landau theory is a phenomenology consistent to thermodynamics where free energy is defined as a functional of an order parameter, composed of van der Waals’ penality and the double-well potential. This free energy is not formulated by the Toland duality because the double-well potential is neither convex nor concave. Generally, non-equilibrium mean field equations in phenomenological theories are described by the chemical potential μ = δF(ϕ), where F = F(ϕ) stands for the free energy F = F(ϕ) defined by the order parameter ϕ = ϕ(x). These equations are classified into model (A), model (B), and model (C) equations [153, 171]. In more details, if  ⊂ Rn , n = 2, 3, is a bounded domain with smooth boundary ∂, then the order parameter ϕ = ϕ(x, t) is a function of the position x ∈  and the time t > 0 indicating the status of the material, and F = F(ϕ) is a quantity determined by ϕ. Thus F = F(ϕ) is regarded as a © Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_4

111

112

4 Phenomenology

functional of ϕ = ϕ(x, t), and the system moves toward the equilibrium, making F(ϕ) decrease. Here, the chemical potential, δF(ϕ) is defined by   d ψ, δF(ϕ) = F(ϕ + sψ) ds s=0

(4.1)

similarly to (1.31). As in the previous cases, this  ,  is usually identified with the L 2 inner product. Model (A) equation is formulated by ϕt = −KδF(ϕ), where K is a positive quantity, possibly associated with ϕ. Then, it holds that d F(ϕ) = − dt

 

KδF(ϕ)2 ≤ 0.

Model (B) equation, on the other hand, is described by ϕt = ∇ · (K∇δF(ϕ)) , In this case, we obtain   d ∂ ϕ= K δF(ϕ) = 0, dt  ∂ ∂ν

  ∂ K δF(ϕ) = 0. ∂ν ∂

d F(ϕ) = − dt

 

K |∇δF(ϕ)|2 ≤ 0.

Thus, model (A) and model (B) equations stand for the thermodynamically closed system and thermodynamically-materially closed system, respectively. A non-trivial time periodic solution, for example, is not permitted to both of them. The stationary state is actually defined by the zero free energy consumption; it is δF(ϕ) = 0 in the model (A) equation, while  δF(ϕ) = 0 constrained by



ϕ=λ

in the model (B) equation, where λ is a prescribed constant indentified with an eigenvalue. More precisely, stationary states of model (A) and model (B) equations are defined by   d F(ϕ + sψ) = 0, ∀ψ ds s=0

4.1 Non-convex Evolution

113

and     d = 0, ∀ψ, ψ = 0, ϕ = λ, F(ϕ + sψ) ds   s=0 respectively. Similarly, the linearized stability of the stationary state ϕ means   1 d2 Q(ψ, ψ) ≡ F(ϕ + sψ) > 0, ∀ψ = 0 2 2 ds s=0 in the model (A) equation, while  Q(ψ, ψ) > 0, ∀ψ = 0,



ψ=0

in the model (B) equation. If model (A) equation or model (B) equation is well-posed globally in time, the associated semi-flow is completely continuous, and the set of stationary solutions, denoted by E, is bounded in a suitable Banach space, then there is a global attractor denoted by A. This A is connected, comprises of the unstable manifold of E (and of the union of the unstable manifolds of the elements in E if any of them is hyperbolic), and attracts the orbit as t ↑ +∞ uniformly in the initial value contained in a bounded set, see [151, 269]. We hereby describe several free energies and model equations derived from them. Helmholtz’ Free Energy As we have mentioned in Sect. 1.3, Helmholtz’ free energy  F(u) = α



u(log u − 1) −

1 2

 ×

G(x, x )u(x)u(x )dxdx

induces the mean field equation of many self-gravitating particles, where u = u(x, t) denotes the particle density. If the absolute temperature α is equal to 1, and the potential G = G(x, x ) is the Green’s function to (1.28), then we obtain the simplified system of chemotaxis (1.8) as a model (B) Eq. (1.32) with K = u. Its stationary state is described by (3.11), that is   v 1 e in , −v = λ  v − || e

  ∂v  = 0, v = 0, ∂ν ∂ 

using the field component, and in two space dimension, the quantized blowup mechanism of this state implies that of the non-equilibrium, see Sect. 1.2.

114

4 Phenomenology

Allen-Cahn Equation Ginzburg-Landau’s free energy,  F(ϕ) =



ξ2 |∇ϕ|2 + W (ϕ) dx 2

(4.2)

induces the Allen-Cahn equation [4] ϕt = K(ξ 2 ϕ − W (ϕ))

in  × (0, T )

(4.3)

in phase separation as a model (A) equation, where ξ > 0 is a constant associated with the intermolecular force, ϕ = ϕ(x, t) is the order parameter, K > 0 is a constant, and W (ϕ) =

2 1 2 ϕ −1 4

is the double-well potential. Usually, this F(ϕ) is associated with all ϕ ∈ H 1 (), and then the natural boundary condition  ∂ϕ  =0 ∂ν ∂

(4.4)

is furthermore imposed to (4.3). Thus, the Allen-Cahn equation, (4.3) and (4.4), is formulated by a semilinear parabolic equation of the second order. Then, the stationary state is described by −ξ ϕ = ϕ − ϕ , 2

3

 ∂ϕ  = 0, ∂ν ∂

and its stability is equivalent to the positivity of the first eigenvalue of the self-adjoint operator in L 2 (), A = −ξ 2  − 1 + 3ϕ2 , with the domain

∂ψ 2 = 0 on ∂ . D(A) = ψ ∈ H () | ∂ν Here, any non-constant stationary solution ϕ is linearly unstable if  is convex. Actually, this property is the case of the general (single) semilinear parabolic equation with the Neumann boundary condition [49, 236].

4.1 Non-convex Evolution

115

In the one-space dimensional case, furthermore, each ω-limit set, see Sect. 3.3, is composed of one element [235, 435]. Actually, due to the parabolic Liouville property [11, 127, 237], and the dynamics of the semi-linear parabolic equation in one-space dimension has several remarkable profiles. For example, there are the Morse-Smale property [10, 162], the meandric permutation concerning the stationary solution [125], and the hetero-clinic cascade [112, 422]. The other aspect of such equations is the strong order preserving property. It is derived also from the strong maximum principle and is valid even to the multi-space dimension [169, 238, 341]. Related aspects of the dynamics of the semilinear parabolic equations are described by [152, 174, 220] including the multi-space dimensional case. The first term of the integrand of F(ϕ) of (4.2) is written as    ∂ϕ 2  .  ξ |∇ϕ| =  ∂(ξ −1 x)  2

2

It is associated with the surface tension and is called van der Waals’ penalty. Here, the parameter 0 < ξ  1 is determined in accordance with the molecular distance. The term W (ϕ), on the other hand, is called the double-well potential because ϕ = ±1 are its bi-stable critical points of the ordinary differential equation ϕ˙ = κ(ϕ − ϕ3 ). The other critical point ϕ = 0 of W = W (ϕ) is unstable, and, consequently, ϕ = ϕ(·, t) is rapidly separated into the regions {x ∈  | ϕ(x, t) = ±1} with the interface {x ∈  | −1 < ϕ(x, t) < +1} of O(ξ)-thickness, and then this interface moves slowly subject to their curvatures, that is, there are fast and slow dynamics [40, 86, 179, 315]. In fact, if ϕ = ϕ(x) is regular, then this free energy is equal to  F(ϕ) =





−∞

ds

{ϕ=s}

ξ 2 |∇ϕ| +

W (s) dS |∇ϕ|

by the co-area formula. This value is estimated from below by  F∗ (ϕ) = 2ξ 2



−∞

W (s)H n−1 ({ϕ = s}) ds

with the equality if and only if ∂ϕ = W (s)1/2 ∂(ξ −1 ν)

on {ϕ = s},

where ν denotes the outer normal vector on {ϕ = s} from {ϕ < s} to {ϕ > s} and H n−1 is the (n − 1)-dimensional Hausdorff measure. Thus, in the equilibrium this ϕ is almost separated into ϕ = ±1, and the interface thickness is of O(ξ).

116

4 Phenomenology

To illustrate the relation between the mean curvature flow, we take K = 1 in (4.3), and assume ϕ(x, t) ≈ φ (d(x, t)/ξ) near the interface, denoted by t , where d(x, t) is the signed-distance function. Then, a rough calculation implies dt ≈ d, where d and dt are the mean curvature of t and the velocity of Mt , respectively. Thus, we otain the suggestion of the convergence of ϕ(x, t) to the mean curvature flow as ξ ↓ 0 of (4.3). In the level set approach [67, 103, 104], actually, this mean curvature flow is described by t = {u(·, t) = 0} with u = u(x, t) satisfying  ut = |Du| div

Du |Du|



  Du ⊗ Du = trace I − . |Du|2

In spite of the formal singularity, this equation is well-posed in the sense of viscosity solutions [76], and the above mentioned convergence is justified by the comparison principle under appropriate time scalings [18, 66, 102, 269]. Also, we obtain efficient numerical schemes from this observation [100, 124, 272, 285]. Cahn-Hilliard Equation Landau-Ginzburg’s free energy induces also the Cahn-Hilliard equation [47] ϕt = −K(ξ 2 ϕ − W (ϕ))   ∂ 2 (ξ ϕ − W (ϕ)) = 0 ∂ν ∂

in  × (0, T )

concerning the phase separation as a model (B) equation. Similarly to the above case of the Allen-Cahn equation, usually we impose  ∂ϕ  = 0, ∂ν ∂ furthermore, assuming that F(ϕ) is associated with all ϕ ∈ H 1 () which means that ϕt = −K(ξ 2 ϕ − W (ϕ)) in  × (0, T )   ∂ (ϕ, ϕ) = 0. ∂ν ∂

(4.5)

4.1 Non-convex Evolution

117

In this case, the stationary state ϕ is defined by 1 −ξ ϕ = ϕ − ϕ − ||  ϕ = λ, 2



3

ϕ − ϕ dx, 3



 ∂ϕ  =0 ∂ν ∂ (4.6)



and its linearized stability means the positivity of the first eigenvalue of the selfadjoint operator in L02 (), A = −ξ 2  + 1 − 3ϕ2 with the domain 

 ∂ψ  = 0, ψ = 0 . D(A) = ψ ∈ H 2 () | ∂ν ∂  If ϕ = ϕξ attains the minimum of Fξ defined by Fξ (ϕ) =



ξ2  2

+∞,

|∇ϕ|2 + W (ϕ) dx,

ϕ ∈ H 1 (), otherwise,







then it is a solution to (4.6). Here, we obtain -convergence of Fξ to F0 in L 1 () as ξ ↓ 0, see [244, 345], where F0 (ϕ) =

2

3 Per {ϕ

+∞,

= −1} ,

ϕ ∈ BV (), W (ϕ) = 0 a.e., otherwise,







BV () denotes the space of functions of bounded variation,

 1 |∇v| < +∞ BV () = v ∈ L () |  

 1 n |∇v| = sup v∇ · g | g ∈ C0 () , g∞ ≤ 1 , 

and Per (A) = means that





 |∇χA |

is the perimeter of the measurable set A, see [147] which

vξ → v as ξ ↓ 0 in L 1 ()



lim inf Fξ (vξ ) ≥ F0 (v) ξ↓0

and any v ∈ L 1 () admits ξk ↓ 0 and {vk } such that vk → v in L 1 (),

lim Fξk (vk ) = F0 (v).

k→∞

118

4 Phenomenology

In this case, any ξk ↓ 0 admits ξk ⊂ {ξk } such that ϕξ converges to a miminizer k of F0 in L 1 () from the genral theory [78]. The limiting problem of (4.5) as ξ ↓ 0 is described by the Mullins-Sekerka-HeleShaw equation, 

v = 0 in

( \ t ) × {t},

t∈(0,T )

 ∂v  =0 ∂ν ∂

   1 ∂v + on t × {t}, v = γkt , V = 2 ∂ν − t∈(0,T )

where kt , γ > 0, and V are the mean curvature of the interface t , a constant, and the normal velocity of t , respectively [2, 296, 347]. The first factor of the micro-phase separation is the spinodal decomposition caused by the composition fluctuation of sine-like waves. It occurs to the blended material when the temperature becomes low. Its rough description is obtained by taking the equation on the whole space, and linearizing it around the constant solution ϕ, that is   in Rn × (0, T ). K −1 wt =  −ξ 2 w + W

(ϕ)w Thus putting w(x, t) = exp (ık · x + Kσt), we obtain   σ = |k|2 −W

(ϕ) − ξ 2 |k|2 , where σ > 0 corresponds to this phenomenon of spinodal decomposition. This range is the case of W

(ϕ) < 0, that is − √1 < ϕ < √1 , and (− √1 , √1 ) is called the 3 3 3 3 spinodal interval, and then it holds that |k| = 2

β±

 β 2 − 4ξ 2 σ , 2ξ 2

where β = −W

(ϕ). The value σ, on the other hand, attains the maximum if |k|2 =

1 −2 βξ . 2

(4.7)

This formula suggests that unstable wave-like fluctuation with the length λ = 2π k =  8 β πξ in the one-dimensional case, and simultaneously, irregular patterns caused by the mixture of a variety of waves for the two-dimensional case [46]. The second mechanism of phase separation is the nucleation, whereby several crystal seeds gather under the presense of the condensate of the solution, caused by super-saturation. It is associated with the intervals I− = (−1, − √1 ) and 3

4.1 Non-convex Evolution

119

I+ = ( √1 , 1) called the metastable region, and if the mean value of the order para3 meter is ϕ ∈ I− , then under the local but sufficiently strong perturbation induces a small region with ϕ ∈ I+ surrounded by that with ϕ ∈ I− , see [3, 21]. Ohta-Kawasaki’s Free Energy Ohta-Kawasaki’s free energy [273],  F(ϕ) =



2 ξ2 σ   |∇ϕ|2 + W (ϕ) + (−)−1/2 ϕ dx 2 2

(4.8)

is concerned with the micro-phase separation in diblock copolymers, where σ > 0 is a parameter proportional to the inverse length of the polymer chain and v = (−)−1 u if and only if (3.10). If the region is separated into ϕ = ±1 with the interface O(ξ), then the third term of (4.8) is not negligible because the variation derived from ϕ becomes large. This term of Ohta-Kawasaki’s free energy is essentially the same as that of Helmholtz’ free energy defined by (1.29), F(u) =

1 2

 

u(log u − 1) −

 1 (−)−1 u, u . 2

It can be made small by the rapid oscillation that guarantees the convergence to ϕ of ϕ in H 1 () which suggests a number of local minima of this functional. The free energy (4.8) induces the Nishiura-Ohnishi equation [270]   αϕt = − ξ 2 ϕ − W (ϕ) − σ(ϕ − ϕ)  ∂  2 ξ ϕ − W (ϕ)  = 0 ∂ν ∂ as a model (B) equation, where α = K −1 . Similarly to the former cases, we impose  ∂ϕ  = 0, ∂ν ∂ using all ϕ ∈ H 1 () to derive δF(ϕ), and then it holds that 



ϕt = − ξ ϕ − W (ϕ) − σ(ϕ − ϕ), 2

  ∂ (ϕ, ϕ) = 0. ∂ν ∂

120

4 Phenomenology

The stationary state is described by 1 −ξ 2 ϕ = ϕ − ϕ3 − ||   ∂ϕ  = 0, ϕ = λ. ∂ν ∂ 

 

ϕ − ϕ3 dx + σ(−JL )−1 ϕ

Then, the linearized stablity of this stationary state means the positivity of the first eigenvalue of the self-adjoint operator A in L02 () defined by Aψ = −ξ 2 ψ − ψ + 3ϕ2 ψ + σ(−JL )−1 ψ with the domain  ∂ψ  2 D(A) = ψ ∈ H () | ∂ν 



 ∂

= 0,



ψ=0 .

We have also -convergence of the functional F = Fξ,σ of (4.8) under several scalings of (ξ, σ), see [286, 287].

4.2 Gradient and Skew-Gradient Systems There are several systems derived from Lagrangian and skew-Lagrangian without duality. In this case, the stationary state is not reduced to the single equation. In the gradient system, the Lagrangian L = L(u, v) acts as a Lyapunov function. If the combination of model (A)–model (A) equations ut = −Lu , τ vt = −Lv , for example, is adopted, then it holds that d L(u, v) = − dt

 

Lu (u, v)2 + τ −1 Lv (u, v)2 dx ≤ 0,

where τ > 0 is a constant. The stationary state (u, v) is defined by Lu (u, v) = Lv (u, v) = 0, and its linearized stability is described by the positivity of the self-adjoint operator  A=

Luu (u, v) Luv (u, v) Lvu (u, v) Lvv (u, v)



4.2 Gradient and Skew-Gradient Systems

121

in L 2 ()2 , provided that L = L(u, v) is C 2 because the linearized equation is given by       d u 1 0 u . (4.9) + JA = 0, J = v 0 τ −1 dt v This operator A is the same as the Hessian of L and the linearly stable stationary solution derived from this Lagrangian is dynamically stable. If Luv (u, v) = 0, then this linearized stability is equivalent to the component-wise positivities of the operators Luu (u, v) and Lvv (u, v), Luu (u, v) > 0, Lvv (u, v) > 0.

(4.10)

For the Toland duality with C 2 Lagrangian L(u, v) = F ∗ (u) + G(v) − v, u, this stability means δ 2 F ∗ (u) > 1 and δ 2 G(v) > 1 as quadratic forms. In the skew-gradient system using the skew-Lagrangian L = L(u, v), that is ut = −Lu , τ vt = Lv , the stationary state is similarly defined by Lu (u, v) = Lv (u, v) = 0, while the linearized equation is indicated by (4.9) for  A=

 Luu (u, v) Luv (u, v) . −Luv (u, v) Lvv (u, v)

In this case, the linearized stablity of (u, v) means that any eigenvalues of A is in the right-half space, or, equivalently, Re (JAw, w)J = Re (Aw, w) > 0, ∀w =

  u

= 0, v

using the complex-valued trial functions {u, v} and the L 2 -inner product  (w1 , w2 ) =



u1 u2∗ + v1 v2∗ dx, wi =



ui vi



where z∗ denotes the complex-conjugate of z ∈ C and (w1 , w2 )J = (J −1/2 w1 , J −1/2 w2 ).

, i = 1, 2,

(4.11)

122

4 Phenomenology

Since (u, v) is real-valued, it holds that Re (Aw, w) = u, Luu (u, v)u + v, Lvv (u, v)v  u1∗ u2 + v1∗ v2 dx w1 , w2  = 

and, therefore, condition (4.11) is equivalent to (4.10) and such a stationary state is dynamically stable if L = L(u, v) is C 2 , see [426, 427]. Eguchi-Oki-Matsumura Equation An example of the gradient system without duality is the Eguchi-Oki-Matsumura equation concerning phase separation of alloys [93]. It is a combination of model (B) and model (A) equations, using the Lagrangian  L(u, v) =



1 ξ2 |∇u|2 + |∇v|2 + f (u, v) dx, 2 2

where f (u, v) =

a 2 b 2 b 4 g 2 2 u − v + v + u v 2 2 4 2

with τ , a, b, b , g > 0 constants, stands for the bulk-energy and u and v are the concentration of the main component and the order parameter, respectively. Thus, we obtain   ∂ Lu (u, v) = 0, τ ut = ∇ · ∇Lu (u, v), vt = −Lv (u, v), ∂ν ∂ and hence d dt

 

u = 0,

d L(u, v) = − dt

 

τ −1 |∇L(u, v)|2 + vt2 dx ≤ 0,

where τ > 0 is a constant. Using all u ∈ H 1 () and v ∈ H 1 () in deriving Lu and Lv , we obtain   ∂ (u, v) = 0 ∂ν ∂ as a natural boudary condition. Then the stationary state is described by   ∂u  −u + au + guv = constant, = 0, u=λ ∂ν ∂   ∂v  −ξ 2 v − bv + b v 3 + gu2 v = 0, = 0. ∂ν ∂ 2

4.2 Gradient and Skew-Gradient Systems

123

This system is difficult to reduce to single equations on u or v. There arise, however, multiple stationary solutions [156]. Gierer-Meinhardt Equation Yanagida [426, 427] formulated the Gierer-Meinhardt equation and the Fitz HughNagmo equation, see Sect. 4.3, as skew-gradient systems. The former is concerned with the morphogenesis [139] and is a combination of model (A) equations rat = −La , qτ ht = Lh , where r, q, τ > 0 are constants. It is derived from the skew-Lagrangian  L(a, h) =



rε qD |∇a|2 − |∇h|2 − H(a, h) dx 2 2

with q r H(a, h) = − a2 + rσa + ap+1 hq + h2 2 2 in the case of p + 1 = r,

q + 1 = s,

(4.12)

where ε, D, σ > 0 are constants. If all a ∈ H 1 () and h ∈ H 1 () are taken to derive La and Lh , then we obtain ap ar at = ε2 a − a + q + σ, τ ht = Dh − h + s h h   ∂ (a, h) = 0. ∂ν ∂

(4.13)

The shadow system is obtained by making D → +∞ in the second equation. More precisely, in this case h = h(t) is independent of x, and then it holds that 1 1 at = ε a − a + a /h , τ ht = −h + s · || h   ∂ (a, h) = 0 ∂ν ∂ 2

p



q



ar (4.14)

 1 by operating ||  ·, where σ = 0 is assumed for simplicity. Then its stationary state is defined by ap ε a − a + q = 0, h 2

  ∂a  1 s+1 = 0, h = ar ||  ∂ν ∂

124

4 Phenomenology

see [198, 268]. In the case of r = p + 1 there is a variational functional J(v) =

1 2

 

ε2 |∇a|2 + a2 dx −

1 (1 − γ)r

1−γ

 

ar

r defined for v ∈ H 1 (), where γ = s+1 . General case other than (4.12) is not formulated by the skew-gradient system, while there is a case of space homogenization where the ODE part takes the form of a Hamilton system, see [188]. We have spiky stationary solutions, slow dynamics of spikes, and Hopf bifurcation to (4.13), see [80, 412, 413]. The skew-gradient system can thus be involved by the top-down self-organization, while the stable stationary solutions take a role of the bottom-up self-organization.

Skew-Gradient Systems with Duality If the skew-Lagrangian is defined by the free energy using Kuhn-Tucker duality, then the stationary states split into the particle and the field components provided with the structure of dual variation. This property is a special case with the convexity of the functional, and the global dynamics near the stationary state is not so complicated. Let X be a Banach space over R, and F, G : X → (−∞, +∞] be proper, convex, lower semi-continuous functionals. The Lagrangian in the Toland duality is defined by L(x, p) = G(x) + F ∗ (p) − x, p for (x, p) ∈ X × X ∗ , see Sect. 3.2, where X ∗ denotes the dual space and F ∗ is the Legendre transformation of F: F ∗ (p) = sup {x, p − F(x)} . x∈X

It is associated with the “free energy” J ∗ (p) =



F ∗ (p) − G ∗ (p), +∞,

p ∈ D(F ∗ ) otherwise

and the “field functional” J(x) =

G(x) − F(x), +∞,

x ∈ D(G) otherwise

through J ∗ (p) = inf L(x, p), J(x) = inf∗ L(x, p). x∈X

p∈X

4.2 Gradient and Skew-Gradient Systems

125

Then we obtain the unfolding-minimality (3.37)–(3.39). Furthermore, p and x are linearly stable critial points of J ∗ and J, respectively, if and only if (x, p) is a linearly stable critical point of L, and the former two conditions are equivalent each other by p ∈ ∂G(x) ∩ ∂F(x) and x ∈ ∂G ∗ (p) ∩ ∂F ∗ (p). Skew-Lagrangian with duality, on the other hand, is defined by L(x, p) =

F ∗ (p) − G(x) + x, p , +∞,

p ∈ D(F ∗ ), x ∈ X otherwise

Then, putting J(x) = G(x) + F(−x), J ∗ (p) = F ∗ (p) + G ∗ (p), we obtain a similar structure. In other words, we say that (x, p) ∈ X × X ∗ is a critical point of L if 0 ∈ Lx (x, p), 0 ∈ Lp (x, p), which is equivalent for x and p to be critical points of J and J ∗ , respectively: p ∈ ∂F(−x) ∩ ∂G(x), x ∈ −∂F ∗ (p) ∩ ∂G ∗ (p).

(4.15)

These two relations of (4.15) are equivalent each other, and if one of them is satisfied then it holds that L(x, p) = J ∗ (p) = −J(x). We have the unfolding L|x∈∂G ∗ (p) = J ∗ ,

L|p∈∂F(−x) = −J

and the“minimality” in the sense of J ∗ (p) ≤ L(x, p) ≤ −J(x) for any (x, p) ∈ X × X ∗ . A critical point (x, p) is called a saddle if L(x, p) ≤ L(x, p) ≤ L(x, p)

(4.16)

for any (x, p) in a neighborhood of (x, p). It is linearly stable if there is ε0 > 0 such that any ε ∈ (0, ε0 /4] admits δ > 0 such that x − xX < ε0 , p − pX ∗ < ε0 L(x, p) − L(x, p) < δ, L(x, p) − L(x, p) < δ



x − xX < ε p − pX ∗ < ε.

126

4 Phenomenology

These conditions of stability split component-wisely, and (x, p) is a linearly stable saddle of L if x and p are linearly stable critical points of J and J ∗ , respectively, and they are equivalent each other. More precisely, these u and v are linearly stable if there exists ε0 > 0 such any ε ∈ (0, ε0 /4] admits δ > 0 such that x − xX < ε0 , J(x) − J(x) < δ



x − xX < ε

and p − pX ∗ < ε0 , J ∗ (p) − J ∗ (p) < δ



p − pX ∗ < ε,

respectively. The skew-gradient system p˙ ∈ −Lp (x, p), τ x˙ ∈ Lx (x, p)

(4.17)

derived from the skew-Lagrangian L = L(x, p) with duality, defined above, has similar properties to (3.57). Thus in the presense of the Hilbert space Y satisfying (3.58), the solution (x(t), p(t)) in (3.59) satisfies lim sup h↓0

lim sup h↓0

1 ∗ F (p(t)) − F ∗ (p(t − h)) ≤ − x, p˙  − ˙p(t)2Y h 1 {G(x(t)) − G(x(t − h))} ≤ ˙x , p − τ ˙x (t)2Y , h

and, therefore, lim sup h↓0

lim sup h↓0

1 {L(x(t), p(t)) − L(x(t), p(t − h))} ≤ − ˙p(t)2Y h 1 {−L(x(t), p(t)) + L(x(t − h), p(t))} ≤ −τ ˙x (t)2Y . h

Thus, if (x, p) is a linearly stable saddle point of L and (4.17) is locally well-posed in its neighborhood in the sense of (3.59), then this skew-gradient system is well-posed globally in time there, (x, p) is dynamically stable, and it holds that (3.56).

4.3 Semi-unfolding-Minimality As is described in the previous paragraph, if L is a skew-Lagrangian with duality, then both J ∗ and J are convex, and hence any critical point of L is a saddle. Several other systems, however, are provided only with the semi-duality, and admit multiple stationary solutions. In this paragraph, we describe the Lagrangian and

4.3 Semi-unfolding-Minimality

127

the skew-Lagrangian combined with convex and non-convex functionals, which are found in several phase field models provided with the double-well potential, see Sect. 6. Semi-Lagrangian First, taking a Banach space X over R, we define the Lagrangian with semi-duality by L(x, p) =

F ∗ (p) + G(x) − x, p , +∞,

x ∈ D(G) otherwise,

where G : X → (−∞, +∞] is proper, convex, lower semi-continuous, while F ∗ : X ∗ → [−∞, +∞] is arbitrary. The semi-critical point (x, p) is defined by Lp (x, p) = 0 or, equivalently, p ∈ ∂G(x).

(4.18)

Here, p ∈ ∂G(x) is equivalent to x ∈ ∂G ∗ (p), and then it holds that G ∗ (p) − x, p = −G(x). Thus, we obtain the semi-unfolding-minimality inf L = L|x∈∂G ∗ (p) = J ∗ , x

where ∗

J (p) =



F ∗ (p) − G ∗ (p), +∞,

p ∈ D(F ∗ ) otherwise.

(4.19)

Given a critical point of J ∗ , denoted by p, we define its infinitesimal stability similarly, that is p ∈ X ∗ of δJ ∗ (p) = 0 is infinitesimally stable if there is ε0 > 0 such that any ε ∈ (0, ε0 /4] admits δ > 0 such that p − pX ∗ < ε0 , J ∗ (p) − J ∗ (p) < δ



p − pX ∗ < ε,

128

4 Phenomenology

and such p is dynamically stable. More precisely, if p is a linearly stable critical point of J ∗ , then any ε > 0 admits δ > 0 such that if (x(t), p(t)) ∈ X × X ∗ (0 ≤ t < T ) satisfies t ∈ [0, T ) → p(t) ∈ X ∗ t ∈ [0, T ) → L(x(t), p(t)) ∈ R p(0) − pX ∗ < δ,

continuous non-increasing

then it holds that sup p(t) − pX ∗ < ε.

t∈[0,T )

We have ˆ p) L(x, p) = J ∗ (p) + L(x, ∗ ˆ p) = G (p) + G(x) − x, p ≥ 0 L(x, ˆ p) = 0 holds if and only if x ∈ ∂G ∗ (p), or, equivalently, p ∈ ∂G(x). Given and L(x, a critical point p ∈ X ∗ of J ∗ , we take x ∈ ∂G ∗ (p), and now define its hyper-linear stability. Thus x ∈ ∂G ∗ (p) is hyper-linearly stable if there is ε0 > 0 such that any ε ∈ (0, ε0 /4] admits δ > 0 such that x − xX < ε0 ˆ p) − L(x, ˆ p) < δ L(x, p − pX ∗ < δ



x − xX < ε.

(4.20)

Consequently, if p is a linearly stable critical point of J ∗ and x ∈ ∂G ∗ (p) is hyperlinearly stable, then any ε > 0 admits δ > 0 such that t ∈ [0, T ) → (x(t), p(t)) ∈ X × X ∗

continuous

t ∈ [0, T ) → L(x(t), p(t)) ∈ R x(0) − xX + p(0) − pX ∗ < δ

non-increasing



sup x(t) − xX + p(t) − pX ∗ < ε.

t∈[0,T )

4.3 Semi-unfolding-Minimality

129

Semi-Skew-Lagrangian Skew-Lagrangian with semi-duality is defined by L(x, p) =

F ∗ (p) − G(x) + x, p , p ∈ D(F ∗ ) +∞, otherwise,

(4.21)

where G : X → (−∞, +∞] is proper, convex, lower semi-continuos, while F ∗ : X ∗ → [−∞, +∞] is arbitrary. A semi-critical point (x, p) of L is defined by (4.18), and there is semi-unfolding-minimality, sup L = L|x∈∂G ∗ (p) = J ∗

(4.22)

x

for the free energy defined by J ∗ (p) = F ∗ (p) + G ∗ (p).

(4.23)

Linearized stability of a critical point p ∈ X ∗ of J ∗ is defined by (4.20). Here, we note that the multiple existence of such p can occur because F ∗ may not be convex. A stronger concept of the robust stability of p indicates the existence of ε0 > 0 such that any ε ∈ (0, ε0 /4] admits δ > 0 such that x − xX + p − pX ∗ < ε0 , L(x, p) − J ∗ (p) < δ



p − pX ∗ < ε.

Although there is neither the Lyapunov function nor the anti-Lyapunov function in the skew-gradient system, such p is stable as far as x(t) stays nearby x, where (x, p) is a semi-critical point of L so that x ∈ ∂G ∗ (p). Thus, the driving force of leaving such (x, p) lies in the x component. We emphasize that this “semi-stability” is different from the (full)-stability of (x, p) defined in Sect. 4.2 for C 2 skew-Lagrangian. Since ˆ p), L(x, ˆ p) = G ∗ (p) + G(x) − x, p , L(x, p) = J ∗ (p) − L(x, the hyper-linear stability of x ∈ ∂G ∗ (p) is defined similarly, where p is a critical point of J ∗ defined by (4.23). It holds that ˆ p) < δ L(x, p) − J ∗ (p) = J ∗ (p) − J ∗ (p) − L(x, and, therefore, if p ∈ X ∗ is a linearly stable critical point of J ∗ defined by (4.23) and x ∈ ∂G ∗ (p) is hyper-linearly stable, then p is robust stable. The skew-gradient system derived from this semi-skew-Lagrangian, on the other hand, has the properties

130

4 Phenomenology

t ∈ [0, T ) → (x(t), p(t)) ∈ X × X ∗ t ∈ [0, T ) → L(x, p(t)) ∈ R t ∈ [0, T ) → L(x(t), p) ∈ R

continuous non-increasing if x − xX < ε0 non-decreasing if p − pX ∗ < ε0

with ε0 > 0, and, therefore, if x ∈ G ∗ (p) is hyper-linearly stable, then (x(t), p(t)) stays nearby (x, p) as far as p(t) − pX ∗ < ε0 is kept. Thus if p ∈ X ∗ is a linearly stable critical point of J ∗ defined by (4.23) and x ∈ ∂G ∗ (p) is hyper-linearly stable, then (x, p) is dynamically stable, and, in this way, the stationary solution and its full-stability split into each component in the skewgradient system with semi-duality. If L(x, p) is C 2 , then this condition is reduced to (4.10). FitzHugh-Nagumo Equation The FitzHugh-Nagumo equation concerning nerve impluse [115, 255] is a combination of model (A) equations ut = −Lu (u, v), τ vt = Lv (u, v), where τ > 0 is a constant. It is derived from the skew-Lagrangian  L(v, u) =



ξ2 σ |∇u|2 + W (u) − |∇v|2 + uv dx 2 2

defined for

 u ∈ H (), v ∈ H (), 1

1



v = 0,

where ξ, σ > 0 are constants and W (u) =

2 1 2 u −1 . 4

Thus, we obtain 1 ut = ξ u − W (u) − v, τ vt = σv + u − ||    ∂ (u, v) = 0, v = 0. ∂ν  ∂ 2

 

u

in  × (0, T ) (4.24)

The above skew-Lagrangian is provided with the semi-duality, and (4.21) holds with  2 ξ |∇u|2 + W (u) dx, u ∈ X = H 1 () F ∗ (u) =  2

4.3 Semi-unfolding-Minimality

131

and σ G(v) = ∇v22 , v ∈ X, 2

 

v = 0.

Then the free energy is defined by (4.19), that is ∗



J (u) =



 ξ2 σ |∇u|2 + W (u) dx + (−)−1 u, u , 2 2

which is nothing but Ohta-Kawasaki’s free energy F(u) defined by (4.8). The stationary state of (4.24) is, in particular, given by δF(u) = 0 and v = (−JL )−1 u. The former is equivalent to −ξ 2 u = u − u3 + σ(−)−1 u in ,

 ∂u  = 0, ∂ν ∂

and we can confirm the semi-unfolding-minimality sup

v,



 v=0

L = L|v=(−)−1 u = F.

(4.25)

Since the FitzHugh-Nagumo equation (4.24) is skew-gradient, the linearized stability of the stationary state (u, v) is reduced to the positivities of Luu (u, v) and Lvv (u, v). The latter positivity is obvious because Lvv (u, v) is nothing but − with the domain 

 ∂ψ  2 ψ ∈ H () | = 0, ψ=0 , ∂ν ∂  and, therefore, its linearized stability is described by the positivity of the first eigenvalue of AFHN = −ξ 2  − 1 + 3u2 with the domain 

∂ψ  2 =0 . D(AFHN ) = ψ ∈ H () | ∂ν ∂ The first eigenvalue of this operator is denoted by μFHN (u).

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4 Phenomenology

The stationary state u is the same as that of the model (A) equation derived from Ohta-Kawasaki’s free energy. In the latter case, its linearized stability is indicated by the positivity of the first eigenvalue of AOK defined by AOK ψ = −ξ 2 ψ − ψ + 3u2 ψ + σ(−JL )−1 ψ with the domain 

∂ψ  2 =0 . D(AOK ) = ψ ∈ H () | ∂ν ∂ Thus if its first eigenvalue is denoted by μOK (u), then we obtain μOK (u) > μFHN (u). This relation, combined with the semi-unfolding (4.25), indicates that the instability around a stationary state (u, v) of the FitzHugh-Nagumo equation satisfying μOK (u) > 0 > μHFN (u) occurs to v at the begining, casting the driving force to the self-organization.

4.4 Kuhn-Tucker Duality Skew-Lagrangian with duality is regarded as a special case of the Kuhn-Tucker duality [94, 124, 398], and this paragraph is devoted to the description of the latter. Let X be a Banch space over R and regard L = L(x, p) : X × X ∗ → [−∞, +∞] as a skew-Lagrangian. In the context of game theory, L stands for the price which X and X ∗ wish to increase and decrease, respectively. Then α = supx inf p L(x, p) indicates the best price of X when X ∗ takes his own best strategy against X. Similarly, β = inf p supx L(x, p) is the best price of X ∗ when X takes his own best strategy against X ∗ . It is obvious that α ≤ β, but a sufficient condition to α = β is the exsitence of the saddle (x, p) ∈ X × X ∗ defined by (4.16). Then von Neumann’s theorem guarantees the existence of such a saddle, provided that x ∈ X → L(x, p) is proper, concave, upper semi-continuous, and coercive from above for any p ∈ X ∗ , and simultaneously, p ∈ X ∗ → L(x, p) is proper, convex, lower semi-continuous, and coercive from below for any x ∈ X, see [94, 360] for instance. If the cost function ϕ = ϕ(x, y) : X × X → [−∞, +∞]

4.4 Kuhn-Tucker Duality

133

is given, then the principal and the dual problems are defined by (P) inf ϕ(x, 0), x∈X

(P∗ ) sup −ϕ∗ (0, q), q∈X ∗

where ϕ∗ (p, q) =

sup

(x,y)∈X×X

{x, p + y, q − ϕ(x, y)}

denotes the Legendre transformation of ϕ(x, y). Then the duality theorem of KuhnTucker guarantees the same value of (P) and (P∗ ), provided that y ∈ X → (y) = inf ϕ(x, y) x

is proper, convex, lower semi-continuous. In fact, we have ϕ∗ (0, q) = sup {y, q − ϕ(x, y)} x,y∈X

= sup {y, q − (y)} = ∗ (q) y∈X

and hence sup −ϕ∗ (0, q) = sup −∗ (q) = ∗∗ (0)

q∈X ∗

q∈X ∗

= (0) = inf ϕ(x, 0) x∈X

by Fenchel-Moreau’s duality. Given the cost function ϕ = ϕ(x, y), we define the skew-Lagrangian by L(x, q) = sup {y, q − ϕ(x, y)} , y∈X

that is the Legendre transformation of y ∈ X → ϕx (y) = ϕ(x, y). If this mapping is proper, convex, lower semi-continuous, then we obtain ϕ(x, y) = ϕ∗∗ x (y) = sup {y, q − L(x, q)} , q∈X ∗

(4.26)

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4 Phenomenology

similarly, and, therefore, the principal problem is described by inf ϕ(x, 0) = − sup inf∗ L(x, q).

x∈X

x∈X q∈X

We have, on the other hand, ϕ∗ (p, q) = sup {x, p + L(x, q)}

(4.27)

x∈X

and hence sup −ϕ∗ (0, q) = − inf∗ sup L(x, q).

q∈X ∗

q∈X x∈X

Using the duality theory mentioned above, thus, we obtain Kuhn-Tucker’s saddle point theorem. Theorem 4.4.1 If  = (y) and ϕx = ϕx (y) defined above are proper, convex, lower semi-continuous, and if the principal and the dual problem have the solutions x and p, respectively, then (x, p) is a saddle of L. Skew-Lagrangian with Duality If p → L(x, p) is proper, convex, lower semi-continuous for each x, coversely, then the cost function ϕ = ϕ(x, y) defined by ϕ(x, y) = sup {y, p − L(x, p)} p

satisfies (4.26), and then equality (4.27) holds true. In this case, the principal and the dual problems are formulated by (P) inf ϕ(x, 0) = − sup inf L(x, q) x





x

q

(P ) sup −ϕ (0, q) = − inf sup L(x, q), q

q

x

and, in particular, these problems have the same value if L = L(x, p) has a saddle, (4.16). The skew-Lagrangian with duality in Sect. 4.2 is described by L(x, p) = F ∗ (p) − G(x) + x, p with F, G : X → (−∞, +∞], proper, convex, lower semi-continuous. In this case, the cost function ϕ(x, y) and its Legendre transformation ϕ∗ (p, q) are defined by

4.4 Kuhn-Tucker Duality

135

ϕ(x, y) = sup {y, p − L(x, p)} = F(y − x) + G(x) p



ϕ (p, q) = sup {x, p + y, q − ϕ(p, q)} = F ∗ (q) + G ∗ (p + q), x,y

and, therefore, the principal and the dual problems are described by (P) inf ϕ(x, 0) = inf {G(x) + F(−x)} = inf J(x) x x x

(P∗ ) sup −ϕ∗ (0, q) = inf F ∗ (q) + G ∗ (q) = inf J ∗ (q). q

q

q

Concerning the existence of the saddle, we can apply the theorem of von Neumann, and then obtain the following theorem. Theorem 4.4.2 If X is a Banach space over R, and F, G : X → (−∞, +∞] are proper, convex, lower semi-continuous, then the skew-Lagrangian L(x, p) = F ∗ (p) − G(x) + x, p has a saddle if x → G(x) − x, p and p → F ∗ (p) − x, p are coercive for each (x, p). In this case, the associated variational problems inf x J(x) and inf p J ∗ (p) have the same value, where J(x) = G(x) + F(−x), J ∗ (p) = F ∗ (p) + G ∗ (p).

Linear Programing Theorems 4.4.1–4.4.2 are applicable to the linear programing, where X = Rn and (d, p) ∈ X × X ∗ with d, r ≥ 0. The last relation means that any components of d and r are non-negative. In more precise, for a non-void closed convex set K ⊂ Rn , its indicator function is denoted by 0, x∈K 1K (x) = +∞, otherwise. Then we define the skew-Lagrangian by L(x, q) = −r · x − q · (d − Ax) − 1x≥0 (x) + 1q≥0 (q). Since sup z · q = q≥0

0, z≤0 +∞, otherwise

136

4 Phenomenology

the cost function is defined by ϕ(x, y) = sup {y · q − L(x, q)} = r · x + 1K (x, y) q

for K = {(x, y) | x ≥ 0, Ax ≥ d + y} with the Legendre transformation ϕ∗ (p, q) = sup {x · p + y · q − ϕ(x, q)} x,y

=

sup

{x · (p − r) + (y − Ax + d) · q + (Ax − d) · q}

x≥0, y−Ax+d≤0

= 1q≥0 (q) − q · d + sup(p − r + t Aq) · x =

x≥0

−qd, if p − r + t Aq ≤ 0, q ≥ 0 +∞, otherwise.

Thus, the principal and the dual problems are given by (P) minimize r · x for x ≥ 0, Ax ≥ d (P∗ ) maximize q · d for q ≥ 0, t Aq ≤ r. These problems have the same value from the duality theorem, and the solutions x, q constitute of the saddle of L: L(x, q) ≤ L(x, q) ≤ L(x, q), ∀(x, q). Stokes System The Stokes system − u + ∇p = f , ∇ · u = 0,

u|∂ = 0

(4.28)

is identified with the saddle of the skew-Lagrangian L(u, p) =

1 ∇u22 − (f , u) − (u, ∇p) 2

(4.29)

defined for (u, p) ∈ X × Y , where X = H01 ()n , Y = L 2 (), and (·, ·) denotes the L 2 -inner product. First, (u, p) ∈ X × Y is a weak solution to (4.28) if it satisfies (∇u, ∇v) − (p, ∇ · v) = (f , v), (u, ∇q) = 0, ∀(v, q) ∈ X × Y . Next, (u, p) ∈ X × Y is a saddle of L if and only if L(u, p) ≤ L(u, p) ≤ L(u, p), ∀(u, p) ∈ X × Y .

(4.30)

4.4 Kuhn-Tucker Duality

137

Theorem 4.4.3 The Stokes system (4.28) is equivalent to (4.30). Proof Let (u, p) ∈ X × Y be a (weak) solution to (4.28) and take (u, p) ∈ X × Y arbitrary. Then it follows that L(u, p) − L(u, p) = (u, ∇(p − p)) = 0 and L(u, p) − L(u, p) 1 1 = ∇u22 − ∇u22 − (f , u − u) − (u − u, ∇p) 2 2 1 = (∇u, ∇(u − u)) − (f , u − u) − (u − u, ∇p) + ∇(u − u)22 2 1 = ∇(u − u)22 ≥ 0. 2 Let (u, p) ∈ X × Y be a saddle of L, conversely, and take (u, p) ∈ X × Y , arbitrary. First, L(u, p) ≥ L(u, p) means (u, ∇(p − p) ≤ 0 which implies (u, ∇p) = 0, ∀p ∈ Y because p ∈ Y is arbitrary. Next, we have L(u, p) ≥ L(u, p), ∀u ∈ X which means 1 (∇u, ∇w) − (f , w) − (w, ∇p) + ∇w22 ≥ 0, ∀w ∈ X 2

(4.31)

and, therefore, we obtain (∇u, ∇w) − (f , w) − (w, ∇ · p) ≥ 0, ∀w ∈ X, checking the degree in w of (4.31). Then it follows that (∇u, ∇w) − (f , w) − (w, ∇p) = 0, ∀w ∈ X, and hence (u, p) is a weak solution to (4.28).



The cost function ϕ(p, u) associated with the skew Lagrangian L(p, u) of (4.29) is defined for (u, p) ∈ X × Y ∗ , that is

138

4 Phenomenology

ϕ(u, p) = sup{z, p + L(u, z)} z∈Y

1 = sup{z, p + ∇u22 − (f , u) − (u, ∇z)} 2 z∈Y 1 = ∇v22 − (f , v) + 1{p=−∇·u} (p), 2 and, therefore, ϕ∗ (w, q) = sup{v, w − L(v, q)} =

v∈X

sup {v, w + (p, q) − ϕ(v, p)}

p∈Y , v∈X

1 = sup{v, w − ∇v22 + (f , v) + (v, ∇q)} 2 v∈X for (w, q) ∈ X ∗ × Y . The principal problem is thus given by

1 2 ∇v2 − (f , v) | v ∈ X, ∇ · v = 0 . inf 2

(P)

Theorem 4.4.4 The above (P) is equivalent to (4.28). Proof Defining B : H01 ()n → Y = L 2 () by Bv = ∇ · v, we see that u solves (P) if and only if u ∈ Ker B ⊂ X, (∇u, ∇v) = (f , v), ∀v ∈ Ker B. If σ : X → X ∗ denotes the duality map defined by σf , vX ∗ ,X = (f , v), v ∈ X, then, (4.32) means u ∈ Ker B, u − σf ∈ (Ker B)⊥ = Ran B∗ . The Poincaré inequality, on the other hand, implies ∇v22 ≥ δv22 , v ∈ H01 ()n , ∇ · v = 0

(4.32)

4.4 Kuhn-Tucker Duality

139

and, therefore, Ran B∗ is closed in X ∗ . Thus (p, u) solves (P) if and only if u ∈ Ker B ⊂ X, u − σf = B∗ p for some p ∈ Y , and hence (4.28).



In the abstract formulation, we take L(x, y) = F(x) + (y, Bx)Y for (x, y) ∈ X × Y , where X and Y are Banach and Hilbert spaces over R, F : X → (−∞, +∞] is proper, convex, and lower semi-continuous, and B : X → Y is bounded linear with Ran(B∗ ) closed in X ∗ , see [124] for more details.

4.5 Summary We have examined several mathematical models concerning self-assembly, which are provided with the variational structure derived from the free energy or (skew) Lagrangian. 1. Closed Systems (a) Model (A) equation describes thermally closed system. (b) Model (B) equation describes thermally and materially closed system. 2. (Skew) Gradient Systems (a) Several model (C) equations are formulated as (skew) gradient flows derived from the (skew) Lagrangian. A typical example is the gradient system provided with the Toland duality. (b) If the skew-Lagrangian is provided with a duality, then the dynamics and the structure of the stationary solutions are simple. (c) Skew-gradient system is involved by the dissipative structure, but the dynamics around the stationary solution is controlled by the calculus of variation. 3. Non-convex evolution (a) Several free energies are involved by the double-well potential. Then, the associated (skew-) Lagrangian is provided with the semi-duality. (b) If the (skew)-Lagrangian is provided with semi-duality, then the linearized stability, the robust stability, and the hyper-linear stability of the critial point are defined. (c) A hyper-linearly stable critical point is dynamically stable in the gradient system with semi-Lagrangian. This property is a case of the skew-gradient system with semi-skew-Lagrangian, while a linearly but not hyper-linearly stable semi-crtical point casts the driving force of the far-from-equilibrium.

Chapter 5

Phase Transition

Having model (A) and model (B) equations, we devote this chapter to the physical principles that derive model (C) equations consistent with the non-equilibrium thermodynamics. Several systems of phenomenological equations are thus derived from the free energy and duality. Similar to entropy, free energy is a fundamental concept of thermodynamics in accordance with the thermal equilibrium.

5.1 Thermal Equilibrium Here we describe the theory of equilibrium thermodynamics, and reformulate the free energy from the physical points of view. Quantity of State First, the objects of thermodynamics are the quantity of state, independent of the history of the system. The (absolute) temperature T , the pressure p, and the volume V are the quantities of state of the ideal gas, subject to the state equation f (p, T , V ) = 0. A thermodynamical quantity of state A, therefore, is a function of two variables of p, T , and V , denoted by x and y, satisfying  γ

dA = 0

for any closed path γ in the xy plane, where dA = Xdx + Ydy, X =

∂A ∂A , Y= . ∂x ∂y

© Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_5

141

142

5 Phase Transition

Thus, this one-form Xdx + Ydy is completely integrable, and it holds that ∂Y ∂X = . ∂y ∂x Energy Balance The first law of thermodynamics is the energy balance described by dU = Q − pdV, where the left-hand side indicates the energy variation of the system in accordance with the volume variation in the right-hand side caused by the outer thermal energy Q. Here, it is emphasized that heat or work is not a quantity state. Entropy Increasing The second law of thermodynamics is the entropy increasing. If the process is reversible and d  Q denotes the energy variation, then we can examine  γ

dQ =0 T

for any closed path γ, and thus we can define the entropy variation dS by dS =

dQ T

in this case. This relation allows us to re-fomulate entropy as a state quanity valid even in the irreversible system. Then, the Clausius-Duhem’s inequality dS >

dQ T

(5.1)

is obtained if the process is irreversible. Entropy decomposition is obtained by introducing the inside thermal energy transport d  Q∗ caused by the thermal energy contact, denoted by d  Qir , to the outer system. More precisely, the entropy variation and the inner entropy production arising at this contact are defined by d  Qir de S = T and di S =

d  Q∗ , T

5.1 Thermal Equilibrium

143

respectively, and then inequality (5.1) is replaced by the equality dS = de S + di S.

(5.2)

Thus, the second law of thermodynamics is re-formulated by di S ≥ 0 with the equality if and only if the process is reversible. Free Energy The first and second laws of thermodynamics are thus summarized by dU = d  Q − pdV, dS =

dQ + di S. T

(5.3)

Then, Helmholtz’ free energy A, Gibbs’ free energy G, and the enthalpy H are defined by A = U − TS, G = H − TS, H = U + pV. If the process is iso-thermal and iso-volumetric, then it holds that dA = dU − TdS = −Tdi S ≤ 0 with the equality if and only if it is reversible. Similarly, if the process is iso-thermal and iso-baric, then it holds that dG = dH − TdS = dU + pdV − TdS = −Tdi S ≤ 0 with the equality if and only if it is reversible. In the reversible process of di S = 0, it follows that dU = TdS − pdV, which guarantees dH = TdS + V dp, dA = −SdT − pdV, dG = −SdT + V dp

(5.4)

and, therefore,         ∂U ∂H ∂H ∂U , p=− ; T= , V = T= ∂S V ∂V S ∂S p ∂p S         ∂A ∂A ∂G ∂G S=− , p=− ; S=− , V =− . ∂T V ∂V T ∂T p ∂p T

144

5 Phase Transition

These equalities result in 

     ∂p ∂T ∂V =− , = ∂S V ∂p S ∂S p S         ∂S ∂p ∂S ∂V = , =− , ∂V T ∂T V ∂p T ∂T p ∂T ∂V





called Maxwell’s relation. Here and henceforth, ∗ in ( )∗ indicates the fixed variable in taking derivatives. Chemical Potential In the open system provided with the material transport between the outer system, Gibbs’ free energy is a function of the numbers of moleculars comprising the system G = G(T , p, n1 , n2 , . . .). In this case, dG of (5.4) shifts to dG = −SdT + V dp +



μi dni ,

i



where μi =

∂G ∂ni

 T ,p,nj=i

is called the chemical potential. It is assumed to be a partial molal quantity, which means       ∂U ∂H ∂A = = , μi = ∂ni S,V,nj=i ∂ni S,p,nj=i ∂ni T ,V,nj=i or, equivalently, dU = TdS − pdV +



μi dni

i

dH = TdS + V dp +



μi dni

i

dA = −SdT − pdV +



μi dni .

(5.5)

i

If the system is iso-thermal and iso-baric, then we obtain dG =

 i

μi dni .

(5.6)

5.1 Thermal Equilibrium

145

Therefore, if the material composition is constant, furthermore, then it follows that G=



μi ni

i

and hence



ni dμi = 0.

i

It is called Gibbs-Duhem’s equality. If ξ denotes the extent of reaction, then dni = νi dξ is the variance of the i-component of the material, when the chemical reaction proceeds from ξ to ξ + dξ, where νi denotes the stoichiometric coefficient. Its time variance is defined by dξ dni = νi , dt dt or

 dξk dni = νik dt dt

(5.7)

k

if this component is associated with the other reactions. Equilibrium If two interacting systems A and B are in equilibrium and are isolated totally, then the total entropy, the total inner energy, the total volume, and the total quatntity of i component are constant. SA + SB = constant,

UA + UB = constant

VA + VB = constant,

niA + niB = constant.

From dSA + dSB = 0, it follows that    μiA dUA pA + dVA − dniA TA TA TA i    μiB dUB pB + dVB − dniB = 0. + TB TB TB i

Then, using dUA + dUB = 0, dVA + dVB = 0, dniA + dniB = 0,

146

5 Phase Transition

we obtain        μiA 1 1 pA pB μiB dUA + dVA − dniA = 0, − − − TA TB TA TB TA TB i

which results in TA = TB , pA = pB , μiA = μiB . Thus, the equilibrium of this contact is described by the balance of the temperature, the pressure, and the chemical potential of two systems. In the iso-thermal and iso-baric system, we have (5.6). In the chemical reaction system aA + bB  cC + dD, for example, the free energy variation is given by G = cμC + dμD − aμA − bμB . The chemical potential of ideal solution, on the other hand, is equal to μi = μ0i + RT log ci ,

(5.8)

where R, ci , and μ0i are the gas constant, the concentration, and the normal chemical potential, respectively. Thus in the equilibrium of G = 0, it holds that G0 = −RT log K, K =

[C]ceq [D]deq [A]aeq [B]beq

G0 = cμ0c + dμ0D − aμ0A − bμ0B , where [ ]eq denotes the equilibrium concentration and K is called the equilibrium coefficient of reaction. In the chemical reaction of ν1 R1 + ν2 R2 + · · · + νj Rj → νj+1 p1 + νj+2 p2 + . . . νc pc , we replace −νi by νi for i ≥ j + 1, and define the chemical affinity by A=−

j  i=1

νi μi .

5.1 Thermal Equilibrium

147

The inner entropy production of this process is given by d  Q∗ = Tdi S = −dG = −



μi dni = Adξ

(5.9)

i

from νi dξ = dni .

5.2 Stefan Problem We begin with the two-phased Stefan problem and its mathematical structures by examining first its free-boundary and interface problem, secondly, the reduction to a degenerate parabolic equation associated with the enthalpy, third, the application of the nonlinear semi-group theory to this equation, and finally, the phase field approach of the Ginzburg-Landau theory using order parameter. Level Set Approach Let θ be the relative temparature, and assume that the heat conductor is water and ice if θ > 0 and θ < 0, respectively. Then we obtain the heat equation (1.7), cρθt = ∇ · (κ∇θ)

in {θ = 0}.

(5.10)

The region {x ∈  | θ(x, t) = 0} is comprised of an interface, denoted by t , where the phase transition occurs and the latent heat is exchanged. Usually, the density ρ = ρ(θ) depends continuously on θ ∈ R, but the specific heat c = c(θ) and the conductivity κ = κ(θ) have the discontinuity of the first kind at θ = 0. To describe the motion of t , we take first the level-set approach, and introduce the C 1 function  = (x, t) satisfying t : (·, t) = 0.

(5.11)

Let ν be the outer unit normal vector of t from {θ > 0} at x ∈ t . If x moves νh during the small time interval t, then it holds that (x + νh, t + t) = 0. Taking the infinitesimal approximation of this relation, we obtain (ν · ∇)h + t t = 0

on t .

(5.12)

Meanwhile, h is radiated from the unit area on t as the latent heat, where  = λρ with λ standing for the latent heat per unit weight, and, therefore, NewtonFourier-Fick’s heat energy balance law guarantees the relation

148

5 Phase Transition



∂θ h = − κ ∂ν where

+

[A]+ − = A+ − A− , A± (x) =



t,

lim

y∈{±θ > 0}, y→x

(5.13)

A(y).

Combining (5.12) with (5.13), thus, we obtain the Stefan condition  ∂θ + ∂ · κ ∂ν ∂ν −

t =

on t ,

(5.14)

which comprises of the Stefan problem with (5.10) and (5.11). Enthalpy Formulation Here, we use the Kirchhoff transformation  u=

θ

κ(θ )dθ

0

and the enthalpy H = H(u) defined by

H(u) =

θ

0θ 0

ρ(θ )c(θ )dθ − , u < 0 ρ(θ )c(θ )dθ , u > 0,

which satisfies H  (u) =

ρ(θ)c(θ) , u = 0, H(+0) − H(−0) = . κ(θ)

Concerning u = u(x, t), we obtain ρc · θt · κ κ

∇u = κ∇θ, H(u)t = and, therefore, H(u)t = u



in

(\ t ) × {t}

0 0 denotes the relaxization time. In the classical theory of thermodynamics, however, decrease of the free energy occurs under the constant temperature, and in this sense this model is not valid if u is far from zero. If the system is involved by the quantity other than the order parameter, several model (C) equations are formulated in phenomenology. This section is devoted to the mathematical study of several model (C) equations. Actually, some of them take the Lyapunov function, and then we obtain the semi-unfolding-minimality concerning the order parameter, which provides stability of the linearly stable order parameter. This property is the case of the Penrose-Fife equation for phase transition [297], the coupled Cahn-Hilliard equation for phase separation [7, 297], and the Falk-Pawlow equation for shape memory alloys [106, 107, 293, 295]. Although the well-posedness of these equations have been studied by several authors [39, 436], this variational

152

5 Phase Transition

structure provides a new point of view to the global dynamics of these equations. Historically, the first model (C) equation was the Fix-Caginalp equation described above, concerning non-isothermal phase transition. When this system is thermally open it is provided with 



∂ϕ , u

= 0. ∂ν ∂

(5.24)

In this case, we have  ξ2 d d ∇ϕ22 − W (ϕ) + 2(u, ϕt ) 2 dt dt  1d  u22 + (ϕt , u) = −κ ∇u22 2 dt 2

τ ϕt 22 = −

and, therefore, d dt



1 ξ 2  u22 + ∇ϕ22 + 2 8 4

Thus L(u, ϕ) =

 τ W (ϕ) = − ϕt 22 − κ ∇u22 4  ≤ 0. (5.25)



1 ξ 2  u22 + ∇ϕ22 + 2 8 4

 

W (ϕ)

(5.26)

acts as a Lyapunov function. The functional L = L(u, ϕ) is not provided with the hook term  uϕ, 

and the Fix-Caginalp equation is not a gradient system derived from L(u, ϕ). In the stationary state, we have u=u≡0 by putting ∂t · = 0 in the enthalpy equation    u + ϕ = κu, 2 t

u|∂ = 0,

and, therefore, the stationary ϕ = ϕ is defined by − ξ ϕ = ϕ − ϕ , 2

3

∂ϕ

= 0, ∂ν ∂

putting ∂t · = 0 in the order parameter equation.

(5.27)

5.3 Phase Field Model

153

This elliptic problem concerning the stationary order parameter has the variational structure defined by Ginzburg-Landau’s free energy,  F(ϕ) =



ξ2 |∇ϕ|2 + W (ϕ) dx, 2

ϕ ∈ H 1 (),

and, therefore, it is equivalent to δF(ϕ) = 0 for ϕ ∈ H 1 (). Then we can confirm the semi-unfolding-minimality L(u, ϕ) ≥ L(u, ϕ) =

 F(ϕ). 4

From the argument described in Sects. 3.2 and 4.3, this property implies that (ϕ, u) = (ϕ, u) is dynamically stable in ϕ component if ϕ is a linearly stable critical point of F(ϕ) defined for ϕ ∈ H 1 (). Thermal Analysis If this system is thermally closed, then it holds that

∂ (ϕ, u)

= 0 ∂ν ∂

(5.28)

instead of (5.24). Equality (5.25) is valid even in this case, and L(u, ϕ) of (5.26) is again a Lyapunov function of this system. Total enthalpy, on the other hand, is preserved in this case, and it holds that d dt



 u + ϕ dx = 0. 2 

From the enthalpy equation    u + ϕ = κu, 2 t

∂u

= 0, ∂ν ∂

the stationary u state of this closed system is a constant. This unknown constant u = u is determined by the prescribed total enthalpy, 

 u + ϕ dx = a. 2 

Thus, the stationary state is defined by −ξ ϕ = ϕ − ϕ + 2u, 2

3

 ∂ϕ

 = 0, u || + ϕ=a ∂ν ∂ 2 

154

5 Phase Transition

or, equivalently,

∂ϕ

= 0. ∂ν ∂

    2 a− ϕ , − ξ ϕ = ϕ − ϕ + || 2  2

3

(5.29)

Regarding this a as an eigenvalue, we see that this physically closed stationary state of Fix-Caginalp equation is realized as a nonlinear eigenvalue problem with non-local term similar to (1.38). The problem (5.29) has a variational function ξ2 ∇ϕ22 + Fa (ϕ) = 2

  2  2 a− W (ϕ) + ϕ  || 2  



(5.30)

defined for ϕ ∈ H 1 (), and it is equivalent to δFa (ϕ) = 0, that is this ϕ ∈ H 1 () satisfies

d Fa (ϕ + sψ)

=0 ds s=0 for any ψ ∈ H 1 (). Then the semi-unfolding is obtained by ξ 2  ∇ϕ22 + 8 4



1 W (ϕ) + u2 || 2     2 2  ξ  1 ∇ϕ22 + a− = W (ϕ) + ϕ 8 4  2 || 2   = Fa (ϕ), 4

L(u, ϕ) =

using



 u + ϕ dx = a. 2 

This property implies the semi-minimality L(u, ϕ) = by



 u + ϕ dx = a, 2 

 Fa (ϕ) ≤ L(u, ϕ), 4 

1 ||

2

 

u



1 ||

 

u2 .

Thus, an infinitesimally stable critical function ϕ of Fa is dynamically stable. More precisely, if the self-adjoint operator Aa in L 2 () with the domain

  ∂ψ

1 =0 D(Aa ) = ψ ∈ H () | ∂ν ∂

(5.31)

5.3 Phase Field Model

155

defined by Aa ψ = −ξ 2 ψ + ϕ3 ψ − ϕψ −

1 ||



 (a − ϕ)ψ 2 

is positive, then each ε > 0 admits δ > 0 such that L(u0 , ϕ0 ) − L(u,  ϕ) < δ   ϕ= u0 + ϕ0 dx a = u || + 2  2 



sup ϕ(·, t) − ϕH 1 < ε. t≥0

Enthalpy Analysis If we regard L=

1 ξ 2  u22 + ∇ϕ22 + 2 8 4

 

W (ϕ)

as a functional of H = u + 2 ϕ and ϕ, then the above structure of semi-duality is derived from the formulation of Sect. 4.3. In fact, in this case we obtain the hook term with  2    ξ 2  1 2   ∇ϕ2 + W (ϕ) L(H, ϕ) = H − ϕ + 2 2 2 8 4  =

1 2  ξ 2  H22 − (H, ϕ) + ϕ22 + ∇ϕ22 + 2 2 8 8 4

 

W (ϕ).

Moreover, since  2 ξ 2  3  Lϕ = − H + ϕ − ϕ + ϕ −ϕ 2 4 4 4   2 3 = − ξ ϕ + ϕ − ϕ + 2u 4  LH = H − ϕ = u, 2 the materially closed Caginalp-Fix equation is the combination of model (A) and model (B) equations derived from this Lagrangian, that is α ϕt = −Lϕ , Ht = κ∇ · ∇LH , 4

∂ (LH , ϕ)

= 0. ∂ν ∂

This Lagrangian takes the form 2 L(H, ϕ) = G(ϕ) + F ∗ (H) − H, ϕ 

156

5 Phase Transition

for  ξ2 1  ∇ϕ22 + W (ϕ) + ϕ22 4 2  4  1 F ∗ (H) = H22 , H = a,  

G(ϕ) =

and this F ∗ is regarded as a proper, convex, lower semi-continuous functional defined on L 2 (). These relations imply 2 2 inf L(H, ϕ) = L(H, ϕ) = J(ϕ), H   where J(ϕ) = G(ϕ) − F(ϕ) ϕ ∈ ∂F ∗ (H) ⇔ H ∈ ∂F(ϕ), and it holds that F(ϕ) = H,

sup



 H=a

  2    1 1 a2 ϕ, H − F ∗ (H) =  ϕ − ϕ − · .   ||  2  || 4

Thus we obtain 1 ξ2 ∇ϕ22 + 4 2 1 = Fa (ϕ), 2

J(ϕ) =

and, therefore,

 

W (ϕ) +

 Fa (ϕ) ≤ L(H, ϕ), 4

  2 1  1 a2 || ϕ + · ||  4  ||

 

H = a.

These structures motivate us to examine the hyper-stability of the critical point, see Sect. 4.3. In fact, it holds that L=

 1 1  1 Fa (ϕ) + u22 − u22 = Fa (ϕ) + u − u22 4 2 2 4 2

for u=

     1 1 a− ϕ = u, || ||  2 

and, therefore, the following theorem is obtained [240].

5.3 Phase Field Model

157

Theorem 5.3.1 If ϕ is an infinitesimally stable critical point of Fa defined by (5.30) for ϕ ∈ H 1 (), then each ε > 0 admits δ > 0 such that  u0 − u2 + ϕ0 − ϕH 1 < δ,

 u0 + ϕ0 dx = a 2 

⇒   sup u(·, t) − u2 + ϕ(·, t) − ϕH 1 < ε,

t∈[0,T )

  where (u(·, t), ϕ(·, t)) ∈ C [0, T ), H 1 () × L 2 () is a solution to (5.23)–(5.28), u=

    1 a− ϕ , || 2 

and T > 0 is arbitrary.

5.4 Summary We have described mathematical modelling of phase transition, reviewing the theory of non-equilibrium thermodynamics. 1. The Fix-Caginalp equation is a combination of the enthalpy equation of the twophase Stefan problem and the model (A) equation using a modified GinzburgLandau’s free energy. 2. Since this system is not consistent with the theory of thermodynamics, the Penrose-Fife theory is adopted if the temperature is far-from-equilibrium. Then we obtain the Penrose-Fife equation and the coupled Cahn-Hilliard equation for the non-isothermal phase transition and phase separation, respectively. 3. Theory of non-equilibrium thermodynamics is concerned with the open system provided with the chemical reaction. It formulates the balance laws of the entropy, the inner energy, and the material density per unit volume, using the local equilibirum principle. Consequently, the solution to this system has a different feature from that of model (A), model (B), and model (C) equations. 4. The stationary state of this system of balance laws casts the driving force to the non-equilibrium from the equilibrium, but its control region is restricted to nearfrom-equilibrium because the variational structure is lost in far-from-equilibrium.

Chapter 6

Critical Phenomena of Isolated Systems

Regarding the order parameter as the “field component,” we see the semi-dual variational structure in phase field equations associated with several critical phenomena. Particularly, the nonlinear eigenvalue problem with non-local term arises as the stationary state of the closed system. Recognizing the profile of the total set of stationary solutions and distinguishing their stability and instability are the first step to clarify the process of self-organization. In this chapter we study these variational structures sealed in several mathematical models of self-assembly.

6.1 Non-equilibrium Thermodynamics We begin with the theory of non-equilibrium thermodynamics from the viewpoint of dynamical systems. Although the system of balance laws can describe open systems, local equilibrium principle guarantees the Prigogine paradigm near the stable equilibrium, whereby the non-equilibrium stationary state is defined. In the range where the phenomenological relation is valid, this stationary state thus realizes the minimum entropy production rate. Then the detailed balance law implies Onsager’s reciprocity, and the irreducible process is identified with null dissipation [83, 266]. A useful application of this theory is the understanding of motion of the free boundary associated with a phase transition [407]. Balance Laws First, local equilibrium principle assures that the entropy, the inner energy, and the material desnity per unit volume v are state quantities, denoted by sv = s/v, uv = u/v, and ci = ni /v, 1 ≤ i ≤ n, respectively. Their balance laws are described by ∂sv = −∇ · Js + σ, ∂t

∂ci = −∇ · Ji + νi Jch , ∂t

∂qv = −∇ · Jq . ∂t

© Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_6

(6.1) 159

160

6 Critical Phenomena of Isolated Systems

Here, the first equation of (6.1) indicates the entropy balance, where Js and σ are the entropy flux and the local entropy production rate, respectively, and hence it holds that  di S = σdV. dt In the second equation, Ji and Jch = dξ dt are the diffusion flux and the chemical reaction rate, respectively, and, therefore, νi Jch =

dni dt

stands for the production rate of the chemical material. In the third equation, Jq and qv denote the total heat flux and the local heat quantity, respectively. Dissipation Function Above mentioned local equilibrium principle induces Tdsv = duv −



μi dci

i

by (5.5), if the process is iso-volumetric. It also holds that duv = dqv and dU = d  Q by (5.3), and hence 1 ∂qv  μi ∂Ci ∂sv = − . ∂t T ∂t T ∂t i

Using (6.1), we obtain  μi 1 (−∇ · Ji + νi Jch ). −∇ · Js + σ = − ∇ · Jq − T T i

Here, in the right-hand side, we have Jq 1 1 ∇ · Jq = ∇ · − Jq · ∇ , T T T

μi μi Ji μi ∇ · Ji = ∇ · − Ji · ∇ , T T T

and, therefore, Jq −



i μi Ji T  μ A 1  i + Jq · ∇ + + Jch , Ji · ∇ − T T T

− ∇ · Js + σ = −∇ ·

i

where A = −



i νi μi

is the chemical affinity.

(6.2)

6.1 Non-equilibrium Thermodynamics

161

Taking regards to (6.2), we put Js =

Jq −

 i

μi Ji

T

, σ = Jq · ∇

 μ A 1  i + + Jch . Ji ∇ − T T T i

Then, using ∇

 μ 1 1 μi 1 i = − ∇μi + 2 ∇T , = − 2 ∇T , ∇ − T T T T T

we obtain σ=−

Jq −

 i

μi Ji

· ∇T −

T2

 Ji T

i

· ∇μi + Jch

A T

 Ji Js A = · ∇(−T ) + · ∇(−μi ) + Jch . T T T i

This equality allows us to define the dissipation function  = T σ = Js · ∇(−T ) +



Ji · ∇(−μi ) + Jch A =

i

n+1 

Ji · Xi ,

i=0

where Ji and Xi are regarded as the flux and the general force, respectively. In more detail, J0 = Js is the entropy flux subject to X0 = −∇T with the temperature T , Ji (1 ≤ i ≤ n) is the i-th diffusion flux subject to Xi = −∇μi with the chemical potential μi , and Xn+1 = Jch is the chemical velocity subject to the chemical affinity Xn+1 = A. Entropy Production Rate In the iso-thermal system with diffusion and chemical reaction, we can define the entropy production rate by P=T

di S = dt

 dV ≥ 0

(6.3)

Using the dissipation function. Here, the equality means the equilibrium of the system. We have, on the other hand, =−

n  i=1

Ji · ∇μi +

 k

Ak vk

162

6 Critical Phenomena of Isolated Systems

in this case, and hence it holds that dP = dt

  n+1

∂Xi  ∂Ji dX P dJ P Xi + dV ≡ + . ∂t ∂t dt dt n+1

Ji

i=1

i=1

Using (5.7), then we obtain dX P = dt

 −

 i



=



 i

Ji · ∇

∂μi  ∂Ak + dV vk ∂t ∂t k

∂μi ∂μi  − dV, Ji · ∇ νik vk ∂t ∂t i,k

and, therefore, dX P = dt

 

∇ · Ji

i

∂μi  ∂μi − dV νik vk ∂t ∂t i,k

   ∂μi ∂cj ∂μi ∂cj ∇ · Ji − dV = νik vk ∂cj ∂t ∂cj ∂t i,j

i,j,k

under suitable boundary conditions. Then the second equation of (6.1) guarantees dX P =− dt

  ∂μi ∂ci ∂cj dV. ∂cj ∂t ∂t i,j

This quantity is non-negative around the stable equilibrium by (5.9), and thus, the stationary state near a linearly stable equilibrium is defined by the zero of this value. Phenomenological Relation Near from the equilibrium, the flux is proportional to the associated force; the diffusion flux is the product of the concentration gradient and the diffusion coefficient; the elecrtic current is that of the electro-static potential and the electric conductivity; and the heat flux is that of the thermal gradient and thermal conductivity, and so forth, referred to as Newton-Fourier-Fick’s law or Ohm’s law. These relations are summarized as Ji = Li Xi , or Ji =

 j

Lij Xj ,

6.1 Non-equilibrium Thermodynamics

163

regarding the effect of interaction, where Lij ’s are constant. Then Onsager’s reciprocity relation is indicated by (6.4) Lij = Lji , and (6.3) reduces to =





Ji Xi =

i

Lij Xi Xj ≥ 0.

i,j

Here, the equality indicates the equilibrium again. Thus, we obtain  i

Xi

   dXi dXj dJi dXi = = = Lij Xi Lij Xj Ji dt dt dt dt i,j

and hence

i,j

i

dJ P 1 dP dX P = = , dt dt 2 dt

which is non-negative near a linearly stable equilibrium. Principle of Detailed Balance In the reaction system k1 → A B ← k−1

(6.5)

the phenomenological coefficient L defined by Jch = LA is equal to L=

k1 cA , RT

(6.6)

where cA denotes the equilibrium concentration of A. In fact, this system is described by dcA = −k1 cA + k−1 cB , dt

dcB = k1 cA − k−1 cB , dt

and the chemical reaction rate from A to B is equal to Jch = −

dcB dcA = . dt dt

In the equilibrium, we have k1 cA = k−1 cB ,

(6.7)

164

6 Critical Phenomena of Isolated Systems

where cA and cB are equilibrium concentrations, regarded as local densities. Here, we define the equilibrium constant by K=

cA k−1 = . k1 cB

Fluctuations from the equilibrium of these concentrations are denoted by αA = cA − cA , αB = cB − cB ,    αA   , c  A

where it is assumed that

   αB    1. c  B

Using mass conservation cA + cB = cA + cB and (6.7), we obtain Jch = k1 cA − k−1 cB = k1 αA − k−1 αB = αA (k1 + k−1 ) = k1 αA (1 + K). Chemical affinity, on the other hand, is defined by A = μA − μB because νA = 1 and νB = −1 follow from (6.5). Since μA = μB , we have μ0A + RT log cA = μ0B + RT log cB by (5.8). Then it holds that  αA A = μA − μB = μ0A + RT log cA + RT log 1 + cA  αB −μ0B − RT log cB − RT log 1 + cB  

 αB αA αA αB − log 1 + ∼ RT = RT log 1 + − cA cB cA cB αA = RT (1 + K). cA Thus, we obtain (6.6) by Jch = LA.

(6.8)

6.1 Non-equilibrium Thermodynamics

165

If there is k2 → C, B ← k−2

k3 → C A ← k−3

besides (6.5), then we obtain three chemical reaction rates and three chemical affinities denoted by J1 = k1 cA − k−1 cB , J2 = k2 cB − k−2 cC , J3 = k3 cC − k−3 cA

(6.9)

and A1 = μA − μB , A2 = μB − μC , A3 = μC − μA , respectively. It holds that A1 + A2 + A3 = 0 and the dissipation function is defined by  = J1 A1 + J2 A2 + J3 A3 = (J1 − J3 )A1 + (J2 − J3 )A2 . Thus, the phenomenological relation guarantees J1 − J3 = L11 A1 + L12 A2 , J2 − J3 = L21 A1 + L12 A2 .

(6.10)

In the equilibrium, we have μA = μB = μC = 0 and, therefore, A1 = A2 = 0 and J1 = J2 = J3 . However, principle of detailed balance requires J1 = J2 = J3 = 0

(6.11)

in the equilibrium. This equality implies k1 cA = k−1 cB , k2 cB = k−2 cC , k3 cC = k−3 cA . Using the perturbations αi from the equilibrium ci , we obtain J1 = k1 αA − k−1 αB , J2 = k2 αB − k−2 αC , J3 = k3 αC − k−3 αA by (6.9). Since (6.8) reads A1 =

RT (k1 αA − k−1 αB ), k1 cA

(6.12)

166

6 Critical Phenomena of Isolated Systems

it holds that J1 =

k1 cA A1 , RT

by (6.8) and similarly J2 =

k2 cB k3 cC A2 , J3 = (A1 + A2 ). RT RT

It holds that k1 cA + k3 cC k3 cC A1 + A2 RT RT k3 cC k2 cB + k3 cC J2 − J3 = A1 + A2 , RT RT J1 − J3 =

(6.13)

and hence we obtain L12 = L21 by comparing (6.10) and (6.13). Thus the principle of detailed balance (6.11) implies reciprocity, (6.4).

6.2 Penrose-Fife Theory If the temperature varies, it is preferable to use the equation provided with the increase of entropy other than the decrease of free energy. This property is realized by the Penrose-Fife theory [297], and then the Penrose-Fife and the coupled Cahn-Hilliard equations are obtained for the phase transition and the phase separation, respectively. First, we infer (6.14) f (T ) = inf (e − Ts(e)) e

from A = U − TS and the minimum energy principle, where e, s(e), and f (T ) are the energy density, the entropy density, and the free energy density, respectively. Since e → s(e) is concave, this (6.14) is regarded as the Legendre transformation. The minimum is attained by e = e(T ) satisfying 1 ∂s = , ∂e T and hence it holds that

f (T ) e(T ) = − s (e(T )) . T T

Using (6.15), we obtain d(f (T )/T ) 1 de(T ) ∂s de(T ) = e(T ) + − · = e(T ). d(1/T ) T d(1/T ) ∂e d(1/T )

(6.15)

6.2 Penrose-Fife Theory

167

Thus, the minimum of (6.14) is attained by e=

d(f (T )/T ) , d(1/T )

(6.16)

and, therefore, it holds that e = f − TfT . Writing (6.14) as

(6.17)

 1 f (T ) = inf e · − s(e) , e T T

we obtain s(e) = inf T

e f (T ) − T T

(6.18)

by Fenchel-Moreau’s duality. Furthermore, the minimum of (6.18) is attained by T = T (e), that is, the inverse function of e = e(T ) defined by (6.15). These relations are valid even when f depends on the order parameter ϕ, and we can define the entropy s by (6.18), namely,

e f (T , ϕ) f (T , ϕ) = inf {e − Ts(e, ϕ)} , s(e, ϕ) = inf − . e T T T

(6.19)

If T = T (e, ϕ) attains the second minimum of (6.19), then it holds that 1 ∂s(e, ϕ) = ∂e T (e, ϕ)

(6.20)

similarly to (6.15). Here, we apply the transformation of variables (e, ϕ) → (1/T , ϕ) for s(e, ϕ) =

f e − , f = f (T , ϕ), T = T (e, ϕ), T T

and obtain 

∂ 1 ∂(f /T ) ∂(1/T ) 1 ∂f ∂s(e, ϕ) =e· − · + ∂ϕ ∂ϕ T ∂(1/T ) ∂ϕ T ∂ϕ 1 ∂f (T (e, ϕ), ϕ) . =− T (e, ϕ) ∂ϕ

(6.21)

168

6 Critical Phenomena of Isolated Systems

Using ϕ = ϕ(x), the free energy functional is now defined by 

ξ2 |∇ϕ(x)|2 dx 2   ξ2 |∇ϕ(x)|2 dx = inf [e − Ts(e, ϕ(x))] + 2  e  ξ2 |∇ϕ(x)|2 dx = inf e(x) − Ts(e(x), ϕ(x)) + e(·)  2 = inf {U(e) − TS(e, ϕ)} ,

F(T , ϕ) =

f (T , ϕ(x)) +

e(·)



where U(e) =



e(x)dx



and S(e, ϕ) =



s(e(x), ϕ(x)) −

κ1 |∇ϕ(x)|2 dx 2

are the energy and the entropy functionals, respectively, and κ1 = ξ 2 /T . Here, taking variations of this entropy functional, we obtain ∂s 1 δS(e, ϕ) = (e(x), ϕ(x)) = δe ∂e T (e, ϕ)

(6.22)

and δS(e, ϕ) ∂s ξ2 = (e(x), ϕ(x)) + ϕ(x) δϕ ∂ϕ T (e(x), ϕ(x))  1 ∂f 2 = − (T (e, ϕ), ϕ) + ξ ϕ T (e, ϕ) ∂ϕ 1 ∂f + κ1 ϕ, =− T ∂e

(6.23)

by (6.20) and (6.21), respectively. Concerning the time evolution of the order parameter, model (A) equation is adopted by the entropy increasing, ϕt = K1

δS (e, ϕ). δϕ

(6.24)

6.2 Penrose-Fife Theory

169

On the other hand, model (B) equation is adopted for the energy density evolution by the entropy increasing and the energy conservation,  δS et = −∇ · M2 ∇ (e, ϕ) . δe

(6.25)

Finally, we assume the energy density by e = u(T )ν(ϕ) + λ(ϕ), where λ(ϕ) is a potential, and u(T ) = 21 kT with the Boltzmann constant k and the degree of microscopic state freedom ν(ϕ) per unit volume [297]. In this case, the first term of e indicates the kinetic energy, and it holds that f (T , ϕ) = ν(ϕ)u1 (T ) + λ(ϕ) − Ts0 (ϕ)  u1 (T ) = T u(T )d(1/T )

(6.26)

with the integration constant s0 (ϕ). Thus, we obtain s(e, ϕ) = ν(ϕ)y(T (e, ϕ)) − s0 (ϕ), y(T ) =

u(T ) − u1 (T ) = T



du T

by (6.19), and then (6.24)–(6.25) reads as follows. 

u1 (T )ν  (ϕ) λ (ϕ)  − + s0 (ϕ) + κ1 ϕ ϕt = K1 − T T  1 . ut ν(ϕ) + uν  (ϕ)ϕt + λ (ϕ)ϕt = −∇ · M2 ∇ T

(6.27)

 2 Sometimes ν(ϕ) and s0 (ϕ) are assumed to be a constant and − 41 ϕ2 − 1 , respectively. In this case, it follows that   ϕt = K1 κ1 ϕ + ϕ − ϕ3 − T −1 λ (ϕ)   k 1  Tt + λ (ϕ)ϕt = −∇ · M2 ∇ . 2 T

(6.28)

Alt-Pawlow [6, 7] obtained this equation using the renormalized free energy density f˜ (e, T ) = f (e, T )/T , where the heat flux q = κ(e, T )∇(1/T ) is defined by Newton-Fourier-Fick’s law concerning the energy balance, and the increase of entropy is derived from Clausius-Duhem’s inequality [39].

170

6 Critical Phenomena of Isolated Systems

6.3 Penrose-Fife Equation Penrose-Fife equation (6.28) is formulated by θt + λ(ϕ)t − α(θ) = 0, ϕt − κϕ + g(ϕ) − α(θ)λ (ϕ) = 0   ∂ (α(θ), ϕ) = 0, θ|t=0 = θ0 (x) > 0, ϕ|t=0 = ϕ0 (x), ∂ν ∂

(6.29)

where  ⊂ Rn , n = 1, 2, 3, is a bounded domain with smooth boundary ∂, ϕ and θ > 0 are order parameter and absolute temperature, respectively, 1 λ(ϕ) = a1 ϕ2 + a2 ϕ, g(ϕ) = ϕ3 − ϕ, α(θ) = − , θ and κ > 0, a1 < 0, and a2 are constants, and λ(ϕ) stands for the latent heat. We note that this is a materially and thermodynamically closed system, and, consequently, there is a Lyapunov function, provided with the semi-unfolding-minimality. More precisely, first, the total energy is preserved, d dt and then

 

θ + λ(ϕ) dx = 0,

κ L(θ, ϕ) = ∇ϕ 22 − 2

 

 log θ +



W (ϕ)

acts as the Lyapunov function: d L(θ, ϕ) = − ϕt 22 − ∇α(θ) 22 ≤ 0. dt Again, this functional L(θ, ϕ) is not provided with the hook term, and (6.29) is not a gradient system derived from this functional. In the stationary state, θ = θ > 0 is a constant by (6.29), and ϕ = ϕ is a solution to  ∂ϕ  λ (ϕ) =0 (6.30) = 0, − κϕ + g(ϕ) + ∂ν ∂ θ satisfying

 || θ +



λ(ϕ) = a

(6.31)



for a=



θ0 + λ(ϕ0 ) dx.

(6.32)

6.3 Penrose-Fife Equation

171

Thus, it is formulated as a nonlinear eigenvalue problem with non-local term,  ∂ϕ  =0 ∂ν ∂

|| λ (ϕ)  = 0, −κϕ + g(ϕ) + a −  λ(ϕ)

comparable to (5.29). This problem has a variational structure, and it is the Euler-Lagrange equation of κ Fa (ϕ) = ∇ϕ 22 + 2





a 1 W (ϕ) − || log − || ||  

defined for ϕ ∈

H 1 ()

satisfying







λ(ϕ) < a, where W (ϕ) =



λ(ϕ)

2 1 2 ϕ − 1 . If 4

 a = || θ +

λ(ϕ),



then it holds that κ |∇ϕ 22 + 2 = Fa (ϕ).





L(θ, ϕ) =



W (ϕ) − || log

a 1 − || ||



 

λ(ϕ)

This fact means the semi-unfolding. On the other hand, we obtain  L(θ, ϕ) − L(θ, ϕ) = − provided that



log θ + || log θ,

 

θ + λ(ϕ) dx = a,

and then the semi-minimality L(θ, ϕ) ≤ L(θ, ϕ) follows from Jensen’s inequality 1 ||



 

− log θ ≥ − log

1 ||

 

θ .

To perform the rigorous analysis, we say that [θ, ϕ] : [0, T ] → L 2 () × V is an to (6.29) if

H 1 -solution

e = θ + λ(ϕ) ∈ L ∞ (0, T ; L 2 ()) ∩ H 1 (0, T ; V ∗ )

172

6 Critical Phenomena of Isolated Systems

ϕ ∈ L 2 (0, T ; H 2 ()) ∩ L ∞ (0, T ; V ) ∩ H 1 (0, T ; L 2 ()) 1 θ > 0, α = − ∈ L 2 (0, T ; V ) θ for V = H 1 (), z, et V,V ∗ + (∇z, ∇α) = 0   (z, ϕt ) + κ(∇z, ∇ϕ) + z, g(ϕ) − αλ (ϕ) = 0 a.e. t ∈ (0, T ) for each z ∈ V , and (e, ϕ)|t=0 = (θ0 + λ(ϕ0 ), ϕ0 ). Comparison theorem and standard parabolic regularity are not valid to (6.29), and, therefore, deriving θ > 0 in the classical sense needs rather strong regularities of the solution. Here, we use the fact that the inverse function of α = α(θ) is realized as a maximum monotone graph ρ(s) =

− 1s , s < 0 +∞, s ≥ 0

in R × R. Then realizing the relation α = −1/θ in a function space, we obtain the following theorem [180]. Theorem 6.3.1 If 0 < θ0 = θ0 (x) ∈ L 2 (), log θ0 ∈ L 1 () 1 α0 = − ∈ V, ϕ0 = ϕ0 (x) ∈ H 2 (), θ0 then there is a unique H 1 solution to (6.29) for any T > 0. Using the above mentioned well-posedness, we confirm the generation of a dynamical system, which is useful to classify the asymptotic profile of the solution [180]. Theorem 6.3.2 System (6.29) generates a weakly continuous dynamical system in the space X = {(θ, ϕ) ∈ L 2 () × H 2 () |

 ∂ϕ  1 = 0, θ > 0, log θ ∈ L 1 (), ∈ V }.  ∂ν ∂ θ

6.3 Penrose-Fife Equation

173

We have α ∈ C([0, ∞); L 2 ()), ϕ ∈ C([0, ∞); V ) lim ∇α(·, t) 2 = lim wt (·, t) 2 = lim θt (·, t) V ∗ = 0 t↑+∞ t↑+∞ t↑+∞   sup α(·, t) V + ϕ(·, t) H 2 < +∞ t≥0

for α(·, t) = α(θ)(·, t). Therefore, the orbit O = {(α(·, t), ϕ(·, t))}t≥0 , described by (α, ϕ), is continuous and compact in L 2 () × V , and in particular, it has a nonempty, connected, and compact ω-limt set ω(α0 , ϕ0 ) composed of stationary solutions. More precisely, we define ω(α0 , ϕ0 ) = {(α, ϕ) | ∃tk ↑ +∞ such that  (α(·, tk ), ϕ(·, tk )) → (α, ϕ) in L 2 () × V and call (α, ϕ) ∈ L 2 () × V a stationary solution if and only if θ = − α1 > 0 is a constant and it holds that (6.30)–(6.32) for ϕ = ϕ. The set of stationary states, denoted by E, is discrete if n = 1, see [226]. For the other case, convergence to the stationary solution of (α(·, t), ϕ(·, t)) in L 2 () × V is not obvious. Here, the linearized stability of the stationary ϕ = ϕ is indicated by the positivity of the self-adjoint operator Aa in L 2 () with the domain (5.31) defined by   ||  λ (ϕ)ψ || λ (ϕ) 3  ψ+  Aa ψ = −κψ + 3ϕ − 1 +  2 . a −  λ(ϕ) a −  λ(ϕ) In this case, we obtain its dynamical stability as well as the asymptotic stability [180]. Theorem 6.3.3 If ϕ is a linearly stable critical point of Fa , and θ > 0 is a constant determined by (6.31) for ϕ = ϕ, then each ε > 0 admits δ > 0 such that     1  

∇(ϕ0 − ϕ) 2 < δ,  log θ0 − log θ < δ ||   a= (θ0 + λ(ϕ0 )) 

implies     1  sup ∇(ϕ(·, t) − ϕ) 2 < ε, sup  log θ(·, t) − log θ < ε. t≥0 t≥0 || 

(6.33)

174

6 Critical Phenomena of Isolated Systems

As is mentioned, each linearly stable critical point of Fa is isolated in V , and we have in L 2 () × V (α(·, t), ϕ(·, t)) → (α, ϕ) as t ↑ +∞ under the assumption of the above theorem, where α = − 1 with θ > 0. θ Classification of the stable stationary states, however, is not complete. We can regard L as a functional of (e, ϕ), using the energy density e = θ + λ(ϕ), L(e, ϕ) =

κ

∇ϕ 22 − 2

 

 log (e − λ(ϕ)) +



W (ϕ).

Then, (6.29) is formulated by a combination of model (A)—model (B) equations: ϕt = −Lϕ , et = ∇ · ∇Le ,

  ∂ (Le , α) = 0. ∂ν ∂

In this case, we obtain inf L(e, ϕ) = L(e, ϕ) = Fa (ϕ), e = λ(ϕ) − e

1 ||

 

λ(ϕ) +

a ||

which confirms the semi-unfolding-minimality  Fa (ϕ) ≤ L(e, ϕ),



e=a

again. However, this L is not formulated as the convex-non-convex functional described in Sect. 4.3, and the hyper-stability to guarantee θ-stability is not certain to hold in spite of the second relation of (6.33).

6.4 Coupled Cahn-Hilliard Equation The coupled Cahn-Hilliard equation describes non-isothermal phase separation using the absolute temperature θ = θ(x, t) > 0 and the order parameter ϕ = ϕ(x, t) ([7, 297]). In this case, we use the combination of model (B) - model (B) equations, and thus it holds that   fϕ 1 , et = −M2  ϕt = M1  −K1 ϕ + θ θ    fϕ  ∂ θ−1 , ϕ, −κ1 ϕ + = 0, (6.34) ∂ν θ ∂

6.4 Coupled Cahn-Hilliard Equation

175

where M1 , M2 , and K1 are positive constants. In fact, Penrose-Fife’s relations (6.20), (6.21), and (6.17) are summarized by  S(e, ϕ) =



s(e(x), ϕ(x)) −

∂s 1 = , e = f − θfθ , ∂e θ

fϕ ∂s =− , ∂ϕ θ 

and, therefore, S(e, ϕ) =

K1 |∇ϕ|2 dx 2



K1 |∇ϕ|2 + fθ dx. 2

Then, (6.34) is obtained by δS δS ϕt = −M1  , et = −M2  , δϕ δe

∂ ∂ν



  δS δS , , ϕ  = 0. δϕ δe ∂

Thus this system is associated with two conservation laws and the entropy increasing. In fact, first, (6.34) describes a kinetically and thermo-dynamically closed system, and we obtain   d d e= ϕ = 0. (6.35) dt  dt  Next, we have 

 

ϕt

fϕ −K1 ϕ + θ



     fϕ 2  = −M1 ∇ −K1 ϕ + θ    ϕt f ϕ K1 d

∇ϕ 22 + = 2 dt  θ

and  

 ft − θ t fθ et + θ(fθ )t = θ θ      2 d  −1  = M2 fθ , ∇θ  + dt  

ϕt f ϕ = θ



and hence it holds that d dt



K1

∇ϕ 22 + 2

 



     fϕ 2  = −M1 ∇ −K1 ϕ θ    2   −M2 ∇θ−1  dx ≤ 0.

(6.36)

176

6 Critical Phenomena of Isolated Systems

Thus, the total heat energy and the total order parameter are preserved, 

 

f − θfθ dx = a,



ϕ = c,

and the entropy acts as the Lyapunov function, re-formulated by K1

∇ϕ 22 + L(θ, ϕ) = 2

 

fθ .

The temperature is constant in the stationary state. It is denoted by θ = θ > 0, and then it follows that  1 ∂ϕ  =0 −K1 ϕ + fϕ (θ, ϕ) = constant, ∂ν ∂ θ   f (θ, ϕ) − θfθ (θ, ϕ) dx = a, ϕ = c. (6.37) 



Here, we use the free energy density of [297] described by f (θ, ϕ) = −cv (θ log θ + δ) −

α 2 ϕ − βϕ + γ − θs0 (ϕ) 2

ϕ4 with concave s0 (ϕ), say, s0 (ϕ) = − , where cv , α, β, δ, and γ are positive con4 stants. In this case, it holds that f − θfθ = cv θ −

α 2 ϕ − βϕ + γ − δcv , 2

and the above stationary θ is prescribed by  cv || θ = (δcv − γ) || + Thus writing A(ϕ) =



α 2 ϕ + βϕ dx + a. 2

α 2 ϕ + βϕ + δcv − γ, 2

we obtain cv θ =

a + ||

 

A(ϕ)

and f (θ, ϕ) = −cv θ log θ − A(ϕ) − θs0 (θ) + constant,

(6.38)

6.4 Coupled Cahn-Hilliard Equation

which guarantees

177

fϕ (θ, ϕ) = −A (ϕ) − θs0 (ϕ).

(6.39)

Relations (6.37), (6.38), and (6.39) are summarized by   cv  A (ϕ) 1 cv || A (ϕ)    + s0 (ϕ) − − s (ϕ) −K1 ϕ = a +  A(ϕ) a +  A(ϕ) ||  0   ∂ϕ  = 0, ϕ=c (6.40) ∂ν ∂  similarly to (6.30)–(6.31). Then, we see that (6.40) is the Euler-Lagrange equation of    K1

∇ϕ 22 − cv || log a + Fa (ϕ) = A(ϕ) − s0 (ϕ) 2   −cv || log (cv ||) 

defined for ϕ ∈ H 1 (),



 ϕ = c, a +



A(ϕ) > 0.

(6.41)

On the other hand, for the stationary θ = θ defined by (6.38), it holds that L(θ, ϕ) =

K1

∇ϕ 22 + 2

 

fθ (θ, ϕ)

with fθ (θ, ϕ) = −cv (log θ + 1) − s0 (ϕ) 

 cv || log θ = cv || log a + A(ϕ) − log(cv ||) . 

This property implies the semi-unfolding, L(θ, ϕ) = Fa (ϕ). Finally, we have fθ (θ, ϕ) − fθ (ϕ, θ) = −cv (log θ − log θ),

1 ||



by (6.38), while Jensen’s inequality implies 1 ||



1 − log θ ≥ − || 

 

θ = − log θ.



θ=θ

178

6 Critical Phenomena of Isolated Systems



Thus, we obtain



fθ (θ, ϕ) − fθ (θ, ϕ) dx ≥ 0,

and hence the minimality L(θ, ϕ) ≥ Fa (ϕ)

(6.42)

for ϕ satisfying (6.41). These structures are sufficient to motivate the study of stationary solutions from the dynamical point of view. We obtain an analogous result to Theorem 6.3.3 if the friction term μut is added to the right-hand side of the first equation to (6.34), see [181].

6.5 Shape Memory Alloys Hysteresis of the shape memory alloys is modelled by [106, 107] for the onedimensional case of  = (0, 1), where u, ε, σ, e, q, h, and g denote the displacement, the strain, the stress, the inner energy, the heat flux, the load density, and the heat source density, respectively. More precisely, momentum and energy balances described by utt − σx + μxx = h(x, t), et + qx − σεt − μεxt = g(x, t) are coupled with the structural relation ε = ux , σ =

∂f ∂f , e = f − θfθ , , μ= ∂ε ∂εx

(6.43)

where f = f (ε, εx , θ) denotes the free energy density defined by γ f = f0 (θ) + α1 (θ − θ1 )f1 (ε) + f2 (ε) + ε2x 2  θ f0 (θ) = −cv θ log − 1 + c˜ , f1 (ε) = ε2 θ2 f2 (ε) = −α2 ε4 + α3 ε6 ,

(6.44)

and furthermore, Newton-Fourier-Fick’s law is adopted: q = −κθx . Here, the thermodynamical relation  ∂A A = U − TS, S = − ∂T V

6.5 Shape Memory Alloys

179

in Sect. 5.1 is used in the last equation of (6.43), see (6.17). Then we obtain  utt −

∂f ∂ε



 + γ∂x4 u

  ∂ (u, uxx , θ) ∂ν

= h, et − κθxx −

x

∂

 ∂f ∂f εxt = g εt − ∂ε ∂εx

= 0.

(6.45)

Since   ∂f ∂f ∂f εxt + εt + θt ∂ε ∂εx ∂θ ∂ et = ft − θt fθ − θ fθ , ∂t ∂ f (ε, εx , θ) = ∂t



it holds that et −



 ∂f ∂f ∂ εxt = −θ fθ , εt − ∂ε ∂εx ∂t

and the second equation of (6.45) is described by θ

∂ fθ + κθxx = g. ∂t

This equation is equivalent to cν θt − κθxx − α1 θ(ε2 )t = g because fθ = f0 (θ) + α1 f1 (ε) = −cν log

θ + α1 ε2 , θ2

and, therefore, (6.45) means 



  + γ∂x4 u = h, cν θt − κθxx − α1 θ ε2 = g t x   ∂ (u, uxx , θ) = 0. ∂ν ∂

utt −

∂f ∂ε

(6.46)

See [39, 432] for more on physical backgrounds and the well-posedness of (6.45). Higher-Dimensional Model Let  ⊂ Rn , n = 2, 3, be a bounded domain with smooth bioundary ∂, to extend 1 the model (6.46). Let As = (A + t A) be the symmetric part of the matrix A, and 2

180

6 Critical Phenomena of Isolated Systems

∂ ·. Deformation of the elastic body is indicated by the vector u = (ui ), and ∂xi  1 ui/j + uj/i . the strain ε = (εij (u)) stands for ε(u) = (∇u)s , the tensor εij (u) = 2 This strain is associated with the strain by Fooke’s law, using the Lamé constants λ, μ: μ > 0, nλ + 2μ ≥ 0. (6.47)

·/i =

Thus we define the fourth-order tensor A = (aijk ) by aijk = λδij δk + μ(δik δj + δi δjk ), Aε = λ(tr ε)I + 2με to put Q(u) = ∇ · Aε(u) = μu + (λ + μ)∇(∇ · u). Furthermore, θ > 0 and F denote the temperature and the free energy density, respectively. Then the model [293] is given by c(ε, θ)θt − κθ = θF/θε : εt + γAεt : εt + g κ utt − γQut + QQu = ∇ · F/ε + h in  × (0, T ), 4

(6.48)

where γ ≥ 0 and κ > 0 stand for the viscosity coefficient and the conductivity, respectively,  εt = ε(ut ), F/ε =

∂F ∂εij

, F/θε = ij

     ∂ F/ε , aij : bij = aij bij , ∂θ i,j

and c(ε, θ) = cv − θF/θθ (ε, θ) denotes the specific heat. Putting h = 0 and g = 0 in (6.48), we provide the initial and boundary conditions   ∂θ  u, Qu, = 0, ∂ν ∂

(u, ut , θ)|t=0 = (u0 (x), u1 (x), θ0 (x)).

(6.49)

Generally F = F(ε, Dε, θ), but first, see [295], we take the case F = F(ε, θ)

(6.50)

although the model (6.50) does not cover (6.49). By (6.50) the first equation of (6.48) is reduced to ∂ (6.51) cv θt − κθ = θ F/θ + γAεt : εt ∂t

6.5 Shape Memory Alloys

181

and then it holds that   d cv log θ − F/θ dx = κθ−1 θ + γθ−1 Aεt : εt dx dt    = κθ−2 |∇θ|2 + γθ−1 Aεt : εt dx. 

Since (6.47) implies Aa : a = λ(tr a)2 + 2μ|a|2 ≥ (λ + 2nμ)(tr a)2 ≥ 0 for any symmetric matri a we obtain the entropy non-decreasing, d dt

 

cv log θ − F/θ dx ≥ 0.

(6.52)

From the second equation of (6.48) and the boundary condition, on the other hand, we obtain

κ κ d 1 2 2

˙u 2 + Qu 2 = (utt , ut ) + (Qu, Qut ) dt 2 8 4  = γ(Qut , ut ) − F/ε : ∇ut , 

where (6.51) is applicable to the second term on the right-hand side. Thus, it holds that      d F/ε : ∇ut = F/ε : ∇εt = F − F/θ , θt dt        d F − F/θ θ dx + θ F/θ t = dt     d = F − F/θ θ + cv θ dx − γ Aεt : εt dt   which results in the energy conservation, d dt



F − F/θ θ + cv θ dx   −Aεt : ∇ut + Aεt : εt dx = γ {(Qut , ut ) + (Aεt , εt )} = γ   =γ −Aεt : εt + Aεt : εt dx = 0. (6.53)

1 κ

˙u 22 + Qu 22 + 2 8





182

6 Critical Phenomena of Isolated Systems

The model (6.48) is based on the mass, momentum, and energy conservation laws indicated by ρt + ρ∇ · v = 0, ρvt − ∇ · σ = ρb, σ = t σ |v|2 )t + ∇ · (−v · σ + q) = ρb · v ρ(e + 2

(6.54)

where ρ, v = (vi ), σ = (σij ), b = (bi ), e, and q = (qi ) denote the mass density, the velocity, the strain tensor, the outer force, the inner energy, and the energy flux, ∂ respectively, and ·t stands for the material derivative + v · ∇. ∂t First, symmetry of the stress tensor implies v · (∇ · σ) = ∇ · (v · σ) − σ : (∇v)s , and, therefore, we obtain ρ(

|v|2 )t − ∇ · (v · σ) + σ : (∇v)s = ρb · v 2

from the second equation of (6.54). Then, the third equation implies ρet + ∇ · q − σ : (∇v)s = 0.

(6.55)

Assuming the small strain, we put ρ = 1 and v = ut . Hence the material derivative is ∂ϕ . In particular, ε = (∇u)s implies (∇v)s = εt , and, therefore, replaced by ∂t : ϕt = ∂t the second equation of (6.54) and (6.55) are reduced to utt − ∇ · σ − b = 0, et + ∇ · q − σ : εt = 0,

(6.56)

while (6.56) implies (e +

|ut |2 )t + ∇ · (−ut σ + q) − b · ut = 0. 2

Thus equations (6.56) represent the balances of the first moment and the energy. Next, from the thermodynamical law f = e − θs

(6.57)

s = −f/θ , θ = e/s ,

(6.58)

we obatin

6.5 Shape Memory Alloys

183

where f = f (ε, Dε, θ), Dε = (eij/k ), is the free energy density of Helmholtz. The first equality of (6.58) is derived by regarding the inner energy e = e(s) as a Legendre transformation of −f = −f (θ), which results in e(ε, Dε, s) = sup{θs + f (ε, Dε, θ)}.

(6.59)

θ

The specific heat ratio is also given by c = e/θ = (f + θs)/θ = θs/θ = −θf/θθ .

(6.60)

δe δf We have = by (6.59). δε δε Here we assume that the stress tensor σ and the energy flux q are associated with a third order tensor denoted by h in such a way as in σ=

δe 1 + θ(h − f/Dε ) · ∇ + σ v , q = q0 − εt · h. δε θ

(6.61)

The term σ v on the right-hand side of the first equation of (6.61) indicates the viscostress tensor, and the terms q0 and εt · h are the stationary heat flux and non-stationary flux, respectively. Then laws of Hooke and Fourie are applied to q0 and σ v , respectively: (6.62) σ v = γAεt , q0 = −k∇θ, and, then, (6.62) implies the disserpation inequality εt :

1 σv + ∇ : q0 ≥ 0. θ θ

Since s = −f/θ and ut = v it holds that st + ∇ · ψ − λu · (utt − ∇ · σ − b) − λθ {(e + +∇ · (−ut σ + q) − b · ut } = εt : where ψ=

|ut |2 )t 2

1 σv + ∇ : q0 ≥ 0 θ θ

1 (q0 + εt (f/Dε − h)), λu = −λθ ut , λθ = 1/θ. θ

So far, we have the following models of the free energy density: 1. energic case (shape memory alloys) [105, 108]: f/θDε = 0 2. entropic case (polymer body) [8]: (f /θ)/θDε = 0. For these cases we take h in (6.61) appropriately.

(6.63)

184

6 Critical Phenomena of Isolated Systems

In the energic case we put h = f/Dε

(6.64)

which results in ψ = q0 /θ. Then (6.63) with (6.56) realizes the inequality of ClausiusDuhem, 1 q0 σv = εt : + ∇ · q0 ≥ 0. st + ∇ · (6.65) θ θ θ Since f = f (ε, Dε, θ), furthermore, it holds that σ=

δf + σ v = f/ε − ∇ · f/Dε + σ v , δε

(6.66)

while the first equation of (6.56) is reduced to utt − γ∇ · (Aεt ) + ∇ · (∇ · f/Dε ) = ∇ · f/ε + b.

(6.67)

For the second equation of (6.56) we use et = (f + θs)t = f/ε : εt + f/Dε : ∇εt + θst st = (−f/θ )t = −f/θε : εt − f/θDε : ∇εt − f/θθ θt derived from (6.57) and (6.58) for f = f (ε, Dε, θ), which results in f/ε : εt + f/Dε : ∇εt + ∇ · q = σ : εt + θf/θε : εt + θf/θθ θt .

(6.68)

Then (6.68) is reduce to −θf/θθ θt + ∇ · (q0 − εt · h) = σ : εt − (f − θf/θ )/ε : εt − (f − θf/θ )/Dε : ∇εt

(6.69)

by (6.61) and f/θDε = 0. Here we use (6.64) and (6.66) to obtain −θf/θθ θt + ∇ · q0 = ∇ · (εt · f/Dε ) + σ : εt − (f − θf/θ )/ε : εt − (f − θf/θ )/Dε : ∇εt = (σ − (f − θf/θ )/ε + ∇ · f/Dε ) : εt + (f/Dε − (f − θf/θ )Dε ) : ∇εt = θf/θε : εt + θf/θDε : ∇εt + σ v : εt = θf/θε : εt + σ v : εt and then it follows that − θf/θθ θt − ∇ · (k∇θ) = θf/θε : εt + γAεt : εt

(6.70)

6.5 Shape Memory Alloys

185

from (6.62). In (6.70), the term −θf/θθ = c(ε, θ) is the specific heat ratio determined by (6.60). We thus end up with (6.67) and (6.70), that is utt − γ∇ · (Aεt ) + ∇ · (∇ · f/Dε ) = ∇ · f/ε + b −θf/θθ θt − ∇ · (k∇θ) = θf/θε : εt + γAεt : εt

(6.71)

in the energic case. We obtain (5.25), putting f (ε, Dε, θ) = −cv θ log θ + F(ε, θ) +

κ |∇ · Aε|2 8

(6.72)

and b = 0 in (6.71). In fact we have

∇ · Aε = aijk εk/j ,

1 |∇ · Aε|2 2

/Dε

1 |∇ · Aε|2 2

/εpq/r

= εk/j akji apqri

= A(∇εA)

in this case, and, therefore, κ κ κ κ A(∇ε(u)) = AQu, ∇ · f/Dε = ∇ · (AQu) = Aε(Qu) 4 4 4 4 κ κ ∇ · (∇ · f/Dε ) = ∇ · (Aε(Qu)) = QQu 4 4

f/Dε =

by ε(Qu) = (∇Qu)s . In the entropic case (f /θ)/θDε = 0

(6.73)

we use h = 0 in (6.61), to replace (6.66) by f/ε f/Dε δ f σv σv σ = ( )+ = −∇ · + . θ δε θ θ θ θ θ The first equation of (6.56) then turns to be utt − γ∇ · (Aεt ) + ∇ · (θ∇ ·

f/Dε ) = ∇ · f/ε + b, θ

(6.74)

while the second equation of (6.56) has been reduced to (6.69). Here we have furthermore (f − θf/θ )/Dε = 0

186

6 Critical Phenomena of Isolated Systems

by (6.73), and then (6.69) implies −θf/θθ θt + ∇ · q0 = (σ − (f − θf/θ )/ε ) : εt f/Dε = θf/θε : εt − θ∇ · : εt + σ v : ε. θ Hence we obtain − θf/θθ θt − ∇ · (k∇θ) = θf/θε : εt − θ∇ ·

f/Dε : εt + γAεt : εt θ

(6.75)

by (6.62). In the entropic case thus we reach (6.74) and (6.75), that is utt − ν∇ · (Aεt ) + ∇ · (θ∇ ·

f/Dε ) = ∇ · f/ε + b − θf/θθ θt − ∇ · (k∇θ) θ f/Dε : εt + γAεt : (6.76) = θf/θε : εt − θ∇ · εt . θ

Semi-duality The model (6.50) realizes the entropy increasing (6.52) and the energy conservation (6.53). These properties are valid to the general case. Hence the stationary states are formulated by the nonlinear eigenvalue problems with non-local terms with semiunfolding-minimality between non-stationary solutions. In the energic case (6.71) we assume f (ε, Dε, θ) = H(ε, θ) +

κ |∇ · Aε|2 8

(6.77)

which contains (6.72) and the other model studied by [434], κ 1 f (ε, Dε, θ) = − cv θ2 + F(ε, θ) + |∇ · Aε|2 . 2 8

(6.78)

In this case, assuming b = 0, we can reduce (6.71) to κ QQu = ∇ · H/ε 4 −θH/θθ θt − kθ = θH/θε : εt + γAεt : εt in  × (0, T ). utt − γQut +

Then we provide with the initial and boundary conditions (6.49). From the first equation of (6.79) and the boundary condition, we have 1d κd

ut 22 +

Qu 22 = γ(Qut , ut ) + (∇ · H/ε , ut ) 2 dt  8 dt  =−



γAεt : ∇ut + H/ε : ∇ut dx = −



γAεt : εt + H/ε : εt dx

(6.79)

6.5 Shape Memory Alloys

187

and hence   1d κd d

ut 22 +

Qu 22 + H dx = −γAεt : εt + H/θ θt dx 2 dt 8 dt dt     d H/θ θ dx − θ(H/θ )t + γAεt : εt dx. = dt   It thus holds that   1d κd d

ut 22 +

Qu 22 + H − H/θ θ dx = − θ(H/θ )t + γAεt 2 dt 8 dt dt     : εt dx = − θH/θε : εt + θH/θθ θt + γAεt : εt dx = kθ = 0 



and the total energy is given by E(u, ut , θ) =

1 κ

ut 22 + Qu 22 + 2 8

 

H(ε(u), θ) − H/θ (ε(u), θ)θ dx.

The total free energy is given by the second equation of (6.77). In fact, assuming θ > 0, we write θ γ = H/θε : εt + Aεt : εt −H/θθ θt − k θ θ this equation to obtain d dt



 

H/θ dx =

 

H/θε : εt + H/θθ θt dx = −



k

   ∇θ 2 γ  + Aεt : εt dx ≤ 0,  k =− θ  θ  

the decreasing of

θ γ + Aεt : εt dx θ θ (6.80)

 W (u, θ) =



H/θ (ε(u), θ).

(6.81)

To formulate the stationary state of (6.49) and (6.79), next, we put ut = θt = 0. From the boundary condition, the function θ is a constant in space-time denoted by θ, which is determined by the total energy defined by the initial value, b = E(u0 , u1 , θ0 ). Hence the stationary deformation u = (ui ) satisfies κ b = Qu 22 + 8

 

H(ε(u), θ) − H/θ (ε(u), θ) dx

(6.82)

188

6 Critical Phenomena of Isolated Systems

and

κ 2 Q u = ∇ · H/ε (ε(u), θ) in , 4

(u, Qu)|∂ = 0.

(6.83)

Since (6.83) is the Euler-Lagrange equation of the functional Jθ (u) =

κ

Qu 22 + 8

 

H(ε(u), θ), u ∈ H01 ∩ H 2 ()

(6.84)

the stationary problem is to find u ∈ H01 ∩ H 2 () and θ > 0 satifying δJθ (u) = 0, b =

κ

Qu 22 + 8

 

H(ε(u), θ) − H/θ (ε(u), θ) dx

(6.85)

for b ∈ R defined by (6.82). If θ → −f (θ) is proper, convex, lower semi-continuous then we have H(ε, θ) − H(ε, θ) ≥ H/θ (ε, θ)(θ − θ). In the non-stationary state, therefore, it holds that  1 κ 2 2 H(ε(u), θ) − H/θ (ε(u), θ)θ dx b = Qu 2 + ut 2 + 8 2   κ ≥ Qu 22 + H(ε(u), θ) − H/θ (ε(u), θ)θ dx = Jθ (u) − θW (u, θ) 8  which is written as θW (u, θ) + b ≥ Jθ (u).

(6.86)



Then, since Jθ (u) = b +



H/θ (ε(u), θ)θ = b + θW (u, θ)

(6.87)

we obtain the semi-duality between W (u, θ) and Jθ (u). Thus, any infinitesimally stable critical point u = u(x) of Jθ = Jθ (u) defined for u ∈ H01 ∩ H 2 () is dynamically stable by (6.86) and (6.87), see [389]. Theorem 6.5.1 Let the dynamical system (6.79) with (6.49) be well-posed in u ∈ H01 ∩ H 2 () local-in-time, and let θ → −H(·, ·, θ) in (6.77) be convex. Assume, furthermore, that θ > 0 is a constant, u is an infinitesimally stable critical point of (6.84), and the initial value satisfies E(u, 0, θ) = b ≡ E(u0 , u1 , θ0 ).

(6.88)

6.5 Shape Memory Alloys

189

Then, any ε > 0 admits δ > 0 such that

∇(u0 − u) H 1

     +  F/θ (ε(u), θ0 ) − F/θ (ε(u), θ) dx  < δ 

(6.89)

implies     sup ∇(u(·, t) − v) H 1 +  F/θ (ε(u), θ(t)) − F/θ (ε(u), θ) dx  < ε. 

t≥0

Several important cases take the form H(ε, θ) = F0 (θ) + G(θ)F1 (ε) + F2 (ε)

(6.90)

G(θ) = θ, θ → −F0 (θ) strictly convex

(6.91)

in (6.77). If

furthermore, the stationary problem is formulated explicitly for each b. Furthermore, if F0 , F1 , and F2 are real-analytic in (6.90), then any local minimum of Jθ is infinitesimally stable [294]. First, from (6.85), (6.90), and (6.91) it follows that   κ 1 2 ˆ b − Qu 2 − F2 (ε(u)) F0 (θ) = || 8 

(6.92)

for Fˆ 0 (θ) = F0 (θ) − θF0 (θ). Here, the inverse function Fˆ 0−1 : (a− , a+ ) → (0, +∞) is defined for F0 ((0, +∞)) ≡ (a− , a+ ) because (6.91) guarantees that θ ∈ (0, +∞) → Fˆ 0 (θ) is strictly increasing. Hence (6.92) implies θ = Fˆ 0−1



  1 κ 2 b − Qu 2 − F2 (ε(u)) || 8 

(6.93)

Next, since (6.90) and (6.91), we have H/ε (ε, θ) = θF1/ε (ε) + F2/ε (ε). Therefore, the stationary state u = u of (6.83) is written as δJ2 (u) = θδJ1 (u), where



κ F1 (ε(u)), J2 (u) = Qu 22 + J1 (u) = 8 

(6.94)  

F2 (ε(u)).

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6 Critical Phenomena of Isolated Systems

The stationary problem is thus formulated by (6.93) and (6.94), through the total energy b defined by the initial value. It is nothing but the Euler-Lagrange equation δJb (u) = 0, u ∈ Vb defined for 1 b − J2 (u) + J1 (u),  = 1/Fˆ 0−1 Jb (u) = − || ||

b − J2 (u) 1 2 Vb = u ∈ H0 ∩ H () | ∈ (a− , a+ ) . || 

(6.95)

By (6.95) we have F0 (θ) = −(Fˆ 0 (θ)), θ ∈ R, while  is concave. Hence we obtain    1 1  ˆ F (θ) ≥ − (6.96) F0 (θ) . ||  0 ||  by Jensen’s inequality. Relations (6.90) and (6.91), on the other hand, imply 

 W (u, θ) =



H/θ (ε(u), θ) =



F0 (θ) + F1 (ε(u)) dx

(6.97)

and  1 κ

ut 22 + Qu 22 + H − H/θ θ dx 2 8   1 = ˙u 22 + J2 (u) + Fˆ 0 (θ) 2 

b=

(6.98)

From (6.96) and (6.98) it follows that   1 1 1 Fˆ 0 (θ) + W (u, θ) ≥ − J1 (u) ≥ Jb (u), || ||  ||

(6.99)

while (6.97) implies θ = Fˆ 0−1 ⇒



  1 κ 2 b − Qu 2 − F2 (ε(u)) || 8 

1 W (u, θ) = Jb (u). ||

(6.100)

From (6.99) and (6.100), we see that any infinitesimally stable critical point u of Jb (u), u ∈ Vb , and θ defined by (6.95), the stationary solution (u, θ) is dynami-

6.5 Shape Memory Alloys

191

cally stable in u component, that is a result analogous to Theorem 6.5.1. We obtain, furthermore,     d2 1 d2  Jb (u + sw) = J (u + sw) + Ru [w, w] 2 θ ds2 ds θ|| s=0 s=0  

2 1 ˆ −1 b − J2 (u) Ru [w, w] = [∇ · F (ε(u))] · w , F 1/ε ||2 0 ||  and hence the infinitesimal stability of u on Jb implies that of Jθ , and the converse is not true [294, 386]. The semi-duality is observed also in the entropic case. Here we assume H(ε, θ) κ f = + |∇ · Aε|2 θ θ 8 and reduce (6.76) to κ ∇ · (θAε(Qu)) 4 = ∇ · H/ε + b − θH/θθ θt − kθ κ = θH/θε : εt − θAε(Qu) : εt + γAεt : εt in  × (0, T ). 4

utt − γQut +

Furthermore, let

(6.101)

  ∂θ  u, Qu, =0 ∂ν ∂

be the boundary condition. The energy functional arises similarly to the energic case. First, since  (∇ · (θAε(Qu), ut ) = −

 

θAε(Qu) : ∇ut = −



θAε(Qu) : εt

we obtain  1d κ

ut 22 − θAε(Qu) : εt = γ(Qut , ut ) + (∇ · H/ε , ut ) 2 dt 4    =− γAεt : ∇ut + H/ε : ∇ut dx = − γAεt : εt + H/ε : εt dx 



from the first equation of (6.101). Hence it follows that    1d κ d

ut 22 − θAε(Qu) : εt + H= −γAεt : εt + H/θ θt dx 2 dt 4  dt     d H/θ θ − θ(H/θ )t + νAεt : εt dx = dt  

192

6 Critical Phenomena of Isolated Systems

which implies   1d κ d θAε(Qu) : εt + H − H/θ θ dx

ut 22 − 2 dt 4  dt    =− θ(H/θ )t + γAεt : εt dx = − θH/θε : εt + θH/θθ θt     κ κ +γAεt : εt dx = kθ − θAε(Qu) : εt dx = − θAε(Qu) : εt . 4 4   The total energy is thus defined by E = E(u, ut , θ) =

1

ut 22 + 2

with

 

H(ε(u), θ) − H/θ (ε(u), θ)θ dx

dE = 0. dt

The total free energy, similarly, is derived from the second equation of (6.101). In fact, assuming θ > 0, we have −H/θθ θt − k

θ ν κ = H/θε : εt + Aεt : εt − Aε(Qu) : εt , θ θ 4

and, therefore,    d θ γ − Aεt : εt H/θ dx = H/θε : εt + H/θθ θt dx = −k dt  θ θ    κ θ γ κ + Aεt : εt + Qu · Qut dx + Aε(Qu) : εt dx = − k 4 θ θ 4      ∇θ 2 γ  + Aεt : εt dx − κ d Qu 2 . k  =− 2  θ θ 8 dt  Hence it holds that  W (u, θ) =



H/θ (ε(u), θ) +

κ

Qu 22 , 8

dW ≤ 0. dt

From the above structure, the stationary state is (u, θ) ∈ H 4 () × R+ satisfying κ 2 θQ u = ∇ · H/ε (ε(u), θ) in , (u, Qu)|∂ = 0 4  b = H(ε(u), θ) − H/θ (ε(u), θ) dx 

(6.102)

6.5 Shape Memory Alloys

193

for b = E(u0 , u1 , θ0 ). The first equation of (6.102) is the Euler-Lagrange equation associated with the functional  κ 2 Jθ (u) = θ Qu 2 + H(ε(u), θ), u ∈ H01 () ∩ H 2 (), 8  and this functional takes the semi-unfolding-minimality, (6.86) and (6.89), between the total free energy defined as above, that is  1 H(ε(u), θ) − H/θ (ε(u), θ)θ dx

ut 22 + 2   ≥ H(ε(u), θ) − H/θ (ε(u), θ)θ dx = Jθ (u) − θW (u, θ).

b=



Also, the second equation of (6.102) implies Jθ (u) − θW (u, θ) = b, and, therefore, the infinitesimal stability implies the dynamical stability, similarly to Theorem 6.5.1. One-Dimensional Case Without viscosity, the model (5.25) loses the parabolic regularity. In one-space dimension, however, the dispersive regularity induces global-in-time well-posedness. The method of time map is also useful to clarify the structure of sationary states in this case. Thus we come back the case n = 1 and γ = 0 in (6.48). We assume G(θ) = θ and take an analogous model to (6.46), utt + uxxxx = (f1 (ux )θ + f2 (ux ))x , θt − θxx = f1 (ux )θuxt in  × (0, T ) (ux , uxxx , θx )|∂ = 0, (u, ut , θ)|t=0 = (u0 (x), u1 (x), θ0 (x)) (6.103) where  = (0, 1). The boundary condition on u may be replaced by (u, uxx )|∂ = 0 in (6.103). This model is used to describe the deformation of shape memory alloys with martensitic phase transition with Fi , Fi = fi , i = 1, 2, where F1 = α1 ε2 , F2 (ε) = α3 ε6 − α2 ε4 − α1 θc ε2 , see [105]. The other model similar to (6.103) is concerned with the visco-elastic deformation, that is utt + uxxxx − uxxt = (f1 (ux )θ + f2 (ux ))x θt − θxx − |uxt |2 = f1 (ux )θuxt in  × (0, T )

(6.104)

194

6 Critical Phenomena of Isolated Systems

provided with the Neumann or Dirichlet boundary condition [170, 343]. While the principal part of (6.104), utt + uxxxx = θx , θt − θxx = uxt , is parabolic, the left-hand side of (6.103) is a system of Boussinesq and diffusion equations. For the first equation of (6.103) we use (∂t + ı∂x2 )(∂t − ı∂x2 ) = ∂t2 + ∂x4 to apply the Strichartz estimate on the Schrödinger equation

e±ıt∂x g L4 (0,T ;L4 ) ≤ C g L2  t     e±ı(t−s)∂x2 f (s)ds ≤ C f L4/3 (0,T ;L4/3 )   4 0 L (0,T ;L 4 )  t     e±ı(t−s)∂x2 f (s)ds ≤ C f L4/3 (0,T ;L4/3 ) .   2

L ∞ (0,T ;L 2 )

0

(6.105)

see [33, 110]. The parabolic maximal regularity [306], on the other hand, is applicable to the second equation. Then the following theorem is obtained [432]. Theorem 6.5.2 If fi , i = 1, 2, are C 2 functions, (u0 , u1 , θ0 ) ∈ H 2 × L 2 × L 1 , and u0 |∂ = 0, then there is T = T ( u0 H 2 , u1 2 , θ0 1 ) > 0 such that the problem (6.103) admits the unique solution (u, θ) = (u(·, t), θ(·, t)), 0 ≤ t < T , satisfying u ∈ C([0, T ]; H 2 ) ∩ L 4 (0, T ; W 2,4 ), ut ∈ L ∞ (0, T ; L 2 ) ∩ L 4 (0, T ; L 4 ) 4

4

θ ∈ C([0, T ]; L 1 ), θx ∈ L 3 +β (0, T ; L 3 +β ), where 0 < β < 1/6. If θ∗ = inf θ0 > 0, F2 (ε) ≥ −C, −∞ < ε < +∞, 

furthermore, it holds that T = +∞ and sup{ ut (·, t) 2 + uxx (·, t) 2 + θ(·, t) 1 } ≤ C. t≥0

Moreover, each T > 0 admits ω > 0 such that θ ≥ θ∗ exp(−ωt) in  × (0, T ).

6.5 Shape Memory Alloys

195

If f1 (0) = f2 (0) = 0

(6.106)

holds in (6.103), we obtain the conservation of momentum other than that of energy. These quantities are defined by 

1

M(ut ) =

ut 0

E(u, ut , θ) =

1 1

ut 22 + uxx 22 + 2 2



1

F2 (ux ) + θ dx = 0

0

and it holds that d M(ut ) = dt

 0

1

x=1 utt = [f1 (ux )θ + f2 (ux )]x=0 =0

(6.107)

from the boundary condition. Similarly, we obtain 1d 1d

ut 22 +

uxx 22 = −(f1 (ux )θ + f2 (ux ), uxt ) 2 dt 2 dt  1   d 1 d 1 θt − θxx dx − F2 (ux ) = − θ + F2 (ux ) dx =− dt 0 dt 0 0 where ( , ) denotes the L 2 inner product, and, therefore, dM dE = = 0. dt dt

(6.108)

Equality (6.108) prescribes M(u1 ) = a, E(u0 , u1 , θ0 ) = b, while (6.107) is not used in the later argument. We can observe the semi-duality as in the general model. First, the stationary state is formulated by θt = ut = 0, which implies θxx = 0,

θx |∂ = 0.

Thus, θ = θ ≥ 0 is a constant detemined by b, that is E(u, 0, θ) = b

(6.109)

where u = u(x) is the stationary state of u. Hence u = u solves uxxxx = (θf1 (ux ) + f2 (ux ))x ,

(ux , uxxx )|∂ = 0.

(6.110)

196

6 Critical Phenomena of Isolated Systems

Since 1 θ + uxx 22 + 2



1

F2 (ux ) = b

(6.111)

0

follows from (6.109), we can reduce (6.110) to

  1 1 b − uxx 22 − F2 (ux ) f1 (ux ) + f2 (ux ) 2 0 x (ux , uxxx )|∂ = 0.

uxxxx =

(6.112)

This (6.112) is the Euler-Lagrange equation associated with the functional 

1

Jb (ux ) = 0

  1 1 2 F1 (ux ) − log b − uxx 2 − F2 (ux ) 2 0

defined for ux |x=0,1 = 0,

1

uxx 22 + 2

In the case of 1

u1 22 + 2



1



1

F2 (ux ) < b.

(6.113)

(6.114)

0

θ0 > 0,

(6.115)

0

the initial value u = u0 ∈ H 2 satisfies (6.114). Now we put  J1 (ux ) =

1

F1 (ux ), J2 (ux ) =

0

1

uxx 22 + 2



1

F2 (ux )

0

to obtain δJ2 (ux ) = θδJ1 (ux ),

ux |∂ = 0

(6.116)

by (6.112). Since (6.111) implies θ = b − J2 (ux ), the problem (6.116) is reduced to δJ2 (ux ) = −(b − J2 (ux ))δJ1 (ux ),

ux |∂ = 0.

(6.117)

Since Jb (ux ) = J1 (ux ) − log(b − J2 (ux )) is derived from (6.113), the problem (6.117) means δJb (ux ) = 0. To show the second law of thermodynamics, we assume θ∗ = inf θ0 > 0

(6.118)

6.5 Shape Memory Alloys

197

and derive

1 (θt − θxx ) = F1 (ux )t θ

from the second equation of (6.103). Then it follows that d dt



1



1

F1 (ux ) − log θ dx = −

0

0

and, therefore,

 W (ux , θ) =

1

θxx =− θ



1 0

(

θx 2 ) ≤ 0, θ

F1 (ux ) − log θ dx

0

casts the total free energy. We note also that (6.118) implies (6.115). Without the hook term in W (ux , θ), we cannot find the Toland duality in (6.103), but, there actually arises the semi-unfolding-minimalilty between Jb . In fact, if (u, θ) solves (6.103) with E = b then it follows that  1 1 1 F2 (ux ) + θ dx

ut 22 + uxx 22 + 2 2 0  1 1 ≥ uxx 22 + F2 (ux ) + θ dx 2 0

b=

which implies the semi-minimality 

1

W (ux , θ) ≥ 

0

1

θ

0

1



 F1 (ux ) − log

0

  1 1 F1 (ux ) − log b − uxx 22 − F2 (ux ) = Jb (ux ) 2 0

by Jensen’s inequality. Semi-unfolding also holds for J2 (ux ) < b, and θ > 0 defined by (6.109) satisfies 

1

W (ux , θ) =

F1 (ux ) − log θ   1  1 1 = F1 (ux ) − log b − uxx 22 − F2 (ux ) = Jb (ux ). 2 0 0 0

Under the assumption of Theorem 6.5.2, the functional v ∈ Vb → Jb (v) ∈ R  1 1 1 2 Vb = {v ∈ H0 () | vx 2 + F2 (v)dx < b} 2 0

(6.119)

198

6 Critical Phenomena of Isolated Systems

is defined. Since θ∗ > 0, the set Vb = ∅ is open in H01 (). We say that the critical point v ∈ Vb of Jb (v), v ∈ Vb , is infinitesimally stable if there is ε0 > 0 such that any ε ∈ (0, ε40 ] admits δ > 0 such that

(v − v)x 2 < ε0 , Jb (v) − Jb (v) < δ



(v − v)x 2 < ε.

By virtue of Theorem 6.5.2, such u is dynamically stable in u-component. In fact, let (u, θ) = (u(·, t), θ(·, t)) be the solution to (6.103) and put b = E(u0 , u1 , θ0 ). Given the infinitesimally stable critical point v ∈ Vb of Jb (v), v ∈ Vb , we put θ = b − J2 (v) > 0. Then, since the above properties assure Jb (v) = W (v, θ), W = W (ux , θ), W (ux , θ) ≤ W (u0x , θ0 ) Jb (ux ) ≤ W (ux , θ) it holds that Jb (ux ) − Jb (v) ≤ W (u0x , θ0 ) − Jb (v) = W (u0x , θ0 ) − W (v, θ).

(6.120)

From (6.120), if there arises

(u0x − v)x 2 < ε0 , W (u0x , θ0 ) − W (v, θ) < δ

(6.121)

at the initial state, then it follows that Jb (ux (·, t)) − Jb (v) < δ, t ≥ 0.

(6.122)

The first equation of (6.121), combined with (6.122), implies

(ux (·, t) − v)x 2 < ε

(6.123)

at t = 0. Then inequality (6.123) is kept for all t ≥ 0 because of ux ∈ C([0, +∞); H01 ) and ε ≤ ε0 /4. Thus we obtain the following theorem [388] where the assumption (6.124) requiers u1 2 1 for u1 = ut |t=0 by E(u0 , u1 , θ0 ) = b = E(v x , 0, θ). Theorem 6.5.3 Besides the assumption of Theorem 6.5.2, suppose (6.106) and (6.118). Let b = E(u0 , u1 , θ0 ) and take the open set Vb = ∅ in H01 by (6.119). Assume, furthermore, that v ∈ Vb is an infinitesimally stable critical point of Jb = Jb (v), v ∈ Vb , and let  1 1 θ = b − v x 22 − F2 (v) > 0. 2 0

6.5 Shape Memory Alloys

199

Then, any ε > 0 admits δ > 0 such that  

(u0x − v)x 2 < δ, 

0

implies

1

  log θ0 − log θ < δ

  sup (ux (·, t) − v)x 2 < ε, sup  t≥0

t≥0

0

1

(6.124)

  log θ(·, t) − log θ < ε.

The parabolic system (6.104) takes the same structure. In the case of the Dirichlet boundary condition, (u, uxx , θx )|∂ = 0,

(6.125)

for example, the total energy is defined by 1d 1d

ut 22 +

uxx 22 + uxt 22 = −(f1 (ux )θ + f2 (ux ), uxt ) 2 dt 2 dt  1  d 1 θt − θxx − |uxt |2 dx − F2 (ux ) =− dt 0 0  d 1 =− θ + F2 (ux ) dx + uxt 22 , dt 0 namely, E(u, ut , θ) =

1 1

ut 22 + uxx 22 + 2 2



1

F2 (ux ) + θ dx

0

is conserved. The stationary state of (6.104) is similar to that of (6.103), and, then, we obtain a similar stability to Theorem 6.5.3, regarding Jb (ux ) as a functional of u ∈ H01 ∩ H 2 according to (6.125). The Neumann boundary condition (ux , uxxx , θx )|∂ = 0 is treated similarliy, if (6.106) is the case. From the general theory [294, 387] if fi , i = 1, 2, are real-analytic, then any local minimum of Jb (v), v ∈ Vb , is infinitesimally stable. Here, linearly degenerate local minima are driving forces to hysteresis, observed in the global bifurcation diagram using b as a pricipal parameter. In the case of n = 1, however, we can show that its Morse index is at most 1, see [386]. Lemma 6.5.1 Assume f1 (v) = 0, v = 0.

(6.126)

200

6 Critical Phenomena of Isolated Systems

If v ≡ 0 is a linearly degenerate local minimum of Jb = Jb (v), v ∈ Vb , then its linearized eigenvalue 0 is simple. The associated eigenfunction, together with v, takes a definite sign in  = (0, 1). Proof We describe the stationary state of (6.103) as the above Euler-Lagrange equation, using b = E(u0 , u1 , θ0 ) and v = ux , that is −vxx = −θf1 (v) − f2 (v), v|∂ = 0  1 1 θ = b − vx 22 − F2 (v). 2 0

(6.127)

Next, assuming the sign-changing solution v = v to (6.127), we obtain v ∈ Vb such that Jb (v) > Jb (v) in its neighbourhood, a contradiction. Hence the local minimum v ∈ Vb , v ≡ 0, of Jb (v), v ∈ Vb has a definite sign. From this property, combined with (6.106) and (6.127), the function v x takes a unique zero in  = (0, 1), see [319]. The linearized operator takes the form L(w) = −wxx + (θf1 (v) + f2 (v))w +



1

f1 (v)w f1 (v)

0

w|∂ = 0.

(6.128)

Let this v is linearly degenerate with the eigenfunction denoted by ψ ≡ 0. In case the non-local term in (6.128) vanishes, that is 

1

f1 (v)ψ = 0

(6.129)

0

we obtain

− ψxx + (θf1 (v) + f2 (v))ψ = 0,

ψ|∂ = 0,

(6.130)

while ψ1 = v x satisfies − ψ1xx + (θf1 (v) + f2 (v))ψ1 = 0

(6.131)

in  by (6.127), and takes a unique zero on [0, 1]. By (6.130) and (6.131), therefore, the sign of ψ = ψ(x) is definite in  = (0, 1) by the comparison theorem of Strum. Hence (6.129) does not arise by (6.126), because the sign of v is also definite. Since 

1

f1 (v)ψ = 0

0

for any eigenfunction ψ ≡ 0, the eigenvalue 0 is simple.



Now we show the main result of this paragraph, see [368, 386] for applications to the study on global bifurcation of stationary states of (6.103).

6.5 Shape Memory Alloys

201

Theorem 6.5.4 Under the assumption of Theorem 6.5.3 and (6.126), let fi = fi (v), i = 1, 2, be real-analytic functions of v ∈ R. Then any local minimum v ≡ 0 of Jb = Jb (v), v ∈ Vb , is infinitesimally stable. Proof Since Jb = Jb (v) is C 2 in v ∈ Vb , we obtain ∞

Jb (v + w) − Jb (v) =

1 μk |(w, ψk )|2 + o( wx 22 ), w ∈ Vb 2 k=0

as wx 2 → 0, where μ0 ≤ μ1 ≤ μ2 ≤ · · · are the linearized eigenvalues, ψk , k = 0, 1, 2, . . ., are the associated L 2 -normalized eigenfunctions, and ( , ) is the L 2 inner product. Since v is a local minimum, it holds that μ0 ≥ 0. If μ0 > 0 then v is linearly and hence infinitesimally stable. From Lemma 6.5.1, if μ0 = 0 it is simple and the associated eigenfunction ψ0 = ψ0 (x) takes a definite sign in  = (0, 1). Hence μ1 > 0, and any ε > 0 admits δ1 > 0 such that Jb (v + w) − Jb (v) < δ1 , wx 2 < 2ε0 ε ⇒

w1x 2 < , w1 = w − (w, ψ0 ). 2 We have, on the other hand, δ2 > 0 satisfying Jb (v + sψ0 ) − Jb (v) < δ2 , |s| < 2ε0



|s|
0, however, (6.132) does not arise, and, thus, v is infinitesimally stable. 

6.6 Summary Several phenomenological equations are provided with the Lyapunov function, and this functional induces a semi-dual variational structure to the stationary state, especially to the field component. The particle component, on the other hand, is sometimes trivial in the stationary state, which guarantees the dynamical stability of linearly stable stationary states. If the system is materially closed, then this stationary state is realized as a nonlinear eigenvalue problem with non-local term. In some cases, we obtain the hyper-linear stability of the field component, which guarantees the stability of the particle component simultaneously.

202

6 Critical Phenomena of Isolated Systems

1. The structure of semi-dual variation is observed, especially in materially closed systems of phenomenology. 2. Among them are the Caginalp-Fix equation, the Penrose-Fife equation, the coupled Cahn-Hilliard equation, and the Falk-Pawlow equation and the stationary state is described by the nonlinear eigenvalue problem with non-local term. 3. Consequently, a stationary field state has a variational functional associated with a non-local term, and a linearly stable stationary state is dynamically stable. 4. Besides the calculus of variation, the method of time map is applicable to the stationary state of these equations if the domain is one-dimensional.

Chapter 7

Self-interacting Fluids

Macroscopic state of particles that constitute self-interacting fluid is formulated by a system of equations provided with self-duality between the particle density and the field distribution. Here, formation of the field is subject to the action in distance and is associated with the Poisson equation. The stationary state is then described by a nonlinear eigenvalue problem with non-local term because of mass conservation, and this problem is provided with the variational structure from the energy conservation law. Then method of scaling detects the critical exponents for mass quantization. This chapter is devoted to the variational and scaling properties of systems of selfinteracting fluid such as the Euler-Poisson equation and the plasma confinement problem.

7.1 Ideal Fluids A solid can be put on a desk on which its some part does not lie. This phenomenon is due to the tangential force. Fluid is a continuum, of which tangential force is absent at the equilibrium. The ideal fluid, on the other hand, is a continuum of which tangential force is absent even when it is moving. Let v = v(x) ∈ R3 , x ∈ R3 , be the velocity of a stationary fluid, and define the local flow {Tt } by (1.2). Thus x(t) = Tt x0 indicates the solution to dx = v(x), dt

x|t=0 = x0 .

(7.1)

If f (x) stands for the observer of particles, then u(x, t) = f (T−t x)

© Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_7

203

204

7 Self-interacting Fluids

indicates the distribution of particles detected at the time −t. From the group property of {Tt }, Tt ◦ Ts = Tt+s , T0 = I d it follows that u(Tt y, t) = f (T−t Tt y) = f (y) for any y ∈ R3 , and, therefore, ∂ ∂ u(Tt y, t) = ∇u(Tt y, t) · Tt y + u t (Tt y, t) ∂t ∂t = v(Tt y) · ∇u(Tt y, t) + u t (Tt y, t),

0=

or

Thus we obtain

∂u + v · ∇u = 0 ∂t

in R3 × (0, T ).

Du = 0 for u(x, t) = f (T−t x) where Dt ∂ D = +v·∇ Dt ∂t

stands for the material derivative, which indicates the differention in t of the transported quantity subject to the velocity v. The Caucy problem of the first order single linear partial differential equation ut +

3  j=1

v j (x)

∂u = 0, ∂x j

u|t=0 = f (x)

(7.2)

is reduced to the ordinary differential equation (7.1), and, thus, the solution u = u(x, t) to (7.2) is given by u(x, t) = f (T−t x). If v = v(x, t) is the velocity of a non-stationary fluid, we define the propagator {T (t, s)}, that is x(t) = T (t, s)ξ if and only if dx = v(x, t), dt

x|t=s = ξ.

Then the accelation vector of this particle x = x(t) is given by     d 2 x  d Dv   v(x(t), t) = = dt 2 t=s dt Dt t=s t=s

(7.3)

7.1 Ideal Fluids

205

and hence the Euler equation arises as the equation of motion, ρ

Dv = ρF − ∇ p, Dt

(7.4)

where ρ, p, and F denote the density, the pressure, and the outer force, respectively. In fact, the left-hand side of (7.4) indicates the mass times accelation vector, while ρF on the right-hand side is compensated by the gradient of the pressure, −∇ p. The other derivation of (7.4) uses the equation of continuity ρt + ∇ · ρv = 0.

(7.5)

This equation describes the mass conservation, d dt



 ω

ρ dx = −

∂ω

ν · j ds,

(7.6)

because j = ρv is the flux of ρ, where ω is an arbitray subdomain with smooth boundary. To derive (7.4), we assume that the rate of change of momentum of a moving piece of ideal fluid is equal to the total force acting on it. This total force is composed of the surface and the volume forces, and, hence it follows that    d ρv d x = − pν d S + ρF d x (7.7) dt T (t,s)ω T (t,s)ω T (t,s)ω where ν denotes the outer unit normal vector. Regarding x = T (t, s)ξ as a transformation of variables (7.8) ξ ∈ R3  → x ∈ R3 , we have Jt (ξ) = 1 + (t − s)∇ · v(ξ, s) + o(t − s), t → s by Liouville’s theorem, see [375] for example, where  Jt = det

∂xi ∂ξ j



denotes the Jacobian. At t = s, therefore, the left-hand side of (7.7) is equal to d dt

 ω

  (ρv)(x(ξ, t), t)|Jt (ξ)| dξ 



D (ρv) + ρv(∇ · v) d x ω Dt    Dρ Dv = + ρ∇ · v v + ρ d x. (7.9) Dt Dt ω

= t=s

206

7 Self-interacting Fluids

Using the equation of continuity (7.5), we obtain  ω

ρ

Dv dx = Dt

 −∇ p + ρF d x

ω

by the divergence formula. Then (7.4) follows because the sub-domain ω is arbitrary. The equation of continuity (7.6) can be derived from d 0= dt

 T (t,s)ω

  ρ(x, t) d x 

(7.10) t=s

which indicates the mass conservation in the Lagrange coordinate. In fact, under the transformation (7.8), equality (7.10) reads as     Dρ d  + ρ∇ · v d x ρ(x(ξ, t), t)|Jt (ξ)| dξ  = 0= dt ω ω Dt t=s  = ρt + ∇ · ρv d x ω

(7.11)

similarly to (7.9). Then (7.5) follows because the subdomain ω is arbitrary. Vorticity In the barotropic fluid the density is a function of the pressure, represented by the state equation ρ = ρ( p). In this case, using  P( p) =

p

dξ ρ(ξ)

in (7.4), we obtain vt + (v · ∇)v = −∇ P + F.

(7.12)

From the thermodynamical law (5.3) with (5.2), it follows that T de S = dU + pd V. The fluid is said to be isentropic if de S = 0. Since V = 1/ρ we have dU = by (7.13) in this case, which means dw =

1 dp ρ

(7.13) p dρ ρ2

(7.14)

p . This w, called the specific enthalpy, is a thermodynamical quantity. ρ Hence it holds that w = w( p) by (7.14), and also a functional relation beween ρ and for w = U +

7.1 Ideal Fluids

207

p. This relation is indicated as the state equation p = p(ρ). Hence an isentropic fluid is barotropic. If the outer force is a potential, we have F = ∇ϕ with a scalar field ϕ in (7.12). Hence it holds that Dv = −∇ Q, Q = P − ϕ (7.15) Dt which implies ωt + ∇ × (v · ∇)v = 0, ω = ∇ × v. Since (v · ∇)v =

1 ∇|v|2 − v × ω 2

(7.16)

and ∇ × (v × ω) = (ω · ∇)v − (v · ∇)ω + v∇ · ω − ω∇ · v = −(v · ∇)ω + (ω · ∇)v − ω∇ · v it holds that

Dω = (ω · ∇)v − (∇ · v)ω. Dt

(7.17)

Equation (7.5) means Dρ = −(∇ · v)ρ, Dt and, therefore, we obtain D Dt

    ω ω = ·∇ v ρ ρ

(7.18)

by (7.17). From (7.3) we have DX = (X · ∇)v, Dt for the matrix X (t, s) =

X |t=s = I

∂ T (t, s)ξ, which implies ∂ξ

    ω ω (x(t), t) = X (t, s) (ξ, s), x(t) = T (t, s)ξ ρ ρ

(7.19)

by (7.18). Equality (7.19) is called the Cauchy integral. Then the vortex theorem of Lagrange, ∇ × v|t=0 = 0 ⇒ ∇ × v = 0, ∀t (7.20)

208

7 Self-interacting Fluids

follows. Thus, barotropic ideal fluids under the potential force are classified into with or without vorticities. Circulation along the oriented Jordan curve C ⊂ R3 is denoted by 

 v · dx =

(C) =

M

C

ω · d S,

where M is a closed surface satisfying ∂M = C and d S stands for the vector surface element. A tube composed of the stream lines of ω = ∇ × v by T ⊂ R3 is called a vortex tube. Then the Stokes formula implies Helmholtz’ circulation theorem, that is if Ci , i = 1, 2, are oriented Jordan curves surrounding T it holds that (C1 ) = (C2 ). If M ⊂ R3 is a closed surface such that ∂M = C, then it holds that       D d  v · dx = (v · d x) dt C(t,s) Dt C t=s  t=s      Dv D  · dx + (d x) = v· Dt C Dt C t=s t=s for C(t, s) = T (t, s)C. From (7.15) it follows that   Dv · d x = − ∇ Q · d x = 0, C Dt C while

implies

D dx = v · d v· Dt



 v· C

Dx Dt

 = v · dv =

1 d|v|2 2

D (d x) = 0. Dt

We thus end up with the circulation theorem of Kelvin, d dt

 C(t,s)

  v · d x 

= 0.

(7.21)

t=s

If we put M(t, s) = T (t, s)M then (7.21) reads as the vortex theorem of Helmholtz,    d ∇ × v · d S = 0. dt M(t,s) t=s Then it follows that (7.20) again.

7.1 Ideal Fluids

209

Conservation Laws Here we assume rapid decays at ∞ and also sufficient regularities of all physical quantities. First, (7.5) implies ρ ≥ 0 from ρ|t=0 ≥ 0. The total mass conservation,  M=

R3

ρ,

also follows from this equation. Assuming (7.12) with F = 0, we now derive the total energy conservation  E=

R3

ρ 2 |v| + Q(ρ) d x, 2

Q (ρ) =

p (ρ) . ρ

(7.22)

In fact, in this case, equation (7.12) is reduced to (ρv)t + ∇ · ρv ⊗ v + ∇ p = 0

(7.23)

by (7.5) where v ⊗ v = (v i v j ) for v = (v j )1≤ j≤3 . Here we have    ∂ j (ρv i v j ) v i = − ρv j v i ∂ j v i 3 3 R  R  1 1 1 j 2 |v|2 ∇ · (ρv) = − |v|2 ρt =− ρv ∂ j |v| = 2 R3 2 R3 2 R3 

 R3

[∇ · ρv ⊗ v] · v =

and 

 R3

(ρv)t · v =

 ρt |v| + ρvt · v d x = 2

R3

1 ρt |v|2 + ρ∂t |v|2 d x, 2 R3

which results in 

 1 |v|2 ρt + ρ∂t |v|2 d x [(ρv)t + ∇ · ρv ⊗ v] · v = 2 R3 R3  1 d |v|2 ρ. = 2 dt R3

Next, we have   ∇p ·v = R3

R3

 ∇ Q(ρ) · ρv =

Then (7.22) follows from (7.23)–(7.25).

R3

Q (ρ)ρt =

d dt

(7.24)

 Q(ρ). R2

(7.25)

210

7 Self-interacting Fluids

Hamilton Formalism Without vortices, the barotropic fluid is called irrotational. If F = 0 Eq. (7.12) is reduced to   1 2 vt = −∇ P + |v| 2 by (7.16) and ω = ∇ × v = 0. Then we obtain Bernoulli’s law,   1 2 ∇ −ψt + P + |v| = 0, 2 using the vector potential, v = −∇ψ. Hence we may suppose 1 ψt = P + |v|2 2 together with P( p) = Q (ρ). The total energy is then written as  E(ρ, ψ) =

R3

ρ Q(ρ) + |∇ψ|2 d x 2

and it holds that δE = −∇ · ρ∇ψ = ∇ · ρv, δψ

δE 1 = P + |v|2 . δρ 2

The irrotational barotropic fluid without outer force is thus formulated by the Hamilton system δE δE , ρt = − . (7.26) ψt = δρ δψ The total energy conservation,

dE = 0, now follows from (7.26). dt

7.2 Gas Dynamics We continue the formal argument, see [371] for rigorous results. Kinetic Formalism This formulation is due to [301]. Here we use the second moment in the Lagrange coordinate, |x − tv|2 . First, we have (|x − tv|2 ρ)t = |x − tv|2 ρt − 2(x − tv) · (v + tvt )ρ  ∇ · |x − tv|2 ρv = |x − tv|2 ∇ · ρv + ρv j ∂ j (xi − tv i )2 i, j

7.2 Gas Dynamics

211

with ρv j ∂ j (xi − tv i )2 = 2ρv j (xi − tv i )(δi j − t∂ j v i ). Hence it follows that  ρv j ∂ j (xi − tv i )2 = 2(x − tv) · (v − t (v · ∇)v)ρ. i, j

By (7.5) and (7.4) with F = 0, thus we obtain (|x − tv|2 ρ)t + ∇ · |x − tv|2 ρv = −2t (x − tv) · ρ(vt + (v · ∇)v) = 2t (x − tv) · ∇ p. (7.27) In (7.27) we use x · ∇ p = ∇ · x p − np, n = 3

(7.28)

and also v · ∇ p = p (ρ)v · ∇ρ = p (ρ)(∇ · ρv − ρ∇ · v) p (ρ)ρ + p (ρ) = − pt − ∇ · ( p (ρ)ρv) + v ·∇p p (ρ) which results in

p (ρ)ρ v · ∇ p = pt + ∇ · ( p (ρ)ρv). p (ρ)

If the state equation is descibed by p = Aργ , it follows that v ·∇p = and

Aργ , Q(ρ) = γ−1

A > 0, 1 < γ < 2

(7.29)

γ pt + ∇ · pv γ−1 γ−1

(7.30)

 E=

R3

ρ 2 Aργ |v| + d x. 2 γ−1

In (7.29), A > 0 is a constant determined by the total entropy of the system, and 1 < γ < 2 is the adiabatic constant. The isentropic fluid is assumed here, while γ = 5/3, 7/5, . . . when the gas is mono-atomic, bi-atomic, and so on, and γ = 4/3 in the case of the excellent radiational pressure [51].

212

7 Self-interacting Fluids

Equalities (7.27), (7.28), and (7.30) are summarized as (|x − tv|2 ρ)t + ∇ · |x − tv|2 ρv     2   2t p 2 2γt 2 −2 n− = ∇ · 2t x p − vp − tp γ−1 γ−1 t γ−1

(7.31)

which implies d dt



2t 2 p dx + |x − tv| ρ + γ−1 R3



2

If γ ≥ 1 +



R3

n(γ − 1) − 2 2t 2 p · = 0. t γ−1

(7.32)

2 , therefore, it follows that n   d d 2t 2 p ≤− |x − tv|2 ρ dt R3 γ − 1 dt R3

2 , on the other hand, we have n R3    d α d 2t 2 p 2t 2 p − =− |x − tv|2 ρ dt R3 γ − 1 t R3 γ − 1 dt R3  for α = 2 − n(γ − 1) ∈ (0, 2), and, therefore, p = O(t −2+α ). In any case it

and hence

p = O(t −2 ). If 1 < γ < 1 +

holds sthat

 R3

R3

p = o(1), t ↑ +∞.

(7.33)

Irrotational Blowup Using (7.26) and (7.33), we can show that the irrotational flow cannot survive in the energy class. For this purpose we write (7.26) as ρt = −∇ · ρv, ψt =

1 2 Aγργ−1 |v| + , v = −∇ψ 2 γ−1

(7.34)

and assume ψ0 = ψ|t=0 ≥ 0 without loss of generality. Then we obtain ρ, ψ ≥ 0 with    d ρ Aγργ dx ρψ = ρt ψ + ρψt d x = − |v|2 + dt R3 2 γ−1 R3  γ+1 = −E + Aργ = −E + o(1) γ − 1 R3 which is impossible by E > 0.

7.2 Gas Dynamics

213

A similar phenomenon arises if the gas has a positive mass and zero velocity at ∞. Thus we have the following theorem of blowup [371], see [339, 423] for the other cases. Theorem 7.2.1 There is no irrotational global-in-time classical solution to (7.4)– (7.5) with (7.29) if the initial value (ρ0 , v0 ) satisfies ρ0 > 0 in R3 , v0 = 0, ρ0 = ρ, |x| > R,

(7.35)

and E < 0, where ρ > 0 and R > 0 are constants and  E=

γ

R3

Aρ0 Aργ ρ0 |v0 |2 + − dx 2 γ−1 γ−1

stands for the defect energy. Proof Since system (7.4)–(7.5) is hyperbolic, it is provided with the propery of finite propagation [338, 339]. It holds that v(x, t) = 0, ρ(x, t) = ρ, |x| > R + σt

(7.36)

by (7.35), where σ = (γργ−1 )1/2 . Hence the defect energy and the defect mass defined by  E=

R3

Aργ ρ 2 Aργ |v| + − d x, 2 γ−1 γ−1

 M=

R3

ρ − ρ dx

(7.37)

converge and are independent of t. Here we use   2γt 2 vp (|x − tv|2 ρ)t + ∇ · |x − tv|2 ρv = ∇ · 2t x( p − p) − γ−1  2    2t ( p − p) 2 − t ( p − p), p = Aργ −2 n− γ−1 γ − 1 t derived from (7.31). By an analogous argument as above, then we obtain  R3

p − p d x ≤ O(t − min(2,2−α) ),

(7.38)

recalling α = 2 − n(γ − 1), n = 3. In particular, it holds that  E≤ by (7.36) and (7.37).

R3

ρ 2 |v| d x + o(1) 2

(7.39)

214

7 Self-interacting Fluids

Turning to (7.34), we may assume ψ0 = 0, |x| > R, for ψ0 = ψ|t=0 . Then it follows that Aγργ−1 t, |x| > R + σt. (7.40) ψ(x, t) = γ−1 Equality (7.40) justifies   

Aγργ−1 ρ Aγρ γ−1 t dx = ρ ρ ψ− − |v|2 + − ργ−1 d x γ−1 2 γ−1 R3 R3  γ−1 Aγρ Aγργ−1 ≤ −E + M + o(1) (ρ − ρ) d x + o(1) = −E − γ−1 R3 γ − 1

d dt



together with (7.38) and (7.39). Hence we obtain   Aγργ−1 Aγργ−1 t dx + Mt ρ ψ− γ−1 γ−1 R3    Aγργ−1 Aγργ−1 = t + ρ ψ− (ρ − ρ) t d x 3 γ−1 γ−1 R Aγργ ρψ − = t d x. γ−1 R3 

− Et + o(t) ≥

(7.41)

It follows, on the other hand, that ψt ρt ≥

Aγργ t γ−1

from (7.34), and hence ρψ −

Aγργ ∂ t ≥ ρψ − ψt ρt = 2ρψ − (ψρt) + ψρt t γ−1 ∂t ∂ ∂ {ψ(ρ − ρ)t} − ρ (ψt) + ψρt t = 2ρψ − ∂t ∂t

 Aγργ ∂ ∂ Aγργ−1 {ψ(ρ − ρ)t} − ρ t t − 2t + ψρt t. = 2ρψ − ψ− ∂t ∂t γ−1 γ−1

Then we have  |x| 5/2 contains the equilibrium, to be defined in the next paragraph, in the case of γ = 6/5. The assumption s > 5/2, on the other hand, is optimal in this approach of quasilinear symmetric hyperbolic system, in the sense that it comes from the fact that H  (Rn ) is an algebra if  > n/2 and  ∈ / N.

7.3 Self-gravitating Fluids

219

The above well-posedness, however, is not satisfactory. First, if

(w, v) ∈ C 1 [0, T ) × Rn , R1+n , n = 3 is a non-trivial radially symmetric solution to (7.48) with the support ρ(·, t) compact for each t ∈ [0, T ), then we obtain T < +∞. This fact is proven with the Lagrange coordinate, see [231]. Next, the space (7.52) is not necessarily physically reasonable if ρ ≥ 0 takes zero somewhere, that is the existence of vaccume, see the next paragraph. Second Moment Concerning (7.44) in n-space dimension, we have the weak form

d dt

d dt 



 Rn

Rn

ρϕ = R

ρv · ψ =

n

ρv · ∇ϕ

ρv ⊗ v · ∇ψ + p∇ · ψ d x    1 + ψ(x) − ψ(x ) · ∇(x − x )ρ ⊗ ρ d xd x (7.53) n n 2 R ×R Rn

for ϕ ∈ C0∞ (Rn ) and ψ ∈ C0∞ (Rn )n , where ρ ⊗ ρ = ρ(x, t)ρ(x , t). Then (7.53) implies d2 dt 2



 Rn

ρϕ =

ρv ⊗ v · ∇ 2 ϕ + pϕ d x    1 + ∇ϕ(x) − ∇ϕ(x ) · ∇(x − x )ρ ⊗ ρd xd x . (7.54) 2 Rn ×Rn Rn

Putting ϕ = |x|2 in (7.54) is justified if ρ = ρ(x, t) and ρv = (ρv)(x, t) have compact support, and are continuous in x ∈ R3 and continuously differentialble in t ∈ (0, T ). In this case, since ∇ 2 |x|2 = 2I, |x|2 = 2n, x · ∇(x) = −(n − 2)(x) it holds that d2 dt 2



 |x| ρ = 2

Rn

2ρ|v|2 + 2np d x − (n − 2) ∗ ρ, ρ    n−2 p d x + 2(n − 2)E (7.55) (4 − n)ρ|v|2 + 2 n − = γ−1 Rn Rn

220

7 Self-interacting Fluids

by (7.45). If n = 3 and 4/3 ≤ γ < 2 we obtain d2 dt 2

 R3

ρ|x|2 ≥ 2E

and hence the support radius of ρ(·, t) diverges to +∞ as t ↑ +∞, see [231, 233, 339]. If 2 (7.56) n ≥ 4, 1 < γ ≤ 2 − , E < 0, n next, the solution cannot exist global-in-time by (7.55). There actually arises such an initial value because inf{F(ρ) | ρ ≥ 0, ρ1 = λ} = −∞

(7.57)

2 n or γ = 2 − and λ > λ∗ , where λ∗ > 0 is a critical mass 2 n determined by the dimension n. In fact we have F = E if v = 0. holds if 1 < γ < 2 −

Kinetic Formalism Since (7.44) implies (|x − tv|2 ρ)t + ∇ · (|x − tv|2 ρv) = 2t (x − tv) · (∇ p + ρ∇) it holds that

dE + 2t (n − 4)F(ρ) + 2ta dt

 Rn

Aργ =0 γ−1

where a = n(γ − 2) + 2 and  E(ρ, v) =

Rn

|x − tv|2 ρ d x + 2t 2 F(ρ)

2 with F = F(ρ) defined by (7.47). Here, we have a ≥ 0 if and only if γ ≥ 2 − , n and, then, it holds that  F(ρ) ≤ by Gronwall’s lemma.

n≥4 O(t −2 ), O(t −1 log2 t), n = 3

(7.58)

7.3 Self-gravitating Fluids

221

Hamilton Formalism The vortex theorem of Lagrange holds also in (7.44) with n = 3. If this fluid is irrotational we take the velocity potential ψ ≥ 0 of v and then (7.34) is replaced by ρt = −∇ · vρ, ψt = Then it follows that  R

1 2 Aγργ−1 |v| + −  ∗ ρ, v = −∇ψ. 2 γ−1

 3

R3

ρψt = R ρt ψ =

3

R3

ρ 2 Aγργ |v| + d x −  ∗ ρ, ρ 2 γ−1  ρv · ∇ψ = − ρ|v|2 R3

and hence d dt

 R3

γ+1 γ−1



3 p −  ∗ ρ, ρ 2 R3 2−γ  ∗ ρ, ρ ≤ o(1) = (γ + 1)F(ρ) − 2

ρψ d x + E =

by (7.58), which results in  R3

ρψ ≤ −Et + o(t), t ↑ +∞.

Thus the irrotational self-gravitating gas fluid cannot exist global-in-time if γ ≥ 2 − 2/n = 4/3 and E > 0. This result is comparable to the non-existence of global-in-time solution including the rotational case, that is (7.56). Equilibrium The classical equilibrium of (7.44) is defined by putting v = 0 and ∂t · = 0. This formulation means Aγ ∇ργ−1 + ∇ = 0 γ−1

in {ρ > 0},

and, therefore, Aγ γ−1 ρ −  ∗ ρ = constant in each component of {ρ > 0} γ−1  ρ=M ρ ≥ 0 in Rn , Rn

(7.59)

222

7 Self-interacting Fluids

Putting q=

1 , u= γ−1



γ−1 Aγ

(γ−1)/(γ−2)

ργ−1 ,  = {u > 0} ,

thus we obtain −u = u q , u > 0 in , u = 0,  ∗ u q = constant on ∂    Aγ 1/(γ−2) q u = M. γ−1 

(7.60)

Here, the constant  ∗ u q on ∂ may be different according as the components of . Conversely, if (7.60) holds, then u −  ∗ u q is harmonic in  and is a constant on the boundary of each component of . This property implies the first equation of (7.60), and hence (7.60) is equivalent to (7.59). Radially Symmetric Equilibrium If u = u(x) satisfies − u = u q , u > 0 in B,

u = 0 on ∂ B

(7.61)

for B = B(0, R), then u = u(|x|) follows from the general theory [135], and, therefore, the first line of (7.60) arises with  = B. It is known, on the other hand, that (7.61) has a solution if and only if 1 < q < (n + 2)/(n − 2), that is 2n/(n + 2) < γ < 2. In this case, furthermore, the solution denoted by u = u R (x) is unique, see, for example, [264]. We have the self-similarity in (7.61). If u = u(x) solves −u = u q , then so does u μ = u μ (x) for μ > 0, where

Thus it holds that

u μ (x) = μ2/(q−1) u(μx).

(7.62)

u R (x) = R −2/(q−1) U (R −1 x)

(7.63)

for the unique solution U = U (y) to (7.61) with B = B(0, 1), that is U = u 1 . We have   q n− 2q u R (x)d x = R q−1 U (y)q dy, B(0,R)

B(0,1)

and, therefore, if q = n/(n − 2) and 1 < q < (n + 2)/(n − 2), then any M > 0 admits R > 0 such that (7.60) has a unique solution for  = B(0, R). Taking zero

7.3 Self-gravitating Fluids

223

extension outside  = B, we obtain the equilibrium (ρ, v) = (ρ, 0) to (7.44) from this u = u(x) ≥ 0 where   Aγ 1/(2−γ) q ρ= u . (7.64) γ−1 Therefore, if γ = 2 − 2/n and 2n/(n + 2) < γ < 2, then (7.44) admits an equilibrium (ρ, v) = (ρ, 0) for each prescribed total mass M > 0. This ρ = ρ(x) ≥ 0 is radially symmetric and has a compact support. Also, its total energy is defined by  Aργ ρ(x)ρ(x ) 1 d xd x E= − n−2 2(n − 2)ωn−1 Rn γ − 1 Rn ×Rn |x − x |  4 − q−1 +n−2 = aR U (y)q+1 dy Rn  4 U (y)q U (y )q − q−1 +n−2 −b R dydy n−2 Rn ×Rn |y − y | 

for

However,

c2 Acγ , b= a= , c= γ−1 2(n − 2)ωn−1



Aγ γ−1



1 2−γ

.

w = ρ(γ−1)/2 ≈ u 1/2

does not belong to H 1 (Rn ) because this u extended 0 outside B = B(0, R) has a derivative gap on ∂ B by the Hopf lemma. In the case of γ = 2 − 2/n, on the other hand, there is λ∗ > 0 such that if λ = λ∗ then the problem (7.60) admits a unique solution for  = B(0, R) with R > 0 arbitrary, and in the other case of λ = λ∗ , there is no radially symmetric solution to (7.60), where   Aγ 1/(γ−2) λ= M. γ−1 Again, the equilibrium defined by this solution is not consistent to the function space described in the previous paragraph for the well-posedness of the Cauchy problem, H s (Rn ), s > 1 + n/2 for (ρ(γ−1)/2 , v). In fact, there is a derivative gap of w = ρ(γ−1)/2 on the interface. If q ≥ (n + 2)/(n − 2), then − v = v q , v > 0 in ,

v = 0 on ∂

(7.65)

admits no solution if  is star-shaped [302]. Radially symmetric solution, on the other hand, exists for any q > 1 if  = {c < |x| < d} ([197]). This solution, however, does not satisfy (7.60) because the values of  ∗ u q on |x| = c and |x| = d are different.

224

7 Self-interacting Fluids

There is, on the other hand, no solution to − u = u q , u > 0

in Rn

(7.66)

if 1 < q < (n + 2)/(n − 2), see [137]. This fact guarantees a priori bounds of the solution to the elliptic and parabolic problems with sub-critical nonlinearity defined on the bounded domain, see [138, 140] and Sects. 12.3 and 11.1, which is proven by the blowup analysis, see Sect. 12.3 for the details of the elliptic case. Equation (7.66), on the contrary, has a unique radially symmetric solution u = u(|x|) if q ≥ (n + 2)/(n − 2). In the case of q > (n + 2)/(n − 2), it holds that 2



|x| q−1 u(|x|) → L =

  1 q−1 2 2 , |x| → ∞, n−2− q −1 q −1

(7.67)

see [149]. This relation implies  Rn

u q = +∞

and, therefore, (7.44) admits no radially symmetric equilibrium with finite total mass, in the case of 1 < γ < 2n/(n + 2). The exponent q = (n + 2)/(n − 2), finally, is associated with the Sobolev imbedding theorem. The solution to (7.66) is classified in this case [42], as it is radially symmetric with respect to some point, provided with the scaling (7.62). The solution v = v(x), thus, takes the form of [n(n − 2)μ2 ](n−2)/4 n vx0 ,μ (x) =  (n−2)/2 , x0 ∈ R , μ > 0. μ2 + |x − x0 |2

(7.68)

Then we see that the total mass defined by this solution is not finite. In particular, there is no radially symmetric equilibrium of (7.44) with finite total mass, also in the case of γ = 2n/(n + 2). In this way, the radially symmetric equilibrium of the EulerPoisson equation (7.44) has different profiles according to γ ∈ (1, 2n/(n + 2)), γ = 2n/(n + 2), γ ∈ (2n/(n + 2), 2) \ {2 − 2/n}, and γ = 2 − 2/n. The clssification (7.68) of the solution to (7.66), q = (n + 2)/(n − 2) was done, first, for the case of

as |x| → ∞. (7.69) v(x) = O |x|2−n In the context of differential geometry, this property means that any metric conformal to the standard metric ds0 on S n with the same mean curvature is a pull-back of ds0 by the conformal transformation on S n , see [271].

7.3 Self-gravitating Fluids

225

Although radial symmetry of the general solution to (7.66) does not follow in the super-critical case q > (n + 2)/(n − 2), see [150, 437], the total set of entire radial solutions has several remarkable structures, see [264] and the references therein. Variational Equilibrium We re-formulate the equilibrium using the calculus of variation. First, as we have seen, the total energy is reduced to  F(ρ) =

Rn

p 1 d x −  ∗ ρ, ρ γ−1 2

(7.70)

in the case of v = 0, and this (variational) equilibrium is defined by v = 0 and δF(ρ) = 0, under the constraint of  ρ ≥ 0, ρ = M. R3

We obtain the semi-unfolding E(v, ρ) ≥ F(ρ) by the definitition, and, therefore, the infinitesimal stability of the variational equilibrium implies its dynamical stability as in Sect. 7.4. The functional (7.70), actually, takes the form Aγ ∗ F − G∗ F= γ−1 for F ∗ (ρ) =

1 γ



ργ , ρ ≥ 0   ρ(x)ρ(x ) 1 d xd x , ρ = M. G ∗ (ρ) = n−2 2(n − 2)ωn−1 Rn ×Rn |x − x | R3 Rn

Here we apply the Toland duality. First, we realize these functionals F ∗ and G ∗ as the Legendre transformation of proper, convex, lower semi-continuous functionals on a Banach space. To this end, we take X = H˙ 1 (Rn ) ⊕ R and define G(μ) = where

1 ∇ξ22 + Mc, μ = ξ ⊕ c ∈ X 2

  2n H˙ 1 (Rn ) = μ ∈ L n−2 (Rn ) | ∇μ ∈ L 2 (Rn ) .

226

7 Self-interacting Fluids

We see that this functional is proper, convex, lower semi-continuous, and  ρ ∈ ∂G(μ)



−μ = ρ in Rn ,

Rn

ρ=M

for (μ, ρ) ∈ X × X ∗ . Furthermore, it holds that G ∗ (ρ) = sup {μ, ρ − G(μ)} μ∈X

=

  1 ξ, ρ − ∇ξ22 + c 1, ρ − Mc 2 ξ∈ H˙ 1 (Rn ), c∈R

=

1 (−)−1 ρ, ρ + χ{ρ,1=M} , ρ ∈ X ∗ , 2

sup

under the agreement that  : H˙ 1 (Rn ) → H˙ 1 (Rn )∗ is an isomorphism and X ∗ → H˙ 1 (Rn )∗ by H˙ 1 (Rn ) → X . Next, we put  γ γ−1 F(μ) = ξ+γ−1 , μ = ξ ⊕ c ∈ X, γ Rn which is also proper, convex, lower semi-continuous. Then we obtain ρ ∈ ∂ F(μ)



ξ+ = ργ−1 ,

(μ, ρ) ∈ X × X ∗ , μ = ξ ⊕ c,

and, in particular, 0 ≤ ρ ∈ L γ (Rn ) follows in this case. These results are summarized by 1 γ n ρ , ρ ≥ 0 ∗ , ρ ∈ X ∗. F (ρ) = sup {μ, ρ − F(μ)} = γ R +∞, otherwise μ∈X Variational equilibrium is now formulated by Aγ ∂ F ∗ (ρ) ∩ ∂G ∗ (ρ) = ∅, ρ ∈ X ∗ , γ−1 and thus it holds that 

γ−1 ξ −μ = ρ in R , ρ = Aγ n

for μ=ξ⊕c ∈



1 γ−1

+

 ,

Rn

Aγ ∂ F ∗ (ρ) ∩ ∂G ∗ (ρ). γ−1

ρ=M

7.3 Self-gravitating Fluids

227

This relation implies 

γ−1 ξ − ξ = Aγ

q



 in R , n

+

Rn

γ−1 ξ Aγ

q +

=M

(7.71)

with q = 1/(γ − 1), and, therefore, the variational equilibrium is defined by 

γ−1 ξ ρ= Aγ

q +

,

using the solution ξ = ξ(x) to (7.71).

1 2−γ Putting ξ = αu and α = γ−1 , we obtain Aγ − u =

q u+

 in R , n

Rn

q u+

= M˜ ≡



Aγ γ−1



1 γ−2

M

(7.72)

and hence (7.60) for  = {x ∈ Rn | u(x) > 0}. Thus, a variational equilibrium is a classical equilibrium. The above ξ = ξ(x), however, is defined on the whole space, has the C 2,θ -regularity, and can be negative somewhere. Therefore, the construction of the solution ξ = ξ(x) to (7.71) from that of ρ = ρ(x) of (7.59), or u = u(x) of (7.60), is not trivial. Therefore, it will be natural to regard (7.71) as a different description of the equilibrium to (7.44). Actually, uniform boundedness and radial symmetry of the solution ξ = ξ(x) to (7.71) with bounded Morse indices are known for 1 < q < (n + 2)/(n − 2) if n ≥ 4 and for 2 < q < 5 if n = 3, see [158, 159]. Self-similar Blowup The Euler-Poisson equation (7.44) with γ = 2 − 2/n is involved by the quantized blowup mechanism, similarly to the Smoluchowski-Poisson equation described in Sect. 1. In fact the free energy functional determing the equilibrium to (7.44),  F(ρ) =

Rn

p 1 −  ∗ ρ, ρ , ρ ≥ 0, γ−1 2

 Rn

ρ = M,

has the scaling homogeneity in this case of γ = 2 − 2/n. More precisely, it holds that   n−2 ρμ = ρ F(ρμ ) = μ F(ρ), ρμ ≥ 0, Rn

Rn

for ρμ (x) = μn ρ(μx) with μ > 0, and this property leads to the quantized blowup mechanism at the equilibrium level [409, 410, 419]. See [376] and Sect. 13.2 for the connection to the two-dimensional mass quantization. There is another

228

7 Self-interacting Fluids

non-equilibrium associated with this free energy. It is formulated by the degenerate parabolic equation derived from the kinetic theory, see Sect. 10.3. Henceforth, we restrict the case n = 3, that is γ = 4/3. First, radially symmetric solution to (7.44) is described by ρ = ρ(r, t), v = ωw(r, t), r = |x| , ω =

x , r

and then it holds that 2 ρt + wρr + ρwr + ρw = 0 r  ρ r ρ(wt + ωw) + pr + 2 ρ(s, t)s 2 ds = 0 r 0 p = Aργ , γ = 4/3

in R3 × (0, T ).

(7.73)

Using (7.73), we can detect the self-similar solution in the form of ρ(r, t) = αy(r/a(t))3 /a(t)3 , v =

a(t) ˙ x, a(t)

for a = a(t) > 0 and y = y(x) satisfying d 2a 3β β A3/2 , μ= = − , α = 2 2 dt a 8 8A3/2  2 d y 2 dy dy  3 + y = μ, y|x=0 = 1, + =0 dx2 r dx d x x=0

(7.74)

for β > 0 and A > 0, see [348]. The first equation of (7.74) is provided with the invariant β a˙ 2 − , E= 2 a √ and if a(0) = a0 > 0, a(0) ˙ = a1 , and a1 < 2λ/a0 , then this a = a(t) has the extinction time T > 0, that is a(T − 0) = 0. The solution y = y(x) to (7.74), on the other hand, has a compact support if |μ|  1, see [230]. Therefore, ρ = ρ(r, t) develops the delta function singularity in finite time. We note that this self-similar solution is not the stationary solution to the backward self-similar transformaion, and, therefore, its blowup rate is of type (II). Blowup Mechanism The ε-regularity holds to (7.44) with γ = 2 − 2/n. In the case of n = 3, γ = 4/3, see [87], if the total mass is sufficiently small then it holds that  √ ( ρv)(·, t) + ρ(·, t)γ ≤ C, 0 ≤ t < T. 2

(7.75)

7.3 Self-gravitating Fluids

229

In fact, Sobolev’s inequality guarantees S0 v26 ≤ ∇v22 , v ∈ H˙ 1 (R3 ) → L 6 (R3 ) with S0 > 0, which implies 1 1 1 2/3 4/3 | ∗ ρ, ρ| ≤ ρ26/5 ≤ ρ1 ρ4/3 2 2S0 2S0 by the duality L 5/6 (R3 ) → H˙ 1 (R3 )∗ and the interpolation θ 5 1−θ + = , 1 γ 6

ρ6/5 ≤ ρ1−θ ρθγ , 1

that is θ = 23 . Using total energy, now we obtain  R3

M 1/3 ρ 2 Aργ |v| + ργγ , dx ≤ E + 2 γ−1 2S0

and hence (7.75) in the case of  M
0 and 0 < ε0  1 independent of t ∈ [0, T ). Similarly, we obtain  2  d   dt 2

R3

   4/3     ρϕ ≤ C 1 + ϕρ1 + ϕ3/4 ρ 4/3

by (7.54) under the same assumptions, and, therefore,

2/3 2/3 ε0 − ϕρ1

 2  d      ≤ C 1 + ϕρ1 . ϕρ 1   + dt 2

(7.81)

232

7 Self-interacting Fluids

Thus, there is δ > 0 such that (ϕρ)(·, t0 )1 < ε0 /4, t0 ∈ [T − δ, T ) ⇒ sup (ϕρ)(·, t)1 < ε0 /2, t∈[t0 ,T )

which results in lim inf (ϕρ)(·, t)1 < ε0 /4 t↑T



lim sup (ϕρ)(·, t)1 < ε0 /2 t↑T

 2  4/3  √     < +∞. lim sup ϕ1/2 ( ρv)(·, t) + ϕ3/4 ρ(·, t)

and

2

t↑T

4/3

Thus we conclude that  ' S= x0 ∈ R3 | lim sup E R (x0 , t) = +∞, lim sup H R (x0 , t) < +∞ R>0

t↑T

t↑T

is finite, and lim inf ρ(·, t) L 1 (B(x0 ,r )) ≥ ε0 /4 t↑T

holds for any x0 ∈ S and 0 < r  1 in (7.44) with n = 3, γ = 4/3, where √  E R (x0 , t) = ( ρv)(·, t) L 2 (B(x

+  p(·, t) L 1 (B(x0 ,R))  ρ(x, t)ρ(x , t) H R (x0 , t) = v(·, t) L ∞ (A R (x0 )) + d xd x . |x − x | A R (x0 )×A R (x0 ) 0 ,R))

7.4 Plasma Confinements To describe the motion of self-gravitating fluid enclosed in a bounded domain  ⊂ Rn with smooth boundary ∂, we take ρt + ∇ · (ρv) = 0, ρ (vt + (v · ∇)v) + ∇ p + ρ∇ = 0  = ρ, p = Aργ in  × (0, T ),

(ν · v, )|∂ = 0.

(7.82)

There arises the non-negativity, the total mass conservation, and the total energy conservation of the solution, that is

7.4 Plasma Confinements

233

 ρ ≥ 0, ρ=M   & 1% ρ 2 p |v| + E= d x − (−)−1 ρ, ρ , γ−1 2  2

(7.83)

similarly. Then the variational equilibrium is formulated by v = 0 and δF(ρ) = 0, where  & 1% p d x − (−)−1 ρ, ρ F(ρ) = 2  γ−1 

defined for ρ ≥ 0,



ρ = M.

This functional is nothing but the Berestycki-Brezis functional [23] concerning the plasma confinement. Damlamian [79] observed the Toland duality in this free boundary problem between the above described formulation and that of Temam [396], where the Nehari principle is involved in the latter case [260, 261]. Thus we obtain a problem relative to (7.72) which arises also in plasma physics [395] in the equilibrium of self-gravitating fluid, that is q

−v = v+ in , v = constant on  = ∂  q v+ = λ

(7.84)



where λ > 0 is a constant. Here we use a different notation from that of the previous paragraph, and v = v(x) in (7.84) is a scalar function, differently from v = v(x, t) in (7.82). The associated variational functions [23, 396] to (7.84) are defined by   1 1 q+1 q ∇v22 − v+ + λv , v ∈ Hc1 (), v+ = λ 2 q +1    q+1 & q 1% ∗ J (u) = u q − (−)−1 u, u , u ≥ 0, u1 = λ, (7.85) q +1  2 J (v) =

where

  Hc1 () = v ∈ H 1 () | v = constant on  .

Henceforth, we describe the sub-critical or critical case, 1 < q ≤

n+2 (n−2)+ .

Then, the

solution v = v(x) to (7.84) is on  from the elliptic regularity, and the Toland duality is examined similarly to the the whole space case described in Sect. 7.3. As is confirmed there, the stationary problem (7.95) with q = n/(n − 2), n ≥ 3, is scaling invariant and actually is provided with mass quantization, see Sect. 13.2. C 2,θ

234

7 Self-interacting Fluids

Dual Variation First, we define the field variational functional, putting G(v) =

1 1 ∇v22 + λv , F(v) = 2 q +1



q+1



v+ , v ∈ X = Hc1 ().

Then it holds that u ∈ ∂G(v) ⇔ (∇v, ∇w) + λw = w, u, ∀w ∈ X ⇔ 1, u = λ, v − v = (−)−1 u

and u ∈ ∂ F(v)



(7.86)

q

u = v+

(7.87)

for (u, v) ∈ X × X ∗ . Thus, the problem (7.84) is equivalent to u ∈ ∂ F(v) ∩ ∂G(v), or q δ J (v) = 0, u = v+ , because q

u = v+ , v ∈ X



0≤u∈L

q+1 q

→ X ∗ .

It holds that D(J ) = X and 1 1 J (v) = G(v) − F(v) = ∇v22 + λv − 2 q +1  and, therefore, the constraint



 

q+1

v+ ,

q

v+ = λ is superfluous to derive (7.84). This property

arises because the Nehari principle, see [360], is involved in the original formulation (7.85). The free energy to which the particle density u ∈ X ∗ is subject is defined by the Legendre transformations. In fact, we have G ∗ (u) = sup {v, u − G(v)} v∈X   1 2 = sup v − v , u + v 1, u − ∇ (v − v )2 − λv 2 v∈X   1 2 v, u − ∇v2 = χ{1,u=λ} + sup 2 v∈H01 () % & 1 = χ{1,u=λ} + (−)−1 u, u 2

7.4 Plasma Confinements

235



and ∗

F (u) = sup {v, u − F(v)} =

+∞,

v∈X

because

q q+1  u

q+1 q

,0≤u∈L otherwise

q+1 q

()

u ∈ ∂ F(v) ⇔ v ∈ ∂ F ∗ (u) ⇔ u = v+ q

for (u, v) ∈ X × X ∗ . Hence the free energy is defined by D(J ∗ ) = D(G ∗ ) ∩ D(F ∗ )   q+1 ∗ q = u ∈ X | 0 ≤ u ∈ L (), u1 = λ

and ∗

J (u) =

q q+1  u

+∞,

q+1 q



1 2

(

) (−)−1 u, u , u ∈ D(J ∗ ) otherwise,

and this variational problem is nothing but Berestycki-Brezis’ formulation to (7.84). If the critical point v = v(x) of the Temam functional J = J (v) defined on X = Hc1 () is not positive-definite on , then the mapping v → u defined by u ∈ ∂ F(v) around v = v. Thus (7.87) is not faithful. The relation u ∈ ∂G(v), on the other hand, induces an isomorphism between u ∈ X ∗ with 1, u = λ and v − v ∈ H01 () for q v ∈ X = Hc1 (). Even in this case, the Nehari constraint  v+ = λ is not efficient to assign v from v − v if supp v ⊂ . Thus, the argument in Sect. 3.2 does not work, but still we have semi-spectral equivalence as is described below. Stable Equilibrium Although the Euler-Poisson equation (7.44) is not formulated by the gradient system, its equilibrium is associated with the duality between the Plasma confinement problem, and the structure of semi-duality controls its dynamical stability. To begin with, again, we confirm the abstract theory of Toland duality described in Sect. 3.2; X denotes a Banach space over R, F, G : X → (−∞, +∞] are proper, convex, lower semi-continuous, and L(u, v) = F ∗ (u) + G(v) − v, u is the Lagrangian. It holds that J ∗ (u) = inf L(u, v) = F ∗ (u) − G ∗ (u) = L(u, v) v∈X

J (v) = inf ∗ L(u, v) = G(v) − F(v) = L(u, v) u∈X

D(J ∗ ) = D(F ∗ ) ∩ D(G ∗ ) ⊂ X ∗ ,

D(J ) = D(G) ∩ D(F) ⊂ X

236

7 Self-interacting Fluids

for v ∈ ∂G ∗ (u) and u ∈ ∂ F(v). We define δ J (v) = 0 and δ J ∗ (u) = 0 by ∂G(v) ∩ ∂ F(v) = ∅ and ∂G(u) ∩ ∂ F ∗ (u) = ∅, respectively. Then it follows that u ∈ ∂G(v) ∩ ∂ F(v)



v ∈ ∂G ∗ (u) ∩ ∂ F ∗ (v).

(7.88)

If one of the conditions of (7.88) is satisfied, then J ∗ (u) = L(u, v) = J (v) follows, and, furthermore, if u ∈ ∂G(v), or equivalently v ∈ ∂G ∗ (u), it holds that J ∗ (u) = L(u, v) ≤ J (v). Thus, we obtain u ∈ ∂G(v)



J ∗ (u) − J ∗ (u) ≤ J (v) − J (v),

(7.89)

u ∈ ∂ F(v)



J (v) − J (v) ≤ J ∗ (u) − J ∗ (u).

(7.90)

and similarly,

In the plasma confinement (7.84), Temam’s functional J (v) of (7.85) is twiceq differentiable by q > 1. If the constraint  v+ = λ is taken into account, then the linearized stability of a critical point v ∈ X = Hc1 () of J = J (v) is defined by the positivity of the self-adjoint operator in L 2 () associated with the quadratic form on X 0 × X 0 defined by    1 d2 2  Q(ϕ, ϕ) = J (v + sϕ) = ∇ϕ2 − c(x)ϕ2 , 2 ds 2  s=0 q−1

where c(x) = qv+

and    c(x)ϕ = 0 . X0 = ϕ ∈ X | 

This condition means    1 c(x)ϕ = 0, ϕ2 = 1 > 0 inf Q(ϕ, ϕ) | ϕ ∈ Hc (), 

and, then, there is ε0 > 0 such that each ε ∈ (0, ε0 /4] admits δ > 0 such that  v − v X < ε0 , ⇒



q

v+ = λ,

v − v X < ε.

J (v) − J (v) < δ

(7.91)

7.4 Plasma Confinements

237 q

From (7.90), if v ∈ X is a infinitesimally stable critical point of J and u = v + , then there is ε0 > 0 such that each ε ∈ (0, ε0 /4] admits δ > 0 such that u = v+ , v − v X < ε0 , J ∗ (u) − J ∗ (u) < δ q

 for v ∈ X =

Hc1 ()

with



v − v X < ε

(7.92)

q



v+ = λ.

The linearized stability of v, on the other hand, may be defined by the positivity of the self-adjoint operator in L 2 () associated with the quadratic form on H01 () × H01 () defined by  Q(ϕ, ϕ) = ∇ϕ22 −



c(x)ϕ2 .

This condition means   inf Q(ϕ, ϕ) | ϕ ∈ H01 (), ϕ2 = 1 > 0

(7.93)

and then (7.92) follows if v ∈ X = Hc1 () satisfies v = v  . We can define, on the other hand, the variational equilibrium of the Euler-Poisson equation (7.82), using the Berestycki-Brezis functional J ∗ (ρ) =

 

& 1% p d x − (− D )−1 ρ, ρ . γ−1 2

For later convenience, we formulate this functional in a slightly different manner from the whole space case, that is J ∗ = F ∗ − G ∗ , where   q+1 q D(F ) = ρ ∈ L () | ρ ≥ 0 q+1 q+1 q A q , ρ ∈ D(F ∗ ) · · ρ ∗  γ−1 q q+1 F (ρ) = +∞, otherwise ∗

and

& * + 1% D(G ∗ ) = ρ ∈ X ∗ | 1, ρ = M , G ∗ (ρ) = (− D )−1 ρ, ρ 2 1 ∈ (1, 5). γ−1   q+1 Aγ 1/γ = (Bρ) q , B = , γ−1

for ρ ∈ X ∗ ⊂ H −1 () = H01 ()∗ and q = Since

A q + 1 q+1 · ρ q γ−1 q

238

7 Self-interacting Fluids

it holds that F(μ) = sup

ρ∈X ∗

% B

−1

&  ρ, Bρ − F ∗ (ρ) =



1 q +1



B −1 μ+

q+1

and, therefore, the dual functional of J ∗ (ρ) is defined by J (μ) = G(μ) − F(μ) =

a 1 ∇μ22 + Mμ − 2 q +1

for μ ∈ X = Hc1 (), where a = (B −1 )q+1 =



γ−1 Aγ

1/(γ−1)

 

q+1

μ+

(7.94)

. Using this functional, q

the variational equilibrium of (7.82) is now defined by v = 0 and ρ = aμ+ , where 

q

− μ = aμ+ in , μ = constant on ∂,

q

aμ+ = M.



(7.95)

Let μ = μ(x) be a solution to (7.95), and (v, ρ) = (0, ρ) be the equilibrium of q (7.82) defined by ρ = aμ+ . Then, the total energy (7.82) is described by  E(v, ρ) =



ρ 2 |v| d x + J ∗ (ρ), 2

and hence it holds that E(v0 , ρ0 ) − E(0, ρ) = E(v, ρ) − E(0, ρ) ≥ J ∗ (ρ) − J ∗ (ρ) for (v0 , ρ0 ) = (v, ρ)|t=0 . Therefore, applying (7.92), we obtain the following theorems concerning the cases that the plasma region occupies  and is enclosed in , respectively. Theorem 7.4.1 Let 2n/(n + 2) ≤ γ < 2 and μ = μ(x) > 0 on  be a solution to (7.95), linearly stable in the sense of (7.91) for c(x) = aqμq−1 . Then, each ε > 0 admits δ > 0 such that     γ−1 γ−1  v0 2 + ρ0 − ρ  1 < δ, ρ0 ≥ 0, ρ0 = M H

⇒      sup v(·, t)2 + ρ(·, t)γ−1 − ργ−1 

t∈[0,T )

 H1



< ε,

where (v, ρ) = (v(·, t), ρ(·, t)) is a solution to (7.82) satisfying (7.83) and

ρ(·, t)γ−1 ∈ C [0, T ), H 1 () . Here, ρ = aμq and T > 0 is arbitrary.

(7.96)

7.4 Plasma Confinements

239

Theorem 7.4.2 Let 2n/(n + 2) ≤ γ < 2 and μ = μ(x) be a solution to (7.95) such q−1 that supp μ ⊂  and linearly stable in the sense of (7.93) for c(x) = aqμ+ . Then, each ε > 0 admits δ > 0 such that     γ−1 γ−1 v0 2 + ρ0 − ργ−1  1 < δ, ρ0 ≥ 0, supp ρ0 ⊂  H ⇒       sup v(·, t)2 + ρ(·, t)γ−1 − ργ−1  1 < ε, supp ρ(·, t)γ−1 ⊂ , H

t∈[0,T )

where (v, ρ) = (v(·, t), ρ(·, t)) is a solution to (7.82) satisfying (7.83) and (7.96). q Furthermore, ρ = aμ+ and T > 0 is arbitrary.

7.5 Related Models This paragraph is devoted to several remarks on the Euler-Poisson equation and the plasma confinement. Neumann Case Let  ⊂ Rn , n ≥ 3, be a bounded domain with smooth boundary ∂, and take n+2 , and X = H 1 (), 1 < q ≤ n−2   1 λ 1 q+1 2 ∇v G(v) = v, F(v) = v , v ∈ X. 2+ ||  2 q +1  + Then it holds that λ ||



w = w, u , ∀w ∈ X  1 ⇔ 1, u = λ, v − v = (−)−1 u || 

u ∈ ∂G(v) ⇔ (∇v, ∇w) +



where w = (−)−1 u if and only if 1 −w = u − ||

 

u,

  ∂w  = 0, w = 0. ∂ν ∂ 

The field variational functional is now defined by J (v) = G(v) − F(v),

D(J ) = X.

(7.97)

240

7 Self-interacting Fluids

Then we obtain δ J (v) = 0, or equivalently u ∈ ∂G(v) ∩ ∂ F(v) with u ∈ X ∗ , if and only if  1 q v = (−)−1 u. u = v+ , v ∈ X = H 1 (), v − ||  Thus δ J (v) = 0 means −v =

q v+

λ − , ||

  ∂v  = 0, v = 0. ∂ν ∂ 

Given u ∈ X ∗ , next, we have G ∗ (u) = sup {v, u − G(v)} v∈X      1 1 1 λ = sup v − v, u + v · 1, u − ∇v22 − v ||  ||  2 ||  v∈X   1 2 v, u − ∇v2 + 1{1,u=λ} (u) = sup 2 v∈X, v=0 

1 = 1{1,u=λ} (u) + (−)−1 u, u, 2 and, therefore, the free energy associated with (7.97) is defined by ∗



q+1 q





D(J ) = u ∈ X | 0 ≤ u ∈ L (), u1 = λ q+1 ( ) q −1 ∗ q − 1 (− ∗ J L ) u, u , u ∈ D(J ) 2 J (u) = q+1  u +∞, otherwise. These variational functionals induce the Euler-Poisson equation in the form of ρt + ∇ · (vρ) = 0, ρ (vt + (v · ∇)v) + ∇ p + ρ∇ = 0  1  = ρ − ρ, p = Aργ in  × (0, T ) ||     ∂  ν · v, = 0, =0 ∂ν  ∂



(7.98)

for 2n/(n + 2) ≤ γ < 2. We obtain  ρ ≥ 0, ρ=M   & ρ 2 p 1% |v| + E= d x − (−)−1 ρ, ρ γ−1 2  2

(7.99)

7.5 Related Models

241

and the variational equilibrium is derived from J ∗ (ρ) =

 

& p 1% − (−)−1 ρ, ρ , γ−1 2

q

that is ρ = aμ+ for − μ =

q aμ+

  ∂μ  = 0, μ=0 ∂ν ∂ 

M − in , ||

(7.100)

1/(γ−1) where a = γ−1 . Aγ The linearized stability of this equilibrium is defined by 



inf Q(ϕ, ϕ) | ϕ ∈ H (), 1



 ϕ = 0, ϕ2 = 1 > 0,



where Q(ϕ, ϕ) =

∇ϕ22





(7.101)

q−1

c(x)ϕ2 , c(x) = aqμ+ .

Then we obtain the following theorem. Theorem 7.5.1 Let 2n/(n + 2) ≤ γ < 2 and μ = μ(x) be a solution to (7.100), q−1 linearly stable in the sense of (7.101) for c(x) = aqμ+ . Then, each ε > 0 admits δ > 0 such that     γ−1 γ−1  v0 2 + ρ0 − ρ  1 < δ, ρ0 ≥ 0, ρ0 = M ⇒ sup

t∈[0,T )

H



    v(·, t)2 + ρ(·, t)γ−1 − ργ−1 

 H1



< ε,

where (v, ρ) = (v(·, t), ρ(·, t)) is a solution to (7.98) satisfying (7.99) and (7.96). q Furthermore, ρ = aμ+ and T > 0 is arbitrary. Navier-Stokes-Poisson Equation There are several technical difficulties in mathematical study of the Euler equation for both compressible and incompressible cases [222, 223]. Theorems 7.4.1–7.5.1 also are not satisfactory because well-posedness of the problem is only established in H s (R3 , R4 ) with s > 5/2 for (ρ(γ−1)/2 , v). Obviously, this function space has a serious discrepancy in the function space describing stability, H 1 (R3 ) × L 2 (R3 , R3 ) for (ργ−1 , v). We can construct, on the other hand, weak solutions to the Navier-StokesPoisson equation in more moderate function spaces by the method of compensated compactness [111]. We continue to take the case n = 3.

242

7 Self-interacting Fluids

This system is then associated with the Lamé constants λ > 0 and σ in nσ + 2λ ≥ 0, n = 3, and is described by ρt + ∇ · (ρv) = 0 ρ (vt + (v · ∇)v) + ρ∇ + ∇ p = λv + (σ + λ)∇(∇ · v)  1  = ρ − ρ, p = Aργ in  × (0, T ) ||     ∂  = 0,  = 0. v, ∂ν  

∂

(7.102)

In this case, if 3/2 = n/2 < γ < 2, there is a solution [92, 204], denoted by (v, ρ) = (v(·, t), ρ(·, t)), satisfying

  2 0, +∞; H01 () 0 ≤ ρ ∈ C∗ [0, +∞), L γ () , v ∈ L loc   2γ γ+1 ρv ∈ C∗ [0, +∞), L () such that 

& 1% ρ 2 p 1,1 |v| + d x − (−)−1 ρ, ρ ∈ Wloc [0, T ) γ−1 2  2  dE + λ ∇v22 + (σ + λ) ∇ · v22 ≤ 0, ρ(·, t) = M, dt 

E=

see [204]. We formulate the variational equilibrium, using ∗



J (ρ) =



& 1% p d x − (−)−1 ρ, ρ γ−1 2 

defined for γ

0 ≤ ρ ∈ L (),



ρ = M.

(7.103)

Although this functional is not twice-differentiable, we can define the infinitesimal stability of its critical function ρ = ρ(x) by the existence of ε0 > 0 such that each ε ∈ (0, ε0 /4] admits δ > 0 such that  ρ ≥ 0, ρ − ργ < ε0 , ⇒

ρ − ργ < ε.



 ρ=



ρ,

J ∗ (ρ) − J ∗ (ρ) < δ (7.104)

7.5 Related Models

243

If lim sup ρ(·, s)γ ≤ ρ(·, t)γ

(7.105)

s↓t

for any t ≥ 0, then t ∈ [0, +∞) → ρ(·, t) ∈ L γ () is right-continuous. In this case, the above ρ is dynamically stable and each ε > admits δ > 0 such that  ⇒



ρ, v0 2 + ρ0 − ργ < δ * + sup v(·, t)2 + ρ(·, t) − ργ < ε.

ρ0 ≥ 0,



ρ0 =



t≥0

Inequality (7.105), however, has not yet been established in the above described renormalized solution to (7.102). Murakami-Nishihara-Hanawa Equation System (7.44) with the heat radiation taken into account arises as a hydrodynamical approximation in the theory of star formation. It is described by ρt + ∇ · ρv = 0, ρ (vt + (v · ∇)v) + ∇ p + ρ∇ = 0 ρ(εt + (v · ∇)ε) + p∇ · v = ∇ · ν∇T,  = ρ, ν = ν0 T n /ρm (z + 1)k B p p = Aργ , (7.106) T = = (γ − 1)ε in R3 × (0, T ) μ ρ where ε, T , k B , μ, γ, and Z are the specific internal energy, the temperature, the Boltzmann constant, the mean atomic mas, the specific heat ratio, and the ionization state, respectively. Similarly to (7.44), there is a type (II) self-similar blowup, see [248]. To take the first mathematical approach, we assume the case of ν0 = 0. Then, system (7.106) is reduced to ρt + ∇ · ρv = 0, et + ∇ · ev = 0 (ρv)t + ∇ · ρv ⊗ v + ∇ p + ρ∇ = 0  = ρ,

p = (γ − 1)eγ−1 ρ

for ε = eγ−1 , and, therefore,   ρ, E = e, M= R3

R3

 H=

R3

in R3 × (0, T )

(7.107)

1 ρ 2 p |v| + d x −  ∗ ρ, ρ 2 γ−1 2

are the quantities conserved. Now we define the variational equilibrium by the critical state of the variational functional  1 (7.108) F(ρ, e) = ρeγ−1 d x −  ∗ ρ, ρ 2 R3

244

7 Self-interacting Fluids

constrained by



 e ≥ 0,

e = E, ρ ≥ 0,

R3

First, Fe = 0 implies ρe

γ−2

R3

ρ = M.

 = constant,

and hence 1

e = Eρ 2−γ /

R3



e=E

1

R3

ρ 2−γ .

Next, Fρ = 0 implies  ∗ ρ = eγ−1 + constant,

 R3

ρ = M.

Thus, it holds that γ−1

E γ−1 ρ 2−γ ∗ρ= + constant, ρ ≥ 0, 1 γ−1 2−γ ρ 3 R

 R3

ρ = M.

(7.109)

To define the equilibrium on bounded domain, we replace ∗ by (−)−1 provided with the Dirichlet boundary condition in (7.108). Problem (7.109) has a similarity 2−γ

to (7.59) except for the non-local term. Similarly we obtain ρ = v+γ−1 for v = v(x) satisfying − v = E −γ+1 With γ = 1 +

1 q



1



v+γ−1

γ−1



2−γ

v+γ−1 ,

v|∂ = v ∈ R,

2−γ



v+γ−1 = M.

(7.110) and v = cw, c > 0, problem (7.110) has the dimensionless form

comparable to (7.84), that is − w =

q−1 w+ ,

 w|∂ = w ∈ R,

q



w+ = λ

(7.111)

unless q = 2, that is γ = 3/2, where λ = M (q−1)/(q−2) E −q+2 . The variational functionals are then defined by  1 1 q 2 w + λw , w ∈ Hc1 () J (w) = ∇w2 + 2 q  +   & 1 1% u γ − (− D )−1 u, u , u ≥ 0, u = λ, J ∗ (u) = γ  2 

7.5 Related Models

where γ = 1 + to (7.111), q =

245

1 q−1 . Then, we can detect the critical exponent for mass quantization 2n n−2 , see [366, 383]. A closely related problem is

−v = |v|

4 n−2

 v in ,

v|∂ = 0,

2n



|v| n−2 = λ,

see [350].

7.6 Summary We have observed semi-duality and the quantized blowup mechanism in the fundamental equations of fluid dynamics. 1. Ideal barotropic fluids are classified into with or without vorticities. We have a Hamilton formalism in the irrotational compressible Euler equation which results in the finite time blowup of the density. 2. There is a duality between self-gravitating fluid and plasma confinement. 3. The Nehari principle is involed by Temam’s variational formulation of the plasma confinement, while there is a semi-correspondence of Morse indices between Berestycki-Brezis’ formulation. Then, several stabilities of the linearized stable equilibrium of the Euler-Poisson equation are obtained by this semi-duality. 4. A quantized blowup mechanism is observed in the non-stationary Euler-Poisson equation of the excellent radial pressure case. There are, however, several other possibilities of the blowup mechanism due to the lack of well-posedness of the problem or that of blowup criteria.

Chapter 8

Magnetic Fields

The electro-magnetic fields are described by the Maxwell equation. Concerning the static case, first, we have the regularity in a specific components of the solution across the interface. Second, a model of the plasma equilibrium is derived from the MHD equation in Sect. 8.3, that is the queer differential equation involving equi-measurable rearrangement of the unknown function. Finally we describe the dynamical stability of MHD fluid.

8.1 Interface Vanishing A special component of electro-magnetic fields gains a stronger interface regularity than that of the other components. This property is called the interface vanishing [202, 203]. Let  ⊂ R3 be a bounded domain with Lipschitz boundary ∂, and M ⊂ R3 be a Lipschitz surface cutting  transversally. We assume that  = M ∩   = φ,  = + ∪  ∪ − (disjoint union) where ± ⊂  are sub-domains, and the Maxwell equation in magneto-statics holds piecewise: ∇ × B = J, ∇ · B = 0 in ± .

(8.1)

Hence B = t (B 1 (x), B 2 (x), B 3 (x)) and J = t (J 1 (x), J 2 (x), J 3 (x)), x = (x1 , x2 , x3 ), are three dimensional vector fields and ∇ = t (∂1 , ∂2 , ∂3 ) stands for the gradient operator. We recall that × and · are the outer and the inner products in R3 , so that ∇× and ∇· denote the operations of rotation and divergence, respectively. © Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_8

247

248

8 Magnetic Fields

Geselowitz Equation Study on the interface vanishing is motivated by magnetoencephalography, or MEG in short, where the interface M coincides with the boundary ∂ D of smooth bounded domain D ⊂ R3 , indicating a volume conductor. The argument below is based on the layer potential, which is useful to pick up the property. First, the standard theory of MEG assumes ∇ × B = J, ∇ · B = 0 as D (R3 )

(8.2)

where B = B(x) and J = J (x) denote the magnetic field and the total electric current density, respectively. Here we put μ0 = 1 for simplicity, where μ0 is the permeability. In (8.2), J is composed of the neuron current, denoted by J p , evoked inside the volume conductor by the quasi-static stimulation from outside, and the secondary current −σ∇V , that is J = J p − σ(x)∇V

(8.3)

where V = V (x) is the electric voltage and σ = σ(x) is the conductivity. In the simplest model, the conductivity is assumed as  σ(x) =

σI , 0,

x∈D x∈ / D,

(8.4)

where σ I > 0 is a constant. Under these circumstances, how J p determines B is described by the Geselowitz equation [133, 134], σI V (ξ) = − 2

 ∇ · J p (y)(ξ − y) dy

 ∂ −σ I V (y) (ξ − y) d S y , ξ ∈ ∂ D ∂ν y ∂D  J p (y) × ∇(x − y) dy B(x) = − D  V (y)ν y × ∇(x − y) d S y , x ∈ / ∂ D, +σ I D

(8.5)

∂D

where (x) = vector.

1 is the Newton potential and ν denotes the outer unit normal 4π |x|

Layer Potential Here we study (8.5) rather than (8.2), (8.3), and (8.4). We assume that J p is a smooth vector field defined on D satisfying J p × ν = 0 on ∂ D. The first term on the right-hand side of the second equation of (8.5) is then equal to

8.1 Interface Vanishing

249

 G(x) =

∇ × J p (y)(x − y) dy. D

Taking 0 extension of J p to D c , we see that the above G = G(x) is C 1 in the whole space R3 by the standard elliptic regularity. If V is smooth on ∂ D, the second term on the right-hand side of (8.5) is equal to σ I F where  F(x) =

∂D

ν y × ∇V (y)(x − y) d S y .

This F(x) is nothing but the first layer integral of ν × ∇V . Therefore, it holds that [F]+ − = 0 on ∂ D

(8.6)

where [A]+ − = A+ − A− for A+ (ξ) =

lim

x→ξ, x∈R3 \D

A(x),

A− (ξ) =

lim

x→ξ, x∈D

A(x), ξ ∈ ∂ D.

These properties imply [B]+ − = 0 on ∂ D, and hence the continuity of B across ∂ D. It holds also that   ∂F + = −ν × ∇V on ∂ D, (8.7) ∂ν − which provides with an additional regularity to B, see [132] for the above used properties of the first layer integral, (8.6) and (8.7). Turning to the interface vanishing, first, we have  + p [ν × ∇ × B]+ − = ν × J − ν × σ(x)∇V − = σ I ν × ∇V by ∇ × B = J in R3 ,

 J p × ν ∂ D = 0.

Second, it holds that + [ν × ∇ × B]+ − = [ν × ∇ × σ I F]−

by B = G + σ I F, [∇G]+ − =0

(8.8)

[ν × ∇ × F]+ − = ν × ∇V on ∂ D.

(8.9)

and, therefore,

250

8 Magnetic Fields

From the identity (ν · ∇)F + ν × ∇ × F = ∇(ν · F) − (∇ · ν)F equalities (8.7) and (8.9) imply [∇(ν · F)]+ − = 0, and, therefore, we have [∇(ν · B)]+ − = 0 on ∂ D

(8.10)

by (8.8). Relation (8.10) indicates that the normal component of B has C 1 regularity in the whole space R3 , in spite of the discontinuity of J across ∂ D. Localization To localize the above result we require several function spaces [146]. Let D ⊂ R3 be a bounded domain with Lipschitz boundary ∂ D and ν be the unit normal vector to ∂ D. Then for p ∈ [1, ∞], L p (D) denotes the standard L p space on D provided with the norm · L p (D) . The Sobolev space W m, p (D) is now defined by

W m, p (D) = u ∈ L p (D) | ∂ α u ∈ L p (D), |α| ≤ m , m = 1, 2, . . . where ∂ α = ∂xα11 ∂xα22 ∂xα33 for the multi-index α = (α1 , α2 , α3 ). We put H m (D) = W m,2 (D). Furthermore, we say that u belongs to H m+σ (D), σ ∈ (0, 1), if u ∈ H m (D) and   |∂ α u(x) − ∂ α u(y)|2 d xd y < +∞, |∀α| = m, n = 3. |x − y|n+2σ D D The space H s () is defined similarly with n = 2, through the local chart of , where s ∈ [0, 1] and  ⊂ ∂ D is a relatively open connected set. Then, we set H −s () = H0s () , where H0s () denotes the closure in H s () of the space composed of Lipschitz continuous functions on  with compact supports. Thus, we have H0s () = H s () if  ⊂ ∂ D is a closed surface, and in particular, it holds that  1/2 H 1/2 (∂ D) = H0 (∂ D). Then the trace theorem assures H 1 (D)∂ D ∼ = H 1/2 (∂ D). We also put

H (div, D) = u ∈ L 2 (D, R3 ) | ∇ · u ∈ L 2 (D)

H (rot, D) = u ∈ L 2 (D, R3 ) | ∇ × u ∈ L 2 (D, R3 ) . Here and henceforth, (·, ·) D and ((·, ·)) D denote the L 2 (D) and L 2 (D, R3 ) inner products, respectively, and ·, ·∂ D and ·, ·∂ D the duality pairing between H −1/2 1/2 (∂ D)− H 1/2 (∂ D) = H0 (∂ D), and H −1/2 (∂ D, R3 )− H 1/2 (∂ D, R3 ), respectively. We shall use the following facts [146].

8.1 Interface Vanishing

251

Lemma 8.1.1 Each v ∈ H (div, D) admits the trace ν · v|∂ D ∈ H −1/2 (∂ D), and Green’s formula ((v, ∇ϕ)) D + (∇ · v, ϕ) D = ν · v, ϕ∂ D holds for any ϕ ∈ H 1 (D). Lemma 8.1.2 Each v ∈ H 1 (rot, D) admits the trace ν × v|∂ D ∈ H −1/2 (∂ D, R3 ), and the Stokes formula ((∇ × v, w)) D − ((v, ∇ × w)) D = ν × v, w∂ D holds for any w ∈ H 1 (D, R3 ). Turning to (8.1), we put ± = ∂± ∩ M for the boundaries ∂± of ± . Hence + and − coincide as sets, but are regarded as the parts of the boundaries of + and − , respectively. Henceforth, ν denotes the outer unit normal vector to − so that −ν is the outer unit normal vector to + . Unless otherwise stated, if M is C k,1 , k = 0, 1, 2, . . ., then C k,1 extension of ν to  is taken. Furthermore, given a function A(x) on ± , we set [ A]+ − = A+ − A− on , where A± (ξ) = lim x→ξ, x∈± A(x), ξ ∈ , are taken in the sense of the traces to ± . We suppose that B and J are in L 2 (± , R3 ), and satisfy (8.1), which means that these equations hold piecewisely in ± as D (± ), that is  ±





 B ·∇ ×C =



J · C, ∀C ∈ C0∞ (± , R3 )

B · ∇ϕ = 0, ∀ϕ ∈ C0∞ (± ).

Unless otherwise stated, again, these vector fields B and J ∈ L 2 (± , R3 ) are identified with the elements in L 2 (, R3 ). Let M be Lipschitz continuous in (8.1), and B, J ∈ L 2 (± , R3 ). Then we obtain B ∈ H (rot, ± ) ∩ H (div, ± ), and hence ν × B|± ∈ H −1/2 (± , R3 ), ν · B|± ∈ H −1/2 (± )

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8 Magnetic Fields

are defined. Thus it holds that B|± ∈ H −1/2 (± )3 . Furthermore, [ν × B]+ − =0 [ν · B]+ − =0

⇔ ⇔

∇ × B = J ∈ L 2 (, R3 ) ∇ · B = 0 ∈ L 2 ()

(8.11)

as D (). If both relations of (8.11) are satisfied, then B ∈ H 1 (, R3 ) follows. In + 1 3 fact, (8.11) implies [B]+ − = 0 on , while B ∈ H (, R ) is equivalent to [B]− = 0 on  for B ∈ H 1 (± , R3 ). Namely, we have the following lemma proven by Green’s formula. Here, we note that the trace p|± is taken in H 1/2 (± ) if p ∈ H 1 (), because M is Lipschitz continuous. Lemma 8.1.3 If M is C 0,1 , p ∈ H 1 (± ), and [ p]+ − = 0 on , then it holds that p ∈ H 1 (). The fundamental interface vanishing is stated as follows [203]. Theorem 8.1.1 Let M be C 1,1 , and assume that B ∈ H 1 (, R3 ) and J ∈ 2 (). H (rot, ± ) satisfy (8.1). Then it holds that ν · B ∈ Hloc Since B ∈ H 1 (, R3 ) is assumed, it solves (8.1) in  as distributions, that is   B ·∇ ×C = J · C, ∀C ∈ C0∞ (, R3 )   B · ∇ϕ = 0, ∀ϕ ∈ C0∞ (). 

We note that J ∈ H (rot, ± ) belongs to J ∈ H (rot, ) if and only if [ν × J ]+ − =0 on  as the element in H −1/2 (, R3 ), and if this condition is satisfied it holds that − B = ∇ × J ∈ L 2 (, R3 ) as D (),

(8.12)

because ∇ × B = J ∈ H (rot, ), ∇ · B = 0 ∈ L 2 () as D () 2 is valid. Equation (8.12) then implies B ∈ Hloc (, R3 ) from the standard elliptic regularity. Theorem 8.1.1 says, in contrast, that even if ν × J has an interface on  = M ∩ , the normal component ν · B of B gains the regularity in one rank, thanks to the solenoidal condition ∇ · B = 0. We note also that the assumption of Theorem 8.1.1 implies the interface vanishing of ν · J . In fact, since (8.1) holds in  as we have seen, it follows that

∇ · J = ∇ · (∇ × B) = 0 as D (). Then, (8.13) implies J ∈ H (div, ), and hence [ν · J ]+ − = 0 on .

(8.13)

8.1 Interface Vanishing

253

The proof of Theorem 8.1.1 relies on the following lemma. Lemma 8.1.4 If M is C 0,1 and p ∈ H 1 (), then [ν × (∇ p)]+ − = 0 on 

(8.14)

holds. Proof From the assumption p ∈ H 1 (), we have ∇ × (∇ p) = 0 as D (),

(8.15)

and hence ∇ p ∈ H (rot, ). Then, ν × ∇ p|± ∈ H −1/2 (± , R3 ) are well-defined, and (8.14) follows from the Stokes formula and (8.15).  The following lemma is a direct consequence of Lemmas 8.1.3 and 8.1.4 applied to p = ν · B. Lemma 8.1.5 If M is C 0,1 , B ∈ H 1 (± , R3 ), and [ν · B]+ − = 0 on , then [(ν × ∇)(ν · B)]+ − = 0 on 

(8.16)

as an element in H −1/2 (, R3 ). The proof of the following lemma is given later. Lemma 8.1.6 If M is C 0,1 and B ∈ H 1 (± , R3 ) satisfies ∇ · B = 0 in ± , [ν × B]+ − = 0 on  then [(ν · ∇)(ν · B) + (∇ · ν)(ν · B)]+ − = 0 on 

(8.17)

holds as an element in H −1/2 (). Then Lemmas 8.1.5 and 8.1.6 imply the following lemma. Lemma 8.1.7 If M is C 0,1 and B ∈ H 1 (, R3 ) satisfies ∇ · B = 0 in , then it holds that [∇(ν · B)]+ − = 0 on 

(8.18)

as an element in H −1/2 (, R3 ). Proof In fact, we have (8.16) and [(ν · ∇)(ν · B)]+ − = 0 on  by (8.17) and B ∈ H 1 (, R3 ). Then, (8.18) follows.



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8 Magnetic Fields

Now we give the following proof. Proof of Theorem 8.1.1 We recall that M is C 1,1 , B ∈ H 1 (, R3 ) satisfies (11.18), and J ∈ H (rot, ± ). Hence it follows that ∇ · B = 0, ∇ × B = J as D (). Henceforth, the differentiation is always taken in the sense of distributions in the domain. First, we have −B = ∇ × J ∈ L 2 (± , R3 ) in ± from the assumption which implies (ν · B) = (ν) · B + 2∇ν : ∇ B + ν · B ∈ L 2 (± ) in ± where T · S =

i, j

ti j si j for T = (ti j ) and S = (si j ). Then we obtain 2 (± ) in ± (ν · B) ∈ L loc

(8.19)

by ν ∈ C 1,1 (). 2 () by Regarding (8.19), we define f ∈ L loc f = (ν · B)|±

in ± .

We have ν · B ∈ H 1 () and (8.18) by Lemma 8.1.7, and, therefore, 2 (ν · B) = f ∈ L loc () in  2 by Green’s formula. Then, the elliptic regularity guarantees ν · B ∈ Hloc ().



To prove Lemma 8.1.6 we introduce the tangential parts of ∇ and B, that is ∇τ = ∇ − ν(ν · ∇),

B τ = B − ν(ν · B).

Lemma 8.1.8 If M is C 0,1 , then it holds that ∇ · B = ∇τ · B τ + (ν · ∇)(ν · B) + (∇ · ν)(ν · B) − [(ν · ∇)ν] · B τ . Proof Since ν ∈ C 0,1 , we can put A = C = ν in the identity ∇(A · C) = (A · ∇)C + (C · ∇)A + A × (∇ × C) + C × (∇ × A)

(8.20)

8.1 Interface Vanishing

255

which implies 0 = ∇(ν · ν) = 2(ν · ∇)ν + 2ν × (∇ × ν) by ν · ν = 1. Then we obtain [(ν · ∇)ν] · ν = [−ν × (∇ × ν)] · ν = det (ν, ∇ × ν, ν) = 0

(8.21)

by (A × B) · C = det (A, B, C). Now, by the definition it holds that ∇τ · B τ =





∂ j − ν j (ν · ∇)

j

=

B j − ν j (ν · B)







∂ j B j − ∂ j ν j (ν · B) − ν j (ν · ∇)B j + ν j (ν · ∇) ν j (ν · B)

j

=



∂ j B j − (∂ j ν j )(ν · B) − ν j ∂ j (ν · B) − ν j (ν · ∇)B j

j



+ ν j (ν · ∇) ν j (ν · B) = ∇ · B − (∇ · ν)(ν · B) − (ν · ∇)(ν · B)

−ν j (ν · ∇)B j + ν j (ν · ∇) ν j (ν · B) + j

where ν = t (ν1 , ν2 , ν3 ). Then, (8.21) implies j

=



ν j (ν · ∇) ν j (ν · B)

(ν j )2 (ν · ∇)(ν · B) + (ν · B)ν j (ν · ∇)ν j



j

= (ν · ∇)(ν · B) and



ν j (ν · ∇)B j =

j



(ν · ∇)(ν j B j ) − B j (ν · ∇)ν j



j

= (ν · ∇)(ν · B) − B · [(ν · ∇)ν] = (ν · ∇)(ν · B) − B τ · [(ν · ∇)ν] . From these relations, we obtain ∇τ · B τ = ∇ · B − (∇ · ν)(ν · B) − (ν · ∇)(ν · B) + [(ν · ∇)ν] · B τ and the proof is complete.



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8 Magnetic Fields

Lemma 8.1.9 Let M be C k,1 , k = 0, 1, 2, . . ., and v = v(x) be a vector field defined in . Then, v τ ∈ H k (, R3 ) if and only if ν × v ∈ H k (, R3 ). Similarly, if f = f (x) is a scalar field defined in , then ∇τ f ∈ H k (, R3 ) if and only if (ν × ∇) f ∈ H k (, R3 ). Proof Since (ν × v) × ν = v − ν(ν · v) = v τ , the first part is obvious. The second part follows from ν · (ν × ∇) f = ν · ∇τ f = 0

(8.22)

and ν × {(ν × ∇) f } = ν × {ν × (∇ f )} = ν(ν · ∇) f − ∇ f = −∇τ f. Here, (8.22) is obtained by ν · (ν × ∇) f = ν · (ν × (∇ f )) = 0 ν · ∇τ f = ν · (∇ f − ν(ν · ∇) f ) = 0, 

and the proof is complete. We are ready to give the following proof.

Proof of Lemma 8.1.6 From the assumption it follows that ν × B ∈ H 1 (, R3 ), and therefore, Proposition 8.1.9 implies B τ ∈ H 1 (, R3 ), or 



+ −

= 0 on .

(8.23)

Equality (8.23) implies [(ν × ∇)B τ ]+ − = 0 on  by Proposition 8.1.4, and therefore, we obtain 

∇τ B τ

+ −

= 0 on 

(8.24)

again by Proposition 8.1.9. Then (8.24) implies 

∇τ · B τ

+ −

= 0 on .

(8.25)

We have, on the other hand, ∇ · B = 0 in ± . Therefore, (8.17) follows from (8.23), (8.25), and (8.20). 

8.1 Interface Vanishing

257

Stokes Equation Theorem 8.1.1 is applicable to the the stationary Stokes equation − v = ∇ p + f, ∇ · v = 0 in ±

(8.26)

where v = t (v 1 (x), v 2 (x), v 3 (x)) denotes the velocity of fluid, p = p(x) the pressure, and f (x) = t ( f 1 (x), f 2 (x), f 3 (x)) the external force. Using the vorticity formulation, we can show the following theorem [203]. Theorem 8.1.2 Let M be C 2,1 , v ∈ H 2 (, R3 ), p ∈ H 1 (), and f ∈ H 1 (± , R3 ) 2 satisfy (8.26), and [ν · ∇ p]+ − = 0 hold on . Then the condition (ν · ∇) (ν × v) ∈ 3 2 1 3 3 H (, R ) implies v ∈ Hloc (, R ) and p ∈ Hloc (). In the standard regularity associated with the above theorem, the conditions v ∈ 3 (, R3 ) and p ∈ H 2 (, R3 ), p ∈ H 1 (), and f ∈ H 1 (, R3 ) imply v ∈ Hloc 2 1 Hloc () in (8.26). Theorem 8.1.2 says that without H regularity of f across , the (H 3 , H 2 ) interface of (v, p) can exist only to the normal derivative of p or to the second normal derivatives to the tangential component of v. Here, the trace ν · ∇ p|± ∈ H −1/2 (± ) is justified under the assumption of the theorem. The condition f ∈ H (div, ) can take place of the assumption [ν ·

∇ p]+ −



∂p = ∂ν

+ −

= 0 on .

(8.27)

In fact, (8.27) implies (8.26) in  similarly. Therefore, the assumption f ∈ H (div, ) implies −p = −∇ · ∇ p = ∇ · f ∈ L 2 () in , 2 and hence ∇ p ∈ H (div, ). This property implies (8.27) and also p ∈ Hloc () from the elliptic regularity. In other words, if (8.26) holds in  with v ∈ H 2 (, R3 ), p ∈ H 1 (), and f ∈ L 2 (, R3 ), then the condition

f ∈ H 1 (± , R3 ) ∩ H (div, ) implies H 2 interface vanishing of p, and therefore, H 3 interface of v can occur only in the second normal derivative of the tangential component. Theorem 8.1.3 Let M be C 2,1 and v ∈ H 2 (, R3 ), p ∈ H 1 (), and f ∈ 2 (). Furthermore, L 2 (, R3 ) satisfy (8.26). Then f ∈ H (div, ) implies p ∈ Hloc 3 (, R3 ). f ∈ H 1 (± , R3 ) with (ν · ∇)2 (ν × v) ∈ H 1 (, R3 ) implies v ∈ Hloc The above interface vanishing theorems are optimal in the sense that we have v ∈ H 2 (, R3 ), p ∈ H 2 (), and f ∈ H 1 (± , R3 ) ∩ H (div, ) satisfying (8.26), / H 1 (, R3 ), see [203]. (8.27), and (ν · ∇)2 (ν × v) ∈

258

8 Magnetic Fields

Systems on Manifold The above argument can be described by differential forms [187]. Let (M, g) be an N -dimensional compact orientable Riemannian manifold with ds 2 = gi j d x i d x j in local coordinate. The cotangent and tangent bundles are denoted by T ∗ M and T M, respectively, and let  p M, 0 ≤ p ≤ N be the set of p-forms on M. We take the inner product (X, Y ) = g(X, Y ) for X, Y ∈ T M. We put also ω  = gi j ω j

∂ ∈ TM ∂x i

for ω = ωi d x i ∈ T ∗ M to define the inner product on T ∗ M by (ω, η) = g(ω  , η  ), ω, η ∈ T ∗ M and also the dual operator  : T M → T ∗ M of  : T ∗ M → T M by g(ω  , η) = g(ω, η  ), ω ∈ T ∗ M, η ∈ T M. Let d :  p M →  p+1 M and δ = ∗d∗ :  p+1 M →  p M be the exterior derivative and the co-derivative, respectively, where ∗ :  p M →  N − p M is the Hodge operator. Then the system d B = J, δ B = 0, d E = 0, δ E = ρ

(8.28)

stands for the static Maxwell equation, where B ∈ 1 M and E ∈ 1 M are one forms on M. To formulate interface to (8.28), let  ⊂ M and H ⊂ M be a smooth connected open submanifold and a smooth submanifold with codimension 1, respectively. We assume  =  ∩ H  = ∅ and divide  into + ∪  ∪ − , where ± ⊂ M are open submanifolds. L 2 -theory is adapted, and H m ( p ) denotes the set of p-forms on  of which coefficients are in W 2,m (0 ), the Sobolev space on , where p = 0, 1, 2 and m = 0, 1, 2, . . .. First, (8.28) is taken in . It is described by d F = J, δ F = ρ in 

(8.29)

for F ∈ H 1 (1 ), which may be B or E. Second, piecewise regularity of the right-hand sides on (8.29) is assumed, that is J ∈ H 1 (δ, 2 ± ), ρ ∈ H 1 (0 ± ).

(8.30)

Here and henceforth, J ∈ H 1 (δ, 2 ± ) means J ∈ L 2 (2 ± ) with δ J ∈ L 2 (1 ± ) in the sense of distributions in ± .

8.1 Interface Vanishing

259

The results are concerned on the interface regularity in accordance with the traces J |± ∈ H 1/2 (2 ) and ρ|± ∈ H 1/2 (0 ) where ± = ± ∩ . In fact, d J = 0 in ± follows from (8.29), and, therefore, we have J ∈ H 1 (2 ± ) by (8.30). It follows also that d F|± ∈ H −1/2 (2 ± ) from the Gauss-Stokes formula. Let ν ∈ T M be the outer unit normal vector on − , and put F ν = (ν  , F)ν  ,

Fτ = F − Fν

where ν  = (ν). Then it holds that ρ|+ ⇒ − =0 + J |− = 0 ⇒

d F ν |+ − =0 τ + d F |− = 0.

(8.31)

The proof is parallel to the flat case. Under the first and the second assumptions of (8.31) it holds that F ν ∈ L 2 (1 ) and F τ ∈ L 2 (1 ), respectively, in the sense of distributions, where  = dδ + δd denotes the Laplacian. Then the elliptic 2 2 (1 ) and F τ ∈ Hloc (1 ), respectively. regularity guarantees F ν ∈ Hloc

8.2 Plasma Equilibrium Here we use the magnetic field B = μH and the total current density J = ∇ × H , to rewrite (8.56) as ∇ p = J × B, ∇ · B = 0, ∇ × B = μJ

in .

(8.32)

Under the cylindrical coordinate x = r cos ϕ, y = r sin ϕ, z = z, we decompose B = Br er + Bϕ eϕ + Bz ez for ⎛

⎛ ⎛ ⎞ ⎞ ⎞ cos ϕ − sin ϕ 0 er = ⎝ sin ϕ ⎠ , eϕ = ⎝ cos ϕ ⎠ , ez = ⎝ 0 ⎠ 0 0 1 which means ⎞ Br cos ϕ − Bϕ sin ϕ B = ⎝ Br sin ϕ + Bϕ cos ϕ ⎠ . Bz ⎛

Then we obtain ∇·B=

∂ Br Br 1 ∂ Bϕ ∂ Bz + + + = 0. ∂r r r ∂ϕ ∂z

(8.33)

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8 Magnetic Fields

Poloidal Magnetic Field If B satisfies

∂ Bϕ = 0, it follows that ∂ϕ ∂ ∂ (r Br ) + (r Bz ) = 0 ∂r ∂z

which guarantees the stream function ψ = ψ(r, ϕ, z) satisfying 1 ∂ψ , r ∂z

Br = −

Bz =

1 ∂ψ . r ∂r

(8.34)

Equalities (8.33)–(8.34) now imply     1 ∂ 2 ψ 1 ∂ψ 1 ∂2ψ ∂2ψ ∂ Bϕ − + 2 er − + eϕ ∇×B = − ∂z r ∂ϕ∂r r ∂r 2 r ∂r ∂z 2   ∂ Bϕ 1 ∂2ψ 1 + + Bϕ + 2 ez , ∂r r r ∂ϕ∂z and, therefore, ∂ Bϕ 1 1 ∂2ψ + 2 , μJϕ = − μJr = − ∂z r ∂ϕ∂r r ∂ Bϕ 1 ∂2ψ 1 μJz = + Bϕ + 2 ∂r r r ∂ϕ∂z



1 ∂ψ ∂2ψ ∂2ψ − + ∂r 2 r ∂r ∂z 2



(8.35)

from the third equation of (8.32) where J = Jr er + Jϕ eϕ + Jz ez . Thus it holds that     1 ∂ ∂2ψ −1 ∂ψ Jϕ = − r r + . μr ∂r ∂r ∂z 2 We have ∇ · J = 0 from the third equation of (8.32). Therefore, if there is f = f (r, ϕ, z) such that μJr = −

∂ Jϕ = 0 again, ∂ϕ

1∂f 1∂f , μJz = . r ∂z r ∂r

From (8.35)–(8.36) it follows that 1 ∂2ψ 1 ∂ (r Bϕ − f ) = 2 , r ∂z r ∂ϕ∂r

1 ∂2ψ ∂ (r Bϕ − f ) = − . ∂r r ∂ϕ∂r

(8.36)

8.2 Plasma Equilibrium

If

261

∂B f ∂ψ = 0 we have = 0. Then we may assume Bϕ = . It follows that ∂ϕ ∂ϕ r B=−

f 1 ∂ψ 1 ∂ψ er + eϕ + ez r ∂z r r ∂r

which implies 1 ∂ψ 1 ∂ψ f 1 ∂ψ er + ez , er × B = ez − eϕ r ∂r r ∂z r r ∂r 1 ∂ψ f ez × B = − eϕ − er . r ∂z r

eϕ × B =

(8.37)

Thus we obtain J × B = Jr er × B + Jϕ eϕ × B + Jz ez × B       Jr ∂ψ Jr Jz Jz ∂ψ Jϕ ∂ψ Jϕ ∂ψ − f er − + eϕ + f + ez . = r ∂r r r ∂r r ∂z r r ∂z In other words, in the poloidal case of ∂B = 0, ∂ϕ

∂J = 0, ∂ϕ

∂p = 0, ∂ϕ

the MHD equilibrium (8.32) is reduced to Jϕ ∂ψ Jz ∂p = − f, ∂r r ∂r r ∂p Jr Jϕ ∂ψ = f + ∂z r r ∂z

∂p Jr ∂ψ Jz ∂ψ = + =0 ∂ϕ r ∂r r ∂z (8.38)

with ∂ψ 1∂f 1 1∂f 1 ∂2ψ , μJz = , μJϕ = − (r −1 )− r ∂z ∂r ∂r r ∂z 2 r ∂r 1 ∂ψ f 1 ∂ψ B=− er + eϕ + ez r ∂z r r ∂r

μJr = −

(8.39)

where ψ = ψ(r, z) and f = f (r, z). Grad-Shafranov Equation The first equation of (8.32) implies J · ∇ p = B · ∇ p = 0. Therefore, ∇ f and ∇ψ are parallel to ∇ p by (8.38)–(8.39), which justifies the functional relations p = p(ψ),

f = f (ψ).

(8.40)

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8 Magnetic Fields

We have, on the other hand, Jϕ Jr f Jz f ∇ψ = − er + ez r r r

∇p − and

Jz 1 ∂f = 2 , r μr ∂r

Jr 1 ∂f =− 2 r μr ∂z

by (8.38) and (8.39), respectively, and hence ∇p =

Jϕ 1 ∇ f 2. ∇ψ + r 2μr 2

(8.41)

Equations (8.40)–(8.41) imply the Grad-Shafranov relation Jϕ = r

dp 1 df2 + dψ 2μr dψ

and (8.38)–(8.39) are reduced to the Grad-Shafranov equation   ∂ψ ∂ ∂2ψ 1 df2 dp 1 r (r −1 )+ + . = r − μ ∂r ∂r ∂z 2 dψ 2μr dψ

(8.42)

(8.43)

Queer Differential Equation Here we define V = V (ψ) by



V (ψ) =

r dr dz ψ(r,z)>ψ

to regard the adiabatic quantity  A= p·

dV dψ



as a known function,  where 1 < γ < 2 stands for the specific-heat ratio. Using  dψ γ , we obtain p(ψ) = A(ψ) dV     dp dψ γ−1 1 d 2 ψ d A dψ γ + γA · dψ = dψ dψ d V dV dV 2 dV     d A dψ γ−1 dψ γ−2 d 2 ψ = + γA (8.44) dV dV dV dV 2 in (8.43). The quantity

8.2 Plasma Equilibrium

q(ψ) =

263

d dψ

 ψ(r,z)>ψ

Bϕ dr dz =



d dψ

ψ(r,z)>ψ

f r dr dz r2

is also regarded as a known function, called the safety coefficient. The statistical mean of 1/r 2 concerning the micro-canonical measure, k1 (V )

−1

 =

V (ψ(y))=V

1 1 · 2 dσ y |∇V (ψ(y))| r

is also regarded as a known function. Applying the co-area formula, we obtain  q(ψ) =

ψ(y)=ψ

k1 (V ) = 

f dσ y = r 2 |∇ψ| 1

dσ y V (ψ(y))=V r 2 |∇V |

 V (ψ(y))=V

1 f dσ y · r 2 |∇V | ddψV

.

Hence it follows that k1 (V )q(ψ)

dψ = f dV

(8.45)

under the agreement of  q(ψ) ≈

V (ψ(y))=V

dσ y 2 r |∇V |

·

f dψ dV

.

Then (8.45) implies df2 dψ d V d = 2k1 (V )q(ψ) · dψ d V dψ d V

 k1 (V )q(ψ)

dψ dV

 .

(8.46)

By (8.44) and (8.46), Eq. (8.43) is reduced to the queer differential equation − μ−1 ∗ ψ = F(V, ψ, ψ  , ψ  ) where   ∂2 −1 ∂ r · + 2  ∂r ∂z   γ−1  d A dψ dψ γ−2 d 2 ψ F =r + rγ A dV dV dV dV 2   1 d dψ + k1 (V )q(ψ) k1 (V )q(ψ) . μr dV dV ∗

∂ =r ∂r

(8.47)

264

8 Magnetic Fields

In (8.47) we regard V as the monotone decreasing rearrangement of ψ. Simpler forms of F are studied mathematically, see [246, 334, 397, 406].

8.3 MHD Fluids The model used in the magnetic hydrodynamics is composed of the equation of continuity (7.5) and that of motion (7.4) with ρF = −μH × (∇ × H ), where H and μ are the magnetic flux density and the permeability, respectively. Thus B = μH stands for the magnetic field. If this fluid is not isentropic, we have A = es in the DA state equation (7.29) where s denotes the entropy subject to = 0. We use the Dt Maxwell equation ∇×E =−

∂B , ∇ · B = 0, ∂t

furthermore, where the electric field E = −v × B is associated with the fluid motion. Thus it holds that ρt + ∇ · ρv = 0, ρ(vt + (v · ∇)v) + ∇ p + μH × (∇ × H ) = 0 st + (v · ∇)s = 0, p = ργ es Ht − ∇ × (v × H ) = 0, ∇ · H = 0 in  × (0, T ) (ρ, v, s, H )|t=0 = (ρ0 , v0 , s0 , H0 ) (8.48) where  ⊂ R3 is a bounded domain with smooth boundary. Assuming that this  is a perfect conductor, we impose the boundary condition v · ν = 0, E × ν = 0 which is reduced to v · ν = H · ν = 0 in the case of H0 · ν = 0, see [428] including the other cases. Since ∇ · H = 0 we have 1 1 H × (∇ × H ) = −(H · ∇)H + ∇|H |2 = −∇ · H ⊗ H + ∇|H |2 2 2 by (7.16). If the fluid is isentropic, then we obtain μ |H |2 ) = 0 2 in  × (0, T )

ρt + ∇ · ρv = 0, (ρv)t + ∇ · (ρv ⊗ v − μH ⊗ H ) + ∇( p + Ht − ∇ × (v × H ) = 0, ∇ · H = 0, p = Aργ (v · ν, H · ν)|∂ = 0, (ρ, v, H )|t=0 = (ρ0 , v0 , H0 )

(8.49)

by (8.48), where A > 0 is a constant. If H = 0 then (8.49) describes the motion of ideal gas, that is (7.5), (7.4) with F = 0, and (7.29), provided with the Dirichlet boundary condition for the normal component of the velocity.

8.3 MHD Fluids

265

In the first equation of (8.49) we have ρ ≥ 0 if ρ0 ≥ 0 and hence the total mass conservation  d ρ=0 (8.50) dt  by v · ν = 0 on ∂. We use the second equation in the form of (ρv)t + ∇ · ρv ⊗ v + ∇ p + μH × (∇ × H ) = 0. Here, similarly to (7.24)–(7.25) it holds that 

1 d [(ρv)t + ∇ · ρv ⊗ v] · v = 2 dt   p d ∇p ·v = dt  γ − 1 

 

|v|2 ρ

by the boundary condition and the state equation. Now we use the triple product, for example, (ν × H ) · (v × H ) = −[H × (v × H )] · ν = 0, to deduce 

  [H × (∇ × H )] · v = v · [H × (∇ × H )] = (∇ × H ) · (v × H )       1 d H · ∇ × (v × H ) = H · Ht = |H |2 . = 2 dt   

Hence there arises the total energy conservation d dt

 

1 2 p μ |v| ρ + + |H |2 d x = 0. 2 γ−1 2

(8.51)

Equilibrium The stationary state is described by ∇ · ρv = 0, ρ(v · ∇)v + ∇ p + μH × (∇ × H ) = 0 ∇ × (v × H ) = 0, ∇ · H = 0, (ν · v, ν · H )|∂ = 0.

(8.52)

In the equilibrium we have v = 0, and, therefore, (8.52) is reduced to   μ ∇ p + |H |2 = μ∇ · H ⊗ H, ∇ · H = 0, 2

ν · H |∂ = 0.

(8.53)

If the plasma region indicated by  p = {x ∈  | p(x) > 0} is enclosed in , the interface boundary condition 

p+

+ μ |H |2 = 0, [ν · H ]+ − = 0 on  2 −

(8.54)

266

8 Magnetic Fields

arises naturally, where  p = + , − =  \  p , and  = ∂ p . Furthermore, we put A± (ξ) =

lim

x∈± →ξ

A(x), [A]+ − (ξ) = A+ (ξ) − A− (ξ)

μ for ξ ∈ . Here, p + |H |2 stands for the total pressure, and the first equality of 2 (8.54) indicates the interface vanishing condition of Kadomatsev. To extend (8.54) for rough functions, let H (div, ) = {v ∈ L 2 (, R3 ) | ∇ · v ∈ L 2 ()} and v ∈ Hloc (div, ) when v ∈ H (div, ω) holds for any sub-domain ω ⊂⊂ . Furthermore, if  is Lipschitz continuous, then any f ∈ W 1, p (± ), 1 ≤ p < ∞, takes the trace f |∂± ∈ W 1−1/ p, p (). Therefore, if p ∈ W 1,1 (± ) and H ∈ H 1 (± , R3 ) satisfies   μ ∇ p + |H |2 = μ∇ · H ⊗ H, ∇ · H = 0 2  + μ p + |H |2 = [ν · H ]+ − = 0 on  2 −

in ± (8.55)

then it holds that p+

μ 1,1 (), |H |2 ∈ Wloc 2

1 H ∈ Hloc (div, )3 .

We have, furthermore, 2 + [ν · (ν · H ⊗ H )]+ − = [(ν · H ) ]− = 0 on  1 and ν · (ν × H ⊗ H ) = 0, and, hence, ∇ · H ⊗ H ∈ L loc (, R3 ). Thus the interface vanishes in (8.55) and there arises

  μ ∇ p + |H |2 = μ∇ · H ⊗ H, ∇ · H = 0 2

in 

(8.56)

as distributions. Transformation Group and Stability Without interface, the plasma equilibrium on the whole space is described by γ

∇ p + μH × (∇ × H ) = 0, ∇ · H = 0, p = Aρ ,

 R3

ρ = M.

(8.57)

8.3 MHD Fluids

267

2 We can take a weak form of (8.56) in  = R3 for H ∈ L loc (R3 , R3 ) and 0 ≤ ρ ∈ γ 3 L loc (R ), that is

  μ ∇ p + |H |2 = μ∇ · H ⊗ H, ∇ · H = 0 2 Here we assume 1,γ

1 H ∈ Hloc (R3 , R3 ), 0 ≤ ρ ∈ Wloc (R3 )

to define the weak solution to (8.57), that is  (∇ p + μH × (∇ × H )) · δv = 0, ∇ · H = 0, R3

p = Aργ

(8.58)

for any smooth (δρ, δv) with compact support satisfying δρ + ∇ · ρδv = 0.

(8.59)

In fact, the second equation of (8.58) implies the total mass conservation d ds

 R3

  g∗s ρ

=0

s=0

for the one-parameter family {g s } of diffeomorphisms g s : R3 → R3 , |s|  1, satisfying   d s  dg s  = δv, = δρ. g ρ ds s=0 ds ∗ s=0 Next, using the Helmholtz decomposition δv = δ H + ∇δq, ∇ · δ H = 0,

(8.60)

we take the one parameter family {h s } of volume preserving diffeomorphisms h s : R3 → R3 , |s|  1 such that  d s  = δ H. (8.61) h∗ H  ds s=0 Then it follows that      d 1 ∗ 2 h |H |  = [H × (∇ × H )] · δ H = [H × (∇ × H )] · δv ds 2 R3 s R3 R3 s=0 from ∇ · H = 0, see Theorem II.2.2 of [12] for the first equality.

268

8 Magnetic Fields

We have, on the other hand, ∇ργ = γργ−1 ∇ρ as distributions by ρ ∈ Wloc (R3 ), and, therefore,     p(ρ + sδρ)  1 d = p(ρ)δρ = ∇ p · δv. ds R3 γ − 1 s=0 γ − 1 R3 R3 1,γ

The first equation of (8.58), thus, means   d =0 F(ρ + sδρ, h s∗ H ) ds s=0

(8.62)

where F = F1 (ρ) + F2 (H ) for 1 F1 (ρ) = γ−1



μ Aρ , F2 (H ) = 2 γ

R3

 R3

|H |2 . 1,γ

Summing up, we define that (ρ, H ) in 0 ≤ ρ = ρ(x) ∈ Wloc (R3 ) ∩ L 1 (R) and 1 H = H (x) ∈ Hloc (R3 , R3 ) is a plasma equilibrium as follows. Namely, given compactly supported smooth (δρ, δv) satisfying (8.59), we take the Helmholtz decomposition (8.60) and the one parameter family {h s } of volume preserving diffeomorphisms h s : R3 → R3 , |s|  1 such that (8.61). Then it holds that (8.62) for (ρ, H ) = (ρ, H ). We can define, on the other hand, the infinitesimal stability of (ρ, H ) ∈ L γ (R3 ) × 2 L (R3 , R3 ), ∇ · H = 0, using (8.50) and (8.51). Thus, each ε > 0 admits δ > 0 such that F1 (ρ) < F1 (ρ) + δ, F2 (H ) < F2 (H ) + δ ⇒ ρ − ρ γ < ε, H − H 2 < ε

(8.63)

for (ρ, H ) ∈ L γ (R3 ) × L 2 (R3 , R3 ), ∇ · H = 0. This (ρ, H ) is dynamical stable in X = L γ (R3 ) × L 2 (R3 × R3 ), if the dynamical system is generated in C([0, T ), X ).

8.4 Summary We have studied static and kinetic magnetic models. 1. There is an interface vanishing in the static magnetic equation. Hence the normal component gains the regularity of one more rank. There is a formulation using differential forms. 2. The plasma equilibrium is reduced to a queer differential equation under several physical assumptions. 3. MHD fluids are described by the model provided with the dual variation in accordance with transformation groups.

Chapter 9

Boltzmann-Poisson Equation

The equilibrium statistical mechanics provides the derivation of the mean field limit of many self-interacting particles in the equilibrium state. In this section, first, we describe the theory of statistical mechanics in accordance with the vortex system.

9.1 Point Vortex If the vector field ⎞ v 1 (x) v = ⎝ v 2 (x) ⎠ ∈ R3 , x = (x1 , x2 , x3 ) ∈ R3 v 3 (x) ⎛

denotes the velocity of the fluid, its rotation, ⎛ ∂v 3 ⎜ ∇ ×v =⎜ ⎝

∂ x2 ∂v 1 ∂ x3 ∂v 2 ∂ x1

− − −

∂v 2 ∂ x3 ∂v 3 ∂ x1 ∂v 1 ∂ x2

⎞ ⎟ ⎟, ⎠

stands for the rigid-like movement of this fluid, and is called the vorticity. To understand this formula, let O ⊂ R3 be a rigid body moving with a fixed point, denoted by the origin. We take an ortho-normal basis moving with O, and let {i(t), j (t), k(t)} be its position at the time t. Since O is rigid, it holds that i(t) · j (t) = j (t) · k(t) = k(t) · i(t) = 0 i(t) · i(t) = j (t) · j (t) = k(t) · k(t) = 1

© Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_9

269

270

9 Boltzmann-Poisson Equation

and, therefore, i  · j + i · j  = j  · k + j · k = k · i + k · i  = 0 i  · i = j  · j = k  · k = 0.

(9.1)

Describing {i  (t), j  (t), k  (t)} by {i(t), j (t), k(t)}, i  = c11 i + c12 j + c13 k j  = c21 i + c22 j + c23 k k  = c31 i + c32 j + c33 k,

(9.2)

we obtain c23 + c32 = c31 + c13 = c12 + c21 = 0 c11 = c22 = c33 = 0 by (9.1). Thus (9.2) is reduced to c3 j −c2 k i = +c1 k j  = −c3 i k  = c2 i −c1 j

(9.3)

using c1 = c23 = −c32 , c2 = c31 = −c13 , c3 = c12 = −c21 , and the vector ⎛

⎞ c1 ω = − ⎝ c2 ⎠ c3 is called the angular velocity. If x(t) denotes the position vector at the time t moving with O, then it follows that x(t) = x1 (0)i(t) + x2 (0) j (t) + x3 (0)k(t), ⎛

⎞ x1 (0) x(0) = ⎝ x2 (0) ⎠ . x3 (0)

where

This formula implies x  (0) = x1 (0)i  (0) + x2 (0) j  (0) + x3 (0)k  (0) ⎛ ⎞ −c2 (0)x3 (0) + c3 (0)x2 (0) = ⎝ −c3 (0)x1 (0) + c1 (0)x3 (0) ⎠ = ω × x|t=0 , −c1 (0)x2 (0) + c2 (0)x1 (0)

9.1 Point Vortex

271

and, therefore, v=

dx =ω×x dt

(9.4)

at t = 0. Relation (9.4) is valid to each t, and, therefore, this rigid body is infinitesimally rotating along ω with the speed |ω| in the direction where ω, x, and v form a righthanded coordinate system. It implies also ∇ × v = 2ω, and thus, the rotation ∇ × v of the velocity v is twice of the angular momentum if the fluid moves like a rigid body. This relation is what we have described at the begining of this paragraph. Euler Equation The velocity v = v(x, t) of non-viscous incompressible fluid is subject to the Euler equation of motion and the equation of continuity described by (7.4) and (7.5), respectively, where ρ is a constant. This equation comprises of vt + (v · ∇)v = −∇ p, ∇ · v = 0 in R3 × (0, T )

(9.5)

when the outer force F is zero, where p denotes the pressure. If the vorticity of v is denoted by ω = ∇ × v, then (9.11) implies ωt + (v · ∇)ω = (ω · ∇)v, ∇ · v = 0 in R3 × (0, T ).

(9.6)

Here, we consider the two-dimensional case, described by v 3 = 0, v 1 = v 1 (x1 , x2 , t), v 2 = v 2 (x1 , x2 , t). Then it holds that

⎛ ω =∇ ×v =⎝

∂v 2 ∂ x1

0 0 −

⎞ ∂v 1 ∂ x2

⎠,

∂v and, therefore, using the two-dimensional scalar field ∂v ∂ x1 − ∂ x2 , still denoted by ω, we obtain ωt + (v · ∇)ω = 0, ∇ · v = 0 in R2 × (0, T ), 2

1

or, equivalently, ωt + ∇ · (vω) = 0, ∇ · v = 0 in R2 × (0, T ).

(9.7)

272

9 Boltzmann-Poisson Equation

Regarding ∇ ·v =

∂v 1 ∂v 2 + = 0, ∂ x1 ∂ x2

we define the scalar field ψ = ψ(x1 , x2 ), called the stream function, by v1 =

∂ψ ∂ψ , v2 = − . ∂ x2 ∂ x1

This relation means v = ∇ ⊥ ψ for  ⊥

∇ = and then it holds that ω=

∂ ∂ x2 − ∂∂x1

,

∂v 2 ∂v 1 − = −ψ. ∂ x1 ∂ x2

(9.8)

Thus (9.7) is reduced to ωt + ∇ · (ω∇ ⊥ ψ) = 0, −ψ = ω in R2 × (0, T ).

(9.9)

If  ⊂ Rn , n = 2, 3, is a bounded domain with smooth boundary ∂, the Euler equation (9.5) is taken place of vt + (v · ∇)v = −∇ p, ∇ · v = 0 in  × (0, T ) ν · v|∂ = 0.

(9.10)

In the case that  ⊂ R2 is simply-connected, we can define the stream function ψ = ψ(x, t) by v = ∇ ⊥ψ using the solenoidal condition ∇ · v = 0. Then the boundary condition of (9.10) is recovered by ψ = constant on ∂. Thus (9.10) is reduced to ωt + ∇ · (ω∇ ⊥ ψ) = 0

ψ(·, t) = G(·, x  )ω(x  , t)d x  in  × (0, T ) 

(9.11)

similarly to (9.7), where G = G(x, x  ) denotes the Green’s function for − D .

9.1 Point Vortex

273

Vortex Equation The problem (9.11) has a similar structure to the simplified system of chemotaxis, that is (3.22) and takes the weak form



d 1 ϕω = ρϕ · ω ⊗ ω (9.12) dt  2 × for ϕ ∈ C 1 () with ϕ|∂ = 0, where ω ⊗ ω = (ω ⊗ ω)(x, x  , t) = ω(x, t)ω(x  , t) ρϕ = ρϕ (x, x  ) = ∇x⊥ G(x, x  ) · ∇ϕ(x) + ∇x⊥ G(x.x  ) · ∇ϕ(x  ). Using

(x) =

1 1 log , |x| 2π

we obtain G(x, x  ) = (x − x  ) + K (x, x  ), and, therefore,





×



+

ρϕ ω ⊗ ω =

×

= lim ε↓0

+

×

∇ ⊥ (x − x  ) · ∇ϕ(x) − ∇ϕ(x  ) ω ⊗ ω

 ∇x⊥ K (x, x  ) · ∇ϕ(x) + ∇x⊥ K (x, x  ) · ∇ϕ(x  ) ω ⊗ ω

×\{|x−x  |0 V1

9.2 Boltzmann Relation

275

under the dilation of the gas with T = 0, where V = V2 − V1 > 0. Let us consider the boxrooms of which volumes are equally v0 . Then, their numbers are M1 = V1 /v0 and M2 = V2 /v0 before and after the dilation, respectively. The numbers of divisions of moleculars into these boxrooms are MN 1 W1 = 1 = N! N!



V1 v0

N

MN 1 , W2 = 2 = N! N!



V2 v0

N ,

where N = N A is the Avogadro constant. Thus we obtain S =

W2 R log , N W1

which results in S = k log W,

(9.17)

where k = R/N A is the Boltzmann constant. This W is called the thermal weight factor. It thus indicates the number of microscopic states admitted by the prescribed macroscopic state. This macroscopic state is divided, furthermore, into many mezoscopic states, labeled by i = 1, 2, . . ., and then we obtain W =

 g ni i , ni ! i

where gi and n i denote the numbers of microscopic states and of particles, respectively, taking the mezoscopic state i. Here, each microscopic state takes f i = n i /gi particles from the principle of equal a priori probabilities. Using Stirling’s formula log n! ∼ n(log n − 1), n → ∞, we obtain S=k



(n i log gi − log n i !) ∼ k

i

=k

 i

= −k

(n i log gi − n i (log n i − 1))

i

( f i gi log gi − f i gi (log f i + log gi − 1))

 i



gi f i (log f i − 1).

276

9 Boltzmann-Poisson Equation

If f = f (x, p) indicates the mean density of particles with the phase variable (x, p), then we replace gi by xi pi / h using microcanonical statistics and uncertainty principle, where h denotes the Planck constant. This formulation implies S=−

k h



f (log f − 1)d xd p.

9.3 Ensemble Ensemble is a fundamental concept of statistical mechanics, indicating the equivalent class of the microscopic states that have the same macroscopic profile. Then, the mission of the equilibrium statistical mechanics is to define their probability measure in the equilibrium using the principle of equal a priori probabilities. There are micro-canonical, canonical, and grand-canonical ensembles, associated with the materially and kinetically closed, materially closed, and open systems, respectively. The micro-canonicial ensemble is prescribed by N , V , and E, the canoical ensemble is prescribed by N , V , and T , and the grand-canonical ensemble is prescribed by V , T , μ, respectively, where N , V , E, T , and μ stand for the number of particles, the volume, the energy, the temperature, and the chemical potential, respectively. Micro-canonical Statistical Mechanics First, micro-canonical statistical mechanics is concerned with the kinetically and materially closed system. Such a system is described by the Hamilton system ∂H dqi = , dt ∂ pi

∂H dpi =− , 1 ≤ i ≤ N, dt ∂qi

(9.18)

where (q1 , . . . , q N ) ∈ R3N and ( p1 , . . . , p N ) ∈ R3N denote the general position and the general momentum of self-interacting particles, respectively, and H = H (q1 , . . . , q N , p1 , . . . , p N ) is the Hamiltonian (Sect. 12.1). We obtain

d H (q(t), p(t)) = 0, dt

and, therefore, the total energy H is a constant, denoted by E, while each point in the phase space ( p1 , . . . , p N , q1 , . . . , q N ) ∈ = R6N is regarded as a microscopic state. Thus the micro-canninical ensemble is the equivalent class of with the equivalent relation defined by this total energy. Here, the co-area formula guarantees dx = d E ·

d(E) , |∇ H |

9.3 Ensemble

277

where d x = dq1 . . . dq N dp1 . . . dp N , and d(E) is the surface element on {x ∈ | H (x) = E} , and, therefore, the probability measure of the micro-canonical ensemble is defined by

d(E) d(E) 1 · μ E,N (d x) = , (E) = (9.19) (E) |∇ H | {H =E} |∇ H | for each E ∈ R, from the principle of equal a priori probabilities. Thus

 f  =  f  E,N = f (x)μ E,N (d x) {H =E}

indicates the phase mean of the physical quantity f over {H = E}. From the ergodic hypothesis, this value  f  is assumed to be equal to the time mean along the orbit, 1 f (x) = lim T →∞ T



T

f (Tt x)dt,

0

for d(E)-a.e. in x ∈ {H = E}, where {Tt } denotes the semi-group associated with (9.18). This value f (x) is regarded as an observable macroscopic datum, and the ergodic hypothesis asserts the practical efficiency of the equilibrium statistical mechanics. Canonical Statistical Mechanics Canonical statistical mechanics is concerned with the materially closed system with prescribed temperature. To assign the probabilities of such ensembles, first, we take a kinetically and materially closed system of n-particles provided with the Hamiltonian H . We use the coordinate (x1 , . . . , x2n ) ∈ = R6n instead of ( p1 , . . . , pn , q1 , . . . qn ) for simplicity, and decompose this phase space into = 1 ⊕ 2 by (x1 , . . . , x2N ) ∈ R6N and (x2N +1 , . . . , x2n ) ∈ R6n−6N . Prescribing the total energy by H = E, we obtain the sets of microscopic states constrained by H = E in 1 and 2 , denoted by G 1 and G 2 , respectively. Let μ1 (E 1 ) be the probability of the microscopic states of G 1 of which energies are E 1 . Then, from the principle of equal a priori probabilities it follows that μ1 (E 1 ) =

1 (E 1 )2 (E − E 1 ) , (E)

(9.20)

where (E), 1 (E 1 ), and 2 (E 2 ) denote the numbers of the microscopic states of

, 1 , and 2 with the total energies E, E 1 , and E 2 , respectively.

278

9 Boltzmann-Poisson Equation

We assume N  n, and regard G 1 and G 2 as a materially closed equilibrium system and a heat bath, respectively, where the temperature T is prescribed. In this case, E 1 = E 1∗ attains the maximum of μ1 (E 1 ), and it holds that 0=

∂1 (E 1 ) 2 (E − E 1 ) 1 (E 1 ) ∂2 (E − E 1 ) · + · ∂ E1 (E) (E) ∂ E1

by (9.20), and hence ∂ ∂ log 1 (E 1 ) = log 2 (E 2 ), ∂ E1 ∂ E2 where E 2 = E − E 1 . Here, we confirm that E 1 = E 1∗ is prescribed and that the temperatures of G 1 and G 2 are equal because they are in the equilibrium. Thus ∂ log 2 (E 2 ) ∂ E2 is a constant determined by T . Regarding (9.17) and (6.15), now we infer  S2 = k log 2 ,

∂ S2 ∂E

 = V

1 , T

and conclude ∂ log 2 (E 2 ) = β, 2 (E 2 ) = constant × eβ E 2 , ∂ E2 where β = 1/(kT ) is called the inverse temperature. Still E 1 = E 1∗ is prescribed, and this guarantees that μ1 (E 1 ) is proportional to e−β E 1 by (9.20). Writing μβ,N (d x), H , and for μ1 (E 1 ), E 1 , and 1 , respectively, thus we obtain the probability measure of the canonical ensemble, called the Gibbs measure, μβ,N (d x) =

e−β H d x , Z (β, N )

Z (β, N ) =



e−β H d x.

(9.21)

If we define the entropy, the inner energy, and the free energy of the macrostate by

S = −k



μ(log μ − 1)d x,

E=



H μ(d x),

and F = −T S + E, respectively, then the above particle density μ = μβ,N (d x) is the minimizer of F = F(μ) defined for the probability measure μ = μ(d x) on . In fact, Euler-Lagrange equation for this variational problem is described by

9.3 Ensemble

279

log μ + β H = constant, and then the minimizer μ = μβ,N (d x) is defined by (9.21), using μ( ) = 1. We note that in the case of model (B) equation derived from Helmholtz’ free energy, the Hamiltonian describing inner energy is associated with the particle density through the self-interaction potential, and then (9.21) induces an elliptic eigenvalue problem involving exponential nonlinearity and non-local term, see Sect. 1.4. Grand-Canonical Statistical Mechanics Grand-canonical statistical mechanics is concerned with the open system with prescribed temperature and pressure. To assign the probabilities of such ensembles, we follow the argument of canonical statistical mechanics. Thus we take a kinetically and materially closed system of n-particles provided with the Hamiltonian H , use the coordinate (x1 , . . . , x2n ) ∈ = R6n to indicate these particles, decompose the phase space into = 1 ⊕ 2 by (x1 , . . . , x2N ) ∈ R6N and (x2N +1 , . . . , x2n ) ∈ R6n−6N , prescribe the total energy by H = E, and obtain the microscopic states in 1 and

2 , denoted by G 1 and G 2 , respectively. Then, we take the probability μ1 (E 1 , n 1 ) of the microscopic states of G 1 of which energies and particle numbers are E 1 and n 1 , respectively. From the principle of equal a priori probabilities, now it follows that μ1 (E 1 , n 1 ) =

1 (E 1 , n 1 )2 (E − E 1 , n − n 1 ) , (E, n)

(9.22)

where (E, n), 1 (E 1 , n 1 ), and 2 (E 2 , n 2 ) denote the total quantities of microscopic states of , 1 , and 2 with the total energies E, E 1 , and E 2 , and the particle numbers n, n 1 , and n 2 , respectively. We assume N  n, and regard G 1 and G 2 as an open equilibrium system and a particle bath, respectively, where the temperature T and the pressure p are prescribed. In this case, (E 1 , n 1 ) = (E 1∗ , n ∗1 ) attains the maximum of μ1 (E 1 , n 1 ), and, therefore, (9.22) implies ∂ ∂ E 1 1 (E 1 , n 1 )

1 (E 1 , n 1 ) ∂ ∂n 1 1 (E 1 , n 1 ) 1 (E 1 , n 1 )

= =

∂ ∂ E 2 2 (E 2 , n 2 )

2 (E 2 , n 2 ) ∂ ∂n 2 2 (E 2 , n 2 ) 2 (E 2 , n 2 )

where E 2 = E − E 1 and n 2 = n − n 1 . The first relation of (9.23) implies ∂ log 2 (E 2 , n 2 ) = β ∂ E2

,

(9.23)

280

9 Boltzmann-Poisson Equation

similarly. The chemical potential, on the other hand, is defined by  μ=

∂G ∂n



 = −T T, p

∂S ∂n

 E,V

from G = H − T S and H = E + pV , and, therefore, it holds that ∂ log 2 (E 2 , n 2 ) = −ζ ∂n 2 for ζ = βμ by the second relation of (9.23). This property implies μ1 (E 1 , n 1 ) = constant × eζ n 1 −β E 1 , and, consequently, the probability measure of the grand-canonical ensemble is defined by μβ,ζ (n, d x) =

eζ n−β H d x , (β, ζ ) = (β, ζ )



eζ n−β H d .

If there are a variety of particles, then it holds that eζ n k −β H d x μβ,ζ (n k , d x) = , k = 1, 2, . . . (β, ζ ) 

(β, ζ ) = eζ n k −β H d k , k

k

where k = R6k and d k denotes the volume element.

9.4 Turbulence One aspect of turbulence is illustrated as a status of the fluid provided with a large number of vortex points. Two-dimensional vortices macroscopically stable for a log time period are observed often. This phenomenon is mentioned in the context of ordered structure, and motivated by this observation, Onsager [284] initiated the equilibrium statistical mechanics of many vortex points. More precisely, twodimensional vortex points of the perfect fluid are described by the Hamilton system (9.15), whereby the above mentioned theory of equilibrium statistical mechanics is applied. Its physical validity, on the other hand, is examined in connection with the ergodic hypothesis [95].

9.4 Turbulence

281

In this case, the phase space is replaced by n and the probability measure of the ensemble is denoted by μn = μn (d x1 , . . . , d xn ), where n  1 is the number of vortex points and X n = (x1 , . . . , xn ) ∈ n . n (x ) by We can define ρ1,i i

n (xi )d xi ρ1,i

=

n−1

μn (d x1 . . . d xi−1 d xi+1 d xn ),

which is independent of i = 1, . . . , n from the principle equal a priori probabilities. This ρ1,i (xi ) is denoted by ρ1 (x), and is called the one-point reduced probability density function, or (reduced) “pdf” in short. Similarly, the k-point reduced probability density function is defined by

ρkn (x1 , . . . , xk )d x1 . . . d xk =

n−k

μn (d xk+1 , . . . , d xn ).

(9.24)

The stationary n-vortex points ω = ωn (x) with the intensities equal to the same value α > 0 is described by ωn (x)d x =

n 

αδxi (d x)

i=1

and, therefore, its phase mean is defined by ωn (x) =

n

 i=1

n

αδ(xi − x)μn (d x1 . . . d xn ) = nαρ1n (x),

using the one-point pdf ρ1n = ρ1n (x). In the high-energy limit, we assume n → ∞ with αn = 1, α 2 n 2 E˜ = E, and α 2 n β˜ = β, where E˜ and β˜ are the energy and the inverse temperature of the nvortex system, respectively, and E and β are constants. In this case, the mean field ρ = ρ(x) is defined by lim ωn (x) = ρ(x) = lim ρ1n (x),

n→∞

n→∞

(9.25)

282

9 Boltzmann-Poisson Equation

which is expected to satisfy the mean field equation, e−βψ ρ =  −βψ , ψ = e



G(·, x  )ρ(x  )d x  .

(9.26)

Thus two-dimensional stationary turbulence is provided with the dual variation between the “particle density” ρ and the “field” ψ. If Eq. (9.26) admits a unique solution, conversely, then the limiting process (9.25) is justified in the sense of measure [44]. This property is actually the case of β > −8π . Furthermore, there is propagation of chaos indicated by ρkn



ρ ⊗k

in the sense of measure, where ρkn (x1 , . . . , xk ) is defined by (9.24) and ρ ⊗k (x1 , . . . , xk ) =

k 

ρ(xi ).

i=1

In the general case, it holds that

ρkn



ρ ⊗k ξ(dρ)

passing to a subsequence with a probability measure ξ(dρ), and if ρ is in the support of ξ , then it is a minimizer of a free energy functional [44, 167]. Actually, these results are obtained in the context of equilibrium statistical mechanics using the Hamiltonian (9.16). Although the above β is called the inverse temperature, it is not associated with the physical temperature, and there is a possibility of negative β. In fact, using the micro-canonical probability measure (E) defined by (9.19), we obtain ∂ log (E) β= ∂E by (9.17) and (6.15). We have, on the other hand,

(E) ≡

{H −∞,

see [279, 337]. The mean field equation (9.37)–(9.42), on the other hand, is formulated by   v e λ e−v   , v|∂ = 0. − v = − (9.43) v −v 2 e e This equation was derived by the above described method [185] and also by the other method of direct calculation [304] using the micro-canonical ensemble. From the latter method, we also obtain (9.32), see [318]. Inequality (9.39) is improved by its dual form [70, 275, 336, 337]. To describe the result, let  be a compact Riemannian surface. First, the extremal value λ is defined by λ>λ



λ −∞. 

Then it holds that ⎧ ⎫ ⎪ ⎪ ⎨ 8π P(K ) ⎬ ± λ = inf  | K Borel set, K ⊂ I ∩ supp P ± ± ±  2 ⎪ ⎪ ⎩ ⎭ α P(dα) K±

for I+ = [0, 1], I− = [−1, 0], see [313]. Next, we have %

&



inf Jλ (v) | v ∈ H (), 1

if inf supp P > 1/2, see [384].



v = 0 > −∞

(9.44)

9.4 Turbulence

291

Stochastic Intensity It is shown by the method of minimal free energy using the canonical ensemble [262] that if the intensities of the vortices are independent random variables α ∈ [−1, 1] subject to the the same distribution P(dα), then the mean field equation is defined by  −αβψ P(dα) [−1,1] αe  (9.45) , ψ = (− D )−1 ρ. ρ= −αβψ P(dα) e [−1,1]  Thus in the neutral case of (9.42), slightly different from (9.43), it holds that 

ev − e−v  v −v e + e



−v = λ 

,

v|∂ = 0,

and the associated variational functional is Jλ (v) =

1 ∇v22 − λ log 2





ev +



e−v



defined for v ∈ H01 (). This case is also derived from the principle of maximal entropy [95]. In fact, in this case we put ˜ x˜i = (xi , α i ) ∈  × [−1, 1] =  because the intensity αi = α i α is also a random variable besides the position xi at each vortex point. The Hamiltonian is described by n n n 1   1  i 2 α i α j G(xi , x j ) + α 2 (α ) R(xi ), H n ( X˜ n ) = α 2 2 2 i=1 j=1, j =i

i=1

˜ n into and, therefore, dividing X˜ n = (x˜1 , . . . , x˜n ) ∈  ˜ k, X˜ k = (x˜1 , . . . , x˜i ) ∈ 

˜ n−k , X˜ n−k = (x˜k+1 , . . . , x˜n ) ∈ 

we obtain H n ( X˜ n ) = H k ( X˜ k ) + H n−k ( X˜ n−k ) +

k n   i=1 j=k+1

α i αα j αG(xi , x j ).

292

9 Boltzmann-Poisson Equation

The canonical measure, on the other hand, is described by μn (d X˜ n ) =

˜

e−β H

n (X ˜ n)

d X˜ n

˜ Z (n, β)

,

Z (n, β) =

˜

˜n 

e−β H

n (X ˜ n)

d X˜ n ,

and, therefore, the k-point reduced pdf is defined by

ρkn =

˜ n−k



·

˜ n−k 

μn (d X˜ n−k ) = ˜

e−β H

n−k ( X ˜

Using

n−k )

˜

˜ k)

e−β H ( X Z (n) ˜

e−βα

˜ βn

k

2 k n i j i=1 j=k+1 α α G(xi ,x j )

d X˜ n−k = Z (n − k)e n−k H

n−k ( X ˜

n−k )

d X˜ n−k .

μn−k (d X˜ n−k ),

we deduce ρkn =

Z (n − k) −β˜ H k ( X˜ k ) e · Z (n)

 n ˜ βk n−k ˜ ˜ 2 k e n−k H ( X n−k ) e−βα i=1 j=k+1 G(xi ,x j ) μn−k (d X˜ n−k ), · ˜ n−k 

and, in particular, ! " β(α 1 )2 Z (n − 1) exp − R(x1 ) Z (n) 2n ⎡

n n   β exp ⎣ α i α j G(xi , x j ) · 2n(n − 1) ˜ n−k  i=2 j=2, j =i ⎡ ⎤  n n   β β − (α i )2 R(xi ) · exp ⎣− α j G(xi , k j )⎦ μn−1 (d X˜ n−1 ). 2n(n − 1) n

ρ1n (x˜1 ) =

i=2

j=2

Thus assuming ρ1n  ρ1 and the propagation of chaos, we obtain " !

1 α 1 α 2 G(x1 , x2 )ρ1 (x˜1 )ρ1 (x˜2 )d x˜1 d x˜2 ρ1 (x˜1 ) = Z −1 exp 2 × ˜  ˜ " !

α 2 G(x1 , x2 )ρ1 (x˜2 )d x˜2 · exp −βα 1 ˜ 

9.4 Turbulence

293

for Z = limn→∞

Z (n) Z (n−1) ,

which guarantees

 exp −βα 1 ˜ α 2 G(x1 , x2 )ρ1 (x˜2 )d x˜2  . ρ1 (x˜1 ) =  1 2 ˜ exp −βα  ˜ α G(x 1 , x 2 )ρ1 ( x˜ 2 ) d x˜ 1  This equality means

exp βα 1 ψ(x) ρ1 (x˜1 ) =  1 ˜ exp βα ψ(x) d x˜ 1 

for the mean field stream function

ψ(x1 ) = α 2 G(·, x2 )ρ1 (x˜2 )d x˜2 , ˜ 

(9.46)



and hence − ψ =

[−1,1]

α 1 ρ1 (x˜1 )P(dα 1 )

(9.47)

because α i (i = 1, 2, . . .) has the same distribution functions. Then, (9.45) holds by (9.46) and (9.47). The mean field equation (9.45) implies log ρα + β(− D )−1 ρ = constant e−αβψ  ρα =  , −αβψ P(dα) [−1,1]  e

where ρα ≥ 0,

(9.48)



[−1,1]

ρα P(dα) = 1, ρ =

[−1,1]

αρα P(dα).

(9.49)

Then, (9.48) is the Euler-Lagrange equation for



I (⊕ρα ) =

 [−1,1]

ρα (log ρα − 1)P(dα)d x +

/ β. (− D )−1 ρ, ρ 2

constrained by (9.49). The relation (9.45) also implies  − ψ = 

[−1,1] αe

[−1,1]



−αβψ P(dα)

e

−αβψ

P(dα)

,

ψ|∂ = 0

(9.50)

294

9 Boltzmann-Poisson Equation

which is provided with the variational function "  !

1 1 1 J (ψ) = ∇ψ22 + log e−αβψ P(dα) + 2 β β  [−1,1] defined for ψ ∈ H01 (). Then, introducing the Lagrangian "

!

1 1 L(⊕ρα , ψ) = − ρα (log ρα − 1)P(dα) + ∇ψ22 − ψ, ρ , β  [−1,1] 2 we obtain the unfolding L|

−αβψ e −αβψ P(dα) [−1,1]  e

ρα = 

(

)

1 L|ψ=(− D )−1 ρ = − I. β

= J,

Using v = βψ and λ = −β, we can write (9.50) as  − v = 

[−1,1] ωe



[−1,1]

αv P(dα)

e

αv

P(dα)

,

v|∂ = 0,

(9.51)

with the variational function 1 Jλ (v) = ∇v22 − λ log 2

 !



[−1,1]

e

αv

" P(dα)

(9.52)

defined for v ∈ H01 (). The following theorem [262], obtained similarly to (9.39), guarantees the existence of the solution to (9.51) for λ < 8π . Theorem 9.4.2 It holds that inf

v∈H01 ()

J8π (v) > −∞

(9.53)

for Jλ = Jλ (v) defined by (9.52). Proof Given v ∈ H01 ()\ {0}, we apply the Pohozaev-Trudinger-Moser inequality (9.41) to v2 1 ∇v22 + 4π α 2 · , αv ≤ 16π ∇v22 where α ∈ [−1, 1] and C > 0 is a constant determined by . This inequality implies  !

log



[−1,1]

e

αv

" P(dα)



1 ∇v22 + K 16π

for v ∈ H01 (), with the constant K determined by . Then (9.53) follows.



9.4 Turbulence

295

Contraly to the deterministic case (9.35), this value λ = 8π is optimal for Jλ (v), v ∈ H01 () defined by (9.52) to be bounded from below. Actually we have the following theorem [314]. Theorem 9.4.3 If  is a compact Riemannian surface and Jλ be as in (9.52) with sup supp |P| = 1, then the value λ defined by (9.44) is 8π .  Proof We put E = {v ∈ H 1 () |  v = 0} and suppose sup supp P = +1 without loss of generality. Let λ > 8π . Since Fontana’s inequality [118] is sharp it holds that inf Jλ0 (v) = −∞,

Jλ0 (v) =

v∈E

1 ∇v22 − λ log 2



ev .

Using veαv ≥ v for α > 0 and v ∈ R, we obtain d dα



eαv =





veαv ≥



v = 0, v ∈ E,

and, therefore,



log

[−1,1] 



≥ log



eαv d x P(dα) ≥ log



[1−δ,1] 

eαv d x P(dα)

e(1−δ)v + log P[1 − δ, 1]

for 0 < δ < 1. Let 0 < δ  1 and w = (1 − δ)v ∈ E. Then it holds that

1 1 2 Jλ (v) ≤ · ∇w − λ log ew + C1,δ 2 2 (1 − δ)2  % &

1 1 2 2 w + C1,δ . ∇w = − λ(1 − δ) log e 2 (1 − δ)2 2  For 0 < δ  1 we have λ˜ = λ(1 − δ)2 > 8π , and hence inf v∈E Jλ (v) = −∞.



9.5 Summary We have studied the stationary mean field turbulence using equilibrium statistical mechanics. 1. Two-dimensional vortex point system is described by a Hamilton system. 2. The equilibrium statistical mechanics formulates the probability measure of ensembles, which are equivalent classes of the microscopic states that have the same macroscopic profiles using the principle of equal a priori probabilities. Then, the ergodic hypothesis asserts its practical validity.

296

9 Boltzmann-Poisson Equation

3. Micro-canonical, canonical, and grand-canonical ensembles are defined for materially and kinetically closed, materially closed, and open systems, respectively. In connection with the non-equilibrium thermodynamics, the micro-canonical setting is concerned with the total energy conservation and the entropy increasing, the canonical setting is concerned with the temperature conservation and Helmholtz’ free energy decreasing, and finally, the grand-canonical setting is concerned with the pressure conservation and Gibbs’ free energy decreasing. 4. We obtain the mean field equation of high-energy limit of two- dimensional vortex systems by several arguments. This equation is provided with the dual variation between the vortex point distribution and the stream function. Using the latter, it is described by the elliptic eigenvalue problem with the exponential nonlinearity competing two-dimensional diffusion, whereby the quantized blowup mechanism is observed.

Chapter 10

Particle Kinetics

This chapter is devoted to several mathematical modelings and analysis used in non-equilibrium statistical mechanics. The first two sections are concerned with the macroscopic description of the non-stationary mean field equation based on the microscopic and mezoscopic overviews, respectively. There we use the KramersMoyal expansion and the adiabatic limit. The last two sections are devoted to the principle of maximum entropy production. There, the Smoluchowski-Poisson equation (1.8) is re-formulated with its relatives in higher space dimensions.

10.1 Kramers-Moyal Expansion The first equation of (1.8) is called the Smoluchowski equation. Here we describe the modeling of semi-conductor physics. First, master equation describes the particle transport using the conditional probability. Second, the Kramers equation is obtained by the Kramers-Moyal expansion. Third, the adiabatic limit of this equation is called the Smoluchowski equation, whereby the conditional probability limit casts the particle distribution. The last drift-diffusion equation takes the form used in mathematical biology in the context of chemotaxis [5, 168, 290, 292, 346, 349]. Here we assume the space dimension equal to 1 to make the description simple. Master Equation Let P(x2 , t2 | x1 , t1 ) be the conditional existence probability of a particle at the position x = x2 and the time t = t2 that was at x = x1 for t = t1 . Then it holds that P(x2 , t1 + t | x1 , t1 ) = Fδ(x2 − x1 ) + t · W (x1 → x2 )

© Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_10

(10.1)

297

298

10 Particle Kinetics

for 0 < t  1, where the first and the second terms of the right-hand side describe the state variation and the transient probability, respectively. Using  d x2 P(x2 , t2 | x1 , t1 ) = 1, we obtain  F + t

d x  W (x1 → x  ) = 1

and, therefore, 



P(x2 , t1 + t | x1 , t1 ) = δ(x2 − x1 ) 1 − t







d x W (x1 − x )

+t · W (x1 → x2 ).

(10.2)

Now, we apply Chapman-Kolmogorov’s equation  P(x3 , t3 | x1 , t1 ) =

P(x3 , t3 | x2 , t2 )P(x2 , t2 | x1 , t1 ) d x2

to (10.2), and then it follows that P(x3 , t2 + t | x1 , t1 )     = d x2 δ(x3 − x2 ){1 − t d x  W (x2 → x  )} + t · W (x2 → x3 )    ·P(x2 , t2 | x1 , t1 ) = 1 − t d x  W (x3 − x  ) P(x3 , t2 | x1 , t1 )  + d x2 W (x2 → x3 )P(x2 , t2 | x1 , t1 )t.

This equality implies 1 [P(x3 , t2 + t | x1 , t1 ) − P(x3 , t2 | x1 , t1 )] t   = − d x  · W (x3 → x  )P(x3 , t2 | x1 , t1 ) + d x2 W (x2 → x3 )P(x2 , t2 | x1 , t1 ),

and, therefore, ∂ P(x, t | x1 , t1 ) = − ∂t +

 

d x  W (x → x  )P(x, t | x1 , t1 ) d x  W (x  → x)P(x  , t | x1 , t1 ).

(10.3)

10.1 Kramers-Moyal Expansion

299

This formula is called the master equation, and describes the time variation of the existence probability P controlled by the transient probability W , see [375]. Moment Expansion The right-hand side on (10.3) is described by  −

dy W (x → (x + y))P(x, t | x1 , t1 )  + dy W ((x − y) → x)P(x − y, t | x1 , t1 ).

Here we apply the formal Taylor’s expansion, f (x + a) =

∞  ak k=0

k!

  ∂ k f (x) = exp(a∂x ) f (x),

to the second term. Thus it holds that  dy W ((x − y) → x)P(x − y, t | x1 , t1 )    = dy exp(−y∂x ) W (x → (x + y))P(x, t | x1 , t1 ), and hence ∂ P(x, t | x1 , t1 ) ∂t    = − dy 1 − exp(−y∂x ) W (x → (x + y))P(x, t | x1 , t1 )  =

dy

∞  1 (−y∂x )k W (x → (x + y))P(x, t | x1 , t1 ) k! k=1

∞  1 = (−∂x )k Ck (x)P(x, t | x1 , t1 ) k! k=1

for  Ck (x) =

W (x → (x + y))y k dy.

Equality (10.4) is called the Kramers-Moyal expansion.

(10.4)

300

10 Particle Kinetics

Now we apply (10.2) and obtain  P(x + y, t + t | x, t)y k dy     δ(y){1 − t d x  W (x → x  )} + t · W (x → (x + y)) y k dy =  = t W (x → (x + y))y k dy by k ≥ 1. Thus it holds that  1 P(x + y, t + t | x, t)y k dy t↓0 t  1 = lim [P(x + y, t + t | x, t) − P(x + y, t | x, t)] y k dy, t↓0 t

Ck (x) = lim

and henceforth,  [P(x + y, t + t | x, t) − P(x + y, t | x, t)] y k dy is denoted by 

[x(t + t) − x(t)]k

x(t)=x

because it indicates the mean value of the k-th moment constrained by x(t) = x. Thus we obtain 1  . [x(t + t) − x(t)]k t↓0 t x(t)=x

Ck (x) = lim

(10.5)

Kramers Equation The Langevin equation describes the motion of a particle subject to the fluctuationfriction and is formulated by dv dx = v, m = −mγv + R(t) + m F(x), dt dt

(10.6)

see [91], where m, γ, R(t), and F(x) denote the mass, the friction coefficient, the fluctuation, and the outer force, respectively. Then we take the Kramers-Moyal expansion of the master equation concering the conditional probability P(x, v, t | x1 , v1 , t1 ). This expansion (10.4) and (10.5) now reads as

10.1 Kramers-Moyal Expansion

301

∂ P(x, v, t | x1 , v1 , t1 ) ∂t ∞

 1 [(−∂x )k , (−∂v )k ] Ck (x, v)P(x, v, t | x1 , v1 , t1 ) = k! k=1 1  |x(t + t) − x(t)|k , Ck (x, v) = lim t↓0 t 1  |v(t + t) − v(t)|k . lim t↓0 t

(10.7)

Here, we replace (10.6) by v(t) = v(s)e

−γ(t−s)

 +

t

e

−γ(t−τ )



s



t

x(t) = x(s) +

 R(τ ) + F(x(τ )) dτ m

v(τ )dτ , t ≥ s,

s

and then it follows that 1 x(t + t) − x(t) v(t)=v, x(t)=x = v t↓0 t 1  lim =0 [x(t + t) − x(t)]2 t↓0 t v(t)=v, x(t)=x lim

(10.8)

from the second equation. Next, it holds that  v(t + t) = (1 − γt)v(t) + t

t+t

e−γ(t+t−τ )

R(τ ) dτ m

+F(x(t))t + o(t), or v(t + t) − v(t) = (−γv(t) + F(x(t))) t  t+t R(τ ) dτ + o(t) e−γ(t+t−τ ) + m t

(10.9)

by the first equation. If the fluctuation is a white noise, we have R(τ ) = 0, and, therefore, 1 v(t + t) − v(t) v(t)=v, x(t)=x = −γv + F(x). t↓0 t lim

(10.10)

302

10 Particle Kinetics

Similarly, we obtain  [v(t + t) − v(t)]

2

= 2(γv(t) + F(x(t)))t ·   + 

t+t

e−γ(t+t−τ )

t

t+t

t

e

−γ(t+t−τ )

2 R(τ )  dτ  + o(t) m

R(τ ) dτ m

and hence 1  [v(t + t) − v(t)]2 t↓0 t v(t)=v, x(t)=x  t+t  t+t 1 1 = lim e−γ(2t+2t−t1 −t2 ) R(t1 )R(t2 ) dt1 dt2 . · t↓0 t m 2 t t lim

Since the correlation of the white noise R(t) is defined by R(t1 )R(t2 ) = 2Dδ(t1 − t2 ) with the diffusion coefficient D > 0, it follows that 1 1 · t↓0 t m 2



t+t



lim

t

t+t

e−γ(2t+2t−t1 −t2 ) R(t1 )R(t2 ) dt1 dt2 =

t

2D . m2

We thus obtain lim

t↓0

1 2D |v(t + t) − v(t)|2 v(t)=v,x(t)=x = 2 , t m

(10.11)

and, therefore,   C1 (x, v) = (v, −γv + F(x)) , C2 (x, v) = 0, 2D/m 2 by (10.8), (10.10), and (10.11). Hence (10.4) is reduced to the Kramers equation,   ∂ ∂ ∂ D ∂2 P(x, v, t | x1 , v1 , t1 ) = − v + (−F(x) + γv) + 2 2 ∂t ∂x ∂v m ∂v (10.12) ·P(x, v, t | x1 , v1 , t) by neglecting the higher terms of k ≥ 3. This equation is called the Fokker-Planck equation if it is uniform in x:   ∂ D ∂ ∂P = −F + γv + 2 P. ∂t ∂v m ∂v

(10.13)

10.1 Kramers-Moyal Expansion

303

Smoluchowski Equation If K and T denote the force and the temperature, respectively, then we obtain F = and D = mkT . In this case, the Kramers equation (10.12) is described by

K m



kT ∂ P ∂P K ∂P ∂P ∂ vP + =− −v +γ ∂t m ∂v ∂x ∂v m ∂v

∂ kT ∂ P K kT ∂ P kT ∂ 2 P ∂P =γ vP + − P+ − −v ∂v m ∂v γm γm ∂x m ∂v∂x ∂x



1 ∂ kT ∂ P K kT ∂ P ∂ − vP + − P+ =γ ∂v γ ∂x m ∂v γm γm ∂x

K kT ∂ P ∂ P− . − ∂x γm γm ∂x Using q=x+

v , γ

p = v,

we obtain ∂ 1 ∂ ∂ = − , ∂v γ ∂q ∂p

∂ ∂ = , ∂x ∂q

and, therefore,

∂P K ∂A ∂ kT ∂ P = −γ − P− ∂t ∂p ∂q γm γm ∂q K kT ∂ P kT ∂ P − P+ . A = vP + m ∂v γm γm ∂x Assuming

p P q − , p, t dp ∼ f (x, t)|x=q− p γ γ



 p p P q − , p, t dp ∼ K (x) f (x, t)|x=q− p , K q− γ γ γ 

now we obtain the Smoluchowski equation 1 ∂ ∂f = ∂t γm ∂x  for f (x, t) =

P(x, v, t)dv.

−K f + kT

∂f ∂x

(10.14)

304

10 Particle Kinetics

10.2 Kinetic Model The Kramers equation (10.12) can describe the motion of a mean field with many particles. First, in a system of particles xi = xi (t), 1 ≤ i ≤ N of identical mass m > 0, let K = K (x, x  ) be the potential of internal force at x created by a particle at x  . Here, K :  ×  → R is a measurable function satisfying K (x, x  ) = K (x  , x) by the action-reaction law. Then it follows that ⎤ ⎡  d xi dvi = vi , m = −m∇xi ⎣V (xi ) + m K (x j , xi )⎦ dt dt

(10.15)

j=i

where V = V (x) stands for the potential of the outer force. Let μ0N (d xdv) = m



δxi (0) (d x) ⊗ δvi (0) (dv),

(xi , vi )|t=0 = (xi (0), vi (0))

i

and assume the convergence ∗ − lim μ0N (d xdv) = f 0 (x, v)d xdv, 0 ≤ f 0 = f 0 (x, v) ∈ L 1 (Rn × Rn ) N →∞

under the constant total mass M, that is m = M/N . Then, under the global-in-time existence of the solution to (10.15) for any initial data, we obtain the convergence ∗ − lim μtN (d xdv) = f (x, v, t) d xdv N →∞

0 ≤ f (·, t) = f (x, v, t) ∈ L 1 (Rn × Rn ) for μtN (d xdv) = m



δxi (t) (d x) ⊗ δvi (t) (dv).

i

Furtheremore, there arises the Vlasov equation f t = −∇v · v f + ∇v · ∇x (U + V ) f where f = f (x, v, t) denotes the particle density at the position x, the velocity v, and the time t and  U (x, t) = K (x, x  ) f (x  , v, t)d x  , Rn

see [34, 263].

10.2 Kinetic Model

305

Provided with F = −∇x (U + V ) for   U (x, t) = K (x, x  )ρ(x  , t) d x  , ρ(x, t) = Rn

Rn

f (x, v, t) dv,

(10.16)

the Kramers equation (10.12) takes the form f t = −∇x · v f + ∇v · ∇x (U + V ) f + γ∇v · (v f + ∇v f ) f |t=0 = f 0 (x, v) ≥ 0.

(10.17)

The position-velocity flux in (10.17) is formulated by F=

−v f (∇x (U + V ) + γv) f + γ∇v f

.

Therefore, if (10.17) is taken in (x, v, t) ∈  × Rn × (0, T ), we require f v · ν|∂ = 0

(10.18)

and replace (10.16) by  U (x, t) =



K (x, x  )ρ(x  , t) d x  , ρ(x, t) =

 Rn

f (x, v, t) dv,

(10.19)

where  ⊂ Rn is a bounded domain with smooth boundary ∂, V = V (x), x ∈ , is a smooth function, see [417, 418]. Equation (10.17) with (10.16) is actually derived as the mean field limit of many particles subject to the friction-fluctuation. The assocaited stochastic equation for (10.17) with (10.16) thus takes the form d xi = vi dt dvi = −∇x (Ui (xi , t) + V (xi ))dt − γvi dt + (2γkT )1/2 dW (t)  Ui (·, t) = m K (x j (t), ·) (10.20) j=i

where k, T , γ, and W (t) are the Boltzmann constant, absolute temperature, friction coefficient, and the normalized white noise, respectively. The Kramers equation (10.12) then arises as f t = −∇x · v f + ∇v · (∇x (U + V ) + γv) f + γkT v f with (10.16). Equation (10.17) is now obtained by the scaling, x → x(1/kT )1/2 , V → V /kT , and K → K /kT .

306

10 Particle Kinetics

Adiabatic Limit At the adiabatic limit γ ↑ +∞ we assume

dv = 0 in (10.20) which guarantees dt

1 vi dt = d xi = − ∇x (Ui + V )dt + (2kT /γ)1/2 dW (t). γ Hence we obtain γρt = ∇x · ρ∇x (U + V ) + kT ρ as N → ∞ with (10.16) and  M=

Rn

ρ(x, t) d x.

Putting x → x/(kT )1/2 and t → γt, we reach the generalized SmoluchowskiPoisson equation ρt = ∇x · ρ∇x (U + V ) + ρ in Rn × (0, T )  U (x, t) = K (x, x  )ρ(x  , t) d x  .

(10.21)

Rn

On the bounded domain , (10.21) takes the form ρt = ∇x · ρ∇x (U + V ) + ρ in  × (0, T ) ν · [ρ∇x (U + V ) + ∇x ρ]|∂ = 0  K (x, x  )ρ(x  , t) d x  . U (x, t) = 

(10.22)

Maxwell-Boltzmann Distribution It is expected that the solution f = f (x, v, t) to (10.17) with (10.18) and (10.16) 2 takes the form ρ(x, t)π −n/2 e−v /2 as t ↑ +∞, where ρ = ρ(x, t) is a solution to 2 (10.22) and π −n/2 e−v /2 is called the Maxwell-Boltzmann distribution. Here, we assume that f = f (x, v, t) > 0 is sufficiently regular and satisfies  ×Rn

  (1 + v 2 ) f + f log f (x, v, t) d xdv < +∞.

First, it is easy to see that the total mass conservation  f (·, t)1 =  f 0 1 ≡ M.

(10.23)

10.2 Kinetic Model

307

Second, writing (10.17) as

2 f t = −∇x · v f + ∇v · ∇x (U + V ) + γ∇v log(ev /2 f ) f, we have  ×Rn

  2 f t U + V + log(ev /2 f )

+ [∇x · v f − ∇v · f ∇x (U + V )] (U + V + log(ev   2 2   f ∇v log(ev /2 f ) ≤ 0 = −γ

2 /2

f )) d xdv (10.24)

×Rn

because U = U (x, t) and V = V (x) are independent of v. Here, the second term on the left-hand side of (10.24) is equal to  ×Rn



=

{−v f · ∇x (U + V + log f + f ∇x (U + V ) · ∇v log(ev

×Rn

2 /2

f )}

−v · ∇x f + ∇x (U + V ) · ∇v f d xdv.

The first term on the left-hand side of (10.24), on the other hand, is nothing but 

1 f t log f + f t v 2 + f t U + f t V d xdv 2   d 1 2 = f log f + v f + V f d xdv + f t U, dt 2 ×Rn ×Rn ×Rn

while we obtain  ft U ×Rn   · K (x, x  ) f t (x, v, t) f (x  , v  , t) d xd x  dvdv  = ×Rn ×Rn   1 d = · K (x, x  ) f (x, v, t) f (x  , v  , t) d xd x  dvdv  2 dt ×Rn ×Rn  1 d = U f. 2 dt ×Rn Thus,  F( f ) =

×Rn

1 1 f (log f − 1) + v 2 f + V f + U f d xdv 2 2

308

10 Particle Kinetics

acts as the free energy, and it follows that d F( f ) = −γ dt

 ×Rn

f |∇v (ev

2 /2

f )|2 ≤ 0.

(10.25)

If the equality holds on the right-hand side of (10.25) then it follows that ev

2 /2

f (x, v, t) = π −n/2 f s (x, t)

with f s = f s (x, t) independent of v. Then it follows that v f + ∇v f = 0 and (10.17) is reduced to   f ts = −v ∇x f s + f s ∇x (U + V ) .

(10.26)

Regarding (10.23), we define the stationary state by  f s = f s (x) > 0,



f s = M, log f s + U + V = constant,

that is the generalized Boltzmann-Poisson equation e−U −V , V (x) = −U −V e

fs = M

 

K (x, x  ) f s (x  ) d x  in .

Global-in-time existence of the weak solution to (10.17) with (10.16) on Rn × R, n ≤ 3, is known to K (x, x  ) = (x −x  ), where  = (x) stands for the fundamental solution to −, see [405]. Fokker-Planck-Boltzmann Equation The Boltzmann equation is the description of the motion of ideal gasses in the kinetic level. Using the wave number ξ in stead of the velocity v as the independent variable it takes the form f t + ξ · ∇ f = Q( f, f ),

f |t=0 = f 0 (x, ξ) ≥ 0.

(10.27)

Hence the friction and the potential terms in (10.17) disappear, while Q( f, f ) stands for the collision term,

10.2 Kinetic Model

309

 Q[ϕ, ϕ](ξ) =

Rn

 dξ∗

 S n−1

 ϕ(ξ  )ϕ(ξ∗ ) − ϕ(ξ)ϕ(ξ∗ ) B(ξ − ξ∗ , ω) dω

ξ  = ξ  (ω, ξ, ξ∗ ) ≡ ξ − [(ξ − ξ∗ ) · ω]ω ξ∗ = ξ∗ (ω, ξ, ξ∗ ) ≡ ξ + [(ξ − ξ∗ ) · ω]ω where S n−1 = {ω ∈ Rn | |ω| = 1} and 1 (Rn × S n−1 ), 0 ≤ B = B(z, ω) ∈ L loc

B = B(|z|, |z · ω|)

denotes the collision kernel. The Fokker-Planck perturbation of (10.27) is now given by f t + ξ · ∇x f − νξ f = Q( f, f ),

f |t=0 = f 0 (x, ξ) ≥ 0

(10.28)

where ν > 0 is a constant. Then we have a weak solution, called the renormalized solution, global-in-time. The following formal calculations provide with several a priori estimates of the approximate solution [90]. First, total mass conservation is valid as  d f d xdξ = 0. (10.29) dt Rn ×Rn Second, for ψ = ψ(ξ) ∈ C0 (Rn ) it holds that    1 ψ Q[ϕ, ϕ]dξ = dξdξ∗ {ϕ(ξ  )ϕ(ξ∗ ) − ϕ(ξ)ϕ(ξ∗ )} n n n n−1 4 R R ×R S ·{ψ(ξ) + ψ(ξ∗ ) − ψ(ξ  ) − ψ(ξ∗ )} B(ξ − ξ∗ , ω) dω. (10.30) Extending (10.30) to ψ = |ξ − a|2 , a ∈ Rn , we obtain  Rn

|ξ − a|2 Q[ϕ, ϕ] dξ = 0

which implies d dt

 Rn ×Rn

|ξ − a|2 f d xdξ

 |ξ − a|2 (ξ · ∇x f − νξ f ) d xdξ =− Rn ×Rn  |ξ − a|2 ξ f d xdξ =ν Rn ×Rn  f d xdξ. = 2nν Rn ×Rn

(10.31)

310

10 Particle Kinetics

Next we have   1 (log ϕ) Q[ϕ, ϕ] dξ = − dξ dξ∗ 4 Rn ×Rn Rn    · {ϕ(ξ  )ϕ(ξ∗ ) − ϕ(ξ)ϕ(ξ∗ )} log[ϕ(ξ  )ϕ(ξ∗ )] − log[ϕ(ξ)ϕ(ξ∗ )] S n−1

·B(ξ − ξ∗ , ω) dω ≤ 0

(10.32)

for 0 < ϕ = ϕ(ξ) ∈ C(Rn ) decaying sufficiently rapid at the infinity. Hence the H theorem arises as   d f (log f − 1) d xdξ = f t log f d xdξ dt Rn ×Rn Rn ×Rn  (ξ · ∇x f − νξ f ) log f d xdξ ≤− n n   R ×R (ξ f ) log f d xdξ =ν n n R ×R f −1 |∇ξ f |2 d xdξ ≤ 0. (10.33) = −ν Rn ×Rn

Finally, recalling (10.32), we have d dt



 Rn ×Rn

|x|2 f d xdξ = −





=

f ∇x · ξ|x| d xdξ = 2

|x|2 (ξ · ∇x f − νξ f ) d xdξ

2

Rn ×Rn

  ≤2

Rn ×Rn

Rn ×Rn

|x|2 f d xdξ

Rn ×Rn

1/2   ·

Rn ×Rn

(x · ξ) f d xdξ 1/2 |ξ|2 f d xdξ

and hence  sup

t∈[0,T )

|x|2 f d xdξ ≤ C(1 + T )

Rn ×Rn

by (10.31). Boltzmann Equilibrium Revisited Here we develop a formal argument for (10.27). If f = f (ξ, t) is independent of x, the H theorem is described by the non-increasing of  H( f ) =

Rn

f (log f − 1) dξ.

10.2 Kinetic Model

311

By (10.31) with ν = 0, we have f = f (·, t) ∈ X 0⊥ in L 2 (Rn ), where X 0 denotes the set of polynomials with the degree up to the quadratic. Since the orbit runs on X 0⊥ , the equilibrium f = f (ξ) > 0 is formulated by  δ H ( f )[g] =

Rn

g log f dξ = 0, ∀g ∈ X 0⊥

which means log f ∈ X 0 . Hence we obtain the Maxwell-Boltzmann distribution f ∞ (ξ) = M

1 4πs

n/2

  exp −|ξ − a|2 /4s

(10.34)

with M > 0, a ∈ Rn , and s > 0 determined by 

 M=

Rn

f 0 dξ,

Rn

 |ξ − a|2 f ∞ dξ =

Rn

|ξ − a|2 f 0 dξ,

using the initial distribution f 0 = f 0 (ξ). Then its linearized stability arises as    d2  H ( f + sg) = f −1 g 2 dξ > 0.  n ds 2 R s=0

10.3 Maximum Entropy Production The Smoluchowski-Poisson equation (1.8) arises also in non-equilibrium statistical mechanics in accordance with the Boltzmann entropy [56]. A relative model is the degenerate parabolic equation of which stationary state is provided with the quantized blowup mechanism [376]. There, the parabolic-elliptic system μt = ∇[D∗ · (∇ p + μ∇ϕ)], ϕ = μ

in  × (0, T )

(10.35)

is derived as the hydrodynamical limit of self-gravitating particles. Here, μ = μ(x, t) ≥ 0 is the function describing particle density at (x, t) ∈  × (0, T ),  ⊂ Rn , n ≥ 2, a domain, ϕ = ϕ(x, t) the gravitational potential generated by μ, and p ≥ 0 the pressure determined by the density-pressure relation p = p(μ, θ). If  has the boundary ∂, the null-flux boundary condition (∇ p + μ∇ϕ) · ν = 0

(10.36)

312

10 Particle Kinetics

is imposed with ν denoting the outer unit normal vector so that the total mass  λ=



μ(x, t)d x

is conserved during the evolution. In fact, the density of particles at (x, t) ∈  × (0, T ) moving at the velocity v is denoted by 0 ≤ f = f (x, v, t). It is subject to the transport equation f t + v · ∇x f − ∇ϕ · ∇v f = −∇v · j

(10.37)

with the general dissipation flux term −∇v · j. This flux term is determined by the maximum entropy production principle, so that f maximizes the local entropy  S=

Rn

s( f (x, v, t))dv under the constraint  μ(x, t) =

Rn

f (x, v, t)dv,

p(x, t) =

1 n

 Rn

|v|2 f (x, v, t)dv.

Averaging f over the velocities v ∈ Rn , and then the passage to the limit of large friction or large time lead to the first equation of (10.35) in the (x, t) space, see [56]. We have, thus, several mean field equations according to the entropy function s( f ) subject to the law of partition of macroscopic states of particles into mezoscopic states, that is the entropies of Boltzmann, Fermi-Dirac, Bose-Einstein, and so forth. System (10.35) with (10.36) is still under-determined, and there are several theories to prescribe the temperature θ. In the cannonical statistics one takes the iso-thermal setting, and hence the temperature θ > 0 is a constant. In the microcannonical statistics, on the other hand, θ = θ(t) > 0 is the function of t, where   1 n pd x + μϕd x E= 2  2  is the prescribed total energy independent of t. First, the transport equation (10.37) is obtained by putting γ = 0 and F = ∇x ϕ in the Kramers equation (10.12). The diffusion coefficient D, furthermore, vanishes here, which is compensated by the general dissipation flux term −∇v · j. Turning to the physical quantities, the total mass is given by  M=

 ρ d x, ρ =

f dv.

(10.38)

The total energy is the sum of kinetic and potential energies, that is E=

1 2

 f |v|2 +

1 2

 ρϕ =

1 2

 f (|v|2 + ϕ) d xdv.

(10.39)

10.3 Maximum Entropy Production

313

The entropy is defined, using the entropy density s = s( f ), by  S= Putting U =

s( f ) d xdv.

(10.40)

v ∇x , we obtain the energy production as ,∇ = ∇v −∇ϕ

 dE 1 E˙ = = f t (|v|2 + ϕ) + f ϕt d xdv dt 2   1 1 = (−∇v · j)(|v|2 + ϕ) − (U · ∇ f )(|v|2 + ϕ) − f ϕt d xdv 2 2  1 = (−∇v · j)|v|2 − (v · ∇x f )ϕ + (∇x ϕ · ∇v f )|v|2 + f ϕt d xdv 2 by (10.39). Since (v · ∇x f )ϕ − (∇x ϕ · ∇v f )|v|2 = ∇x · (v f ϕ) − ∇x ϕ · (v f + |v|2 ∇v f ) = ∇x · (v f ϕ) − ∇x ϕ · ∇v · ((v ⊗ v) f ) it holds that E˙ =

 j ·v+

1 f ϕt d xdv. 2

Similarly, (10.40) implies the entropy production,   dS S˙ = = s ( f ) ft = s  ( f )[−v · ∇x f + ∇x ϕ · ∇v f − ∇v · j] dt  = s  ( f ) j · f − v · ∇x s( f ) + ∇x ϕ · ∇v s( f ) d xdv  = s  ( f ) j · f. The maximum entropy production principle says that j is selected to maximize S˙ | j|2 , furthermore, under the constraint of E˙ = 0. Here we impose the boundedness of 2f to get the Lagrange multiplies β(t) and D such that δ S˙ − β(t)δ E˙ −



2 1 | j| ·δ = 0. D 2f

(10.41)

314

10 Particle Kinetics

  Equation (10.41) means j = D f s  ( f )∇v f − β(t) f v and then it follows that   f t + v · ∇x f − ∇x ϕ · ∇v f = ∇v · D − f s  ( f )∇v f + β(t) f v

(10.42)

from (10.37). Taking the canonical ensemble to see the mean field of particles, we just put β(t) in (10.42) to be a constant. Thus it is actually a function of t if the micro-canonical ensemble is used, which is to be determined by (10.42) and (10.39) with given E. The Kramers equation (10.12) is nothing but (10.42) in the canonical setting, using the Boltzmann entropy s( f ) = − f (log f − 1). Then the fundamental equations of fluid dynamics are obtained by several moments of the general Kramers equation (10.42) in the canoninal setting. First, the zero moment equation is derive just by an integration in v. Using the particledensity ρ defined by the second equality of (10.38) and the particle velocity 1 f v dv, then we obtain the equation of continuity, u= ρ ∂ρ + ∇ · ρu = 0. ∂t

(10.43)

Next, the  first moment arises with the relative velocity w = u − v and stress tensor Pi j =

f wi w j dv. In fact, we multiply v to (10.42) get ∂ ∂ ∂ ∂ϕ (ρu i u j ) + Pi j + ρ (ρu i ) + ∂t ∂x j ∂x j ∂xi  ∂f =D f s  ( f ) + β f vi dv. ∂vi

(10.44)

Using A ( f ) = f s  ( f ), here we obtain   ∂f ∂  fs (f) dv = A( f ) dv = 0, ∂vi ∂vi and, therefore, (10.44) is reduced to the equation of motion, ∂ ∂ ∂ ∂ϕ (ρu i ) + (ρu i u j ) + Pi j + ρ = −Dβρu i . ∂t ∂x j ∂x j ∂xi

(10.45)

If the gasses are under the thermal equilibrium, the right-hand side on (10.45) vanishes. If the fluid is incompressible and isotropic, furthermore, then ρ is a constant and it holds that (10.46) Pi j = pδi j .

10.3 Maximum Entropy Production

315

By (10.43) and (10.46) we obtain (9.5), the Euler equation of motion. Then equation of energy balance arises with the second moment, that is ρ

∂e + p∇ · u = 3Dρ, ∂t

p=

2 1 ρe = 3 3

 f |w|2 dv,

see [58]. If ρ is unknown, we require one more equation other than (10.43) and (10.45). To this end we use the free energy minimum principle at each time. For the moment we drop the variable t. Here we use the Helmholz free energy valid to the canonical ensemble,  1 F = E −TS = (|v|2 + ϕ) f − T s( f ) d xdv (10.47) 2 where T denotes the temperature. Comparing (10.47) with (10.41), we regard β = T −1 as the inverse temperature. At the local  thermal equilibrium, the minimum of f d xdv = λ. Hence this f satisfies

(10.47) is attained under the constraint of the Euler-Lagrange equation s( f ) = β



|v|2 + λ(x) 2

with the Lagrangian multiplier λ(x). Consequently, it follows that 2

|v| f =A β + λ(x) 2

(10.48)

where f = A(σ) denotes the inverse function of σ = s  ( f ). The density ρ and the pressure p satisfy  ρ=

f dv,

1 p= 3



1 f |w| dv, u = ρ 2

 f v dv

with w = u − v, that is  ρ=

f dv,

1 p= 3



2   1  f |v| dv −  f v dv  . 3 2

(10.49)

From (10.48) and (10.49) we obtain a functional relation between ρ and p, that is the state equation indicated by p = p(ρ). (10.50) Thus this fluid is barotropic.

316

10 Particle Kinetics

If the case of isotropic (10.46), Eqs. (10.43)–(10.45) are reduced to Du ∂ρ + ∇ · ρu = 0, ρ = −∇ p − ρ∇ϕ − ξρu ∂t Dt where ξ = Dβ and

(10.51)

∂ D = + u · ∇. In the thermal equilibrium, we obtain Dt ∂t

Du ∂ρ + ∇ · ρu = 0, ρ = −∇ p − ρ∇ϕ, ∂t Dt

p = p(ρ)

(10.52)

by (10.51) and (10.50). System (10.52) is nothing but the compressible Euler equation. In the high friction limit the right-hand side on the second equation (10.51) is zero, which results in the generalized Smoluchowski equation 1 ∂ρ = ∇ · [∇ p + ρ∇ϕ] . ∂t ξ

(10.53)

The state equation (10.50) is determined by (10.48) and (10.49), depending on the choice of the entropy density s = s( f ). It may be the entropies of Boltzman, Fermi-Dirac, Bose-Einstein, and so forth. In the case of the Boltzmann entropy s( f ) = −k f (log f − 1)

(10.54)

Equation (10.50) results in the Boyle-Charles’ law p = kρT,

(10.55)

see [56]. Then the standard Smoluchowski equation arises with (10.53) and (10.55), that is ∂ρ = ∇ · (∇ρ + ρ∇ϕ) ∂t if all the physical constants are 1. The Tsallis entropy is the q-analogue of the Boltz−1 mann entropy, given by s( f ) = ( f q − f ). Particles composing a polytropic q −1 celestial body are subject to this entropy. Then the state equation (10.50) becomes p = K ρ1+γ ,

1 n 1 = + γ q −1 2

(10.56)

10.3 Maximum Entropy Production

317

where K is a constant [56, 57] and n = 3 is the space dimension. From (10.56) and (10.53) we obtain ∂ρ = ∇ · (∇ρm + ρ∇ϕ), ∂t

1 n 1 = + , m−1 q −1 2

(10.57)

putting all the physical constants to be 1.

10.4 Summary We have studied the molecular kinetics using Kramers-Moyal expansion, kinetic theory, and maximum entropy production. 1. Particle equation of the system of chemotaxis is associated with the molecular transport, and non-equilibrium mean field equations are derived from the master equation. 2. The Kramers equation is reduced to several kinetic models and then the Smoluchowski-Poisson equation arises as the adiabatic limit. 3. The Fokker-Planck-Boltzmann equation is an adiabatic limit of the kinetic equation. Then we have global-in-time existence of the renormalized solution. 4. There is an equivalent mezoscopic approach of statistical mechanics using several entropies to describe the motion of the mean field which results in the state equation.

Chapter 11

Parabolic Equations

A degenerate parabolic equation arises in the context of astrophysics whereby the critical exponent is detected from the scaling invariance of the model compatible to the total mass conservation. We describe several fundamental properties of parabolic equations. Then we turn to the variational and scaling structures of this degenerate parabolic equation in accordance with the Tsallis entropy. There arise ε-regularity and the bounded variation in time of the local mass, which results in a structure theorem of the blowup set.

11.1 Semilinear Equations The semilinear parabolic equation with power nonlinearity u t − u = u p ,

u > 0 in Rn × (0, T )

(11.1)

was introduce as a toy model of the incompressible Navier-Stokes equation vt + (v · ∇)v − v = −∇ p, ∇ · v = 0 in Rn × (0, T ).

(11.2)

Scaling and variational structures provided with these equations are used in the mathematical analysis, see, for example [128, 129], concerning (11.2). Equation (11.1), however, is provided with the comparison principle, which clarifies the study on critical exponents, blowup in finite time, blowup in infinite time, blowup rate, blowup profile, complete blowup in accordance with the post-blowup continuation, and so forth [310, 317].

© Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_11

319

320

11 Parabolic Equations

The Cauchy problem to (11.1) is well-posed local-in-time under the appropriate decay of u at x = ∞, and, henceforth, T = Tmax ∈ (0, +∞] denotes the maximum existence time. The case T < +∞ is called the blowup of the solution because then it follows that lim u(·, t)∞ = +∞. t↑T

In 1966, Fujita [123] formulated his exponent in comparison with the ODE part u t = u p , u > 0, which is summarized as follows, see [160, 201] for the critical case p = 1 + n2 . 1. If 1 < p ≤ 1 + n2 , then any non-trivial solution blows-up in finite time. 2. If p > 1 + n2 , then there is a non-trivial solution with T = +∞. Forward Self-similar Transformation The scaling invariance u μ (x, t) = μ2/( p−1) u(μx, μ2 t), μ > 0 of u t − u = u p , u > 0 in Rn × (0, +∞)

(11.3)

induces the forward self-similar transformation 1

v(y, s) = (t + 1) p−1 u(x, t),

y = x/(t + 1)1/2 , s = log(t + 1),

and then it follows that vs − v =

v y · ∇v + + v p , v > 0 in Rn × (0, +∞). 2 p−1

(11.4)

The stationary solution to the rescaled equation (11.4) is called the self-similar solution. If 1 + n2 < p < n+2 n−2 , the solution to (11.3) blows-up in finite time or converges N

to 0 in infinite time with the rate either of the Gauss kernel O(t − 2 ) or that of the − 1 − 1 self-similar solution O(t p−1 ). The latter case of u(·, t)∞ = O(t p−1 ) arises in accordance with the threshold modulus so that this behavior is an exceptional case [265]. In this threshold modulus case, the rescaled solution v = v(·, s) converges to the stationary solution to (11.4), that is the self-similar solution, as s ↑ +∞, see [196]. What happens to p = n+2 n−2 , n ≥ 3, is also studied [176].

11.1 Semilinear Equations

321

We have Lv ≡ −v −

y 1 · ∇v = − ∇ · (K ∇v) 2 K

  for K = K (y) = exp |y|2 /4 , and this L is realized as a self-adjoint operator in L 2 (K ) = L 2 (Rn , K (y)dy) with the domain   H 2 (K ) = v ∈ L 2 (K ) | D α v ∈ L 2 (K ), |α| ≤ 2 . Its first eigenvalue is n/2. Using this value and Kaplan’s argument of taking the L 2 inner product between the rescaled solution v = v(·, s) and the first eigenfunction ϕ1 = ϕ1 (y) > 0 of L normalized by ϕ1 1 = 1, we can provide an alternative proof of the result [123] described above; any non-trivial solution to (11.3) blows-up in finite time if 1 < p ≤ p f , see [96, 195]. The exponent ps = n+2 n−2 , n ≥ 3 is called the Sobolev exponent. The non-trivial stationary solution to the prescaled equation, that is − u = u p ,

u > 0 in Rn

(11.5)

exists if and only if p ≥ ps , see also (7.67). It p f < p < ps , there is a unique positive self-similar solution, that is the stationary solution to (11.4), with appropriate decay at x = ∞, while there is no positive self-similar solution in the case of p ≥ ps . Several other critical expontes have been recognized in connection with the blowup profile, blowup rate, and complete blowup of the solution, see [114] and the references therein. Backward Self-similar Transformation There are vast references concerning the blowup mechanism of the solution to u t − u = u p , u > 0 in  × (0, T ),

u|∂ = 0,

(11.6)

where  ⊂ Rn is a bounded domain with smooth boundary ∂. The main difference of (11.6) between (11.1) is the appearance of the Poincaré inequality, λ1 u22 ≤ ∇u22 , where λ1 = λ1 () > 0 denotes the first eigenvalue of − D : −ϕ1 = λ1 ϕ1 , ϕ1 > 0 in ,

ϕ1 |∂ = 0.

Role of L 2 -norm in the global-in-time behavior of the solution, actually, arises with the variational structure formulated by

322

11 Parabolic Equations

d J (u) = − u t 22 , dt

1 d u22 = −I (u), 2 dt

where J (u) =

1 1 p+1 ∇u22 − u p+1 , 2 p+1

p+1

I (u) = ∇u22 − u p+1 .

A remarkable implication of this fact is T = +∞



sup u(·, t)2 < +∞ t≥0

for any p > 1, see [50, 288]. The comparison principle is also useful, which results in the properties of the (strongly) order preserving and the parabolic Liouville property or the intersection comparison principle [127]. The scaling invariance (12.23) now induces the backward self-similar transformation defined by 1

v(y, s) = (T − t) p−1 u(x, t),

y = (x − x0 )/(T − t)1/2 , s = − log(T − t),

where T > 0 is the blowup time, that is lim u(·, t)∞ = +∞, t↑T

and x0 ∈  is a blowup point. This relation implies vs − v +

v y · ∇v + = v p in 2 p−1

v=0



es/2 ( − {x0 }) × {s}

s>− log T

on



es/2 (∂ − {x0 }) × {s}

s>− log T

and the limiting equations are vs − v +

v y · ∇v + = v p , v > 0 in Rn × (−∞, +∞) 2 p−1

and vs − v + v|∂R+n = 0

v y n × (−∞, +∞) · ∇v + = v p , v > 0 in R+ 2 p−1

(11.7)

11.1 Semilinear Equations

323

if x0 ∈  and x0 ∈ ∂, respectively. Only the constant v =



1 p−1



1 p−1

is admitted

see [141]. as a bounded stationary solution to (11.7) if 1 < p ≤ If 1 < p < n+2 , the global blowup rate is of type (I), that is n−2 n+2 n−2 ,

1

lim sup(T − t) p−1 u(·, t)∞ < +∞, t↑T

  1 p−1 1 and v(·, s) converges to p−1 , the constant solution to (11.7), locally uniformly in Rn as s ↑ +∞, see [141–143]. In particular, any blowup point is of type (I), and aggregation dominates concentration, where aggregation and concentration indicate the growth of local L 1 norm and that of local L ∞ norm, respectively. If  ⊂ Rn , n ≥ 3, is convex, p = n+2 n−2 , T = Tmax < +∞, and lim J (u(·, t)) > −∞, t↑T

(11.8)

the blowup rate is of type (II), 1

lim(T − t) p−1 u(·, t)∞ = +∞. t↑T

The leading blowup profile, furthermore, is the normalized entire stationary solution (11.5), u ∗ = u ∗ (x) > 0, u ∗ (0) = 1, involved by a scaling transformation, and the formation of the second collapse arises at the wedge of the principal parabolic envelope [367]. We obtain, thus, the collision of collapses in this case; compare such profiles to those of the two-dimensional Smoluchowski-Poisson equation (1.127). Actual existence of the solution satisfying (11.8) with T = Tmax < +∞ has not been confirmed, and in this connection, it should be noted that in the sub-critical case 1 < p < n+2 n−2 , it holds always that T < +∞ if and only if lim J (u(·, t)) = −∞, t↑T

see [140, 177]. If  ⊂ Rn , n ≥ 3, is convex and p = there arises the energy quantization T = +∞



n+2 n−2 ,

there is no stationary solution, and

lim J (u(·, t)) = md t↑T

with m = 1, 2, . . ., where d = J (u ∗ ) and u ∗ = u ∗ (x) > 0 is the above described normalized entire stationary solution to (11.5), see [175, 178]. Although (11.5) has a family of solutions in the critical case p = n+2 n−2 , the energy value d = J (u ∗ ) is the same because of the scaling invariance of the problem. This property is actually the origin of the energy quantization to (11.6) with p = n+2 n−2 .

324

11 Parabolic Equations

11.2 Degenerate Equations We take (10.57) on the whole space  = Rn , provided with the gravitational potential ϕ for the self-interaction, that is ϕ = ρ. Normalizing constants again, we arrive at the degenerate parabolic equation ut =

m−1 m u − ∇ · u∇ ∗ u, u ≥ 0 in Rn × (0, T ) m

(11.9)

where

(x) =

1 ωn (n − 2) |x|n−2

(11.10)

with ωn denoting the (n − 1) dimensional volume of the boundary of the unit ball in Rn if n ≥ 3 and 1 1 log 2π |x|

(x) =

(11.11)

if n = 2. Thus, (x) is the fundamental solution to − and

∗ u(x, t) =

(x − x )u(x , t) d x . Rn

If n = 3 and q = 53 in (10.57), we have m = 43 . Equation (11.9) of this exponent, m = 2 − n2 is regarded as a higher-dimensional analogue of the SmoluchowskiPoisson equation associated with the Boltzmann entropy in two-space dimensions, that is (11.12) u t = u − ∇ · u∇ ∗ u, u ≥ 0 in R2 × (0, T ) for = (x) defined by (11.11). Similarly to (1.8), the simplified system of chemotaxis, Eq. (11.12) is provided with the total mass conservation d dt

R2

u=0

(11.13)

and the decrease of the total free energy d F(u) = − u|∇(log u − ∗ u)|2 ≤ 0 2 dt R 1 F(u) = u(log u − 1) −  ∗ u, u. 2 R2

(11.14)

11.2 Degenerate Equations

325

There is also a collapse formation with the quantized mass of the blowup solution in finite time to (11.9), provided that u 0 = u|t=0 ∈ X = L 1 (R2 , (1 + |x|2 )d x) ∩ L ∞ (R2 ) ∩ H 1 (R2 ). In fact, (11.12) is well-posed in this function space X local-in-time, and the viral identity arises as



dI λ2 = 4λ − , dt 2π

I =

|x| u, λ = 2

R2

u.

(11.15)

R2

Relations (11.14)–(11.15) guarantees the blowup threshold λ = 8π . In the case of T = Tmax < +∞, furthermore, it follows that lim sup I (t) < +∞ t↑T

which implies the boundedness of the blowup set in R2 from the ε-regularity and a Chebyshev type inequality. Then we obtain an analogous result of (1.24)–(1.26), so that if T < +∞ then

m(x0 )δx0 (d x) + f (x)d x u(x, t)d x x0 ∈S

in M(R2 ∪ {∞}) as t ↑ T , where S is the blowup set of u, which is finite, m(x0 ) ∈ 8π N, 0 ≤ f = f (x) ∈ L 1 (R2 ) ∩ C(R2 \S), and R2 ∪ {∞} is the one-point compactification of R2 . Equation (11.9) is also a model (B) equation associated with the free energy F(u) =

Rn

1 um −  ∗ u, u . m 2

(11.16)

It is the functional that formulates the equilibrium of the Euler-Poisson equation in Sect. 7.3. In fact, we have

d F(u + sv) δF(u)[v] = = v, u m−1 − ∗ u , ds s=0 where  ,  denotes the L 2 -inner product. Identifying F(u) with u m−1 − ∗ u, we can write (11.9) as 

 m−1 m ∇u − u∇ ∗ u m = ∇ · u∇δF(u) in Rn × (0, T ).

ut = ∇ ·

(11.17)

326

11 Parabolic Equations

From this form of (11.17), it is easy to infer, at least formally, the total mass conservation (11.13), u(t)1 = u 0 1 = λ

(11.18)

and the decrease of the total free energy (11.14), d F(u) = − u |∇δF(u)|2 dt Rn 2 =− u ∇(u m−1 − ∗ u) ≤ 0.

(11.19)

Rn

We have rigorous proof for these properties using weak solutions [354, 356, 379]. Regarding (11.18)–(11.19), we formulate the stationary state by u = λ. (11.20) u m−1 − ∗ u = constant in {u > 0}, Rn

1

Then we assume u = v+m−1 for v = v(x) satisfying q q n − v = v+ in R , v+ = λ,

(11.21)

Rn

where m = 1 + q1 . We obtain, thus, a relative to (7.84) for the bounded domain case, see Sect. 13.2. Problem (11.21) is invariant under the scaling transformation v(x) → vμ (x) = μγ v(μx)

(11.22)

1 n if and only if γ = n − 2 and q = m−1 = n−2 , that is m = 2 − n2 , where μ > 0 is a constant. For this exponent, problem (11.22) admits a family of solutions, each of which is necessarily radially symmetric and has compact support [409]. Then, we recall the results in Sect. 7.3 to define the normalized solution v∗ = v∗ (x) to (11.21) which determines the quantized mass λ∗ > 0 by q q − v∗ = v∗+ , v∗ ≤ v∗ (0) = 1 in Rn , λ∗ = v∗+ . (11.23) Rn

The above solution v∗ = v∗ (x) is unique and radially symmetric. The constant λ∗ = λ∗ (n) > 0 defined by (11.23) is the best constant of Wang-Ye’s dual form of the Trudinger-Moser inequality [379, 409],  m n inf F(u) | 0 ≤ u ∈ L (R ),

Rn

where m = 2 − n2 .

 u = λ∗ > −∞

(11.24)

11.2 Degenerate Equations

327

Two-dimensional analogue of (11.23) is the Boltzmann-Poisson equation − v = ev in R2 ,

R2

ev < +∞

(11.25)

q

to be studied in Sect. 12.7. Differently from (11.25), v+ in (11.23) has a compact support. The other difference is the scaling property of the free energy F(u μ ) = μn−2 F(u)

(11.26)

which refines (11.24) as inf{F(u) | 0 ≤ u ∈ L m (Rn ),

Rn

u = λ∗ } = 0.

(11.27)

The non-stationary problem (11.9), m = 2 − n2 , is also provided with the L 1 preserving self-similar transformation u μ (x, t) = μn u(μx, μn t),

(11.28)

see Sect. 12.1. This transformation induces the backward self-similar transformation v(y, s) = (T − t)u(x, t), y = (x − x0 )/(T − t)1/n , s = − log(T − t) (11.29) and the rescaled equation m−1 m |y|2 v − ∇ · v∇( ∗ v + ) m 2n v ≥ 0 in Rn × (− log T, +∞). vs =

(11.30)

The main difference between 2D Smoluchowski-Poisson equation and higher dimensional degenerate parabolic equation with critical exponent lies on the potential 1 and x · ∇ = −(n − 2) for (x) defined of self-interaction, that is, x · ∇ = − 2π by (11.11) and (11.10), respectively.

11.3 Blowup Threshold Since (11.9) is degenerate, the solution to (11.9) which we handle with is the weak solution [379]. First, given the initial value 1 n 0 ≤ u 0 ∈ L 1 (Rn ) ∩ L ∞ (Rn ), u m 0 ∈ H (R ),

(11.31)

328

11 Parabolic Equations

we take the approximate solution u ε = u ε (x, t) satisfying m−1 (u ε + ε)m − ∇ · (u ε ∇ ∗ u ε ) m = u 0ε , 0 < ε  1,

u εt = u|t=0

in Rn × (0, T )

where n , n + 3] n−1 m u 0ε  p ≤ u 0  p , ∀ p ∈ [1, ∞], ∇u m 0ε 2 ≤ ∇u 0 2 n , ∞). u 0ε → u 0 , ε ↓ 0 in L p (Rn ), ∃p ∈ [ n−1 0 ≤ u 0ε ∈ L 1 ∩ W 2, p (Rn ), ∀ p ∈ [

Then we obtain the following theorem [379], passing through a subsequence of ε ↓ 0. Theorem 11.3.1 Assuming (11.31), we have 0 < T  1 such that (11.9) has a weak solution in the sense that m−1 m u0ξ d x ∇u · ∇ξ − u∇ ∗ u · ∇ξ − uξt d xdt = m Rn ×[0,T ] Rn provided with the properties u ∈ C∗ ([0, T ), L p (Rn )), 1 < p ≤ ∞, regarding

L p (Rn ) = L p (Rn ) , p1 + 1p = 1, ∞ u ∈ L ∞ ([0, T ]; L 1 (Rn )) ∩ L loc ([0, T ); L ∞ (Rn ))

∇u m ∈ L ∞ (0, T ; L 2 (Rn )), ∂t u

m+1 2

∈ L 2 (0, T ; L 2 (Rn ))

∞ ∇ ∗ u ∈ L loc ([0, T ); L 2 (Rn )),

(11.32)

and u(·, t)1 = u 0 1 , t ∈ [0, T ),

(11.33)

where ξ ∈ H 1 (0, T ; L 2 (Rn )) ∩ L 2 (0, T ; H 1 (Rn )) is the test function satisfying ξ(·, t) = 0 for 0 < T − t  1. Furthermore, it holds that u ε u ∗ weakly in L ∞ (0, T ; L q (Rn )), ∀q ∈ (1, ∞]

(11.34)

in the sense of L ∞ (0, T ; L q (Rn )) = L 1 (0, T ; L q (Rn )) , q1 + q1 = 1, passing through a subsequence of the above approximate solutions. If the existence time of the weak solution u denoted by T = Tmax ∈ (0, +∞] is finite, then lim u(·, t)∞ = +∞. t↑T

(11.35)

11.3 Blowup Threshold

329

The above approximate solution u ε = u ε (x, t) is extended as far as u ε (·, t)∞ is bounded, while B ≡ sup u ε (·, t)∞ ≤ t∈(0,T )

u 0 ∞ , 1 − T u 0 ∞

0 < T < u 0 −1 ∞

holds by the L ∞ -energy method [289]. Then we derive several estimates of u ε (·, t) using B uniform in 0 < ε  1 and 0 ≤ t ≤ T to take the process of the passage to the limit. This argument guarantees that the existence time of the weak solution u = u(·, t) is bounded from below by u 0 ∞ , and, consequently, it extends in t as far as u(·, t)∞ is bounded. This scheme is due to [354, 355, 357] where the Bessel potential is used instead of . With the lack of sufficient decay at the infinity of , we use ∇( ∗ u)∞ ≤ C(n, q)(u1 + uq ), n < q ≤ ∞ derived from the decomposition

∗ u = v1 + v2 , v1 = [ · χRn \B(0,1) ] ∗ u, v2 = [ · χ B(0,1) ] ∗ u

(11.36)

and also the Calderón-Zygmund estimate ∇ 2 ( ∗ u) p ≤ C(n, p)u p , 1 < p < ∞. There may be the case that this solution exists after the classical solution expires. We may suspect the profile of such a solution by the example [213] concerning n = 1, m = 3, and the Bessel potential for . Thus given the regular non-negative initial value u 0 = u 0 (x), u 0 ≡ 0 with compact support, we have a unique classical solution local-in-time which breaks down in a finite time Tc satisfying lim ∂x u(·, t)∞ = +∞, lim sup u(·, t) p < +∞, 1 ≤ p ≤ ∞.

t↑Tc

t↑Tc

Henceforth T = Tmax ∈ (0, +∞] denotes the existence time of the weak solution u = u(·, t). We take the case |x|2 u 0 < +∞ (11.37) Rn

to control the behavior of the solution at x = ∞. The next theorem [29, 380] assures the existence of the threshold of λ = u 0 1 for T = +∞. The threshold value λ∗ may be prescribed by the best constant C(n) of the Hardy-Littlewood-Sobolev inequality 2/n

| f, ∗ f | ≤ C(n) f m m  f 1 , m = 2 −

2 . n

(11.38)

330

11 Parabolic Equations

It is actually equal to the one defined by (11.23) for q =

n n−2 .

Theorem 11.3.2 If u 0 = u 0 (x) is the initial value satisfying (11.31), (11.37), and u 0 1 < λ∗ , then T = +∞ holds in (11.9) for m = 2 − n2 , n ≥ 3. Each λ > λ∗ , on the other hand, takes u 0 = u 0 (x) such that (11.31), (11.37), u 0 1 = λ, and T < +∞. The arguments described above and below are justified using the approximate solution. For the proof of Theorem 11.3.2, first, if u 0 1 = λ < λ∗ is the case then lim sup u(·, t)m < +∞

(11.39)

t↑T

holds by (11.19) and (11.27). Now we set up Moser’s iteration scheme which guarantees lim sup u(·, t)∞ < +∞ t↑T

and hence T = +∞ by (11.35). From |x|2 u 0 ∈ L 1 (Rn ), next, the function |x|2 u(·, t) ∈ [0, +∞) t ∈ [0, T ) → Rn

is locally absolutely continuous and it holds that d dt



m−1 |x| u(·, t) = · 2n u m (·, t) − (n − 2)  ∗ u, u m Rn Rn = 2(n − 2)F(u), (11.40) 2

which implies T < +∞ in case F(u 0 ) < 0. Since (11.27) is sharp and inf{F(u) | 0 ≤ u ∈ L (R ), m

n

Rn

u = λ} = −∞

for each λ > λ∗ , we obtain the initial value u 0 = u 0 (x) ≥ 0 with compact support such that F(u 0 ) < 0 and u 0 1 = λ. Theorem 3.2.1 is thus proven.

11.4 Structure of the Blowup Set Since (11.35) arises if T = Tmax < +∞, the blowup set is then defined by S = Rn \ B, B = {x0 ∈ Rn | ∃r > 0 such that lim sup u(·, t) L ∞ (B(x0 ,r )) < +∞}. t↑T

11.4 Structure of the Blowup Set

331

To detect the blowup rate, we write (11.9) as m−1 m u − ∇u · ∇ ∗ u + u 2 , m

ut = to take the ODE part

ζ˙ = ζ 2 . Since ζ (t) = (T − t)−1

(11.41)

we define the type I blowup rate by u(·, t)∞ = O((T − t)−1 ). Then we say that x0 ∈ S is of type I if lim inf (T − t)u(·, t) L ∞ (B(x0 ,r0 )) < +∞, ∃r0 > 0 t↑T

and type II in the other case. We obtain finiteness of the type II blowup points [382]. Theorem 11.4.1 Let u 0 = u 0 (x) be the initial value satisfying (11.31) and (11.37), and assume T < +∞ for the above described weak solution u = u(x, t) to (11.9) with m = 2 − n2 . Then, S is bounded and S I I is finite, where 



S I I = x0 ∈ S | lim(T − t)u(·, t) t↑T

L ∞ (B(x

0 ,r0 ))

= +∞, ∀r0 > 0 .

The first step to prove Theorem 11.4.1 is the ε-regularity [381] which is a localization of the anti-blowup criterion u 0 1 < λ∗ . Theorem 11.4.2 We have ε0 > 0 and C > 0 independent of x0 ∈ Rn and 0 < R  1 such that sup u(·, t) L 1 (B(x0 ,R)) < ε0 ⇒

t∈(0,T )

sup u(·, t) L ∞ (B(x0 ,R/2)) ≤ C

t∈(0,T )

where u = u(x, t) is a weak solution to (11.9). For the proof of Theorem 11.4.2 we use v1 W 1,∞ (Rn ) + v2 W 1,q (Rn ) ≤ C(n, q)u1 , 1 ≤ q
R

u(·, t) ≤ R −2 C(T, u 0 ).

Taking R  1 as C(T, u 0 )R −2 < ε0 , we obtain S ⊂ Rn \ B(0, R) by Theorem 11.4.2.  The constant C in (11.42) is involved by the initial value. This inconvenience, however, is removed by the parabolic regularity concerning local norms of the solution [382]. This property was first noticed to (11.12), see [330].

11.4 Structure of the Blowup Set

333

Theorem 11.4.3 Each r ∈ [2, ∞) admits 0 < εr  1 such that sup u(·, t) L 1 (B(x0 ,2R)) < εr



t∈(0,1)

u(·, t) L r (B(x0 ,R)) ≤ t −1

for 0 < t ≤ 1, where u = u(·, t) is the weak solution to (11.9), x0 ∈ Rn , and R > 0. Given x0 ∈ S and 0 < R  1, we take 0 ≤ ϕ = ϕx0 ,R (x) ∈ C0∞ (Rn ) satisfying supp ϕ ⊂ B(x0 , 2R) and ϕ = 1 on B(x0 , R) and put A(t) =

Rn

ϕu(·, t).

First, it holds that d dt

R

2 ϕu = n

2 m−1 − ∗ u) · ∇ϕ n u∇(u R 2 ≤ u ∇(u m−1 − ∗ u) · u |∇ϕ|2 Rn

Rn

≤ −λ ∇ϕ2∞

d F(u) dt

(11.44)

which means (A )2 ≤ −



∇ϕ2∞ λ

H , 2(n − 2)

H (t) =

Rn

|x|2 u(·, t).

(11.45)

If lim F(u(t)) > −∞ t↑T

(11.46)

is the case, therefore, it follows that

T 0

d dt

R

 ϕu dt ≤ T 1/2 n

T

0

d dt

R

2 1/2 ϕu dt < +∞ n

and hence lim A(t) = lim t↑T

t↑T

Rn

ϕu(·, t)

(11.47)

334

11 Parabolic Equations

exists. Since Lemma 11.4.1 guarantees lim inf A(t) = lim sup A(t) ≥ lim sup u(t) L 1 (B(x0 ,R)) ≥ ε0 , t↑T

t↑T

t↑T

we obtain lim lim inf u(·, t) L 1 (B(x0 ,2R)) ≥ ε0 R↓0

t↑T

for any x0 ∈ S, and hence the finiteness of S by the total mass conservation. In the other case of lim F(u(·, t)) = −∞, t↑T

(11.48)

we have F(u(t0 )) < 0 for some t0 ∈ [0, T ). We may assume t0 = 0 without loss of generality. Equality (11.40) then implies dH ε

u(·, t) = 0

for any ε > 0 which implies S ⊂ {0} by Theorem 11.4.2. Thus we may assume H (T ) > 0 furthermore. Lemma 11.4.2 It holds that A(t ) ≤ A(t) + C(H (t) − H (T ))1/2 .

sup t ∈[t, t+T 2

]

Proof Inequality (11.45) implies t

t

(t − s)A (s)2 ds ≤

∇ϕ2∞ λ (H (t) − H (t )) 2(n − 2)

for 0 ≤ t ≤ t < T by H (t) ≤ 0. Therefore, it holds that

(11.50)

11.4 Structure of the Blowup Set

335

t+t 2 2 t + t 2

A( = A (s)ds ) − A(t) t 2 t+t t 2

−1 ≤ (t − s) ds · (t − s)A (s)2 ds t

log 2 ≤ 2 log 2 ≤ 2

t

∇ϕ2∞ · · λ · (H (t) − H (t )) n−2 ∇ϕ2∞ · · λ · (H (t) − H (T )) n−2

for t ∈ [t, T ). This inequality implies A(

t + t ) ≤ A(t) + C(H (t) − H (T ))1/2 2

for t ∈ [t, T ) and hence (11.50).



Now we use the scaling property (11.28). Lemma 11.4.3 Each r0 > 0 admits t0 ∈ [0, T ) and C > 0 such that u(·, t1 ) L 1 (B(x0 ,r0 )) < ε0 /2

(11.51)

implies sup t∈(t1 + 18 (T −t1 ),t1 + 38 (T −t1 ))

(T − t)u(·, t) L ∞ (B(x0 ,(T −t1 )1/n )) ≤ C

where x0 ∈ Rn and t1 ∈ [t0 , T ).



Proof We have A(t1 ) < ε0 /2 for A(t) = that sup T +t t ∈[t1 , 2 1 ]

(11.52)

Rn

ϕx0 ,r0 u(·, t) by (11.51). Hence it holds

A(t ) < ε0 , t1 ∈ [t0 , T )

(11.53)

for 0 < T − t0  1 by Lemma 11.4.2. Here we use the scaling property (11.28) and take μ > 0 and u(x, ˜ t) by u(x, ˜ t) = μn u(μx + x0 , μn t + t1 ), μn + t1 =

T + t1 . 2

336

11 Parabolic Equations T −t1 2

It holds that μn =

and

m−1 m u˜ − ∇ · (u∇

˜ ∗ u), ˜ u˜ ≥ 0 in Rn × (0, 1) m sup u(·, ˜ t) L 1 (B(0,r0 μ−1 )) < ε0

u˜ t =

(11.54)

t∈(0,1)

by (11.53). Now we use Theorem 11.4.3 and then Moser’s iteration scheme applied to the proof of Theorem 11.4.2. We obtain sup t∈[1/4,3/4]

u(·, ˜ t) L ∞ (B(0,1)) ≤ C1

(11.55)

similary, because r0 μ−1 ≥ 2 holds for 0 < T − t1  1. Inequality (11.55) implies sup t∈(t1 + 18 (T −t1 ),t1 + 38 (T −t1 ))

(T − t1 )u(t) L ∞ (B(x0 ,(T −t1 )1/n )) ≤ C1 . 

Then (11.52) follows for C = 43 C1 . The proof of Theorem 11.4.1 is complete with inf lim lim inf u(·, t) L 1 (B(x0 ,r )) ≥ ε0 /2.

x0 ∈S I I r ↓0

t↑T

(11.56)

In fact, inequality (11.56) implies S I I < +∞ from the total mass conservation of u = u(·, t). Assuming the contrary, we have x0 ∈ S I I , r0 > 0, and t j ↑ T such that u(·, t j ) L 1 (B(x0 ,2r0 )) < ε0 /2,

j = 1, 2, . . . .

Then we obtain sup

y∈B(x0 ,r0 )

u(·, t j ) L 1 (B(y,r0 )) < ε0 /2,

and, therefore, (T − t)u(·, t) L ∞ (B(y,(T −t j )1/n )) ≤ C

sup

t∈(t j + 18 (T −t j ),t j + 38 (T −t j ))

by Lemma 11.4.3, where y ∈ B(x0 , r0 ) is arbitrary. Inequality (11.57) implies sup t∈(t j + 18 (T −t j ),t j + 38 (T −t j ))

(T − t)u(·, t) L ∞ (B(x0 ,r0 )) ≤ C

(11.57)

11.4 Structure of the Blowup Set

337

and hence lim inf (T − t)u(·, t) L ∞ (B(x0 ,r0 )) < +∞, t↑T

that is x0 ∈ S I = S \ S I I , a contradiction.



11.5 Other Properties There is a difference between (11.9) and (11.12) in the time derivatives of the global second moment. The former and the later are associated with the free energy and the total mass, respectively. A technical difficulty in the former is the indefiniteness of the sign of the density of the free energy. We note, however, that the ε-regularity is sufficient to guarantee the uniqueness of the blowup point of radially symmetric solution [356]. This property is similar to the full system of chemotaxis in two space dimension [252, 253]. Taking the weak formulation, d dt



Rn

ϕu =

Rn

m−1 m 1 u ϕ + m 2

Rn ×Rn

ρϕ (x, x )u(x, t)u(x , t)d xd x

with ρϕ (x, x ) = (∇ϕ(x) − ∇ϕ(x )) · (x − x ), we obtain d dt

m−1 2 2 ϕ∞ · um ϕu ≤ m + D ϕ∞ · (n − 2)  ∗ u, u . n m R

Here we have 0

T

u(·, t)m m dt < +∞ ⇔

T

 ∗ u, udt < +∞

(11.58)

0

by (8.27) and (11.19). Therefore, if one of (11.58) is the case the formation of collapse to μ(d x, t) = u(x, t)d x follows similary to (11.12) with S < +∞. We have a formal argument to guarantee this property. Concerning the boundedness of the free energy, first, we note the following property. Theorem 11.5.1 We have F(u(t)) ≥ −C(T − t)−1

(11.59)

with a constant C > 0 independent of t ∈ [0, T ). Proof Formal arguments below are justified by the approximate solution. To begin with, inequality (11.59) is obvious if (11.46). Hence we assume the case (11.48).

338

11 Parabolic Equations

Equality (11.40) now reads; 2(n − 2)F(u) =

d dt

= −2



Rn



Rn

|x|2 u = −

Rn

u∇(u m−1 − ∗ u) · ∇ |x|2

u∇(u m−1 − ∗ u) · x

and then it holds that d dt

Rn

2 |x|2 u ≤ 4

2 m−1 |x|2 u u ∇(u − ∗ u) n n R R d 2 |x| u. = −4 F(u) · dt Rn

We obtain 

dg dt

2 ≤−

1 d2 2 d F(u) = − g , g = H 1/2 , dt 2(n − 2) dt 2

or equivalently, gg

+ (n − 1)(g )2 ≤ 0 which implies d2 gg

− (g )2 log g = ≤ −n dt 2 g2



g g

2

 = −n

d log g dt

or −

d d h ≤ −nh 2 , h = − log g > 0. dt dt

Here we recall (11.49). Then it follows that d −1 h ≤ −n < 0, dt and there exists h(T ) = lim h(t) ∈ (0, +∞] such that t↑T

h(T )−1 − h(t)−1 ≤ −n(T − t), t ∈ [0, T ).

2 ,

11.5 Other Properties

339

Neglecting h(T )−1 ∈ [0, +∞), we obtain h(t)−1 ≥ n(T − t), t ∈ [0, T ), and then it follows that h(t) ≤

1 d 1 =− log {T − t} n(T − t) n dt

or   d H (t) ≥ 0. log dt (T − t)2/n

(11.60) 

Now we obtain (11.40) by (8.27). Condition (11.58) for the formation of collapse u(x, t)d x



m(x0 )δx0 (d x) + f (x)d x in M(Rn ) as t ↑ T

x0 ∈S

S < +∞, m(x0 ) > 0, 0 ≤ f = f (x) ∈ L 1 (Rn )

(11.61)

may be replaced by the existence of a = a(t) > 0, a measurable function in [0, T ) such that

T

a(t)F(u(t))dt > −∞,

0

T

T

0

s

ds a(t)dt

< +∞.

(11.62)

We note that (11.62) is slightly stronger than (11.59). In fact, here we may assume F(u 0 ) < 0. Then it holds that

T





T

a(t)dt 0

A (s)2 ds =

s





T

t

a(t)dt 0

≤ −∇ϕ2∞ λ

A (s)2 ds

0



T

a(t)F(u(t))dt < +∞

0

by (11.40) and (11.45). We obtain t 2 2 |A(t2 ) − A(t1 )|2 = A (s)ds t1  t2  T T ds · ≤ a(t)dt A (s)2 ds T t 0 s a(t)dt 1 s

340

11 Parabolic Equations

for 0 ≤ t1 ≤ t2 < T , and hence the existence of (11.47) which implies the desired property. Now we proceed to the blowup rate, using the rescaled equation (11.30). First, we note the decrease of the free energy. In fact, it is written as in Rn × (− log T, +∞) vs = ∇ · v∇δ Fˆ (v) m 2 v |y| 1 ˆ F(v) = − v dy −  ∗ v, v , 2n 2 Rn m which implies d ˆ F(v) = − ds

Rn

2 v ∇δ Fˆ (v) .

ˆ We have also the relation between F(v) and the second moment. These relations are summarized as

d ds



  2 |y|2 v ∇ v m−1 − ∗ v − ≤0 2n Rn ˆ |y|2 v = 2(n − 2)F(v) + |y|2 v.

d ˆ F(v) = − ds Rn



(11.63)

Rn

Since relation (11.63) implies d ds

Rn

ˆ 0) + |y|2 v ≤ 2(n − 2)F(v

Rn

|y|2 v,

(11.64)

the assumption ˆ 0) + 2(n − 2)F(v

Rn

|y|2 v0 < 0

induces Rn

|y|2 vdy < 0, s  1, a contradiction. Hence we obtain ˆ 0) + 2(n − 2)F(v

Rn

|y|2 v0 ≥ 0.

(11.65)

Inequality (11.65) must be translated as in d Hˆ ˆ = 2(n − 2)F(v) + Hˆ ≥ 0, ds which is nothing but (11.60).

Hˆ =

Rn

|y|2 v

11.5 Other Properties

341

The following theorem is comparable to Theorem 1.10.1. Theorem 11.5.2 If (11.46) holds, then any x0 ∈ S admits the property lim(T − t) u(·, t) L ∞ (B(x0 ,b(T −t)1/n ) = +∞, ∀b > 0. t↑T

Proof We have (11.61) in this case. By (11.44), furthermore, it holds that T d dt ≤ C(T λ)1/2 ∇ϕ∞ . ϕu dt  0 Putting ϕ = ϕx0 ,R , we obtain     ϕx ,R , u(t) − ϕx ,R , μ(d x, T ) ≤ C(T λ)1/2 R −1 (T − t) 0 0

μ(d x, T ) = m(x0 )δx0 (d x) + f (x)d x.

(11.66)

(11.67)

(11.68)

x0 ∈S

Given b > 0, we can take R = b(T − t) for 0 < T − t  1 in (11.68), and then it follows that (11.69) lim lim sup u(·, t) − m(x0 ) = 0. b↑+∞

t↑T

B(x0 ,b(T −t))

Under the transformation (11.29), inequality (11.69) reads; lim lim sup v(·, s) − m(x ) 0 = 0. n−1 − n s b↑+∞ s↑+∞

Since

B(0,be

)

(11.70)

Rn

v(·, s) = λ, s > − log T,

(11.71)

any tk ↑ T admits {sk } ⊂ {sk } for sk = − log(T − tk ) such that v(y, sk )dy ζ (dy) in M(Rn )

(11.72)

ζ (dy) ≥ m(x0 )δ0 (dy)

(11.73)

with

by (11.70). Relations (11.72)–(11.73) imply   lim v(·, sk ) L ∞ (B(0,b)) = +∞, ∀b > 0,

k→∞

and hence (11.66).



342

11 Parabolic Equations

The scaling property, combined with the compactness of the solution sequence, derives another aspect of the finiteness of type II blowup points [382]. Theorem 11.5.3 The set  S∗,ξ = x0 ∈ Rn | ∃y(t) ∈ Rn , ∃b > 0, lim y(t) = x0 , t↑T  lim inf (T − t)u(·, t) L ∞ (B(y(t),b(T −t)1/n )) ≥ ξ t↑T

is finite for each ξ > 0 and hence S∗ =

 ξ >0

S∗,ξ is countable.

Proof Here we develop a formal argument, given ξ > 0. First, the property  S∗,ξ < +∞ is reduced to inf lim lim inf u(·, t) L 1 (B(x0 ,r )) > 0

x0 ∈S∗,ξ r ↓0

t↑T

(11.74)

from the total mass conservation of u = u(·, t). By Theorem 11.4.2 if (11.74) is not the case, we have xk ∈ S∗,ξ , rk > 0, 0 < T − t jk < 1/( jk), such that



B(xk ,4rk )

u(·, t jk ) < min

j, k = 1, 2, . . .

 ε0 1 , . 2 2k

(11.75)

Let k be fixed. Since xk ∈ S∗,ξ , we have yk (t) → xk as t ↑ T and bk > 0 such that lim inf (T − t)u(·, t) L ∞ (B(yk (t),bk (T −t)1/n )) ≥ ξ. t↑T

(11.76)

Then there is jk  1 such that B(yk (t jk ), 3rk ) ⊂ B(xk , 4rk ), and hence

j ≥ jk

 ε0 1 . < min , 2 2k 

sup

y∈B(yk (t jk ),rk )

u(·, t jk ) L 1 (B(y,3rk ))

by (11.75). Then it follows that

(11.77)

11.5 Other Properties

343

sup t∈(t jk + 18 (T −t jk ),t jk + 38 (T −t jk ))

(T − t)u(·, t) L ∞ (B(yk (t jk ),(T −t jk )1/n ))

≤C

(11.78)

from Lemma 11.4.3. We have also sup t∈(t jk , 21 (T +t jk ))

u(·, t) L 1 (B(yk (t jk ),2rk ))
0. t↑T

Therefore, we obtain x0 ∈ S∗,1 if (11.85) is not the case. It is a contradiction, and (11.85) holds. 

11.6 Summary We have studied semilinear and degenerate parabolic equations. 1. In the semilinear parabolic equation, the linear part dominates the nonlinearity in a short time. Then blowup of the solution arises with its L ∞ norm. 2. Several critical exponents are detected to the semilinear parabolic equation in accordance with the stationary and self-similar solutions. 3. For the semilinear parabolic equation with the critical Sobolev exponent, there is an energy quantization to the non-compact solution sequence with bounded total energy. 4. A degenerate parabolic equation formulates the motion of mean field of many self-gravitating particles.

11.6 Summary

347

5. Mass quantization is observed for the stationary solution derived from the canonical setting and the Tsallis entropy. 6. As for the dynamics, there are some differences between the SmoluchowskiPoisson equation in two-space dimensions and the critical exponent of this equation associated with the Tsallis entropy. 7. There arise ε-regularity, the key property of the bounded variation in time of the local mass, and structure theorem of the blowup set in the critical degenerate parabolic equation.

Chapter 12

Gauge Fields

In the first quantization the movement of the particle density is described by a transformation of the Hamiltonian of classical particles. We take this process under the Maxwell gauge, using the Bogomol’nyi structure and the self-duality, to obtain again the exponential nonlinearity competing to the two-dimensional diffusion. Then we can show the quantized blowup mechanism for solutions to this Boltzmann-Poisson equation in two-space dimension. We develop the blowup analysis based on the scaling invariance of the problem and reveals the quantized blowup mechanism through several hierarchical arguments. This process provides with the fundamental motivation of the study on the system of chemotaxis in Chap. 1, that is the nonlinear spectral mechanics.

12.1 Field Theory We begin with the fundamental notion of gauge theory, particularly, covariant derivative and self-duality, and derive the gauged Schrödinger Chern-Simons equation [393, 394]. Classical Mechanics First, if q1 , . . . , q N are particles in R3 with the masses m 1 , . . . , m N , then the Newton equation is described by  ∂ m i q¨i = ∂q j j=i

 mi m j  , − qi − q j 



1 ≤ i ≤ N.

© Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_12

(12.1)

349

350

12 Gauge Fields

This equation means  δ

L = 0,

(12.2)

using the Lagrangian L=

1 2

i

m i q˙i2 −

 mi m j  . qi − q j  j 0, Q, E, B, and V denote the position, the mass, the electric charge, the electric field, the magnetic field, and the kinetic potential, respectively. Here, Q E and Q x˙ × B stand for the electric field action force and the Lorentz force, respectively. These electric and magnetic fields are subject to Maxwell’s equation ∇×E =−

∂B , ∇ · B = 0, ∂t

and, therefore, B = −∇ × A,

E = −∇ +

∂A ∂t

using the vector and the scalar potentials A and , respectively.

12.1 Field Theory

351

Writing A = (Ai )i=1,2,3 and y = m x˙ = (yi )i=1,2,3 , it holds that ∂V y˙i = −Q(E − x˙ × B)i − i ∂x     ∂ ∂ Ai ∂ Ai ∂V j ∂Aj =Q − − − i − Q x˙ i i j ∂x ∂t ∂x ∂x ∂x ∂ A ∂ d Ai ∂V j + Q i − Q x˙ j = −Q − i, dt ∂x ∂x i ∂x and hence we obtain ⎞ ⎛  d ∂ ⎝ (yi + Q Ai ) = x˙ j A j − V ⎠ . Q − Q dt ∂xi j

This equation means (12.3) for L=

1 2 m x˙ + Q − Q x˙ · A − V 2

and N = 1. Thus the general momentum and the Hamiltonian are described by pi =

∂L = yi − Q Ai ∂ x˙ i

and H =



pi x˙ i − L

i

 1 1 2 1 (yi + Q Ai )yi − yi + Q + Qyi Ai − V m 2m m 1 1 2 y − Q + V = ( pi − Q Ai )2 − Q + V, = 2m 2m

=

respectively. Regarding q = x as the general coordinate and putting  = A0 , we re-formulate the Hamiltonian of this system by H = H (q, p, t) =

1 ( pi − Q Ai )2 − Q A0 + V. 2m

First Quantization The particle mass is quantized by the Schrödinger equation. In the first quantization, the electro-magnetic field remains classical, coupled with the vector potential Aμ , μ = 0, 1, 2, 3. Thus we replace E = H (q, p, t) by

352

12 Gauge Fields

t → t, Using the normalization of ı

E → ı h 2π

h ∂ , 2π ∂t

pi →

1 h ∂i . ı 2π

= 1, we obtain

1 ∂ψ =− (∂i − ı Q Ai )2 ψ − Q A0 ψ + V ψ, ∂t 2m

(12.4)

and then the gauge covariant derivatives D μ = ∂μ − ı Q A μ ,

μ = 0, 1, 2, 3,

replace (12.4) by ı D0 ψ = −

1 2 D ψ + V ψ, 2m i

(12.5)

where ∂0 = ∂t . The energy operator is thus described by 1 2 D + (V − Q A0 ) Eˆ = − 2m i and, therefore, we obtain the energy expectation,  E= R =

3

R3

ˆ = ψ Eψ



 1 2 Di ψ + (V − Q A0 )ψ ψ − 2m R3

1 |Di ψ|2 + (V − Q A0 ) |ψ|2 d x 2m

by ∂i (ψ Di ψ) = ∂i ψ Di ψ + ψ∂i Di ψ = Di ψ Di ψ + ψ Di Di ψ = |Di ψ|2 + ψ Di2 ψ. This E is regarded as a Hamiltonian again, denoted by H , and then (12.4) is equivalent to ı

δH ∂ψ = . ∂t δψ

The equivalent formulation (12.5) is obtained by replacing H and ∂t by  H=

R3

1 |Di ψ|2 + V |ψ|2 d x 2m

12.1 Field Theory

353

and the covariant derivative D0 , respectively, that is ı D0 ψ =

δH δψ

.

Chern-Simons Relation The pre-gauged Schrödinger equation is defined by ı

∂ψ 1 =− ψ + V ψ, ∂t 2m

and then there arises the nonlinear Schrödinger equation ı

1 ∂ψ =− ψ − g |ψ|2 ψ, ∂t 2m

(12.6)

if the potential V = −g |ψ|2 is adopted to describe the mean field self-interaction, where g denotes the gravitational constant. Henceforth, we consider the case of two space dimension. The particle density ρ = |ψ|2 = ψψ is subject to ∂ρ ∂ψ ı ∂ψ =ψ +ψ =− (ψψ − ψψ) ∂t ∂t ∂t 2m   ı ∂k ψ∂k ψ − ψ∂k ψ =− 2m or, equivalently, ρt + ∇ · j = 0 for j = ( j k )k=1,2 ,

jk =

ı (ψ∂k ψ − ψ∂k ψ) 2m

which results in the Maxwell equation ∂μ J μ = 0,

(12.7)

where J = (J μ ) = (ρ, j k ) = (ρ, j), denotes the current density.

μ = 0, 1, 2

354

12 Gauge Fields

Equation (12.6) is described by (12.2) for L = ıψ∂0 ψ −

1 g |∂k ψ|2 + |ψ|4 , 2m 2

and we obtain the gauged nonlinear Schrödinger equation ı D0 ψ = −

1 2 D ψ − g |ψ|2 ψ, 2m k

(12.8)

using the gauge covariant derivative Dμ = ∂μ − ı Q Aμ . The current density, on the other hand, is equal to J = (J μ ) = (ρ, J k )k=1,2 for ρ = |ψ|2 and J k =

ı 2m (ψ Dk ψ

− ψ Dk ψ) by

∂μ (ψ1 ψ2 ) = ψ1 Dμ ψ2 + (Dμ ψ1 )ψ2 ,

μ = 0, 1, 2.

Since this J is subject to (12.7), we take the gauge potential Aμ satisfying ∂ν Fμν = −J μ ,

μ = 0, 1, 2,

where Fμν = ∂μ Aν − ∂ν Aμ denotes the rate of change of electromagnetic field. In the Chern-Simons theory, however, this relation is replaced by Fμν =

1 εμνα J α , κ

μ, ν, α = 0, 1, 2,

(12.9)

using the coupling constant κ > 0 and the skew-symmetric tensor εμνα normalized by ε012 = 1. Thus (12.8) with (12.9) comprises of the gauged Schrödinger-Chern-Simons equation, where J = (J μ ), J 0 = |ψ|2 ı (ψ Dk ψ − ψ Dk ψ), k = 1, 2 Jk = 2m Fμν = ∂μ Aν − ∂ν Aμ , and this equation is described by (12.2) for 1 g κ |Dk ψ|2 + |ψ|4 L = − εμνα Aμ ∂ν Aα + ıψ D0 ψ − 2 2m 2 κ 1 g 2 |Dk ψ| + |ψ|4 . = − ε μνα Aμ Fνα + ıψ D0 ψ − 4 2m 2

12.1 Field Theory

355

Once the Lagrangian action density is defined, then the Bogomol’nyi structure and self-duality induce a system of the first order equations and a single equation of the second order, respectively, see below. Bogomol’nyi Equation In two-space dimension, the Lagrangian action density concerning superconductivity is associated with the carrier (A0 , A1 , A2 ) and the Higgs field ϕ, where A0 and Ai (i = 1, 2) are the electric and magnetic gauge fields, respectively, and |ϕ|2 describes the Cooper pair density. In the law temperature of A0 = 0, non-relativistic Ginzburg-Landau density and the retivistic Abelian-Higgs density share the same stationary state involved by the Higgs term and the double-well potential [145, 267], that is  2 2 1 1  λ 2 |ϕ| − 1 d x. (∂ j − ı A j )ϕ + F jk F jk + L(A, ϕ) = 4 8 R2 2 Here, the Higgs field ϕ : R2 → C is regarded as a cross-section of a C-line bundle on R2 , A = (A j ) j=1,2 is the connection, and F jk = ∂ j Ak − ∂k A j is its curvature tensor. From these topological requirements, it follows that n=

1 2π

 R2

F12 ∈ Z

under the assumption of |ϕ| → 1 j

DAϕ = ∂ j ϕ − ı A j ϕ → 0

as |x| → ∞,

(12.10)

and this n is called the vortex number. It is actually the first Chern class of the C-line bundle. If A = 0, then the above L(A, ϕ) describes the Landau-Ginzburg vortices [24]. Putting ϕ = ϕ1 + ıϕ2 , we have  2 1  1 (∂ j − ı A j )ϕ = (∂1 ϕ1 + A1 ϕ2 )2 + (∂1 ϕ2 − A1 ϕ1 )2 2 2  1 + (∂2 ϕ1 + A2 ϕ2 )2 + (∂2 ϕ2 − A2 ϕ1 )2 2

356

12 Gauge Fields

=

1 [(∂1 ϕ1 + A1 ϕ2 ) − (∂2 ϕ2 − A2 ϕ1 )]2 2 1 + [(∂1 ϕ2 − A1 ϕ1 ) + (∂2 ϕ1 − A2 ϕ2 )]2 2 1 + (A1 ∂2 − A2 ∂1 )(ϕ21 + ϕ22 ) 2

and 1 2

 R2

(A1 ∂2 −

A2 ∂1 )(ϕ21

+ ϕ22 )

1 = 2

 R2

F12 (ϕ21 + ϕ22 ).

In the case of λ = 1, therefore, it holds that  1 L(A, ϕ) = [(∂1 ϕ1 + A1 ϕ2 ) − (∂2 ϕ2 − A2 ϕ1 )]2 R2 2 1 + [(∂2 ϕ1 + A2 ϕ2 ) + (∂1 ϕ2 − A1 ϕ1 )]2 2   2 1 1 2 1 F12 + ϕ1 + ϕ22 − 1 + dx + F12 2 2 2 R2 by  2 1 1 2 2 F12 + (ϕ1 + ϕ2 − 1) 2 2 2 1 1 2 1 2 |ϕ| − 1 + F12 (ϕ21 + ϕ22 − 1). = F12 + 2 8 2 Thus we obtain L ≥ nπ with the equality if and only if (∂1 ϕ1 + A1 ϕ2 ) − (∂2 ϕ2 − A2 ϕ1 ) = 0 (∂2 ϕ1 + A2 ϕ2 ) + (∂1 ϕ2 − A1 ϕ1 ) = 0  1 2 ϕ1 + ϕ22 − 1 = 0. F12 + 2

(12.11)

Self-Duality The above described Bogomol’nyi equation [30] is reduced to a single second order nonlinear equation, similarly to the axially symmetric four dimensional self-dual SU (2) Yang-Mills equation [393, 394, 416].

12.1 Field Theory

357

More precisely, the first two relations of (12.11) are regarded as the real and the imaginary parts of ˆ = 0, 2∂ϕ − ı Aϕ where 1 1 Aˆ = A1 + ı A2 , ∂ = (∂1 − ı∂2 ) , ∂ = (∂1 + ı∂2 ) . 2 2 This formulation implies Aˆ = −2ı∂ log ϕ, and in particular, ϕ = ef for some f = f 1 + ı f 2 with f 2 (θ, r ) ≡ f 2 (θ + 2π, r )

modulo 2πZ.

(12.12)

Then, we obtain A 1 = ∂1 f 2 + ∂2 f 1 ,

A2 = −∂1 f 1 + ∂2 f 2 ,

and, therefore, F12 = ∂1 A2 − ∂2 A1 = − f 1 . It holds that − f 1 +

 1  2 f1 e −1 =0 2

and f 1 → 0, ϕ → eı f2

as |x| → ∞

by (12.10) and (12.11). If f 1 has a finite number of singular points denoted by {a1 , . . . , am } ⊂ R2 then ϕ(ak ) = 0 because ϕ(x) is continuous. If n k denotes the order of zero, more precisely, then f 1 (x) ∼

nk log |x − ak |2 2

as x → ak .

358

12 Gauge Fields

Thus writing v = 2 f1 −



n k log |x − ak |2 ,

k

we obtain v = ev − 1 + 4π



n k δ(x − ak )

in R2 .

k

Similarly, it holds that −v = ev − 1 + 4π



n k δ(x − ak )

in R2

k

when λ = −1. Several elliptic equations involving exponential nonlinearity are obtained in stationary self-dual gauge theories in the two-space dimension by similar arguments [429], that is non-relativistic and relativistic super-conductivities in high temperature are described by the gauged Schrödinger and the Chern-Simons densities with the self-dual equations defined by  n k δ(x − ak ) in R2 −v = ±ev + 4π k

and v = ev (ev − 1) + 4π



n k δ(x − ak )

in R2 ,

(12.13)

k

 respectively. The effect of vortex term 4π k δ(x − ak ) is particularly studied in detail [392]. Finally, the doubly periodic second solution to (12.13) without vortices is reduced to the mean field equation   v  e 1 in , − v = λ  v − v = 0, (12.14) ||  e where  = R2 /aZ × bZ, a, b > 0 and λ = 4π, see [391].

12.2 Exponential Nonlinearity Revisited As we have seen, the elliptic eigenvalue problem λev − v =  v e

in ,

v=0

on ∂

(12.15)

12.2 Exponential Nonlinearity Revisited

359

relative to (12.14) describes the stationary mean field of many vortices in the perfect fluid, stationary self-dual gauge field associated with super-conductivity, and stationary state of the chemotaxis system, where  ⊂ R2 is a bounded domain with smooth boundary ∂, λ > 0 is a constant, and v = v(x) ∈ C 2 () ∩ C() is a classical solution. For this problem, several results have been obtained such as the classification of the singular limit [254], uniqueness of the solution [359], singular perturbation [16], and the topological degree calculation [61]. Theorem 12.2.1 If {(λk , vk )} is a family of stationary solutions to (12.15) for (λ, v) = (λk , vk ), and lim vk ∞ = +∞

lim λk = λ0 ∈ [0, ∞),

k→∞

k→∞

(12.16)

  then λ0 = 8πm for some m ∈ N, and there is vk ⊂ {vk } such that vk → 8π

m 

G(·, xk∗ )

(12.17)

k=1

  locally uniformly in  \ S, where S is the blowup set of vk defined by   S = x0 ∈  | ∃xk → x0 such that vk (xk ) → +∞ . This blowup set is composed of m-distinct interior points denoted by   S = x1∗ , . . . , xm∗ ⊂ , satisfying  1 ∇ R(x ∗j ) + ∇x G(x ∗j , xk∗ ) = 0, 1 ≤ j ≤ m, 2 k= j

where G = G(x, x  ) denotes the Green’s function of − D defined by −x G(·, x  ) = δ(· − x  ),

 G(·, x  )∂ = 0, x  ∈ 

and 

  1 log x − x   R(x) = G(x, x ) + 2π is the Robin function.



 x  =x

,

x ∈

(12.18)

360

12 Gauge Fields

We have 



e vk

e

vk

d x  8π

m 

δx ∗j (d x)

j=1

in M(), and furthermore, (12.18) is equivalent for (x1∗ , . . . , xm∗ ) ∈  × · · · ×  to be a critical point of the Hamiltonian H (x1 , . . . , xm ) =

m



j=1

1≤i< j≤N

1 R(x j ) + 2

G(xi , x j ).

This fact means a recursive profile of the vortex mean field hierarchy, recalling (9.16). Besides the stationary state of chemotaxis (1.38), the above described quantized blowup mechanism is actually observed in several problems. The quantized mechanism, conversely, clarifies the structure of the total set of solutions mathematically, and we obtain the following theorems [16, 19, 85, 98, 60, 61, 234, 359, 376] Theorem 12.2.2 If 0 < λ < 8π, then there is a unique solution to (12.15). Theorem 12.2.3 If (x1∗ , . . . , xm∗ ) ∈  × · · · ×  is a non-degenerate critical point of H (x1 , . . . , xm ), then there is a family of solutions {(λk , vk )} to (12.15) satisfying (12.16) for λ0 = 8πm and (12.17) to vk = vk locally uniformly in  \ S. Theorem 12.2.4 Total topological degree of (12.15), denoted by d(λ), is determined by the genus g of  if λ ∈ / 8πN. More precisely, it holds that  d(λ) =

m+1−g m



for 8πm < λ < 8π(m + 1). In particular, if g ≥ 1 and λ ∈ / 8πN, then (12.15) admits a solution Collapse Collision The above described quantized blowup mechanism is still valid without the boundary condition or even to variable coefficients, − v = V (x)ev

in ,

(12.19)

12.2 Exponential Nonlinearity Revisited

361

where  ⊂ R2 is a bounded domain. If v = v(x) is a solution to (12.15), then w = v + log λ − log  ev solves −w = e

w

 in ,



ew = λ.

The following theorem [215] and its proof are, therefore, quite useful in the study of the self-dual gauge field equation with vortex terms. There may arise, however, the collision of collapses, that is multiple bubbles. Thus in Theorem 12.2.1, we exclude the collision of the blowup points and also prescribe the location of blowup points, provided with the boundary condition. Theorem 12.2.5 Let vk = vk (x), k = 1, 2, . . . satisfy −vk = Vk (x)evk , 0 ≤ Vk (x) ≤ C1

 in ,



e vk ≤ C 2 ,

where C1 , C2 > 0 are constants, Vk = Vk (x) is continuous, and Vk → V uniformly on . Then, passing to a sub-sequence, we have the following alternatives: (i) {vk } is locally uniformly bounded in . (ii) vk → −∞ locally uniformly in . (iii) We have a finite set S = {ai } ⊂  and m i ∈ N such that vk → −∞ locally uniformly in  \ S and Vk (x)evk d x 



8πm i δai (d x)

i

in M(). Here, S is the blowup set of {vk } in . For the proof, we use the preliminary version sometimes called the rough estimate [37]. Theorem 12.2.6 Let  ⊂ R2 be a bounded domain, and vk = vk (x), k = 1, 2, . . . satisfy −vk = Vk (x)evk

in 

for Vk = Vk (x) ≥ 0, and assume the existence c1 , c2 > 0 such that   Vk ∞ ≤ c1 , evk 1 ≤ c2 ,

k = 1, 2, . . . .

Then, passing to a sub-sequence, we have the following alternatives: (i) {vk } is locally uniformly bounded in . (ii) vk → −∞ locally uniformly in .

362

12 Gauge Fields

(iii) There is a finite set S = {ai } ⊂  and αi ≥ 4π such that vk → −∞ locally uniformly in  \ S and 

Vk (x)evk d x 

αi δai (d x)

i

in M(). Furthermore, S is the blowup set of {vk } in . Thanks to Theorem 12.2.6, the proof of Theorem 12.2.5 is reduced to the following case, where B = B(0, R) ⊂ R2 and Br = B(0, r ). Theorem 12.2.7 If in B −vk = Vk (x)evk , Vk (x) ≥ 0 Vk → V in C(B), lim max vk = +∞ k→∞

B

lim max vk = −∞, 0 < ∀r < R

k→∞ B\Br



lim

k→∞ B

Vk evk = α,



e vk ≤ C 0 , B

then it holds that α = 8πm for some m ∈ N. This quantization implies the existence of the solution for the disquantized λ, see [214, 353] for details. The origin of this property is the self-similarity of (12.19) described below. If the boundary condition is provided, then the multiple blowup, indicated by m ≥ 2, cannot occur [214, 257, 278], while there is actual collision in the general case [65]. The proof of Theorem 12.2.5 says that the non-compact solution sequence is approximated by a finite sum of rescaled entire solutions. The required estimate of mass from below is standard, using the self-similarity and the classification of the entire solution. The estimate from above or the residual vanishing is, thus, essential. It is obtained by the sup + inf inequality formulated by Theorem 12.6.1 below, combined with the scaling argument. If the boundary condition is provided, this process can be replaced by the profile of the asymptotic symmetry of the solution [214] or the method of the second moment [257, 278] . In more precise, we obtain m i = 1 in Theorem 12.2.5, provided that max vk − min vk ≤ C, ∇Vk ∞ ≤ C, k = 1, 2, . . . . ∂

∂

Similarly, if (12.20) holds for  = B, then it follows that α = 8π and     vk (x) − log    1+ in Theorem 12.2.7.

    2  ≤ C, ∀x ∈ B, k = 1, 2, . . . Vk (0) vk (0) 2  |x| 8 e evk (0)

(12.20)

12.2 Exponential Nonlinearity Revisited

363

Theorem 12.2.5, however, has a variety of implications because it is free from any boundary condition. For instance, even if the original problem is provided with the boundary condition, the rescaled problem loses it. It often happens that this fact causes a trouble in the blowup analysis.

12.3 Scaling Invariance Equation (13.12) or its general form (12.19) is scaling invariant, that is it is invariant under the transformation v μ (x) = v(μx) + 2 log μ, V μ (x) = V (μx), μ > 0.

(12.21)

Such a property, called scaling invariance or self-similarity, is observed in several nonlinear partial differential equations and is used essentially for their mathematical analysis as we have seen in Chap. 1. A typical example is −v = v p , v > 0 for 1 < p < ∞. Thus if v = v(x) is a solution to this equation, then so is vμ (x) = μ2/( p−1) v(μx) for μ > 0. Similarly, if v = v(x, t) is a solution to vt − v = v p , v > 0,

(12.22)

vμ (x, t) = μ2/( p−1) v(μx, μ2 t).

(12.23)

then so is

This structure causes obstruction to construct (weak) solution, while there are applications of renormalization group and blowup analysis. In the blowup analysis, we classify non-compact (approximate) solutions and obtain actual solution by excluding these possibilities. Applications of such argument to the calculus of variation are described in Bahri [13], and for example, −v = v 5 , v > 0,

v|∂ = 0

has a solution if the bounded domain  ⊂ R3 is not contractible, see Sect. 7.3. Another application is the proof of the a priori bound of the solution to the subcritical elliptic equation, which guarantees the topological degree approach to this problem [81, 138].

364

12 Gauge Fields

Theorem 12.3.1 Let  ⊂ Rn is a bounded domain with smooth boundary ∂ and 1 < p < n+2 n−2 . Then, there is C > 0 such that any solution v = v(x) to − v = v p , v > 0 in ,

v|∂ = 0

(12.24)

satisfies v∞ ≤ C.

(12.25)

Here we illustrate the outline with fundamental concept of the blowup analysis. In fact, first, we assume the contrary, the existence of a sequence of the solutions to (12.24), denoted by {vk }, satisfying vk (xk ) = vk ∞ → +∞, xk ∈ . Then we take 2/( p−1)

v˜k (x) = μk

vk (μk x + xk )

for μk = vk (xk )−( p−1)/2 → 0 and obtain p

−v˜k = v˜k , v˜k > 0

in k ,

v˜k |∂k = 0, v˜k (0) = max v˜k (x) = 1, x

where k = μ−1 k ( − {x k }). From the boundary condition and the elliptic estimate, we obtain a subsequence, denoted by the same symbol such that v˜k → v locally uniformly in Rn with smooth v = v(x), and it holds that either − v = v p , 1 = v(0) ≥ v ≥ 0

in Rn

(12.26)

or −v = v p , 1 = v(0) ≥ v ≥ 0

n in R+

v=0

n on ∂R+ ,

(12.27)

n = {x ∈ Rn | x > 0} denotes the half space. where R+ n Problem (12.26), however, admits no solution in the case of 1 < p < n+2 n−2 . The same is true for (12.27) if 1 < p ≤ n+2 , and hence we end up with (12.25). n−2 Ingredients of the blowup analysis are thus summarized as follows:

1. 2. 3. 4.

scaling invariance. classification of the entire solution. control at infinity of the rescaled solution. hierarchical argument.

12.3 Scaling Invariance

365

In the case of (12.15), first, we use complex-geometric structure and linear theory associated with the real analysis. This study comprises of the pre-scaled analysis for the proof of Theorem 12.2.6, see Sects. 12.4, 12.5, and 12.6. Next, method of the moving plane is applied to classify the entire solution, see Sect. 12.7. Finally, control at infinity of the rescaled solution is done by the Harnack type inequality. We thus complete the proof of Theorem 12.2.7 in Sect. 12.8 by the hierarchical argument using (12.21). Self-similarity (12.23) induces forward and backward self-similar transformations to (12.22), see Sect. 11.1. The underlying structure of these transformations takes a fundamental role in the study of the system of chemotaxis, see Chap. 1.

12.4 Liouville-Bandle Theory The problem (12.15) is associated with the complex function theory and the theory of surfaces. In this paragraph, we describe them briefly. Complex Structure If we identify x = (x1 , x2 ) ∈  to z = x1 + ı x2 ∈ C, then − u = eu

(12.28)

means 1 u zz = − eu 4 for z = x1 − ı x2 . This equality implies 1 1 sz = u zzz − u z u zz = − eu u z + u z eu = 0 4 4 for 1 s = u zz − u 2z , 2

(12.29)

and, therefore, s = s(z) is a holomorphic function of z ∈  ⊂ C. Regarding (12.29) as a Riccati equation of u z , we obtain 1 ϕzz + sϕ = 0 2

(12.30)

366

12 Gauge Fields

for ϕ = e−u/2 . Here, we take x ∗ = (x1∗ , x2∗ ) ∈ , and define the fundamental system of the linear equation (12.30), denoted by {ϕ1 (z), ϕ2 (z)} , such that ϕ1 |z=z ∗

 ∂ϕ2  = = 1, ∂z z=z ∗

 ∂ϕ1  = ϕ2 |z=z ∗ = 0, ∂z z=z ∗

(12.31)

where z ∗ = x1∗ + ı x2∗ . This {ϕ1 (z), ϕ2 (z)} forms a system of analytic functions of z ∈ , and it holds that ϕ = e−u/2 = f 1 (z)ϕ1 (z) + f 2 (z)ϕ2 (z)

(12.32)

with some functions f 1 and f 2 of z. These functions are prescribed by the Wronskian. It holds that W (ϕ1 , ϕ2 ) ≡ ϕ1 ϕ2z − ϕ1z ϕ2 = 1, and hence f 1 (z) = W (ϕ, ϕ2 ) = ϕϕ2z − ϕz ϕ2 f 2 (z) = W (ϕ1 , ϕ) = ϕ1 ϕz − ϕ1z ϕ. Here, the left-hand side is independent of z. Taking z = z ∗ on the right-hand side, we obtain f 1 (z) = ϕ(z ∗ , z), f 2 (z) = ϕz (z ∗ , z). Since ϕ is real-valued, there holds that 1 ϕzz + sϕ = 0 2 for s = s(z) defined by s(z) = s(z). Thus this equality is valid to f 1 (z) and f 2 (z), while {ϕ1 , ϕ2 } forms a fundamental system such that ϕ1 |z=z ∗

 ∂ϕ2  = = 1, ∂z z=z ∗

 ∂ϕ1  = ϕ2 |z=z ∗ = 0. ∂z z=z ∗

In particular, f 1 (z) and f 2 (z) are linear combinations of ϕ1 (z) and ϕ2 (z).

12.4 Liouville-Bandle Theory

367

If x ∗ = (x1∗ , x2∗ ) ∈  is a critical point of u, then it holds that   f 1 (z ∗ ) = ϕ(z ∗ , z ∗ ) = e−u/2 

x=x ∗

 ∂ f1 ∗ ∂ −u/2  ∗ ∗ (z ) = ϕz (z , z ) = e  ∗ =0 ∂z ∂z x=x  ∂ e−u/2  f 2 (z ∗ ) = ϕz (z ∗ , z ∗ ) = =0 ∂z x=x ∗     ∂ f2 ∗ 1 1 (z ) = ϕzz (z ∗ , z ∗ ) = − e−u/2  = − e−u/2 u  ∂z 4 8 x=x ∗ x=x ∗   1 = eu/2  , 8 x=x ∗ and, therefore, f 1 (z) = cϕ1 (z),

f 2 (z) =

1 −1 c ϕ2 (z) 8

 for c = e−u/2 x=x ∗ . This relation means f 1 = cϕ1 ,

f2 =

c−1 ϕ2 8

and, therefore, we obtain e−u/2 = c |ϕ1 |2 +

c−1 |ϕ2 |2 8

(12.33)

by (12.32). For the proof of Theorem 12.2.1, let (λ, v) be the solution to (12.15). We apply the above argument replacing u by v and taking x∗ as the maximum point of v. Then, we obtain e−v/2 = c |ϕ1 |2 +

σc−1 |ϕ2 |2 8

similarly, where λ , c = e−v∞ /2 v e 

σ=

and {ϕ1 (z), ϕ2 (z)} is a system of fundamental solutions of (12.30) defined for 1 2 . s(z) = vzz − vzz 2

(12.34)

368

12 Gauge Fields

Given the solution sequence {(λk , vk )}, we exclude the boundary blowup points of {vk }, using the reflection argument [135] and the L 1 boundedness of the right-hand side [81]. Thanks to the maximum principle and the classical Montel’s theorem for holomorphic functions, this boundary estimate implies the compactness of the family of holomorphic functions {sk (z)} defined (12.34) for v = vk and that of the analytic functions {ϕ1k (z), ϕ2k (z)} defined by (12.30) and (12.31). Passing to a subsequence, we obtain ϕ1k → ϕ10 , ϕ2k → ϕ20 locally uniformly in . Here, we have ck = e−vk ∞ → 0, with the relation e−vk /2 = ck |ϕ1k |2 +

σk ck−1 |ϕ2k |2 . 8

Since {vk } is uniformly bounded near ∂ and zero points of the analytic function ϕ20 are discrete, there arise σk ck−1 ≈ 1 and the finiteness of the blowup points of {vk }. Each blowup point is thus isolated, and the classification of the singular limit of the solution (12.17) and (12.18) is obtained by the residue analysis. More precisely, since the limit function s0 (z) of {sk (z)} is holomorphic, the singular point of s0 (z) arising as a blowup point of {vk } is removable. This removable singular point is formally a pole of the second order, and then, (12.17) and (12.18) follow, see [254]. The other proof uses the Pohozaev identity to detect (12.18), see [227]. Geometric Structure Writing ψ1 = c1/2 81/4 ϕ1 and ψ2 = c−1/2 8−1/4 ϕ2 in (12.28), we have W (ψ1 , ψ2 ) = W (ϕ1 , ϕ2 ) = 1 and    −1  1/2  1/2 −1/2 1 1 1 |ϕ1 |2 + c−1 |ϕ2 |2 eu/2 = c 8 8 8 =

1 |ψ1 | + |ψ2 |2 2

,

12.4 Liouville-Bandle Theory

369

and, therefore,   F 

 1/2 1 = = eu/2 2 2 2 8 1 + |F| |ψ1 | + |ψ2 | W (ψ1 , ψ2 )

(12.35)

for F = ψ2 /ψ1 . Thus we can transform (12.15) to  ρ(F)|∂ =

λ 1 · v 8 e

using

1/2

 u = v + log λ − log

with v|∂ = 0, where ρ(F) =

  F  1 + |F|2



(12.36)

ev

.

This ρ(F) stands for the spherical derivative of the meromorphic function F = F(z). More precisely, if C and dσ 2 denote the Riemann sphere with the south and north poles (0, 0, 0) and (0, 0, 1) and its standard metric, respectively, and if τ : C → C ∪ {∞} is the stereographic projection, then, the conformal transformation F = τ −1 ◦ F induces the relation dσ = ρ(F), ds

(12.37)

where ds 2 = d x12 + d x22 stands for the Euclidean metric in C. In particular, ρ(F) is invariant under the O(3) transformation on C. If ω ⊂⊂  is a sub-domain, then the immersed length of F(∂ω) and the immersed area of F(ω) are defined by   1 (∂ω) = ρ(F)ds, m 1 (ω) = ρ(F)2 d x, ∂ω

ω

respectively. If F is injective and F(ω) is homeomorphic to a disc, therefore, it follows that 1 (∂ω)2 ≥ 4m 1 (ω) (π − m 1 (ω)) from the isoperimetric inequality on C. Putting 

 (∂ω) =

∂ω

p 1/2 ds, m(ω) =

ω

pd x

(12.38)

370

12 Gauge Fields

with p = eu , we thus obtain (∂ω)2 ≥

1 m(ω) (8π − m(ω)) 2

(12.39)

by (12.35) and (12.38). Spectral Analysis The above (12.39) is valid without the topological assumption of F(ω). Thus in the non-parametric surface with the Gaussian curvature less than or equal to 1/2, we obtain (12.39) if p = p(x) > 0 is a C 2 function defined on the domain  ⊂ R2 of which boundary is composed of a finite number of Jordan curves, −  log p ≤ p

in ,

(12.40)

and ω ⊂⊂  is a sub-domain. Then, Schwarz’ symmetrization using the metric dσ = p(x)1/2 ds guarantees Bandle’s isoperimetric inequality [15]. Theorem 12.4.1 If  ⊂ R2 is a domain with the boundary ∂ composed of a finite number of Jordan curves and p = p(x) > 0 is a C 2 function satisfying (12.40), then it holds that  λ≡ p < 8π ⇒ ν1 ( p, ) ≥ ν1 ( p ∗ , ∗ ), (12.41) 

where

 ν1 ( p, ) = inf



 |∇v|2 | v ∈ H01 (),



 v2 p = 1

and ( p ∗ , ∗ ) is determined by   ∗ = x ∈ R2 | |x| < 1 ,



λeu u∗ ∗ e

(12.42)

 u ∗ ∂∗ = 0

(12.43)

p∗ = 

and ∗

λeu u∗ ∗ e

− u ∗ = 

in ∗ ,

Here, each λ ∈ (0, 8π) admits a unique solution u ∗ = u ∗λ (x) to (12.43), which guarantees the well-definedness of ν1 ( p ∗ , ∗ ) in (12.41). This u ∗λ (x) is radially symmetric, described explicitly, and satisfies

12.4 Liouville-Bandle Theory

371

lim u ∗λ (x) = 0

uniformly in x ∈ ∗

λ↓0

lim u ∗λ (x) = 4 log

λ↑8π

1 |x|

locally uniformly in x ∈ ∗ \ {0}.

Furthermore, the value ν1 ( p ∗ , ∗ ) is associated with the Laplace-Beltrami operator on C, and, therefore, using the separation of variables and the stereographic projection, we obtain 0 < λ < 4π ⇒ ν1 ( p ∗ , ∗ ) > 1.

(12.44)

More precisely, ν1 ( p ∗ , ∗ ) is realized as the first eigenvalue of the eigenvalue problem − ϕ = νζ0 ϕ

in ∗ ,

ϕ|∂∗ = 0,

(12.45)

where 8ρ ζ0 = ζ0 (x) =  2 2 |x| + ρ and ρ > 0 is a constant determined by  λ=

∗

ζ0 .

Then, putting ϕ(x) = (ξ)eımθ , x = r eıθ , ξ =

ρ − r2 ,  = 1/ν, ρ + r2

(12.46)

we obtain the associated Legendre equation     (1 − ξ 2 )ξ + 2/ − m 2 /(1 − ξ 2 )  = 0, ξρ < ξ < 1 ξ

(1) = 1, (ξρ ) = 0

(12.47)

for ξρ = (ρ − 1)/(ρ + 1), see [15]. This formulation is natural because the LaplaceBeltrami operator on two-dimensional round sphere is immersed into the three dimensional Laplacian in the Cartesian coordinate, which is involved by the associated Legendre equation through the separation of variables using polar coordinates [360]. Thus if  = (ξ) denotes a solution to the first equation of (12.47) for  = 1, m = 0, and (1) = 1, then ν1 ( p ∗ , ∗ ) > 1 is equivalent to (ξ) > 0,

ξρ < ξ < 1.

372

12 Gauge Fields

Since such  is given by P0 (ξ) = ξ, this relation means ξρ > 0, and, therefore, (12.44) follows from λ < 4π



ρ>1

ξρ > 0.



Theorem 12.2.2, on the other hand, is proven by the bifurcation analysis and an a priori estimate. More precisely, we show that the linearized operator is nondegenerate if 0 < λ < 8π because the solution set is compact in this range by Theorem 12.2.1. Actually, the linearized operator concerning (12.15) around the solution v = v is defined by 

  v e ψ ev ψ v  Lψ = −ψ − λ  v −   ·e v 2 e e  with the domain L = H 2 () ∩ H01 () in L 2 (), and the transformation ϕ=ψ−

1 ||

 

ψ

used in Sect. 3.3 induces the eigenvalue problem (3.66). Using p = p(x) defined by λev p = u =  v, e see (3.66), this problem is written as  − ϕ = μ pϕ in ,

ϕ|∂ = constant,

∂

∂ϕ = 0, ∂ν

(12.48)

and this p = p(x) > 0 is provided with the property (12.40). Thus we have  only to show that μ = 1 is not an eigenvalue of (12.48) provided that 0 < λ =



p < 8π.

Since the first eigenvalue of this eigenvalue problem is μ = 0, we show that the second eigenvalue is greater than 1 in this case [359]. This property means 0 < λ < 8π ⇒ μ2 ( p, ) > 1,

(12.49)

where

 μ2 ( p, ) = inf

 

|∇v|2 | v ∈ Hc1 (),



 vp = 0,



 v2 p = 1

12.4 Liouville-Bandle Theory

for

373

  Hc1 () = v ∈ H 1 () | v = constant on ∂ ,

which comprises of the key lemma for the proof of Theorem 12.2.2. For simplicity, we assume that  ⊂ R2 is simply-connected. To derive (12.49), first, we show that the associated eigenfunction to μ2 ( p, ) has exactly two nodal domains. Next, if any of them is not enclosed inside , then we can apply (12.41) with (12.44) to either and obtain (12.49). The same argument is valid if one of them, denoted by ω, is enclosed inside  and it holds that  ω

For the rest case of

p < 4π.

 ω

p ≥ 4π,

we take the doubly-connected domain ωˆ =  \ ω, and obtain the eigenvalue problem −ϕ = μ p(x)ϕ, ϕ > 0 in ω, ˆ

ϕ|∂ ω\∂ = 0, ˆ

ϕ|∂ = constant.

Thus μ = μ2 ( p, ) is the first eigenvalue of this eigenvalue problem, with p = p(x) > 0 satisfying (12.40) and  p < 4π. (12.50) ωˆ

Here, we modify the Schwarz symmetrization on C, using the above described nodal domain property. In this case, (12.49) is reduced to examine an eigenvalue problem of the Laplace-Beltrami operator provided with a modified Dirichlet boundary condition defined on an annular domain of C, which is contained in the chemi-sphere by (12.50). Through the transformation (12.46), this problem is realized as the associated Legendre equation, and, then, (12.49) is shown [359, 360]. This proof of (12.49) is valid even to the multiply-connected  ⊂ R2 , and also to the case of λ = 8π, (if the solution exists), see [52, 376] for details.

12.5 Alexandroff-Bol’s Inequality Using Bol’s inequality and its refinement of Alexandrov’s inequality, we obtain the (bubbled) Harnack principle [360] and the sup + inf inequality [335]. For the moment, we specify the volume and the line elements by d x and ds, respectively.

374

12 Gauge Fields

Bol’s Inequality First, we recall the analytic form of Bol’s inequality described in Sect. 12.4. Theorem 12.5.1 If  ⊂ R2 is a simply-connected domain, p = p(x) > 0 is a C 2 function satisfying −  log p ≤ p

in ,

(12.51)

and ω ⊂⊂  is a sub-domain, then it holds that (∂ω)2 ≥

1 m(ω) (8π − m(ω)) , 2

(12.52)

where  (∂ω) =



∂ω

p 1/2 ds, m(ω) =

ω

pd x.

Inequality (12.52) holds even if  is multiply-connected, when p = p(x) is constant on ∂ (see [19]). Nehari’s isoperimetric inequality [259] described below is the limiting case of (12.51) and (12.52) to the zero Gaussian curvature. Theorem 12.5.2 If  ⊂ R2 is a simply-connected domain, h = h(x) is harmonic in , and ω ⊂  is a su-domain, then it holds that



2 eh/2 ds

∂ω

 ≥ 4π

ω

eh d x.

(12.53)

For the proof of the above theorem, first, we take a harmonic function v = v(x), x = (x1 , x2 ) in  such that f = h +ıv is holomorphic function of z = x1 +ı x2 ∈ . Second, we define the holomorphic function g = g(z), z ∈  by g  = e f /2 , which  2 satisfies g   = eh . Then the quantities 

 ∂ω

e

h/2

ds =

∂ω

  g  ds,



 e dx = h

ω

ω

  2 g  d x

indicate the immersed length of g(∂ω) and the immersed area of g(ω), and (12.53) follows from the isoperimetric inequality on the plane. We now describe the analytic proof [14] of Theorem 12.5.1. Proof of (12.53)⇒ (12.52): We may suppose that ∂ω is C 1 . Define h = h(x) by h = 0

in ω,

h|∂ω = log p

12.5 Alexandroff-Bol’s Inequality

375

and put q = pe−h . This formulation implies −  log q ≤ qeh

in ω,

q|∂ω = 1

(12.54)

by (12.51). Using the right-continuous non-increasing functions 

 K (t) =

{q>t}

qeh d x, μ(t) =

{q>t}

eh d x

defined by {q > t} = {x ∈ ω | q(x) > t}, we obtain 



− K (t) =

{q=t}

qeh ds = t |∇q|

 {q=t}

eh ds = −tμ (t) |∇q|

a.e. t > 1

(12.55)

by the co-area formula [101]. We have, on the other hand, 



{q>t}

(− log q)d x =

|∇q| 1 ds = q t

{q=t}

 {q=t}

|∇q| ds

a.e. t > 1,

using Green’s formula and Sard’s lemma. Thus we obtain 1 t



 {q=t}

|∇q| ds ≤

{q>t}

qeh d x = K (t)

a.e. t > 1

by (12.54), and, therefore,   eh 1 |∇q| ds · t ds t {q=t} {q=t} |∇q|

 2 h/2 ≥ e ds {q=t}  ≥ 4π eh d x = 4πμ(t) a.e. t > 1

− K  (t)K (t) ≥

{q>t}

(12.56)

by Schwarz’ and Nehari’s inequalities. Thus from (12.55) and (12.56) it follows that

 d K (t)2 1 μ(t)t − K (t) + = μ(t) + K (t)K  (t) ≤ 0 dt 8π 4π for a.e. t > 1.

(12.57)

376

12 Gauge Fields

We obtain, on the other hand, K (t + 0) = K (t) ≤ K (t − 0), while  j (t) = K (t) − μ(t)t =

(q − t)eh d x

{q>t}

is continuous: j (t + 0) = j (t) = j (t − 0). Thus (12.57) guarantees ∞  K (t)2 K (1)2 ≤ 0. − j (t) + = j (1) − 8π t=1 8π

(12.58)

Using 

 j (1) =

(q − 1)e d x ≥



h

{q>1}

(q − 1)e d x = m(ω) − h

ω

ω

eh d x

and K (1)2 ≤ m(ω)2 , we deduce m(ω)2 ≤ m(ω) − 8π



1 e dx ≤ 4π ω



h

∂ω

2 e

h/2

ds

=

(∂ω)2 . 4π 

This inequality means (12.52). Similarly to (12.58), we obtain j (t) ≤

K (t)2 8π

(12.59)

for any t > 1. This inequality implies the mean value theorem [360]. We note that the converse is also true in the following theorem. Theorem 12.5.3 If  ⊂ R2 is an open set, p = p(x) > 0 is a C 2 function satisfying (12.51), and B = B(x0 , r ) ⊂⊂ , then it holds that 1 log p(x0 ) ≤ |∂ B|

   1 log p ds − 2 log 1 − p dx . 8π B ∂B +



(12.60)

Proof Using (12.59) for ω = B, we obtain K (t)2 μ(t) ≥ t



1 1 − K (t) 8π

 (12.61)

12.5 Alexandroff-Bol’s Inequality

377

for any 1 < t < t0 = max B q. Furthermore, J (t) =

μ(t) 1 j (t) μ(t) μ(t) − = − − K (t) 8π t t K (t) 8π

is right-continuous: J (t + 0) = J (t), and it holds also that

 1 1 1 j (t) − + J (t − 0) − J (t) = (μ(t) − μ(t − 0)) t K (t) K (t − 0) 8π

 1 1 1 j (t) − + = (μ(t) − μ(t − 0)) t j (t) + μ(t)t j (t) + μ(t − 0)t 8π 

1 j (t) − . = − (μ(t) − μ(t − 0)) · K (t)K (t − 0) 8π Using  K (t − 0) − K (t) =

{q=t}

qeh d x = t (μ(t − 0) − μ(t)) ≥ 0

and j (t) ≥ 0, we obtain  J (t − 0) − J (t) ≤ − (μ(t − 0) − μ(t)) ·

j (t) 1 − 8π K (t)2

 ≤0

by (12.59). On the other hand, we have  1 K  (t) 1 − − μ(t) J (t) = μ (t) K (t) 8π K (t)2  tμ(t) K (t) ≥ μ (t) · − μ(t) · = 0 a.e. t > 1 2 K (t) K (t)2 





by (12.59) and (12.55). These relations imply lim J (t) = lim t↑t0

t↑t0

1 μ (t) = ≥ J (t) = μ(t) K  (t) t0



1 1 − K (t) 8π



for 1 ≤ t ≤ t0 by (12.55), recall t0 = max B q. Here, the right-hand side is estimated from below by K (t)2 t



1 1 − K (t) 8π

2 +

378

12 Gauge Fields

using (12.61). Then, putting t = 1, we obtain    K (1) 2 m 2 1 ≥ 1− ≥ 1− , t0 8π + 8π +  where m =

p d x. B

This inequality means  m 2 1− ≥ t0 = max pe−h ≥ p(x0 )e−h(x0 ) 8π + B or, equivalently,

   1 log p(x0 ) ≤ h(x0 ) − 2 log 1 − pd x . 8π B +

Here, using the mean value theorem to the harmonic function, we obtain h(x0 ) =

1 |∂ B|

 ∂B

h ds =

1 |∂ B|

 ∂B

log p ds, 

and the proof of (12.60) is complete.

Similarly to the harmonic case, the mean value theorem (12.60) implies the following Harnack inequality. Theorem 12.5.4 If B = B(0, R) ⊂ R2 and v = v(x) ∈ C 2 (B) ∩ C(B) satisfies λev 0 ≤ −v ≤  v , v ≥ 0 e

in B

then it holds that   R + |x| λ , x ∈ B. v(0) ≤ v(x) − 2 log 1 − R − |x| 8π + Proof We obtain − log p ≤ p for p = λev /



e

v,

in B

and, therefore, (12.60) is applicable. It holds that

log p(0) ≤

1 |∂ B|

  λ log p ds − 2 log 1 − , 8π + ∂B



(12.62)

12.5 Alexandroff-Bol’s Inequality

379

or, equivalently, v(0) ≤

1 |∂ B|

  λ v ds − 2 log 1 − . 8π + ∂B



(12.63)

Since v(x) ≥ 0 is super-harmonic, on the other hand, it holds that  2π R2 − r 2 1 v(Reıϕ ) dϕ 2 2π 0 R − 2Rr cos(θ − ϕ) + r 2  R −r 1 ≥ v ds, 0 ≤ r < R. R + r |∂ B| ∂ B

v(r eıθ ) ≥



This formula implies (12.62).

From the standard argument, Harnack’s inequality (12.62) implies the Harnack principle [358] stated as follows. Theorem 12.5.5 If  ⊂ R2 is an open set and vk = vk (x), k = 1, 2, . . ., are C 2 functions satisfying λ k e vk 0 ≤ −vk ≤  v , vk ≥ 0 k e

in 

for λk > 0, then passing to a sub-sequence, we obtain the following alternatives, where S = {x0 ∈  | there exists xk → x0 such that vk (xk ) → +∞} . (i) {vk }k is locally uniformly bounded in . (ii) vk → +∞ locally uniformly in . (iii) S = ∅ and S ≤ lim inf k [λk /(8π)]. Alexandrov’s Inequality Alexandrov’s inequality, see [15] for the proof, is a sharp form of Bol’s inequality. First, we introduce the metric dσ 2 = p(x)ds 2 on B = B(0, 1) ⊂ R2 , using p = p(x) > 0 such that C 2 (B) ∩ C(B). This formulation implies that its Gaussian curvature K , total volume m, the boundary length  are defined by K =−

 log p , m= 2p



 pd x,  = B

∂B

p 1/2 ds.

Then, if α = 2π − m + μ (B) > 0 for m+ μ (B) =

 {K >μ}

(K (x) − μ) p d x,

380

12 Gauge Fields

it holds that 2 ≥ (2α − μm)m,

(12.64)

where μ ∈ R. In the case of K ≤ 1/2, it follows that m+ μ (B) ≤



     1 1 1 p dx = −μ − μ m, α = 2π + μ − m 2 2 2 B

for μ < 1/2. Then, (12.64) reads; 2 ≥ m · (4π + (μ − 1)m) . Letting μ ↑ 1/2, we obtain 2 ≥

1 m(8π − m). 2

This formula means Bol’s inequality on B. In the actual application [15], we take μ∗ satisfying 

 {K >μ∗ }

p d x ≤ m/2,

{K μ∗ } = B, {K < μ∗ } = B, and, therefore, {K ≤ μ∗ } = ∅, {K ≥ μ∗ } = ∅. This relation implies a b ≤ μ∗ ≤ , 2 2 provided that b a ≤ K (x) ≤ , 2 2 where a, b > 0 are constants.

12.5 Alexandroff-Bol’s Inequality

381

There are C, D ≥ 0 such that   m m − C, −D p dx = pd x = 2 2 {K >μ∗ } {K μ∗ }  = 4π − 2 (K − μ∗ ) pd x − μ∗ m {K ≥μ∗ }  = 4π − 2 K pd x + 2μ∗ D. {K ≥μ∗ }

On the other hand, we obtain    a a m −D K pd x ≥ pd x = 2 {K 0 be constants. Then, there is C0 > 0 such that −v = V (x)ev , a ≤ V (x) ≤ b  V (x)ev ≤ α0

in B

B

implies v(0) ≤ C0 ,

(12.66)

where v ∈ C 2 (B) ∩ C(B). Proof We can apply (12.65) to p = ev and K = V /2. In fact, the function 

 f (r ) = 4π − 2

{K >μ∗ }∩Br

(K − μ∗ ) p d x − μ∗

p dx Br

is strictly decreasing in r ∈ [0, 1), and, therefore, f (r ) ≥ f (1) = 2α − μ∗ m ≥ γ0 . Putting



r

A(r ) =

dr 





0

∂ B(0,r  )

p ds =

B(0,r )

p d x,

on the other hand, we obtain 



A (r ) = ≥

∂ B(0,r )

1 p ds ≥ 2πr



2 ∂ B(0,r )

p

1/2

ds

1 f (r )A(r ), 0 ≤ r < 1 2πr

(12.67)

by (12.64). This inequality implies  1 A (r ) 1 {log A(r )} dr dr = r0 f (r )A(r ) r0 f (r )  1 log A(r )  log A(1) log A(r0 ) − + f (r ) dr, r0 ∈ (0, 1). = f (1) f (r0 ) f (r )2 r0

1 − log r0 ≤ 2π



1

Since A(r0 ) = πr02 p(0)(1 + o(1)),

  f (r0 ) = 4π − O r02 ,

r0 ↓ 0,

12.5 Alexandroff-Bol’s Inequality

383

it follows that

lim

r0 ↓0

 log A(r0 ) 1 1 A(r0 ) − log r0 = lim log 2 r0 ↓0 4π f (r0 ) 2π r0 1 = (log p(0) + log π) , 4π

and, therefore, v(0) = log p(0) ≤

4π log A(1) − log π + 4π f (1)



1 0

log A(r )  f (r )dr. f (r )2

We have, on the other hand, 

 2 α0 K p dx ≤ a a B B α0 1 log A(r ) + log α0    log A(r )  − f (r ) f (r ) = 2 f (r ) f (r )2 α0    · − f (r ) , ≤ γ0−2 log A(r ) A(r ) ≤ A(1) =

pd x ≤

and 

 (K − μ∗ ) p ds + μ∗ p ds {K >μ∗ }∩∂ Br ∂ Br     b a b − p ds + ≤ 2· p ds 2 2 ∂ Br {K >μ∗ }∩∂ Br 2     3 3 b−a b − a A (r ). ≤ p ds = 2 2 ∂ Br

0 ≤ − f  (r ) = 2

Using the transformation t = 

1 0

A(r ) α0 ,

we thus obtain

log A(r ) · f  (r ) dr ≤ γ0−2 f (r )2

and, therefore, (12.66).



3 b−a 2



1

log 0

1 dt < +∞, t 

If V (x) is restricted to a compact family of C(B), then the blowup analysis based on the classification of the entire solution, see Sect. 12.7, provides with an alternative proof of the above lemma. The conclusion α0 > 4π, on the other hand, is regarded as an improvement of Brezis-Merle’s rough estimate described in the next paragraph under the cost of V (x) ≥ a > 0.

384

12 Gauge Fields

Sup + Inf Inequality Lemma 12.5.1 implies the sup + inf inequality [335], see [36, 59] for the other version. Theorem 12.5.6 If  ⊂ R2 is a bounded domain, K ⊂  is a compact set, and a, b > 0 are constants, then there are c1 = c1 (a, b) ≥ 1 and c2 = c2 (a, b, dist(K , ∂)) > 0 such that − v = V (x)ev , a ≤ V (x) ≤ b

in 

(12.68)

implies sup v + c1 inf v ≤ c2 . 

K

(12.69)

Proof From the standard covering argument, the assertion is reduced to the case  = B(x0 , r ), K = {x0 }, c2 = c2 (a, b, r ). Assuming x0 = 0, we take v(x) ˜ = v(r x) + 2 log r, which satisfies (12.68) for  = B(0, 1) and V˜ (x) = V (r x). Thus we have only to consider the case  = B = B(0, 1) and K = {0}. Actually, we show, more strongly, v(0) +

c1 2π

 ∂B

vds ≤ c2

(12.70)

in this case. For α0 > 4π in Lemma 12.5.1, we define c1 ≥ 1 by 4π(c1 + 1)/c1 = α0 . In the case of 

V ev > 4π(c1 + 1)/c1 , B

we define r0 ∈ (0, 1) by  B(0,r0 )

V ev = 4π(c1 + 1)/c1

and r0 = 1 otherwise. In any case, we obtain  B(0,r0 )

V ev ≤ 4π(c1 + 1)/c1 .

(12.71)

12.5 Alexandroff-Bol’s Inequality

385

Putting v(x) ˜ = v(r0 x) + 2 log r0 and V˜ (x) = V (r0 x), we obtain (12.68) for  = B = B(0, 1) and also  V˜ ev˜ ≤ 4π(c1 + 1)/c1 = α0 . B

This inequality implies v(0) ˜ = v(0) + 2 log r0 ≤ c0

(12.72)

by Lemma 12.5.1. Now, we define c1 G(r ) = v(0) + 2πr  = v(0) + c1

 ∂ B(0,r ) 2π

v ds + 2(c1 + 1) log r

v(r, θ) dθ + 2(c1 + 1) log r.

0

Since v is super-harmonic, we obtain 1 2πr

 ∂ B(0,r )

v ds ≤ v(0)

and, therefore, G(r0 ) ≤ (c1 + 1)v(0) + 2(c1 + 1) log r0 ≤ (c1 + 1)c0

(12.73)

by (12.72). Thus it holds that (12.70) if r0 = 1, by G(1) = v(0) +

c1 2π

 ∂B

v ds.

In the other case of r0 < 1, we obtain (12.71), and, therefore, it holds that  V ev ≥ 4π(c1 + 1)/c1 , r0 ≤ r ≤ 1. (12.74) B(0,r )

Using 



2(c1 + 1) vr (r, θ) dθ + r

0   1 c1 = vr ds + 2(c1 + 1) r 2π ∂ B(0,r )

G (r ) = c1



386

12 Gauge Fields

and 

 vr ds =

∂ B(0,r )

 B(0,r )

v = −

B(0,r )

V ev ,

we obtain G  (r ) ≤ 0,

r0 ≤ r ≤ 1

by (12.74). This inequality implies v(0) +

c1 2π

 ∂B

v ds = G(1) ≤ G(r0 ) ≤ (c1 + 1)c0 

by (12.73), and the proof is complete.

12.6 Pre-scaled Analysis This paragraph is devoted to Brezis-Merle’s theorem [37] concerning −v = V (x)ev , 0 ≤ V (x) ≤ c0  ev ≤ c1 ,

in 



(12.75)

where  ⊂ R2 is a bounded domain. First, we show Brezis-Merle’s inequality. Theorem 12.6.1 If  ⊂ R2 is a bounded domain, f ∈ L 1 (), and −v = f (x) in ,

v = 0 on ∂,

then it holds that  4π 2 4π − δ |v(x)| d x ≤ (diam )2 , exp  f 1 δ 





where 0 < δ < 4π. Proof We take B = B(x0 , R) containing  for R = 21 diam , 0-extension of f (x) outside , and    1 2R v(x) = log ·  f (x  ) d x  .  |x − x | 2π B

12.6 Pre-scaled Analysis

387

Then, it holds that −v = | f |

in R2

  and v ≥ 0 in B by 2R/ x − x   ≥ 1 for x, x  ∈ B, and, therefore, |v| ≤ v from the maximum principle. This inequality implies 

 

exp

    4π − δ 4π − δ |v(x)| d x ≤ exp v(x) d x.  f 1  f 1 

(12.76)

Next, applying Jensen’s inequality, we obtain       f (x  ) 2R 4π − δ 4π − δ  · log exp v(x) = exp dx ·  f 1 |x − x  |  f 1 2π B  2− δ    2π f (x  )  2R dx , ≤   f 1 B |x − x | 

and the right-hand side of (12.76) is estimated from above by    f (x  ) B

 f 1

dx ·

  B

2R |x − x  |

2−

δ 2π

d x.

Since B = B(x0 , R) ⊂ B(x  , 2R) for any x  ∈ B, it holds that  2− δ      2π f (x  )  2R 4π − δ |v(x)| d x ≤ exp dx · dx   f 1  B  f 1 B(x  ,2R) |x − x | 



=

4π 2 4π 2 (2R)2 = (diam )2 , δ δ 

and the proof is complete. We show the following theorem as an application. Theorem 12.6.2 If 

1 v − ∈ L loc (R2 ), V ∈ L p (R2 ), ev ∈ L p ()

and −v = V (x)ev

in R2 ,

388

12 Gauge Fields

then it follows that v + ∈ L ∞ (R2 ), where 1 < p ≤ ∞, max(±v, 0).

1 p

+

1 p

= 1, and v ± =

Proof Given ε ∈ (0, 1/ p  ), we divide V ev ∈ L 1 (R2 ) by V ev = f 1 + f 2 ,  f 1  L 1 (R2 ) < ε,

f 2 ∈ L ∞ (R2 ).

Fix x0 ∈ R2 , and put Br = B(x0 , r ) for simplicity. We apply Theorem 12.6.1 to δ = 4π − 1 and −vi = f i

in B1 , vi = 0

on ∂ B1 , i = 1, 2,

to obtain  exp (|v1 | /ε) ≤ C, B1

and, in particular, v1  L 1 (B1 ) ≤ C. Here and henceforth, C > 0 denotes a constant independent of x0 , possibly changing from line to line. We have, on the other hand, v2  L ∞ (B1 ) ≤ C from the elliptic regularity, and furthermore, v3 = v − v1 − v2 is a harmonic function in B1 . Using the mean value theorem, therefore, we obtain  + v  3

L ∞ (B1/2 )

  ≤ C v3+  L 1 (B ) . 1

Now, we use v3+ ≤ v + + |v1 | + |v2 | and 

+

R2

v ≤



 e R2

v+

  −1 =

v

{v>0}

to conclude  + v  3

L 1 (B1 )

≤C

e ≤

 R2

ev ≤ C

12.6 Pre-scaled Analysis

389

and, therefore,  + v  3

L ∞ (B1/2 )

≤ C.

Finally, in −v = V ev = V ev1 · ev2 +v3 = g, we have ev2 +v3 ∈ L ∞ (B1/2 ), V ∈ L p (B1 ), ev1 ∈ L 1/ε (B1 ) with 1/ε > p  , and, therefore, g L 1+δ (B1/2 ) ≤ C

(12.77)

with δ > 0. Then, the elliptic regularity guarantees the decomposition v = w + h with w L ∞ (B1/2 ) ≤ C g L 1+δ (B1/2 ) ,

(12.78)

where h is harmonic in B1/2 , and, therefore, it holds that  + h 

L ∞ (B1/4 )

  ≤ C v + 

  ≤ C h +  L 1 (B

1/2 )

L 1 (B1/2 )

 + w L 1 (B1/2 ) ≤ C

(12.79)

by the mean value theorem. We obtain  + v 

L ∞ (B1/4 )

≤C

with C > 0 independent of x0 by (12.77)–(12.79), and the proof is complete.



Similarly, we obtain the following theorem called the rough estimate in the previous paragraph. Theorem 12.6.3 Let 1 < p ≤ ∞, c1 , c2 > 0, 0 < ε0 < 4π/ p  with 1p + p1 = 1 be given,  ⊂ R2 a bounded domain, and K ⊂  a compact set. Then, there exists C > 0 such that in  −v = V (x)ev   + |V (x)| ev ≤ ε0 V  p ≤ c1 , v 1 ≤ c2 , 

390

12 Gauge Fields

implies  + v 

L ∞ (K )

≤ C.

Proof From the covering argument, we may assume  = B R and K = B R/2 , where B R = B(0, R). We decompose v = v1 + v2 , where −v1 = V (x)ev

in ,

v1 |∂ = 0

and v2 is harmonic in . First, we apply the mean value theorem to v2 and obtain  + v  ∞ 2 L (B

3R/4 )

  ≤ C R v2+  L 1 (B

R)

  ≤ C R v1  L 1 (B R ) + c2 .

On the other hand, we have    |v1 |  e 

Lp

 +δ

(B R )

≤C

by Theorem 12.6.1, where δ > 0. Similarly to the proof of the previous theorem, this inequality implies v1  L 1 (B R ) ≤ C and hence  + v  2

≤ C.

L ∞ (B3R/4 )

Consequently, it holds that  v e 

Lp

 +δ

(B3R/4 )

≤ C.

Then, we obtain  v V e 

L q (B3R/4 )

≤C

by the assumption. Using v = w1 + w2 with −w1 = V (x)ev

in B3R/4 ,

w1 |∂ B3R/4 = 0,

and w2 = 0 in B3R/4 , we obtain  + v 

L ∞ (B R/2 )

≤ C,

similarly, from the elliptic estimate and the mean value theorem [144]. The proof is complete. 

12.6 Pre-scaled Analysis

391

Now, we show Theorem 12.2.6 in the following extended form. Theorem 12.6.4 ([37]) Let  ⊂ R2 be a bounded domain, and vk = vk (x), k = 1, 2, . . . satisfy −vk = Vk (x)evk

in 

for Vk = Vk (x) ≥ 0, and assume the existence of 1 < p ≤ ∞ and c1 , c2 > 0 such that Vk  p ≤ c1 ,

 v  e k 

p

≤ c2

for k = 1, 2, . . .. Then, passing to a sub-sequence, we have the following alternatives: (i) {vk } is locally uniformly bounded in . (ii) vk → −∞ locally uniformly in . (iii) There is a finite set S = {ai } ⊂  and αi ≥ 4π/ p  such that vk → −∞ locally uniformly in  \ S and Vk (x)evk d x 



αi δai (d x)

i

in M(). Furthermore, S is the blowup set of {vk } in . Proof Since {Vk evk }k is bounded in L 1 (), we obtain Vk evk d x  μ(d x) in M(), passing to a sub-sequence. Then, the set  

= x0 | μ ({x0 }) ≥ 4π/ p  is finite by μ() ≤ c1 · c2 . / , there is ψ ∈ C0∞ () satisfying 0 ≤ ψ ≤ 1, ψ = 1 around x0 , and If x0 ∈  

ψdμ < 4π/ p  .

Then, we obtain R0 > 0 such that  + v  k

L ∞ (B(x0 ,R0 ))

= O(1)

by Theorem 12.6.3. This means S ⊂ , while the converse is obvious.   conclusion / S, then vk+ is uniformly bounded in B(x0 , R0 ) for some R0 > 0, In fact, if x0 ∈ / by the elliptic regularity. and this property implies μ({x0 }) = 0 and hence x0 ∈ / S. Now, we distinguish two cases. Thus μ({x0 }) < 4π/ p  implies x0 ∈

392

12 Gauge Fields

Case 1: = ∅ implies (i) or (ii).

  Again, we may assume  = B R . In this case, vk+ is uniformly bounded in ω = B R/2 . Defining wk by −wk = f k

in ω,

wk = 0

on ∂ω

for f k = Vk evk , we see that {wk } is uniformly bounded in ω because { f k } is bounded in L p (ω). Thus v˜k = vk − wk   is harmonic in ω and v˜k+ is uniformly bounded in ω. From the Harnack principle to the harmonic function, therefore, {v˜k } is locally uniformly bounded in ω, or otherwise v˜k → −∞ locally uniformly in ω. These cases imply (i) and (ii), respectively. Case 2: = ∅ implies (iii). Since is finite, each x0 ∈ admits R > 0 such that ω = B(x0 , R) ⊂⊂  and B(x0 , R) ∩ = {x0 }. From the above argument, {vk } is locally uniformly bounded or otherwise diverges to −∞ locally uniformly, in B(x0 , R) \ {x0 }. In the former case, it holds that vk ≥ −C

on ∂ω.

Defining z k by −z k = f k

in ω,

z k = −C

on ∂ω,

we obtain vk ≥ z k in ω. We have, on the other hand, f k (x)d x  αδx0 (d x) + f (x)d x in M(ω) with α ≥ 4π/ p  , 0 ≤ f ∈ L 1 (ω), and, therefore, z k → z locally uniformly in ω \ {x0 } with z(x) ≥

1 4π 1 log · − O(1), |x − x0 | p  2π

This property implies    +∞ = e p z ≤ lim inf B(x0 ,R)

k

by Fatou’s lemma, a contradiction.

B(x0 ,R)

x ∈ ω \{x0 }.

  p  e p vk ≤ lim inf evk  p < +∞ k

12.6 Pre-scaled Analysis

393

Thus we obtain vk → −∞ locally uniformly in B(x0 , R)\{x0 }, and Vk evk → 0 p

in L loc (\ ) from the convering argument. This conclusion implies μ(d x) =



αi δai (d x),

i



and the proof is complete.

12.7 Entire Solution Classification of the entire solution is an important ingredient of the blowup analysis, where the method of moving plane is useful. The results for −v = v p , v > 0

in Rn

with 1 < p < ∞ and n ≥ 3 are described in Sect. 7.3. A refined approach guarantees also the two-dimensional version formulated as follows [62]. Theorem 12.7.1 If 

− v = ev

in R2 ,

R2

ev < +∞,

(12.80)

then it holds that 



8μ2

v(x) = log  2 1 + μ2 |x − x0 |2

,

x0 ∈ R2 , μ > 0.

We begin with several lemmas for the proof. Lemma 12.7.1 If (12.80), then it holds that  R2

ev ≥ 8π.

Proof We note that v = v(x) is real-anlaytic. In particular,  t

ev = −



 t

v =

∂t

|∇v| , −

d |t | = dt

 ∂t

1 |∇v|

(12.81)

394

12 Gauge Fields

for each t ∈ R, and, therefore,    d − |t | · ev ≥ |∂t |2 ≥ 4π |t | , dt t   where t = x ∈ R2 | v(x) > t . This implies 

d dt

Operating

t

∞

−∞ ·dt,

 −

e

v

2

e

v

2

   ev =2 e · − t ∂ |∇v|  t d |t | · ev ≤ −8πet · |t | . = 2et · dt t 

we obtain 





∞ d |t | dt e |t | dt = 8π et · dt −∞ −∞  ∞   ∞  1 d t = −8π e dt · ev = 8π ∂t |∇v| −∞ −∞ dt t  = −8π ev .

≤ −8π

R2

v

t

R2



The proof is complete. Lemma 12.7.2 We have v(x) 1 →− log |x| 2π

 R2

ev ≤ −4

(12.82)

uniformly as |x| → ∞. Proof We see that w(x) =  is well-defined by R2

1 2π

 R2

(log |x − y| − log |y|) ev(y) dy

ev < +∞. Furthermore, it holds that

w = ev

in R2 ,

w(x) 1 = |x|→∞ log |x| 2π



lim

R2

ev ,

see [64]. Writing u = v + w, we have u = 0,

u(x) ≤ C + C1 log (|x| + 1)

in R2

12.7 Entire Solution

395

because v(x) is bounded from above by Theorem 12.6.3. Then, Liouville’s theorem assures that v is a constant, and the proof is complete.  Method of the second moment described in Sect. 1.7 guarantees  R2

ev ≤ 8π,

(12.83)

see Sect. 13.1. The radial symmetry of v = v(x), particularly the decreasing in r direction, however, is a key factor in the blowup argument developed in Sect. 12.8. The classification (12.81) is thus obtained by the radial symmetry of the solution, v = v(|x − x0 |)

(12.84)

for some x0 ∈ R2 . This property is obtained if v has the axile symmetry of two Q-independent directions, and, therefore, is reduced to the symmetry of v in x1 direction because of the rotation and translation invariance of (12.80). Once radial symmetry (12.84) is obtained, the proof of Theorem 12.7.1 is obvious. Thus (12.80) implies (12.81) and the proof will be complete. To show the symmetry of v in x1 -direction, we assume the maximum of v in x1 < −3, regarding Lemma 12.7.2. Then, we put

λ = {(x1 , x2 ) | x1 < λ} , Tλ = ∂ λ = {(x1 , x2 ) | x1 = λ} x λ = (2λ − x1 , x2 ), wλ (x) = v(x λ ) − v(x), wλ (x) = wλ (x)/g(x) for λ ∈ R, where g(x) = log (|x| − 1). This wλ (x) is well-defined for x ∈ λ with λ < −2, and it follows that −wλ = eψ wλ   2 g wλ = 0, wλ + ∇g · ∇w λ + exp ψ + g g

(12.85) (12.86)

where ψ(x) is between v(x) and v(x λ ). Lemma 12.7.3 There is R0 > 2 independent of λ < −2 such that if x0 attains the negative minimum of w λ in λ , then |x0 | < R0 . Proof Since g −1 = g |x| (|x| − 1)2 log (|x| − 1) we obtain R0 > 2 such that exp v +

g 0 in λ for λ < λ0 and (b) wλ0 ≡ 0 in λ0 .

∂v ∂x1

> 0 in λ0 .

For this purpose, we use ∂v ≥ 0, ∂x1

x 1 ≤ λ0

(12.89)

and wλ ≡ 0



∂v = 0 somewhere on Tλ ∂x1

(12.90)

for λ ≤ λ0 , the latter being obtained by the strong maximum principle and the Hopf lemma.

12.7 Entire Solution

397

To prove (a), first, we show wλ > 0 in λ if λ < λ0 . In fact, if this property is not the case, it holds that wλ0 −δ ≡ 0 for δ > 0 from the strong maximum principle, and, therefore, v(λ0 − 2δ, x2 ) = v(λ0 , x2 ). This conclusion implies ∂v = 0, ∂x1

λ0 − 2δ ≤ x1 ≤ λ0

by (12.89). In particular, we obtain ∂v =0 ∂x1

on Tλ0 −δ

and hence wλ0 −2δ ≡ 0 by (12.90). Repeating the argument, we obtain the independence of v in x1 , a contradiction. Once this property is proven, it holds that ∂wλ 0, ∂x1

x 1 < λ0 ,

∂v and hence ∂x > 0 in λ0 . 1 We have proven that v is strictly increasing in x1 < λ0 , and hence λ0 < −3. We can define w λ in λ for λ < λ0 + 1. We have also proven wλ0 ≥ 0, and, therefore, in case wλ0 ≡ 0, it holds that

w λ0 > 0

in λ0 ,

∂w λ0 0. Proof We obtain xk ∈ B such that vk (xk ) = max vk , xk → 0, vk (xk ) → +∞. B

This relation implies δk = e−vk (xk )/2 → 0 and v˜k (x) = vk (δk x + xk ) + 2 log δk is defined in B(0, R/(2δk )), which satisfies −v˜k = Vk (δk x + xk )ev˜k , v˜k ≤ 0 = v˜k (0)  ev˜k ≤ C0 . B(0,R/(2δk ))

in B(0, R/(2δk ))

12.8 Blowup Analysis

399

Given r > 0, we obtain the well-definedness of {v˜k } in Br = B(0, r ), and Theorem 12.6.4 is applicable to  = Br . The other cases than (i) are impossible, and, therefore, {v˜k } is uniformly bounded in B(0, r ). The standard diagonal argument and the elliptic estimate guarantee the convergence 1,α in Cloc (R2 ),

v˜k → v˜ where 0 < α < 1, and it holds that

˜ −v˜ = V (0)ev˜ , v˜ ≤ 0 = v(0)  ev˜ ≤ C0 .

in R2

R2

In case V (0) = 0, v˜ is a non-negative harmonic function in R2 , and hence is a constant by Liouville’s theorem. This property implies  R2

ev˜ = +∞, 

a contradiction. In the above lemma, we have v(x) ˜ = log  1+



1 V (0) 8

|x|2

2 ,

R2

V (0)ev˜ = 8π

by Theorem 12.7.1. In fact, v(x) = v(x) ˜ + log V (0) satisfies −v = ev in R2 ,

 R2

ev < +∞, max v = v(0) = log V (0). R2

Taking smaller R > 0, we may assume a ≤ Vk (x) ≤ b,

x∈B

(12.92)

for a, b > 0. This condition makes it possible to apply sup + inf inequality, which takes the role of the Harnack inequality or the monotonicity formula. Then, we can exhaust the blowup mechanism by counting the particle concentrations. Lemma 12.8.2 Given a, b > 0, we obtain c1 ≥ 1, c2 > 0 (independent of R > 0) such that −v = V (x)ev , a ≤ V (x) ≤ b

in B R

400

12 Gauge Fields

implies v(0) + c1 inf v + 2(c1 + 1) log r ≤ c2 ∂ Br

for 0 < r ≤ R. Proof The assertion is an immediate consequence of the sup + inf inequality and the rescaling. In fact, we take v(x) ˜ = v(r x) + 2 log r for r ∈ (0, R], and obtain −v˜ = V (r x)ev˜

in B1 .

Theorem 12.5.6 applied to  = B1 and K = {0}, thus implies v(0) ˜ + c1 inf v˜ ≤ c2 B1

or, equivalently, v(0) + 2(c1 + 1) log r + c1 inf v ≤ c2 . Br

Here, we have inf v = inf v Br

∂ Br

because v is super-harmonic, and the proof is complete.



We use the following lemma also, obtained by the Harnack inequality for harmonic functions. Lemma 12.8.3 There is β ∈ (0, 1) such that given C1 , C2 > 0, we have C3 > 0 (independent of R, R0 > 0 in 0 < R0 ≤ R/4) such that −v = V (x)ev , |V (x)| ≤ C1 , v(x) + 2 log |x| ≤ C2

in B R \ B R0

implies sup v ≤ C3 + β inf v + 2(β − 1) log r, 2R0 ≤ r ≤ R/2. ∂ Br

∂ Br

Proof Taking r ∈ [2R0 , R/2], we put v(x) ˜ = v(r x) + 2 log r

12.8 Blowup Analysis

401

and obtain −v˜ = V (r x)ev˜ v(x) ˜ = v(r x) + 2 log (r |x|) − 2 log |x| ≤ C2 + 2 log 2     in B2 \ B1/2 . V (r x)ev˜  ≤ C1 exp (C2 + 2 log 2) Then, we have C4 determined by C1 , C2 such that |w| ≤ C4

in B2 \ B1/2 ,

where −w = V (r x)ev˜

in B2 \ B1/2 ,

w|∂(B2 \B1/2 ) = 0.

The Harnack inequality is valid to the non-negative harmonic function h = w − v˜ + C5 with C5 = C4 + C2 + 2 log 2, we obtain the absolute constant β ∈ (0, 1) such that β sup h ≤ inf h. ∂ B1

∂ B1

The right-hand and the left-hand sides are estimated from above and from below by C5 + C4 − sup v˜ = C5 + C4 − 2 log r − sup v ∂ B1

∂ Br

and   β C5 − C4 − 2 log r − inf v , ∂ Br

respectively. Then, we obtain sup v ≤ (1 − β)C5 + (1 + β)C4 + 2(β − 1) log r + β inf v ∂ Br

and the proof is complete.

∂ Br



We are ready to prove the key inequality [215]. Theorem 12.8.1 Given a, b > 0, C1 > 0, we obtain γ > 0, C2 > 0 independent of 0 < R0 ≤ R/4 such that

402

12 Gauge Fields

−v = V (x)ev , a ≤ V (x) ≤ b v(x) + 2 log |x| ≤ C1

in B R in B R \ B R0

(12.93)

implies ev(x) ≤ C2 e−γv(0) · |x|−2(γ+1)

(12.94)

for 2R0 ≤ |x| ≤ R/2. Proof Lemmas 12.8.2 and 12.8.3 guarantee inf v ≤

∂ Br

c2 1 1 − v(0) − 2(1 + ) log r, 0 < r ≤ R c1 c1 c1

and     β c2 β − v(0) − 2 sup v ≤ c3 + β · + 1 log r, 2R0 < r ≤ R/2, c1 c1 c1 ∂ Br respectively, and, therefore, (12.94) holds for   c2 . γ = β/c1 , C2 = exp c3 + β · c1 

The proof is complete. Here, we give the estimate from below in Theorem 12.2.7.

Lemma 12.8.4 Under the assumption of Theorem 12.2.7, passing to a subsequence, we have m ∈ N, 1 ≤ m ≤ V (0) · C0 /(8π)   j j xk ⊂ B R , lim xk = 0, 0≤ j ≤m−1 k→∞   j j σk , lim σk = +∞, 0≤ j ≤m−1 k→∞

such that j

vk (xk ) =

max vk → +∞

j j j (x ) σ k δk k

(12.95)

B j

i = j B2σ j δ j (xk ) ∩ B2σi δi (xki ) = ∅, k k k k  ∂ j  j j j vk (t x + xk ) < 0, δk ≤ |x| ≤ 2σk δk ∂t t=1

(12.96) (12.97)

12.8 Blowup Analysis

403

 lim

k→∞ B

j

j (x k )

j 2σk δk

Vk evk = lim



k→∞ B

j

j (x k )

j σ k δk

Vk evk = 8π

(12.98)

   j max vk (x) + 2 log min x − xk  ≤ C,

(12.99)

j

x∈B R j

where δk = e−vk (xk )/2 . j

Proof We take xk in the proof of Lemma 12.8.1, denoted by xk0 . Thus it holds that xk0 ∈ B R , vk (xk0 ) = max vk , xk0 → 0, vk (xk0 ) → +∞ v˜k0

→ v˜

in

BR 1,α 2 Cloc (R ),

0 0 holds to a subsequence, then it holds that  (12.111) V˜k ev˜k → β j j B1/2 (x˜k )

12.8 Blowup Analysis

409

by (12.109) and (12.110). In the other case of j lim r˜ k→∞ k

= 0,

we apply the lemma with m = 1. Then, (12.111) follows again. By (12.110) and (12.111), it holds that  lim

k→∞ B4 A



V˜k ev˜k = lim

k→∞ B2 A

V˜k ev˜k =

m−1 

βj,

j=0

which means (12.107). Case 2 The other case. There is non-void J ⊂ {0, 1, . . . , m − 1}, 1 ≤  ≤ m, such that m !

J ∩ Jm = ∅,  = m,     j xk − xki  ≤ Adk ,    j lim xki − xk  /dk = ∞,

J = {0, 1, . . . , m − 1}

=1

k→∞

i, j ∈ J i ∈ J , j ∈ Jm ,  = m.

Taking x ∈ J , we obtain  lim

k→∞ B2 Ad (x ) k

Vk e

vk

 = lim

k→∞ B Ad (x ) k

Vk evk =



βj

j∈J

similarly to the case 1. On the other hand, we can apply the lemma for m = k:  lim

k→∞ B R

Vk evk =

k   =1 j=J

βj =

m−1 

βj.

j=0

The proof is complete.



12.9 Summary We studied two-dimensional stationary mass quantization in detail. 1. A typical profile of self-assembly, the quantized blowup mechanism, is observed in the stationary state of self-gravitating particles, the mean field of turbulence, and particularly, in the self-dual gauge field.

410

12 Gauge Fields

2. This profile is controlled by the exponential nonlinearity competing twodimensional diffusion. 3. This equation is provided with several mathematical structures, complex function theory, theory of surfaces, real analysis, calculus of variation, spectral analysis, and particularly, blowup analysis. 4. Blowup analysis is based on the self-similarity of the problem, and uses hierarchical arguments. 5. For this purpose, it is necessary to apply the methods of second moment, moving planes, or the Harnack inequality to envelope the blowup mechanism.

Chapter 13

Higher-Dimensional Blowup

Mass and energy quantizations are observed even in higher-space dimension. The methods of scaling and duality still work. Here we describe the method of duality applied to higher-dimensional problems, the higher-dimensional stationary mass quantization arising from the scaling invariance, and the general dimension control of the blowup set.

13.1 Method of Duality The following two paragraphs are concerned with the higher-dimensional concentration to the stationary state. Here, we provide preliminary considerations on the method of duality. This method has played a fundamental role in the study of the chemotaxis system, see Sect. 13.1, based on the symmetry of the Green’s function (1.30) resulting in the weak formulation (1.46). Pohozaev Identity Let  ⊂ Rn be a bounded domain with smooth boundary ∂, and v = v(x) be a smooth solution to − v = f (v) in ,

v|∂ = 0,

(13.1)

where f = f (s) is a continuous function of s ∈ R. In this case it holds that 

1 2−n f (v)v dx = nF(v) + 2 2 

 ∂



∂v ∂ν

2

© Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3_13

(x · ν),

(13.2)

411

412

13 Higher-Dimensional Blowup

where  F(s) =

s

f (s )ds .

0

Equality (13.2) is called the Pohozaev identity [302] and has a variety of applications [360]. Regarding v and u = f (v) as the field density and the particle distribution, respectively, we obtain  v(x) =



G(x, x  )u(x  )dx  , u∇v = ∇F(v)

and hence ∇ · (∇F(v) − u∇v) = 0,

(13.3)

where G = G(x, x  ) denotes the Green’s function: −x G(·, x  ) = δx in ,

 G(·, x  )∂ = 0.

Given a C 2 -function ψ = ψ(x) of x ∈ , we have  ∇ · (∇F(v) − u∇v)ψ 0=    ∂ ∂v ∂ψ F(v) − u ψ − F(v) ds = ∂ν ∂ν ∂ν ∂  + F(v)ψ + u∇v · ∇ψ dx. 

Here, the boundary integrals on the right-hand side vanish by F(0) = 0, F  (v) = f (v) = u so that  

F(v)v +

1 2

 ×

ρψ (x, x  )u ⊗ u = 0

by the method of symmetrization, where ρψ (x, x  ) = ∇ψ(x) · ∇x G(x, x  ) + ∇ψ(x  ) · ∇x G(x, x  ).

(13.4)

13.1 Method of Duality

413

Since  

∇ψ(x) · ∇x G(x, x  )(−v(x  ))dx  = ∇ψ · ∇v(x),

it holds that  1 ρψ (x, x  )v(x)v(x  )dxdx  2 ×  = ∇ψ(x) · ∇x G(x, x  )(−v(x  ))(−v(x))dxdx  ×   ∂v + = (∇ψ · ∇v, −v) = − ∇ψ · ∇v ∇(∇ψ · ∇v) · ∇v ∂ν ∂     2  ∂v ∂ψ  + ψij vi vj + ψi vij vj , =− ∂ν ∂ ∂ν   i,j

where wi =

i,j

∂w ∂2w , wij = , and so forth. ∂xi ∂xi ∂xj

Using  Iij =

 

ψi vij vj =



∂

ψi νi vj2 −





vj (ψi vj )i =

∂

 ψi νi vj2 −



ψii vj2 − Iij ,

we have  i,j

1 ψi vij vj = 2 



 ∂

∂v ∂ν

2

∂ψ 1 − ∂ν 2

 

ψ|∇v|2

and, then it follows that  1 ρψ (x, x  )v(x)v(x  )dxdx  2 ×   2   ∂v 1 ∂ψ 1 2 − =− ψ|∇v| + ψij vi vj . 2 ∂ ∂ν ∂ν 2  

(13.5)

i,j

We obtain  

F(v)ψ +

 i,j

1 ψij vi vj = 2 



 ∂

∂v ∂ν

2

1 ∂ψ + ∂ν 2

by (13.4) and (13.5), and hence (13.2), putting ψ = |x|2 .

 

ψ|∇v|2

(13.6)

414

13 Higher-Dimensional Blowup

Concentration Mass Estimate We can show that (12.80) implies (12.83) by the method of symmetrization. For this purpose, we apply (12.82) and confirm the well-definedness of 1 2π

w(x) =

 log R2

1  · ev(x ) dx  |x − x  |

(13.7)

with the estimate |w(x)| ≤ C(1 + log(1 + |x|)).

(13.8)

Then, Liouville’s theorem guarantees, see [305], v = w + constant by (v − w) = 0, (12.82), and (13.8). Writing u = ev ,

(13.9)

we have ∇u = u∇v = u∇w and hence ∇ · (∇u − u∇w) = 0

in R2 .

(13.10)

Equations (13.7), (13.9), and (13.10) comprise of the stationary system of chemotaxis, and therefore, we obtain   1 ϕ(x) · u(x)dx + ρ0 (x, x  )u(x)u(x  )dxdx  = 0 (13.11) 2 R2 ×R2 ϕ R2 for ϕ ∈ C02 (R2 ), where ρ0ϕ (x, x  ) = −

1 (∇ϕ(x) − ∇ϕ(x  )) · (x − x  ) · . 2π |x − x  |2

Using c = c(s) of (1.62), we have δ > 0 determined by M such that  R2

  c(|x|2 ) + 1 u(x)dx ≥ δ

13.1 Method of Duality

415

in case 

v

R2

e =

 R2

u = M > 8π

similarly to [206] or Sect. 1.7. This time, |y|2 /4 factor of the right-hand side of (1.72) does not appear, and we obtain the scaling uμ (x) = μ2 u(μx),

vμ (x) = v(μx) + 2 log μ.

Then it follows that   M= u= uμ 2 R2    R  2 c(|x| ) + 1 uμ (x)dx = R2

R2

  c(μ−1 |x|2 ) + 1 u(x)dx

and therefore, we can find μ 1 satisfying  R2

  c(|x|2 ) + 1 uμ (x)dx < δ,

a contradiction. Two-Dimensional Mass Quantization Revisited Putting  w = v + log λ − log



ev

in (12.15), we obtain − w = ew in ,

 w| = constant,



ew = λ

(13.12)

for  = ∂. If w = w(x) solves (13.12), conversely, then v = w − w is a solution to (12.15). Theorem 12.2.1 then implies the quantized blowup mechanism to (13.12). Theorem 13.1.1 ([376]) Assume that  ⊂ R2 is a bounded domain with smooth

k boundary ∂ and (λk , w ) is a solution sequence to (13.12) satisfying λk → λ0 . Then passing to a subsequence the following alternatives hold: 1. w k ∞ = O(1). 2. sup w k → −∞.

416

13 Higher-Dimensional Blowup

3. λ0 = 8π for some  ∈ N, and there exist xj∗ ∈ , j = 1, . . . , , satisfying (12.18) and xk → xj∗ , such that x = xk is a local maximum point of w k = w k (x),

j w k (xk ) → +∞, w k → −∞ locally uniformly in  \ x1∗ , . . . , x∗ , and j

j

ew dx  k



8πδxj∗ (dx)

in M().

j





Thus S = x1∗ , . . . , x∗ is the blowup set of w k . If we take u = ew in (13.12), then it follows that  ∇u = u∇w, w − w = G(·, x  )u(x  )dx  , 

where G = G(x, x  ) denotes the Green’s function for − provided with the boundary condition ·|∂ = 0. This relation implies 

 

u∇ · ψ +

×

ψ(x) · ∇x G(x, x  )u(x)u(x  )dxdx  = 0

(13.13)

for any ψ ∈ C02 ()2 . Equality (13.13) is the dual weak formulation of (13.12). Using this formula, we can show that if {(w k , λk )} is a solution sequence to (13.12) satisfying ew dx  k



m(x0 )δx0 (dx)

in M()

x0 ∈S

with m(x0 ) > 0, then it holds that m(x0 ) = 8π and (12.18). The proof is similar to [277]. Mass quantization, conversely, implies the residual vanishing, see [278, 364].

13.2 Higher-Dimensional Quantization Problem (13.12) is regarded as a free boundary problem associated with the plasma confinement, where {w > 0} indicates the plasma region [120]. The higher-dimensional mass quantization is observed in an analogous free boundary problem q

−w = w+ in ,  q w+ = λ, 

w = constant on  (13.14)

where  ⊂ Rn , n ≥ 3 is a bounded domain with smooth boundary ∂ = , and n q = n−2 . This problem describes the stationary state of the degenerate parabolic

13.2 Higher-Dimensional Quantization

417

equation derived from the kinetic theory, see Sect. 10.3. It is also the equilibrium self-gravitating fluid equation described by the field component, and the problem of plasma confinement, see Sect. 7.3. First, we obtain vk = w k − wk ≥ 0 in  by the maximum principle similarly to the two-dimensional case. Here, the quantized value m∗ > 0 is defined by  m∗ =

Uq B

for U = U(x) satisfying −U = U q , U > 0 in B,

U = 0 on ∂B

with B = B(0, R). As is examined in Sect. 7.3, this U = U(x) is radially symmetric and exists uniquely for each R > 0, while m∗ is independent of R > 0. In the following theorem, G = G(x, x  ) denotes the Green’s function of − on  with the Dirichlet boundary condition and

R(x) = G(x, x  ) − (x − x  ) x =x , where (x) =

1 ωn (n − 2) |x|n−2

is the fundamental solution to − and ωn is the (n − 1) dimensional volume of the boundary of the unit ball in Rn . n Theorem 13.2.1 ([376, 383])

If  ⊂ R , n ≥ 3 is a bounded domain withn smooth k boundary ∂ and (λk , w ) is a solution sequence to (13.14) with q = n−2 satisfying λk → λ0 , then passing to a subsequence the following alternatives hold: 1. w k ∞ = O(1). 2. sup w k → −∞. j 3. λ0 = m∗  for some  ∈ N, and there exist xj∗ ∈  and xk → xj∗ (j = 1, . . . , ),



∗ where S = x1 , . . . , x∗ ⊂  coincides with the blowup set of w k on  j satisfying (12.18), x = xk is a local maximum point of w k = w k (x), w k j (xk ) → +∞, w k → −∞ locally uniformly in  \ S, and q

w k (x)+ dx 



m∗ δxj∗ (dx)

in M().

j

There are several results on the actual existence of the solution sequence described in the above theorem [410, 411, 419].

418

13 Higher-Dimensional Blowup

Local version comparable to Theorems 3.4.1 and 3.4.2 also holds. Actually, there are ε-regularity, self-similarity, classification of the entire solution, and sup + inf inequality. These structures guarantee the following theorem similarly to the twodimensional case of Theorem 12.2.5. A slight difference to Theorem 13.1.1 is that the entire solution q

−w = w+ , w ≤ w(0) = 1  q w+ < +∞

in Rn

Rn

q

provided with a compact support to w+ dx. This property, however, does not cause any difficulties. Theorem 13.2.2 ([409]) If  ⊂ Rn , n ≥ 3 is a bounded domain and w = w k , k = 1, 2, . . ., satisfies 

q

−w = w+ in ,

q



w+ ≤ C

n and C > 0, then passing to a subsequence, we obtain the following for q = n−2 alternatives.

1. w k is locally uniformly bounded in . 2. w k → −∞ locally uniformly in . j j 3. There exist  ∈ N, xj∗ (j = 1, . . . , ), and xk → xj∗ such that x = xk is a local j

k k k k maximum ∗ point of∗ w = w (x), w (xk ) → +∞, w → −∞ locally uniformly in  \ x1 , . . . , x , and q

w k (x)+ dx 



m∗ nj δxj∗ (dx)

in M(),

j

where nj ∈ N. Here, we emphasize that Theorem 13.2.2 is used in the proof of Theorem 13.2.1, and we recall that Theorem 12.2.1 is proven independently of Theorem 12.2.5. To exclude boundary blowup points in Theorem 13.2.1, we use Kazdan-Warner’s identity. We have readily the finiteness of the blowup set S on . First, we assume x0 ∈ ∂ ∩ S take 0 < R  1 such that ω ∩ S = {x0 }, where ω =  ∩ B(x0 , R). q q+1 Putting f (w) = w+ and F(w) = w+ /(q + 1), we have 

 −w · ∇w = k

ω

k

 f (w )∇w = k

ω

k

 ∇F(w ) = k

ω

∂ω

ν · F(w k ).

(13.15)

13.2 Higher-Dimensional Quantization

419

The right-hand side of (13.15) vanishes for k 1, because w k → −∞ loc. unif. in  \ S, w k = ck → −∞ on ∂. The left-hand side of (13.15) is identically equal to  −

∂ω

∂w k 1 ∇w k + |∇w k |2 ν ds ∂ν 2

and hence 1 2

 ∂ω∩∂

  2   ∇w k  ν =

∂ω∩



2 ∂w k 1   ∇w k + ∇w k  ν ds ∂ν 2

for k 1. Thus we obtain  ∂ω∩∂

 2   ∇w k  = O(1)

(13.16)

because ∂ω ∩  ∩ S = ∅. Second, we have  m(x0 )∇G(·, x0 ) locally uniformly in  \ S ∇w k → x0 ∈S

with m(x0 ) ≥ m∗ for any x0 ∈ S, which contradicts (13.16) by Fatou’s theorem.

13.3 Dimension Control of the Blowup Set Even the general blowup mechanism in high-space dimensions is enclosed by smaller sets. In the final paragraph, we describe a simple case of [377], see also [316] for the other result. First, given 1 ≤ p < n, we put ∗

K p = {f ∈ L p (Rn , R) | ∇f ∈ L p (Rn , R)} for

1 p∗

=

1 p

− n1 , and define the p-capacity of a subset A ⊂ Rn by  Capp (A) = inf

Rn

 |∇f |p | f ≥ 0, f ∈ K p , A ⊂ {f ≥ 1}◦ ,

420

13 Higher-Dimensional Blowup

where B◦ denotes the interior of B ⊂ Rn . This Capp is an outer measure on Rn satisfying Capp (A) ≤ CH n−p (A),

A ⊂ Rn ,

where H s denotes the s-dimensional Hausdorff measure: H s (A) = lim Hδs (A) δ↓0 ⎧ ⎫ ∞ ∞ ⎨ ⎬  Hδs (A) = inf α(s)rjs | A ⊂ B(xj , rj ), rj < δ ⎩ ⎭ j=1

α(s) =

π s/2 ( 2s − 1)



, (s) =

j=1



e−t t s−1 dt.

0

We have H n−p (A) < +∞, 1 < p < n Capp (A) = 0, s > n − p





Capp (A) = 0

H (A) = 0, s

see [101]. Theorem 13.3.1 Let  ⊂ Rn , n ≥ 3 be a bounded open set, T > 0, and u = u(x, t) :  × [0, T ] → (−∞, +∞] be a continuous function satisfying 

D=

D(t) × {t} ⊂  × [0, T ]

0≤t≤T

ut − u ≥ 0

in  × (0, T ) \ D,

(13.17)

where D(t) = {x ∈  | u(x, t) = +∞} and suppose that u = u(x, t) is Lipschitz continuous near ∂ uniformly in t ∈ [0, T ]. Then, it holds that  0

T

Cap2 (D(t))dt ≤

L n () . 2

(13.18)

13.3 Dimension Control of the Blowup Set

421

The second relation of (13.17) is taken in the distributional sense, so that   uϕt + uϕ ≤ 0 ×(0,T )\D

×(0,T )\D

for any ϕ = ϕ(x) ≥ 0 in ϕ ∈ C0∞ ( × (0, T ) \ D). If u is independent of t, then we obtain Cap2 (D) = 0 by (13.18), where D = D(t). If  is convex, on the other hand, we have the boundary estimate for solutions to the semilinear elliptic equation, see [81, 135]. Then, the following theorem follows from the proof of Theorem 13.3.1, where dH (A) denotes the Hausdorff dimension: dH (A) = inf{s ≥ 0 | H s (A) = 0}. Theorem 13.3.2 Let  ⊂ Rn , n ≥ 3 be a bounded convex domain, f = f (u, λ) be smooth in u ≥ 0, and {(uk , λk )}k be a solution sequence to −u = f (u, λ) ≥ 0 in , satisfying

u|∂ = 0

 

f (uk , λk )dx ≤ C, uk ∞ → +∞.

Then, it holds that Cap2 (S) = 0 and in particular, dH (S) ≤ n − 2, where S = {x0 ∈  | ∃xk → x0 such that uk (xk ) → +∞} denotes the blowup set. We have uk → +∞ in the case of

locally uniformly in 

 

f (uk , λk )dx → +∞.

The proof of this fact of entire blowup is similar to the two-dimensional case, see [254] and Sect. 12.4.

422

13 Higher-Dimensional Blowup

13.4 Summary Although the non-stationary blowup mechanism in higher-space dimension is a problem in future, we have several evidences as well as new overviews for mass and energy quantizations. 1. Method of scaling is valid to clarify higher-dimensional blowup mechanism. 2. There is mass quantization for the stationary higher-dimensional state to the free boundary problem with critical exponent. 3. A general dimension control of the blowup set holds in terms of the capacity.

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Index

A Angiogenesis, 51 Angular velocity, 270 Attractor, 113

B Barotoropic, 207 Bounded variation, 117 Boyle-Charles law, 274

C Chemical potential, 111, 144, 146 Chemital affinity, 146 Closed system —materially, 277 —thermally, 112 —thermally-materially, 112, 170, 276 Concentration-compactness principle, 27 Constant —Boltzmann, 275 —Planck, 276 -convergence, 117 Convex, 88 Cost function, 132

D Detailed balance, 165 Direction derivative, 4 Dissipation (open system), 144, 159, 279 —function, 161, 165 Double-well potential, 114 Dual problem, 133

Duality —Fenchel-Moreau, 88 —Hardy-BMO, 103 —Kuhn-Tucker, 90, 132, 133 —Toland, 90, 93, 95 —map, 138

E Energy —balance, 142 —minimum principle, 166 Ensemble, 276 Enthalpy, 143, 148 Entropy, 167, 278 —balance, 160 —decomposition, 142 —density, 166 —dual functional, 100 —flux, 160 —functional, 101 —increase, 142, 168 —production, 142, 159–161 —state quantity, 142 —variation, 142 Equality —Gibbs-Duhem, 145 Equation —Allen-Cahn, 114 —Bogomol’nyi, 356 —Cahn-Hilliard, 116 —Chapmann-Kolmogorov, 298 —Eguchi-Oki-Matsumura, 122 —Euler equation of motion, 205 —Euler-Poisson, 216

© Atlantis Press and the author(s) 2015 T. Suzuki, Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model, Atlantis Studies in Mathematics for Engineering and Science 11, DOI 10.2991/978-94-6239-154-3

441

442 —Falk, 179 —FitzHugh-Nagumo, 130 —Fokker-Planck, 11, 302 —Geselowitz, 248 —Gierer-Meinhardt, 123 —Grad-Shafranov, 262 —Keller-Segel, 6, 47 —Kramers, 302 —Langevin, 11, 300 —Maxwell, 350, 353 —Mullins-Sekerka-Hele-Shaw, 118 —Newton, 349 —Nishiura-Ohnishi, 119 —Smoluchowski, 303 —Stokes, 136 —canonical, 350 —coupled Cahn-Hilliard, 174 —gauge-Shrödinger, 358 —heat, 5 —master, 299 —mean curvature, 116 —nonlinear Schrödinger, 353 —state, 141, 216, 311 —turbulence mean field, 282 —vortex, 273 Equilibrium, 115, 145, 146, 151, 161–163, 165, 283 —concentration, 163, 164 —constant, 164 —far-from, 157 —local, 159, 160 —near-from, 162 —plasma, 221, 223, 225, 237, 238, 241 —statistical mechanics, 277 Ergodic hypothesis, 277

F Faithful, 94 Flow —harmonic heat, 60 —normalized Ricci, 65 Formula —Bochner-Weitzenböck, 67 —Green, 5, 375 —Stirling, 275 —coarea, 115, 276, 282, 375 —divergence, 1, 4, 149 —monotonicity, 62, 399 Free energy —Fix-Caginalp, 151 —Gibbs, 143, 144 —Ginzburg-Landau, 113, 116

Index —Helmholtz, 11, 113, 143, 278, 282, 291 —Ohta-Kawasaki, 119, 131 —Penrose-Fife, 168, 176, 178

G Gauge covariant derivative, 352 Gel’fand triple, 100 Gibbs measure, 277, 278, 280 Gradient system, 99, 120, 122 —skew, 121, 123, 126, 129 Green’s function, 11 —regular part, 14

H Hamiltonian, 350 Harmonic map, 61 Hetero-clinic cascade, 115 High-energy limit, 281 Hook term, 90

I Identity —Fenchel-Moreau, 91 Index, 94 —anti, 94 Inequality —Alexandroff, 380 —Bandle, 370 —Benilan-Crandall, 69 —Bol, 374 —Brezis-Merle, 386 —Chang-Yang, 15 —Clausius-Duhem, 142 —Harnack, 399 —Moser-Fontana, 69 —Moser-Onofri-Hong, 289 —Nehari, 374 —Pohozaev-Trudinger-Moser, 289 —Poincaré, 321 —Shafrir, 381 —Trudinger-Moser, 16 —Wang-Ye, 326 —spherical Harnack, 378 —sup + inf, 384, 399 Inverse tempeature, 278 Isentropic, 206

K Kramers-Moyal expansion, 299

Index L Lagrangian —Kuhn-Tucker, 90 —Toland, 89 —semi-Kuhn-Tucker, 129 —semi-Toland, 127 Law —Bernoulli, 210 —action-reaction, 11 Lemma —Sard, 375 Level set approach, 116, 147 ω-limit set, 98, 99, 115 Linear programing, 135 Liouville property, 115 Lower semi-continuous, 88

M Material derivative, 204 Maximum monotone graph, 149, 150, 172 Mean field —equation, 113, 290, 291 —hierarchy, 11 —stream function, 293 —turbulence, 281 Mendric permutation, 115 Metastable region, 119 Minimality, 91, 125 Model —(A), 112 —(A)-stationary, 112, 113 —(B), 112 —(B)-stationary, 112, 113 —(C), 151 Morse-Smale property, 115 Moving plane, 395 Mushy region, 151

443 Phenomenological relation, 6, 162 Principal problem, 133 Principle —Harnack, 379 —equal a priori probabilities, 274 —equal a priori probability, 275 Propagation of chaos, 282 Proper, 88

Q Quantity of state, 141

R Reciprocity, 166 Reduced pdf, 281 Regularity —ε, 16, 48, 62, 228, 325, 418 —elliptic, 233, 388, 389, 391 —parabolic, 15, 33, 172 —regularity, 227

O Order parameter, 111, 150

S Saddle, 125, 132, 134 Scaling, 14, 24, 43, 363 Second moment, 17 Self-organization, 132 —bottom-up (self-assembly), 15, 124 —top-down (dissipative structure), 159 Self-similar —backward transformation, 24, 322 —blowup, 228, 243 —forward solution, 320 —forward transformation, 320 Shadow system, 123 Singular limit, 15, 359 Space —Zygmund, 104 Spinodal —decomposition, 118 —interval, 118 Statistical mechanics —canonical, 277 —grand-canonical, 279 —micro-canonical, 276 Stefan problem, 148 Stream function, 272 Sub-differential, 90

P Perimeter, 117

T Tangential force, 203

N Newton-Fourier-Fick’s law, 5, 147, 162, 178 Norm —Luxemburg, 105 —L p , 6 Nucleation, 118

444 Theorem —Lagrange, 207 Theory —Chern-Simons, 354 —Ginzburg-Landau, 111, 147, 150 —Penrose-Fife, 166 —dynamical system, 66 —game, 132 —gauge field, 349 —nonlinear semi-group, 147, 150 —phase field, 150 —statistical mechanics, 280 —thermodynamics, 151 —transport, 1 Thermal weight factor, 275 Thermodynamics —equilibrium, 141 —non-equilibrium, 159 Transformation —conformal, 224, 369 —group, 66

Index —Kirchhoff, 148 —Legendre, 88, 89, 93, 136, 166 —scaling, 326, 363 —second Legendre, 88, 101 —self-similar, 14, 327 —SW, 104, 372 Turbulence, 280

U Unfolding, 91, 125

V Van der Waals penalty, 115 Vorticity, 269

W White noise, 301