Emerging Topics On Differential Equations And Their Applications - Proceedings On Sino-japan Conference Of Young Mathematicians : Proceedings on Sino-Japan Conference of Young Mathematicians 9789814449755, 9789814449748

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EMERGING TOPICS ON DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS

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NANKAI SERIES IN PURE, APPLIED MATHEMATICS AND THEORETICAL PHYSICS Editors:

S. S. Chern, C. N. Yang, M. L. Ge, Y. Long

Published: Vol. 1

Probability and Statistics eds. Z. P. Jiang, S. J. Yan, P. Cheng and R. Wu

Vol. 2

Nonlinear Anaysis and Microlocal Analysis eds. K. C. Chung, Y. M. Huang and T. T. Li

Vol. 3

Algebraic Geometry and Algebraic Number Theory eds. K. Q. Feng and K. Z. Li

Vol. 4

Dynamical Systems eds. S. T. Liao, Y. Q. Ye and T. R. Ding

Vol. 5

Computer Mathematics eds. W.-T. Wu and G.-D. Hu

Vol. 6

Progress in Nonlinear Analysis eds. K.-C. Chang and Y. Long

Vol. 7

Progress in Variational Methods eds. C. Liu and Y. Long

Vol. 8

Quantized Algebra and Physics Proceedings of International Workshop eds. M.-L. Ge, C. Bai and N. Jing

Vol. 9

Operads and Universal Algebra Proceedings of the International Conference eds. C. Bai, L. Guo and J.-L. Loday

Vol. 10 Emerging Topics on Differential Equations and Their Applications Proceedings on Sino-Japan Conference of Young Mathematicians eds. H. Chen, Y. Long and Y. Nishiura

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Proceedings on Sino-Japan Conference of Young Mathematicians on Emerging Topics on Differential Equations and their Applications Nankai University, China

Nankai Series in Pure, Applied Mathematics and Theoretical Physics

5 – 9 December 2011

EMERGING TOPICS ON DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS

Vol. 10

Edited by

Hua Chen Wuhan University, China

Yiming Long Nankai University, China

Yasumasa Nishiura Tohoku University, Japan

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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CHENNAI

4/12/12 9:04 AM

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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Nankai Series in Pure, Applied Mathematics and Theoretical Physics — Vol. 10 EMERGING TOPICS ON DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS Proceedings on Sino-Japan Conference of Young Mathematicians Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE The Sino-Japan Conference of Young Mathematicians (SJCYM) “Emerging Topics on Differential Equations and their Applications” organized by Hua Chen, Yiming Long and Yasumasa Nishiura was held at the Chern Institute of Mathematics (CIM) of Nankai University in Tianjin, People’s Republic of China, during December 4th-9th of 2011. The aim of the conference was to provide a forum for presenting and discussing recent trends and developments in differential equations and their applications and to promote scientific exchanges and collaborations among young mathematicians both from China and Japan. The topics discussed in this conference included, for instance, mean curvature flows, KAM theory, n-body problems, flows on Riemannian manifolds, hyperbolic systems, vortices, water waves, and reaction diffusion systems. More than 80 participants attended the conference and 31 talks were presented from both countries. This volume is a collection of the 23 presentations given at SJCYM. We also had a round-table meeting discussing about the promotion of exchange of young mathematicians bi-directionally. We are very grateful to all the members of the Scientific Committee, speakers and participants who made great contributions to and boosted up the conference multilaterally. Finally for the extreme success of our SJCYM, we would like to sincerely thank the strong supports from the Chern Institute of Mathematics (CIM) in Nankai University, Wuhan University, and the Alliance for breakthrough between Mathematics and Sciences of Japan Science and Technology Agency (ABMS-JST).

Hua Chen Yiming Long Yasumasa Nishiura The Organizing Committee of SJCYM

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CONTENTS

Preface

v

A Spectral Theory of Linear Operators on Rigged Hilbert Spaces under Certain Analyticity Conditions H. Chiba

1

Conditional Fredholm Determinant and Trace Formula for Hamiltonian Systems: a Survey X.-J. Hu and P.-H. Wang

12

Initial Value Problem for Water Waves and Shallow Water and Long Wave Approximations T. Iguchi

24

On the Existence and Nonexistence of Maximizers Associated with Trudinger-Moser Type Inequalities in Unbounded Domains M. Ishiwata

41

Computer-assisted Uniqueness Proof for Stokes’ Wave of Extreme Form K. Kobayashi

54

From the Boltzmann H-theorem to Perelman’s W -entropy formula for the Ricci flow X.-D. Li

68

The Spreading of a New Species with Free Boundaries X. Liu and B. Lou

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Recent Progress on Observability for Stochastic Partial Differential Equations Q. L¨ u and Z.-Q. Yin The Nonlinear “Hot Spots” Conjecture in Balls of S2 and H2 Y. Miyamoto

94

109

Mean-Field Models Describing Micro Phase Separation in the Two-Dimensional Case B. Niethammer and Y. Oshita

121

Global Existence of Classical Solutions to Partially Dissipative Quasilinear Hyperbolic Systems P. Qu and C.-M. Liu

132

Time Averaged Properties along Unstable Periodic Orbits of the Kuramoto-Sivashinsky Equation Y. Saiki and M. Yamada

145

Anomalous Enstrophy Dissipation via the Self-Similar Triple Collapse of the Euler-α Point Vortices T. Sakajo

155

Action Minimizing Periodic Solutions in the N -Body Problem M. Shibayama

170

Some Geometric Problems of Conformally Compact Einstein Manifolds Y. Shi

184

Mathematical Modelling and Analysis of Droplet Motion on Obstacles K. Svadlenka

196

Introduction to Varifold and Its Curvature Flow Y. Tonegawa

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Weak KAM Theory in Time-Periodic Lagrangian Systems K.-Z. Wang and J. Yan

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A Note on Resonant Interaction of Rossby Waves in TwoDimensional Flow on a β Plane M. Yamada and T. Yoneda

239

Lp -Solvability of Nonlocal Parabolic Equations with Spatial Dependent and Non-smooth Kernels X.-C. Zhang

247

A Convergence Theorem of K¨ ahler-Ricci Flow Z.-L. Zhang KP Approximation to the 3-D Water Wave Equations with Surface Tension M. Ming, P. Zhang and Z.-F. Zhang

263

271

Smooth Convergence of K¨ahler-Ricci Flow on a Fano Manifold X.-H. Zhu

290

Scientific Committee and Organizing Committee

304

List of All Speakers and Titles of Their Talks

305

Group Photo of the Conference

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A Spectral Theory of Linear Operators on Rigged Hilbert Spaces under Certain Analyticity Conditions Hayato Chiba Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan E-mail: [email protected] A spectral theory of linear operators on rigged Hilbert spaces (Gelfand triplets) is developed under the assumptions that a linear operator T on a Hilbert space H is a perturbation of a selfadjoint operator, and the spectral measure of the selfadjoint operator has an analytic continuation near the real axis. It is shown that there exists a dense subspace X of H such that the resolvent (λ − T )−1 φ of the operator T has an analytic continuation from the lower half plane to the upper half plane for any φ ∈ X, even when T has a continuous spectrum on R, as an X  -valued holomorphic function, where X  is a dual space of X. The rigged Hilbert space consists of three spaces X ⊂ H ⊂ X  . Basic tools of the usual spectral theory, such as spectra, resolvents and Riesz projections are extended to those defined on a rigged Hilbert space. They prove to have the same properties as those of the usual spectral theory.

1. Introduction A spectral theory of linear operators is one of the fundamental tools in functional analysis and well developed so far. Spectra of linear operators provide us with much information about the operators such as the asymptotic behavior of solutions of linear differential equations. However, there are many phenomena that are not explained by spectra. For example, transient behavior of solutions of differential equations is not described by spectra; even if a linear operator T does not have spectrum on the left half plane, a solution of the linear evolution equation dx/dt = T x on an infinite dimensional space can decay exponentially as t increases for a finite time interval. Now it is known that such transient behavior can be induced by resonance poles or generalized eigenvalues, and it is often observed in infinite dimensional systems such as plasma physics,3 coupled oscillators1,8 and Schr¨odinger equations.5,7 In the literature, resonance poles for Schr¨ odinger operators −Δ + V are

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defined in several ways. When a wave operator and a scattering matrix can be defined, resonance poles may be defined as poles of an analytic continuation of a scattering matrix.7 When a potential V (x) decays exponentially, resonance poles can be defined with the aid of certain weighted Lebesgue spaces, which is essentially based on the theory of rigged Hilbert spaces.6 When a potential has an analytic continuation to a sector around the real axis, spectral deformation (complex distortion) techniques are often applied to define resonance poles, see5 and references therein. The spectral theory based on rigged Hilbert spaces (Gelfand triplets) was introduced by Gelfand et al.4 to give generalized eigenfunction expansions of selfadjoint operators. Although they did not treat with resonance poles, the spectral theory of resonance poles (generalized spectrum) of selfadjoint operators based on rigged Hilbert spaces are established by Chiba2 without using any spectral deformation techniques. Let H be a Hilbert space, X a topological vector space, which is densely and continuously embedded in H, and X  a dual space of X  . A Gelfand triplet (rigged Hilbert space) consists of three spaces X ⊂ H ⊂ X  . Let T be a selfadjoint operator densely defined on H. When λ lies on the lower half plane, the resolvent (λ − T )−1 exists and is holomorphic in λ. In general, the resolvent (λ − T )−1 φ for φ ∈ H diverges when λ is on the spectrum σ(T ) ⊂ R. However, for a “good” function φ, (λ− T )−1 φ may exist on σ(T ) in some sense and it may have an analytic continuation from the lower half plane to the upper half plane. The space X consists of such good functions with a suitable topology. Indeed, under certain analyticity conditions given in Sec.2, it is shown in2 that the resolvent has an analytic continuation from the lower half plane to the upper half plane, which is called the generalized resolvent Rλ of T , even when T has the continuous spectrum on the real axis. The generalized resolvent is a continuous operator from X into X  , and it is defined on a nontrivial Riemann surface of λ in general. The set of singularities of Rλ on the Riemann surface is called the generalized spectrum of T . The generalized spectrum consists of the generalized point spectrum, the generalized continuous spectrum and the generalized residual spectrum, which are defined in a similar manner to the usual spectral theory. In particular, a point λ of the generalized point spectrum is called the generalized eigenvalue. If the generalized eigenvalue is not an eigenvalue of T in the usual sense, it is called the resonance pole in the study of Schr¨odinger operators. The generalized eigenfunctions, the generalized eigenspace and the multiplicity associated with the generalized eigenvalue are also defined. The generalized Riesz projection Π is defined through a contour integral of

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Rλ as usual. In,2 it is shown that they have the same properties as the usual theory. For example, the range of the generalized Riesz projection Π around an isolated generalized eigenvalue coincides with its generalized eigenspace. Although this property is well known in the usual spectral theory, our result is nontrivial because Rλ and Π are operators from X into X  , so that the resolvent equation and the property of the composition Π ◦ Π = Π do not hold. If the operator T satisfies a certain compactness condition, the Riesz-Schauder theory on a rigged Hilbert space is applied to conclude that the generalized spectrum consists of a countable number of generalized eigenvalues having finite multiplicities. It is remarkable that even if the operator T has the continuous spectrum (in the usual sense), the generalized spectrum consists only of a countable number of generalized eigenvalues when T satisfies the compactness condition. In much literature, resonance poles are defined by the spectral deformation techniques. The formulation of resonance poles based on a rigged Hilbert space has the advantage that generalized eigenfunctions, generalized eigenspaces and the generalized Riesz projections associated with resonance poles are well defined and they have the same properties as the usual spectral theory, although in the formulation based on the spectral deformation technique, correct eigenfunctions associated with resonance poles of a given operator T is not defined because T itself is deformed by some transformation. The defect of our approach based on a rigged Hilbert space is that a suitable topological vector space X has to be defined, while in the formulation based on the spectral deformation technique, a topology need not be introduced on X because resonance poles are defined by using the deformed operator on the Hilbert space H, not X. Once the generalized eigenfunctions and the generalized Riesz projections associated with resonance poles are obtained, they can be applied to the dynamical systems theory. The generalized Riesz projection for an isolated resonance pole on the left half plane (resp. on the imaginary axis) gives a stable subspace (resp. a center subspace) in the generalized sense. They are applicable to the stability and bifurcation theory1 involving essential spectrum on the imaginary axis. Throughout this paper, D(·) and R(·) denote the domain and range of an operator, respectively. 2. A review of the spectral theory on rigged Hilbert spaces This section is devoted to a review of the spectral theory on rigged Hilbert spaces developed in.2 In order to apply Schr¨odinger operators, assumptions given2 will be slightly relaxed. Let H be a Hilbert space over C and

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H a selfadjoint operator densely defined on H with the spectral measure {E(B)}B∈B ; that is, H is expressed as H = R ωdE(ω). Let K be some linear operator densely defined on H. Our purpose is to investigate spectral properties of the operator T := H + K. Let Ω ⊂ C be a simply connected open domain in the upper half plane such that the intersection of the real ˜ Let I = I\∂ ˜ I˜ be an open axis and the closure of Ω is a connected interval I. interval (see Fig.1). For a given T = H + K, we suppose that there exists

㱅 I Fig. 1.

A domain on which E[ψ, φ](ω) is holomorphic.

a locally convex Hausdorff vector space X(Ω) over C satisfying following conditions. (X1) X(Ω) is a dense subspace of H. (X2) A topology on X(Ω) is stronger than that on H. (X3) X(Ω) is a quasi-complete barreled space. Let X(Ω) be a dual space of X(Ω), the set of continuous anti-linear functionals on X(Ω). The paring for (X(Ω) , X(Ω)) is denoted by  · | · . For μ ∈ X(Ω) , φ ∈ X(Ω) and a ∈ C, we have aμ | φ = aμ | φ = μ | aφ. The space X(Ω) is equipped with the strong dual topology or the weak dual topology. Because of (X1) and (X2), H , the dual of H, is dense in X(Ω) . Through the isomorphism H  H , we obtain the triplet X(Ω) ⊂ H ⊂ X(Ω) ,

(1)

which is called the rigged Hilbert space or the Gelfand triplet. The canonical inclusion i : H → X(Ω) is defined as follows; for ψ ∈ H, we denote i(ψ) by ψ|, which is defined to be i(ψ)(φ) = ψ | φ = (ψ, φ),

(2)

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for any φ ∈ X(Ω), where ( · , · ) is the inner product on H. The inclusion from X(Ω) into X(Ω) is also defined as above. Then, i is injective and continuous. The topological condition (X3) is assumed to define Pettis integrals and Taylor expansions of X(Ω) -valued holomorphic functions. Any complete Montel space, Fr´echet space, Banach space and Hilbert space satisfy (X3) (we refer the reader to9 for basic notions of locally convex spaces, though Hilbert spaces are mainly used in this paper). Next, for the spectral measure E(B) of H, we make the following analyticity conditions: (X4) For any φ ∈ X(Ω), the spectral measure (E(B)φ, φ) is absolutely continuous on the interval I. Its density function, denoted by E[φ, φ](ω), has an analytic continuation to Ω ∪ I. (X5) For each λ ∈ I ∪ Ω, the bilinear form E[ · , · ](λ) : X(Ω) × X(Ω) → C is separately continuous. Due to the assumption (X4) with the aid of the polarization identity, we can show that (E(B)φ, ψ) is absolutely continuous on I for any φ, ψ ∈ X(Ω). Let E[φ, ψ](ω) be the density function; d(E(ω)φ, ψ) = E[φ, ψ](ω)dω,

ω ∈ I.

(3)

Then, E[φ, ψ](ω) is holomorphic in ω ∈ I∪Ω. We will use the above notation for any ω ∈ R for simplicity, although the absolute continuity is assumed only on I. Let iX(Ω) be the inclusion of X(Ω) into X(Ω) . Define the operator A(λ) : iX(Ω) → X(Ω) to be ⎧ √ 1 ⎪ ⎪ E[ψ, φ](ω)dω + 2π −1E[ψ, φ](λ) (λ ∈ Ω), ⎪ ⎪ ⎪ Rλ −ω ⎪ ⎪  ⎨ 1 √ E[ψ, φ](ω)dω (λ = x ∈ I), lim A(λ)ψ | φ = y→−0 R x + ⎪ −1y −ω ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ E[ψ, φ](ω)dω (Im(λ) < 0), λ − ω R (4) for any ψ ∈ iX(Ω) and φ ∈ X(Ω). It is known that A(λ)ψ | φ is holomorphic on the region {Im(λ) < 0} ∪ Ω ∪ I. It is proved in2 that A(λ) ◦ i : X(Ω) → X(Ω) is continuous when X(Ω) is equipped with the weak dual topology. When Im(λ) < 0, we have A(λ)ψ | φ = ((λ − H)−1 ψ, φ). In this sense, the operator A(λ) is called the analytic continuation of the resolvent (λ − H)−1 in the generalized sense. The operator A(λ) plays a central role for our theory. Let Q be a linear operator densely defined on X(Ω). Then, the dual operator Q is defined as follows: the domain D(Q ) of Q is the set of

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elements μ ∈ X(Ω) such that the mapping φ → μ | Qφ from X(Ω) into C is continuous. Then, Q : D(Q ) → X(Ω) is defined by Q μ | φ = μ | Qφ. The (Hilbert) adjoint Q∗ of Q is defined through (Qφ, ψ) = (φ, Q∗ ψ) as usual when Q is densely defined on H. If Q∗ is densely defined on X(Ω), its dual (Q∗ ) is well defined, which is denoted by Q× . Then, Q× = (Q∗ ) is an extension of Q which satisfies i ◦ Q = Q× ◦ i |D(Q) . For the operators H and K, we suppose that (X6) there exists a dense subspace Y of X(Ω) such that HY ⊂ X(Ω). (X7) K is H-bounded and there exists a dense subspace Y  of X(Ω) such that K ∗ Y  ⊂ X(Ω). (X8) i−1 K × A(λ)iX(Ω) ⊂ X(Ω) for any λ ∈ {Im(λ) < 0} ∪ I ∪ Ω. Due to (X6) and (X7), we can show that H × , K × and T × are densely defined on X(Ω) . In particular, D(H × ) ⊃ iY, D(K × ) ⊃ iY and D(T × ) ⊃ iY . When H and K are continuous on X(Ω), (X6) and (X7) are satisfied with Y = X(Ω). Then, H × and T × are continuous on X(Ω) . Recall that K(λ − H)−1 is bounded on H when K is H-bounded. Since A(λ) is the analytic continuation of (λ−H)−1 as an operator from iX(Ω), (X8) gives an “analytic continuation version” of the assumption that K is H-bounded. In,2 the spectral theory of the operator T = H + K is developed under the assumptions (X1) to (X8). However, a Schr¨odinger operator with a dilation analytic potential does not satisfy the assumption (X8). Thus, we ˆ = make the following condition instead of (X8). In what follows, put Ω Ω ∪ I ∪ {λ | Im(λ) < 0}. Suppose that there exists a locally convex Hausdorff vector space Z(Ω) satisfying following conditions: (Z1) X(Ω) is a dense subspace of Z(Ω) and the topology of X(Ω) is stronger than that of Z(Ω). (Z2) Z(Ω) is a quasi-complete barreled space. (Z3) The canonical inclusion i : X(Ω) → X(Ω) is continuously extended to a mapping j : Z(Ω) → X(Ω) . ˆ the operator A(λ) : iX(Ω) → X(Ω) is extended to (Z4) For any λ ∈ Ω, an operator from jZ(Ω) into X(Ω) so that A(λ) ◦ j : Z(Ω) → X(Ω) is continuous if X(Ω) is equipped with the weak dual topology. ˆ j −1 K × A(λ)jZ(Ω) ⊂ Z(Ω) and j −1 K × A(λ)j is con(Z5) For any λ ∈ Ω, tinuous on Z(Ω).

Z(Ω)

⊂

⊃

X(Ω) ⊂ H ⊂ X(Ω) −→

jZ(Ω)

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If we can take Z(Ω) = X(Ω), then (Z1) to (Z5) are reduced to (X1) to (X8), and the results obtained in2 are recovered. In what follows, the extension j of i is also denoted by i for simplicity. Let us show the same results as2 under the assumptions (X1) to (X7) and (Z1) to (Z5). Lemma 2.1. (i) For each φ ∈ Z(Ω), A(λ)iφ is an X(Ω) -valued holomorphic function in ˆ λ ∈ Ω.

(ii) Define the operators A(n) (λ) : iX(Ω) → X(Ω) to be ⎧ 1 ⎪ ⎪ E[ψ, φ](ω)dω ⎪ ⎪ (λ − ω)n ⎪ R ⎪ √ (−1)n−1 dn−1  ⎪ ⎪ ⎪ −1 (n−1)! dzn−1  E[ψ, φ](z), (λ ∈ Ω), +2π ⎪ ⎪ z=λ ⎨ (n)  (5) A (λ)ψ | φ = 1 ⎪ ⎪ √ E[ψ, φ](ω)dω, (λ = x ∈ I), lim ⎪ ⎪ y→−0 R (x + −1y − ω)n ⎪ ⎪ ⎪  ⎪ ⎪ 1 ⎪ ⎪ E[ψ, φ](ω)dω, (Im(λ) < 0), ⎩ n R (λ − ω)

for n = 1, 2 · · · . Then, A(n) (λ) ◦ i has a continuous extension A(n) (λ) ◦ i : Z(Ω) → X(Ω) , and A(λ)iφ is expanded in a Taylor series as A(λ)iφ =

∞

(λ0 − λ)j A(j+1) (λ0 )iφ,

φ ∈ Z(Ω),

(6)

j=0

which converges with respect to the strong dual topology on X(Ω) . (iii) When Im(λ) < 0, A(λ) ◦ iφ = i ◦ (λ − H)−1 φ for φ ∈ X(Ω). ˆ for any Proof. (i) In,2 A(λ)iφ | ψ is proved to be holomorphic in λ ∈ Ω φ, ψ ∈ X(Ω). Since X(Ω) is dense in Z(Ω), Montel theorem proves that A(λ)iφ | ψ is holomorphic for φ ∈ Z(Ω) and ψ ∈ X(Ω). This implies that A(λ)iφ is a weakly holomorphic X(Ω) -valued function. Since X(Ω) is barreled, Thm.A.3 of 2 concludes that A(λ)iφ is strongly holomorphic. (ii) In,2 Eq.(6) is proved for φ ∈ X(Ω). Again Montel theorem is applied to show the same equality for φ ∈ Z(Ω). (iii) This follows from the definition of A(λ).  Lemma 2.1 means that A(λ) gives an analytic continuation of the resolvent (λ − H)−1 from the lower half plane to Ω as an X(Ω) -valued function. Similarly, A(n) (λ) is an analytic continuation of (λ − H)−n . A(1) (λ) is also denoted by A(λ) as before. Next, let us define an analytic continuation of the resolvent of T = H + K. Due to (Z5), id − K × A(λ) is an operator on iZ(Ω). It is easy to verify that id − K × A(λ) is injective if and only if

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id − A(λ)K × is injective on R(A(λ)) = A(λ)iZ(Ω). Definition 2.2. If the inverse (id − K × A(λ))−1 exists on iZ(Ω), define the generalized resolvent Rλ : iZ(Ω) → X(Ω) of T to be Rλ = A(λ) ◦ (id − K × A(λ))−1 = (id − A(λ)K × )−1 ◦ A(λ),

ˆ (7) λ ∈ Ω.

Although Rλ is not a continuous operator in general, the composition Rλ ◦ i : Z(Ω) → X(Ω) may be continuous: Definition 2.3. The generalized resolvent set ˆ(T ) is defined to be the set of ˆ of λ such ˆ satisfying following: there is a neighborhood Vλ ⊂ Ω points λ ∈ Ω  that for any λ ∈ Vλ , Rλ ◦ i is a densely defined continuous operator from Z(Ω) into X(Ω) , where X(Ω) is equipped with the weak dual topology, and the set {Rλ ◦ i(ψ)}λ ∈Vλ is bounded in X(Ω) for each ψ ∈ Z(Ω). The ˆ ˆ(T ) is called the generalized spectrum of T . The generalized set σ ˆ (T ) := Ω\ ˆ (T ) at which id − K × A(λ) is point spectrum σ ˆp (T ) is the set of points λ ∈ σ not injective. The generalized residual spectrum σ ˆr (T ) is the set of points λ∈σ ˆ (T ) such that the domain of Rλ ◦ i is not dense in Z(Ω). The generˆ (T )\(ˆ σp (T ) ∪ σ ˆr (T )). alized continuous spectrum is defined to be σ ˆc (T ) = σ We can show that if Z(Ω) is a Banach space, λ ∈ ˆ(T ) if and only if id − i−1 K × A(λ)i has a continuous inverse on Z(Ω) (Prop.3.18 of 2 ). The next theorem is proved in the same way as Thm.3.12 of,2 in which (X1) to (X8) are assumed. Theorem 2.4 .2 Suppose (X1) to (X7) and (Z1) to (Z5). (i) For each φ ∈ Z(Ω), Rλ iφ is an X(Ω) -valued holomorphic function in λ ∈ ˆ(T ). (ii) Suppose Im(λ) < 0, λ ∈ ˆ(T ) and λ ∈ (T ) (the resolvent set of T in H-sense). Then, Rλ ◦ iφ = i ◦ (λ − T )−1 φ for any φ ∈ X(Ω). In particular, Rλ iφ | ψ is an analytic continuation of ((λ− T )−1 φ, ψ) from the lower half plane to I ∪ Ω for any φ, ψ ∈ X(Ω). Next, we define the operator B (n) (λ) : D(B (n) (λ)) ⊂ X(Ω) → X(Ω) to be B (n) (λ) = id − A(n) (λ)K × (λ − H × )n−1 .

(8)

The domain D(B (n) (λ)) is the set of μ ∈ X(Ω) such that K × (λ − H × )n−1 μ ∈ iZ(Ω). Definition 2.5. A point λ in σ ˆp (T ) is called a generalized eigenvalue (resonance pole) of the operator T . The generalized eigenspace of λ is defined

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by Vλ =



Ker B (m) (λ) ◦ B (m−1) (λ) ◦ · · · ◦ B (1) (λ).

(9)

m≥1

We call dimVλ the algebraic multiplicity of the generalized eigenvalue λ. In particular, a nonzero solution μ ∈ X(Ω) of the equation B (1) (λ)μ = (id − A(λ)K × )μ = 0

(10)

is called a generalized eigenfunction associated with the generalized eigenvalue λ. Theorem 2.6 .2 For any μ ∈ Vλ , there exists an integer M such that (λ − T × )M μ = 0. In particular, a generalized eigenfunction μ satisfies (λ − T × )μ = 0. This implies that λ is indeed an eigenvalue of the dual operator T × . In general, σ ˆp (T ) is a proper subset of σp (T × ) (the set of eigenvalues of T × ),  and Vλ is a proper subspace of the eigenspace m≥1 Ker(λ − T × )m of T × . Let Σ ⊂ σ ˆ (T ) be a bounded subset of the generalized spectrum, which ˆ is separated from the rest of the spectrum by a simple closed curve γ ⊂ Ω.  Define the operator ΠΣ : iZ(Ω) → X(Ω) to be  1 √ Rλ φ dλ, φ ∈ iZ(Ω), (11) ΠΣ φ = 2π −1 γ which is called the generalized Riesz projection for Σ. The integral in the right hand side is well defined as the Pettis integral. We can show that ΠΣ ◦i is a continuous operator from Z(Ω) into X(Ω) equipped with the weak dual topology. Note that ΠΣ ◦ ΠΣ = ΠΣ does not hold because the composition ΠΣ ◦ ΠΣ is not defined. Nevertheless, we call it the projection because it is proved in Prop.3.14 of 2 that ΠΣ (iZ(Ω)) ∩ (id − ΠΣ )(iZ(Ω)) = {0} and the direct sum satisfies iZ(Ω) ⊂ ΠΣ (iZ(Ω)) ⊕ (id − ΠΣ )(iZ(Ω)) ⊂ X(Ω) .

(12)

Let λ0 be an isolated generalized eigenvalue, which is separated from the ˆ Let rest of the generalized spectrum by a simple closed curve γ0 ⊂ Ω.  1 Π0 = √ Rλ dλ, (13) 2π −1 γ0  be a projection for λ0 and V0 = m≥1 Ker B (m) (λ0 ) ◦ · · · ◦ B (1) (λ0 ) a generalized eigenspace of λ0 . The main theorems in2 are stated as follows: Theorem 2.7 .2 If Π0 iZ(Ω) = R(Π0 ) is finite dimensional, then Π0 iZ(Ω) =

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V0 . Note that Π0 iZ(Ω) = Π0 iX(Ω) when Π0 iZ(Ω) is finite dimensional because X(Ω) is dense in Z(Ω). Then, the above theorem is proved in the same way as the proof of Thm.3.16 of.2 Theorem 2.8 .2 In addition to (X1) to (X7) and (Z1) to (Z5), suppose that (Z6) i−1 K × A(λ)i : Z(Ω) → Z(Ω) is a compact operator uniformly in ˆ λ ∈ Ω. Then, the following statements are true. ˆ the number of generalized eigenvalues in (i) For any compact set D ⊂ Ω, D is finite (thus σ ˆp (T ) consists of a countable number of generalized eigenˆ or infinity). values and they may accumulate only on the boundary of Ω ˆp (T ), the generalized eigenspace V0 is of finite dimen(ii) For each λ0 ∈ σ sional and Π0 iZ(Ω) = V0 . ˆr (T ) = ∅. (iii) σ ˆc (T ) = σ Recall that a linear operator L from a topological vector space X1 to another topological vector space X2 is said to be compact if there exists a neighborhood U ⊂ X1 such that LU ⊂ X2 is relatively compact. When L = L(λ) is parameterized by λ, it is said to be compact uniformly in λ if such a neighborhood U is independent of λ. When the domain X1 is a Banach space, L(λ) is compact uniformly in λ if and only if L(λ) is compact for each λ. The above theorem is also proved in a similar manner to the ˆc (T ) = ∅ even if T has the proof of Thm.3.19 of.2 It is remarkable that σ continuous spectrum in H-sense. When we emphasize the choice of Z(Ω), σ ˆ (T ) is also denoted by σ ˆ (T ; Z(Ω)). Now suppose that two vector spaces Z1 (Ω) and Z2 (Ω) satisfy the assumptions (Z1) to (Z5) with a common X(Ω). Then, two generalized ˆ (T ; Z2 (Ω)) for Z1 (Ω) and Z2 (Ω) are defined, respectra σ ˆ (T ; Z1 (Ω)) and σ spectively. Let us consider the relationship between them. Proposition 2.9. Suppose that Z2 (Ω) is a dense subspace of Z1 (Ω) and the topology on Z2 (Ω) is stronger than that on Z1 (Ω). Then, the following holds. ˆ (T ; Z1 (Ω)). (i) σ ˆ (T ; Z2 (Ω)) ⊂ σ (ii) Let Σ be a bounded subset of σ ˆ (T ; Z1 (Ω)) which is separated from the rest of the spectrum by a simple closed curve γ. Then, there exists a point of ˆ (T ; Z1 (Ω)), σ ˆ (T ; Z2 (Ω)) inside γ. In particular, if λ is an isolated point of σ then λ ∈ σ ˆ (T ; Z2 (Ω)).

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Proof. (i) Suppose that λ ∈ / σ ˆ (T ; Z1 (Ω)). Then, there is a neighborhood Vλ of λ such that Rλ ◦ i is a continuous operator from Z1 (Ω) into X(Ω) for any λ ∈ Vλ , and the set {Rλ ◦ iψ}λ ∈Vλ is bounded in X(Ω) for each ψ ∈ Z1 (Ω). Since the topology on Z2 (Ω) is stronger than that on Z1 (Ω), Rλ ◦ i is a continuous operator from Z2 (Ω) into X(Ω) for any λ ∈ Vλ , and the set {Rλ ◦ iψ}λ ∈Vλ is bounded in X(Ω) for each ψ ∈ Z2 (Ω). This proves that λ ∈ /σ ˆ (T ; Z2 (Ω)). (ii) Let ΠΣ be the generalized Riesz projection for Σ. Since Σ ⊂ σ ˆ (T ; Z1 (Ω)), ΠΣ iZ1 (Ω) = {0}. Then, ΠΣ iZ2 (Ω) = {0} because Z2 (Ω) is dense in Z1 (Ω). This shows that the closed curve γ encloses a point of  σ ˆ (T ; Z2 (Ω)). References 1. H. Chiba, A proof of the Kuramoto’s conjecture for a bifurcation structure of the infinite dimensional Kuramoto model, (submitted) 2. H. Chiba, A spectral theory of linear operators on rigged Hilbert spaces under certain analyticity conditions, (submitted) 3. J. D. Crawford, P. D. Hislop, Application of the method of spectral deformation to the Vlasov-Poisson system, Ann. Physics 189 (1989), no. 2, 265–317 4. I. M. Gelfand, N. Ya. Vilenkin, Generalized functions. Vol. 4. Applications of harmonic analysis, Academic Press, New York-London, 1964 5. P. D. Hislop, I. M. Sigal, Introduction to spectral theory. With applications to Schrodinger operators, Springer-Verlag, New York, 1996 6. J. Rauch, Perturbation theory for eigenvalues and resonances of Schrodinger Hamiltonians, J. Funct. Anal. 35 (1980), no. 3, 304-315 7. M. Reed, B. Simon, Methods of modern mathematical physics IV. Analysis of operators, Academic Press, New York-London, 1978 8. S. H. Strogatz, R. E. Mirollo, P. C. Matthews, Coupled nonlinear oscillators below the synchronization threshold: relaxation by generalized Landau damping, Phys. Rev. Lett. 68 (1992), no. 18, 2730–2733 9. F. Tr´eves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967

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Conditional Fredholm Determinant and Trace Formula for Hamiltonian Systems: a Survey Xijun Hu∗ and Penghui Wang† Department of mathematics, Shandong University, Jinan, 250100, China ∗ E-mail: [email protected] † E-mail: [email protected] This note is a brief survey of our results on the stability criteria of S-periodic orbits in Hamiltonian system, by using the conditional Fredholm determinant and the trace formula. Most results appear in the papers.9,10 Keywords: Hamiltonian systems, Conditional Fredholm determinant, Trace formula, Maslov-type index, Linear stability

1. Introduction This is a brief survey on our results on the stability of linear Hamiltonian system by using Conditional Fredholm determinant and trace formula.9,10 Motivated by the studies,2,3,5 we develop the theory of conditional Fredholm determinant and get a generalized Hill’s formula for the S-periodic orbits in Hamiltonian systems. Based on our generalized Hill’s formula, we build up a trace formula which can be consider as a generation of Krein’s trace formula,14,15 moreover, inspired by Krein’s interesting work, we give some new stability criteria by combining our trace formula with the Maslov-type index theory. The survey paper by Yakubovich17 gives a nice introduction to Krein’s work in this field. Consider the linear Hamiltonian system

x(t) ˙ = JB(t)x(t),

(1)

0 −In is symplectic matrix and B(t) is a continuous path In 0 of real symmetric matrices. A basic problem is to find the conditions under which system (1) are stable (all its solutions are bounded for −∞ < t < ∞). Before Krein, this field was not inverstigated too well. The results obtained

where J =

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were mostly connected with the Hill equation d2 y + q(t)y = 0, dt2

(2)

where q(t) = q(t + T ) is periodic, which is deduced by Hill in his studies of the motion of moon. It is obviously, the system

(2) can be reduced

1 0 y˙ to a Hamiltonian system (1) for B(t) = and x = . It is 0 q(t) y remarkable to point out that Lyapunov give the following criteria  T p(t)dt ≤ 4. (3) p(t) > 0, T 0

The above system (1) is scalar. For higher dimensional system(n > 1), Krein considers the system with the following boundary-value problem of positive type x(t) ˙ = λJB(t)x(t)

(4)

x(0) = −x(T ), (5) T with the condition that B(t) ≥ 0 and 0 B(t)dt > 0. Let · · · ≤ λ−2 ≤ λ−1 < 0 < λ1 ≤ λ2 ≤ · · · be the sequence of al eigenvalues of the above boundary-value problem, then 1 = T r(A11 A22 − A212 ), (6) λ2j

T A11 A12 = T1 0 H(t)dt. Based on (6), Krein gives an interesting where A21 A22 stability criteria15 T2 T r(A11 A22 − A212 ) < 1, (7) 2 T under the condition B ≥ 0 and 0 B(t)dt > 0 . As will be seen, our trace formula is a kind of generalization of Krein’s trace formula, and we will give some new stability criteria which generalized (7). It should be pointed out that our trace formula comes from the viewpoint of differential operators. In our papers,9,10 we mainly concentrate on the following boundaryvalue problem z(t) ˙ = JB(t)z(t)

(8)

z(0) = Sz(T ),

(9)

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where S ∈ Sp(2n) ∩ O(2n). Here Sp(2n) is the symplectic group, and O(2n) is the orthogonal group on R2n . Such a boundary-value problem appears naturally in the study of the closed geodesics on Riemannian manifold and N-body problem, please refer to22,23 for details. Obviously, the boundaryvalue problem considered by Krein is a special case of (8)-(9), by taking S = −I2n . In studying the periodic solution of Hamiltonian system, a very successful index theory for such symplectic paths is introduced by Conley and Zehnder4 and developed by Long and others, details can be found in the book.13 For the S-periodic solutions, such an index theory is studied by the first author and Sun,22 and in the same paper, based on the index theory, the stability criteria for N -body problems is given. As an application of our trace formula, we can estimate the relative Morse index, and based on the work of Maslov-type index,13,22 some new stability criteria will be given. The purpose of the paper is to describe the way to get our trace formula, and the application of our trace formula in the study of the stability criteria. And the paper will be organized as follows. In Section 2, we will describe the Hill formula for S-periodic Hamiltonian system, which is the starting point of our trace formula. In section 3, we will list our trace formula, and give the applications of it. 2. Hill Formula for S-periodic Hamiltonian systems In this section, we will introduce Hill’s Formula for S-periodic Hamiltonian systems. At first, we will introduce the most important conception in this section, the conditional Fredholm determinant. 2.1. Trace finite condition and conditional Fredholm determinant We begin with the Fredholm determinant in the classical settings. Details can be found in Simon’s book.16 The Fredholm determinant is a kind of infinite determinant, which is defined for id + C, where C is a trace class operator and id is the identity on a Hilbert space H. Recall that, an operator C is called trace class, if for ∞  1 (C ∗ C) 2 ei , ei  is summarizable. The any orthonormal basis {ei } of H, i=1

Fredholm determinant has many properties of the classical determinant of matrix, such as the multiplicity, for trace class operators C1 , C2 , det ((id + C1 )(id + C2 )) = det(id + C1 ) det(id + C2 ).

(10)

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Moreover, it has the finite analyticity. However, unfortunately, many operators occurred in the Hamiltonian system are not of the forms id + C for trace class operator, but of the form id + F for Hilbert-Schmidt operator F . ∞  ||F ej ||2 is Recall that, an operator F is called Hilbert-Schmidt, if j=1

summarizable for any orthonormal basis {ej }∞ j=1 . Moreover, if F is Hilbert∞  ||F ej ||2 is independent on the choice of the orSchmidt, then the sum j=1

thonormal basis, and ||F ||2 =



∞ 

 12 ||F ej ||

2

is called the Hilbert-Schmidt

j=1

norm of F . To introduce the conditional Fredholm determinant, we recall the regularized determinant det2 . Firstly, it is easy to see that, if F is a HilbertSchmidt operator, then (id + F )e−F has the form id + C for some trace class operator C, and hence the Fredholm determinant det((id + F )e−F ) is well-defined. In this case, we could define the regularized determinant16 by   (11) det2 (id + F ) = det (id + F )e−F , moreover, if ||Fk − F ||2 → 0, then det2 (id + F ) = lim det2 (id + Fk ). k→∞

(12)

Of cause, in many cases, we may assume that Fk are finite-rank operators, and det2 (id + Fk ) = det(id + Fk )e−T rFk .

(13)

It is obvious that, if T rFk tends to some finite number, then the limit of det(id + Fk ) exists, which can be defined as the conditional Fredholm determinant. Now, it is natural to consider when the limit of T rFk exists. This is the reason why we introduce the trace finite condition. First, we will consider a sequence of orthogonal projections {Pk } with the following properties. 1) Range(Pk ) ⊆ Range(Pm ), if k ≤ m, 2) Pk converge to id in the strong operator topology. Next, we will give the definition of trace finite condition. Definition 2.1. Given a sequence of orthogonal projections {Pk } as above, a Hilbert-Schmidt operator F is called to have the trace finite condition

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depending on {Pk }, if the limit lim T r(Pk F Pk ) exists and the limit is N →∞

finite.

Obviously, the trace finite condition is dependent on the choice of {Pk }, however, if the projections are given, then the trace finite condition is welldefined. From the next subsection, we will fix the projections which are induced from the Hamiltonian system. Now, if F is a Hilbert-Schmidt operator with trace finite condition, then ||PN F PN − F ||2 → 0, and hence det2 (id + F ) = lim det2 (id + PN F PN ) N →∞   = lim det (id + PN F PN )e−PN F PN N →∞

= lim det(id + PN F PN )e−T r(PN F PN ) . N →∞

It follows that lim det(id + PN F PN ) = det2 (id + F ) lim eT r(PN F PN ) .

N →∞

N →∞

(14)

Now, we can define the conditional Fredholm determinant as follows. Definition 2.2. Let F be a Hilbert-Schmidt operator with trace finite condition, the conditional Fredholm determinant det(id + F ) = lim det(id + PN F PN ). N →∞

It is easy to see that, if F is a trace class operator, then F has the trace finite condition. Moreover, if F1 and F2 are Hilbert-Schmidt operators with trace finite condition, then F1 + F2 has trace finite condition. And hence, det ((id + F1 )(id + F2 )) = det(id+F1 +F2 +F1 F2 ) is well-defined. It should be pointed out that, many properties of determinant for matrix preserves for conditional Fredholm determinant, such as the multiplicity, det ((id + F1 )(id + F2 )) = det(id + F1 ) det(id + F2 ).

(15)

2.2. Hill formula and its application In this subsection, we come back to the linear Hamiltonian system z(t) ˙ = JB(t)z(t)

(16)

z(0) = Sz(T ),

(17)

where B(t) = B(t)T is temporarily assumed to be a continuous real symmetric path on [0, T ], S ∈ Sp(2n) ∩ O(2n). Recall that, we denote by

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17 d A = −J dt which is densely defined in the Hilbert space E = L2 ([0, T ], C2n ) with the domain

   DS = z(0) = Sz(T )  z(t) ∈ W 1,2 ([0, T ], C2n ) , and the operator B on E is defined by Bz = B(t)z(t), for any z(t) ∈ E. Since B is a continuous real symmetric path, A, B and A − B are all selfadjoint operators. Due to the non-commutativity, the matrices equation (16)-(17) could not be solved directly. To understand the equation, we will study the property of the associated operator (A−B)(A+P0 )−1 , where P0 is the orthogonal projection onto ker A. One of the motivations of our work is the stability of the Hamiltonian system. Recall that, let γ = γz (t) be the fundamental solutions of the Hamiltonian system (16)-(17), that is γ(t) ˙ = JB(t)γ(t),

(18)

γ(0) = I2n .

(19)

Then, z(t) is the solution of (16)-(17) if an only if z(0) ∈ ker(Sγ(T ) − I2n ). Let ϕt be the Hamiltonian flow of the S-periodic solution z is a fixed point of the symplectic map Sϕt . We give the next definition. Definition 2.3. z is called spectrally stable if all the eigenvalues of Sγz (T ) are on the unit circle U, and it is called linear stable, if moreover Sγz (T ) is semi-simple. In studying the stability of closed geodesics and the cyclic type symmetric periodic solution, it is reduced to judge whether the eigenvalues of Sγz (T ) locate in the unit circle, and Definition 2.3 is reasonable in the sense.22,23 Notice that, all the information of A, B is assumed to be clear from the system. To study the stability of the solution, it is natural to ask that, can we read the eigenvalues of Sγ(T ) from the operators A and B directly? This is the starting point of our study. Inspired by,2,3,5 we prove the following theorem, which is the main theorem in this paper. We call it Hill’s formula because of Hill’s work on some special ODE.6 Theorem 2.1. (Hill’s Formula) There is a constant C(S) > 0, which de-

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pends only on S, such that for any ν ∈ C det((A − B − νJ)(A + P0 )−1 ) 1

= C(S)e− 2 where λ = e

νT

T 0

T r(JB(t))dt −n

λ

det(Sγ(T ) − λI2n ),

(20)

, and C(S) is a positive constant depending only on S.

Now, we have to explain the notations in the theorem. Since Sγ(T ) − λI2n is a 2n × 2n-matrix, the determinant det(Sγ(T ) − λI2n ) can be defined unambiguously. Next, we will clarify that det((A − B − νJ)(A + P0 )−1 ) is the conditional Fredholm determinant. To state the trace finite condition, we will fix {Pk }. In what follows, Pk will be chosen as {PN }, where PN are the orthogonal projections onto  WN = ker(A − ν). ν∈σ(A),|ν|≤N

Now, consider the operator (A − B − νJ)(A + P0 )−1 . Obviously, (A − B − νJ)(A + P0 )−1 = id − (P0 + B + νJ)(A + P0 )−1 . Since S ∈ Sp(2n) O(2n), then it is not hard to see that, S is unitarily ∩ V , where equivalent to V ⎛ √ ⎞ e− −1θ1 √ ⎜ ⎟ e− −1θ2 ⎜ ⎟ ⎜ ⎟, for some 0 ≤ θj < 2π. V =⎜ .. ⎟ . ⎝ ⎠ √ e− −1θn θ

Write νj = Tj , then it can be shown that the eigenvalues of A with the S-periodic boundary condition are n

{νi + 2kπ/T }k∈Z, (21) σ(A) = i=1

and each eigenvalue is of multiplicity two. Therefore,the operator (P0 + B + νJ)(A+ P0 )−1 is not trace class, but Hilbert-Schmidt. Fortunately, we have the following proposition [9, Proposition 2.8]. Proposition 2.1. Let 0 = ν1 = · · · = νk0 < νk0 +1 ≤ · · · ≤ νn < eigenvalues of A with S-boundary condition, then

2π T

lim T r(PN B(A + P0 )−1 PN )

N →∞

=2

k0  j=1

0

T

 n 1 + cos T νj T Bjj (t)dt + T Bjj (t)dt. sin T νj 0 j=k0 +1

be the

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That is, for any B ∈ B = C([0, T ], GL(2n, C)), B(A + P0 )−1 has the trace finite condition. From this, we know that (P0 + B + νJ)(A + P0 )−1 also has the trace finite condition. It follows that the conditional Fredholm determinant det((A − B − νJ)(A + P0 )−1 ) is well-defined. It is worth being pointed out that, Denk5 obtained the Hill’s formula for periodic solution of ODE, and Bolotin and Treschev3 studied the Hill’s formula for periodic solution of Lagrangian system. The key-point of the formula (20) is to construct the relationship between the monodromy matrix and the differential operators, and hence, we can read the information of eigenvalues for Sγ(T ) for the differential operators partly. And this can be used to study the spectral stability. An operator is called to be non-degenerate, if its kernel is nontrivial. From Theorem 2.1, we could immediately obtain a criteria to judge the instability for the S-periodic solutions. Corollary 2.1. In the case B(t) = H  (t, z(t)), then z is spectrally unstable if one of the condition satisfies 1) A − B is non-degenerate, and det((A − B)(A + P0 )−1 ) < 0, √

−1π T J

is non-degenerate, and (−1)n det (A − B −  √ −1π −1 < 0. T J)(A + P0 )

2) A − B −

We only explain this for condition 1), and condition 2) is similar. In fact, ¯ since Sγ(T ) ∈ Sp(2n), if λ is an eigenvalue of Sγ(T ), then so are λ−1 , λ −1 ¯ and λ (some of them maybe equal to each other), moreover, they have the same geometric and algebraic multiplicities. If 1 is not an eigenvalue of Sγ(T ), then det(Sγ(T ) − I2n ) < 0 implies that there exists a positive eigenvalue of Sγ(T ) which is bigger than 1, thus the solution is spectrally unstable. For the Lagrangian systems, it was proved in.3 The Maslov-type index can be defined for the solution with S-boundary condition.13,22 This index can be considered as an intersection number of paths in Sp(2n) with the singular set, on the other hand, it is just the relative Morse index I(A, A − B), see section 6 for the detail. The sign of conditional Fredholm determinant depends on that I(A, A − B) is even or odd. We denote PB be the orthogonal projection onto ker(A − B), then det((A − B + PB )(A + P0 )−1 ) = 0. The following Theorem will be proved in Section 6, which build the relationship between the sign of the conditional Fredholm determinant and the relative Morse index. Theorem 2.2. det((A − B + PB )(A + P0 )−1 ) > 0(< 0) if and only if

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I(A, A − B) is even(odd). 3. Trace formula in Hamiltonian systems Let S(2n) be the set of 2n × 2n real symmetric matrices and B = C([0, T ]; S(2n)). For B(t), D(t) ∈ B, we consider the following linear perturbation of (8)-(9), z(t) ˙ = J(B(t) + λD(t))z(t)

(22)

z(0) = Sz(T ).

(23)

By the formula (20), a parameterized formula holds det((A − (B + λD) − νJ)(A + P0 )−1 ) = C(S)e−nνT det(Sγλ (T ) − eνT I2n ).

(24)

This is the foundation of the trace formula. In fact, both sides of (24) are analytic functions on λ. Then, by taking Taylor expansion and comparing the coefficients on both sides of (24), we get the trace formula. To state our result, we give some notations firstly. Let γ0 (t) be the fundamental solution for the system (16-17), that is, γ˙ 0 (t) = JB(t)γ0 (t) with γ0 (0) = I2n . Write ˆ = γ0T (t)D(t)γ0 (t). For k ∈ N, let M = Sγ0 (T ) and D(t)  T  t1  tk−1 ˆ ˆ ˆ k )dtk · · · dt2 dt1 , Mk = J D(t1 ) J D(t2 ) · · · J D(t 0

0

0

and Gk = (M − eνT I2n )−1 M Mk . Theorem 3.1. Let ν ∈ C such that A − B − νJ is invertible. Then for any positive integer m, m !  m  (−1)k =m T r(Gj1 · · · Gjk ) . T r D(A − B − νJ)−1 k j +···+j =m k=1

1

k

Obviously, our trace formula generalize Krein’s trace formula. Motivated by Krein’s work, we will give some new stability criteria by using the above trace formula. For large m, the right hand side of (25) is a little complicated. However, for m = 1, 2, we can write it down more precisely. Corollary 3.1.

  T r[D(A − B − νJ)−1 ] = −T r J 0

T

 γ0T (t)D(t)γ0 (t)dt · M (M − eνT I2n )−1 .

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  T r [D(A − B − νJ)−1 ]2    T T γ0 (t)D(t)γ0 (t)J = −2T r J 

 +T r

0

J

s

γ0T (s)D(s)γ0 (s)dsdt · M (M − λI2n )−1

0 T

γ0T (t)D(t)γ0 (t)dt · M (M − eνT I2n )−1

!2 

0

Moreover, in the case that M = ±I2n , set ω = eνT , then   T r (D(A − νJ − B)−1 )2  T ±ω T r((J γ0T (s)D(s)γ0 (s)ds)2 ). = (1 ∓ ω)2 0

(25)

Let U be the unit circle. For ω ∈ U such that ω = eiθ0 with θ0 ∈ [0, π], let Uω = {eiθ , θ ∈ [−θ0 , θ0 ]}, denote eω (M ) by the total algebraic multiplicities of all eigenvalues of M on Uω . Denote by e(M ) the total algebraic multiplicities of all eigenvalues of M on U. Obviously e(M ) ≥ eω (M ), ∀ω ∈ U. For simplification, we consider the case S = I2n , and the general case is similar. We denote " γ be the fundamental solution with respect to B + D, #=" and M γ (T ). The next theorem is motivated by Krein’s work.15 Theorem 3.2. Suppose M = I2n , D > 0 and  T ω 2 ˆ T r((J ) ≤ 1. D(s)ds) (1 − ω)2 0 Then #)/2 = n. eω (M

(26)

More stability criteria is given in the paper.10 Compare with (7), obviously, Theorem 3.2 is a generalization of Krein’s stability criteria, the proof is based on the trace formula and the Maslov-type index theory for symplectic paths.11,13,22 Masov-type index is a very useful tool in studying the linear stability of periodic orbits in Hamiltonian systems12,13 and it is applied to judge the stability in N-body problems.22,23 Remark 3.1. Recall that, in,1 Atiyah, Patodi and Singer defined a kind of zeta function for self-adjoint elliptic differential operator A(the operator may be not positive). Let {λ} be the eigenvalues for A, then ηA (s) = (signλ)|λ|−s , (27) λ =0



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for Re(s) large, and it can be extended meromorphically to the whole splane. Now, for the differential operator A, if we can take some proper B, D and S in our framework, such that λ are the eigenvalues of A = D−1 (A − B − νJ) is real, then by the trace formula, we can obtain the values for ηA (s) at odd integers. Acknowledgement. The first author is partially supported by NSFC(No.11131004) and the second author is partially supported by NSFC(No. 11101240,No.10831007). References 1. M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry, Math. Proc. Cambridge Philos. Soc., 79(1976)79-99. 2. S.V. Bolotin, The Hill determinant of a periodic orbit, Mosc. Univ. Mech. Bull. 43:3(1988), 7-11. 3. S.V. Bolotin and D.V. Treschev, Hill’s formula, Russian Math. Surveys, Vol.65(2010), No.2, 191-257. 4. C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., Vol.37(1984), 207-253. 5. R. Denk, On the Hilbert-Schmidt operators and determinants correspoinding to periodic ODE systems, Differential and intergral operators (Regensburg, 1995), 57-71, Oper. Theory Adv. Appl., Vol. 102, Birkh¨ auser, Basel, 1998. 6. G. W. Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Math., Vol.8(1886), No.1, 1-36. 7. X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltonian systems with its application to figure-eight orbit, Comm. Math. Phys., Vol. 290(2009), 737-777. 8. X. Hu and S. Sun, Morse index and the stability of closed geodesics, SCIENCE CHINA Mathematics., Vol.53(2010), No.5, 1207-1212. 9. X. Hu and P. Wang, Conditional Fredholm determinant for the S-periodic orbits in Hamiltonian systems, J. Funct. Anal. Vol. 261(2011), 3247-3278. 10. X. Hu and P. Wang, Trace formula and its applications for the S-periodic solution in Hamiltonian systems, submitted. 11. Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149. 12. Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Advances in Math., 154 (2000), 76-131. 13. Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math., Vol.207, Birkh¨ auser. Basel. 2002. 14. M.G. Krein, Foundations of the theory of λ-zones of stability for a canonical system of linear differential equations with periodic coefficients, Amer. Math. Soc. Transl.(2) Vol. 120(1983), 1-70.

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15. M.G. Krein, On tests for the stable boundedness of solutions of periodic canonical systems, PrikL Mat. Mekh., 19, Issue 6, 641-680 (1955). 16. B. Simon, Trace ideals and their applications. Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, 2005. 17. V.A.Yakubovich, On M.G.Krein’s work in the theory of linear periodic Hamiltonian systems, Ukrainian Mathematical Journal, Vol.46, Nos.1-2,1994.

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INITIAL VALUE PROBLEM FOR WATER WAVES AND SHALLOW WATER AND LONG WAVE APPROXIMATIONS T. IGUCHI Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan E-mail: [email protected] In this article we explain the well-posedness of the initial value problem for water waves, and the shallow water and long wave approximations for water waves. Especially, we explain that the solution of the full water wave problem can be approximated by the solution of the shallow water equations, the Green– Naghdi equations, the KdV equation, the Kawahara equation, the forced KdV equation, and the Benjamin–Ono equation in some scaling regimes. Keywords: Water wave; shallow water; long wave; KdV; Benjamin–Ono

1. Introduction In this article we consider the full water wave problem and relations with approximate equations. In the theory of water waves, we are mainly interested in a large scale motion of the water like tides or tsunamis. In such a case, the Reynolds number is so large and the effect of the viscosity becomes negligible. Moreover, the water is a typical example of an incompressible fluid. Therefore, we usually adopt the incompressible Euler equations as a model for water waves. The main external force is the gravity and this is a potential force. In fact, the wave on the water surface propagates due to the gravity as the main restoring force. Lagrange’s vortex theorem implies that the vorticity is never produced nor annihilated for such a fluid motion, so we usually assume that the flow is irrotational. Also we sometimes take into account the effect of the surface tension on the water surface. To summarize, the water wave problem is formulated as a free boundary problem of the incompressible Euler equations with the irrotational condition under the gravitational field. More precisely, the problem is formulated as follows. Let x = (x1 , x2 , . . . , xn ) be the horizontal spatial variables and xn+1 the vertical spatial variable. We denote by X = (x, xn+1 ) = (x1 , . . . , xn , xn+1 )

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the whole spatial variables. We will consider a water wave in (n + 1)dimensional space and assume that the water region Ω(t) at time t, the water surface Γ(t), and the bottom Σ are of the forms   Ω(t) =  X = (x, xn+1 ) ∈ Rn+1 ; b(x) < xn+1 < h +η(x, t) , Rn+1 ; xn+1 = h  + η(x, t) , Γ(t) =  X = (x, xn+1 ) ∈ n+1 ; xn+1 = b(x) , Σ = X = (x, xn+1 ) ∈ R where h is the mean depth of the water. The functions b and η represent the bottom topography and the surface elevation, respectively. The motion of the water is described by the velocity v = (v1 , . . . , vn , vn+1 ) and the pressure p of the water satisfying the equations  $  ρ v t + (v · ∇X )v + ∇X p = −ρgen+1 , (1) in Ω(t), t > 0, divX v = 0, rotX v = O where ρ is the density of the water, g is the gravitational constant, ∇ = (∂1 , . . . , ∂n), ∇X = (∂1 , . .. , ∂n , ∂n+1 ), divX v = ∂1 v1 + · · · + ∂n+1 vn+1 , and rotX v = 12 (∂j vi − ∂i vj ) 1≤i,j≤n+1 . It is assumed that both ρ and g are positive constants. The dynamical and kinematical boundary conditions on the water surface are given by $ p = p0 − σH, (2) on Γ(t), t > 0, ηt − v · (−∇η, 1) = 0 where p0 is an atmospheric pressure, σ is the surface  tension coefficient, and H is the twice mean curvature, that is, H = ∇ · √ ∇η 2 . Here, p0 1+|∇η|

is a constant and σ is a non-negative constant. The kinematical boundary condition on the bottom is given by v·N = 0

on Σ,

t > 0,

(3)

where N is a normal vector on the bottom, that is, N = (−∇b, 1). These are the basic equations for the water wave problem. The basic equations (1)–(3) can be transformed equivalently in the following way. In view of the third equation in (1), we see that the velocity v has a single-valued potential: v = ∇X Φ, where Φ is called the velocity potential and determined uniquely up to an arbitrary function of t. By choosing Φ appropriately, Φ and η satisfy the following equations ΔX Φ = 0 %

in

Ω(t),

t > 0,

Φt + 12 |∇X Φ|2 + gη − σρ H = 0, ηt − (−∇η, 1) · ∇X Φ = 0 on Γ(t),

(4)

t > 0,

(5)

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∂n+1 Φ − ∇b · ∇Φ = 0

on Σ,

t > 0,

(6)

where Δ = + ··· + and ΔX = Δ + The equations (4)–(6) is equivalent to (1)–(3). In fact, once we obtain the solution (Φ, η) of(4)–(6),  1 2 by putting v = ∇X Φ and p = p0 − ρ Φt + 2 |∇X Φ| + g(xn+1 − h) , we see that (v, p, η) is a solution of (1)–(3). We note that both of (1)–(3) and (4)–(6) are free boundary problems. Because of complexities of the problem, several simplified models have been proposed. One of them is the shallow water model of finite amplitude, and the corresponding approximate equations are the shallow water equations   $ ηt + ∇ · (h + η − b)u = 0, (7) ut + (u · ∇)u + g∇η = 0, ∂12

∂n2

2 ∂n+1 .

where u = (u1 , . . . , un ) represents the velocity in the horizontal directions. The shallow water equations are used, for example, to simulate the propagation of tsunamis and can be derived, at least formally, from (1)–(3) or (4)–(6) in the shallow water limit δ=

mean depth h → 0. wave length λ

(8)

The Green–Naghdi equations are known as a higher-order approximation of the water waves in this shallow water regime. In the two-dimensional case with flat bottom, the Green–Naghdi equations have the form   $ ηt + (h + η)u x = 0,   (9) ut + uux + gηx = 13 (h + η)−1 (h + η)3 (uxt + uuxx − u2x ) x . Another famous simplified model is the Korteweg-de Vries (KdV) equation &

1 h3 σh 3 g 1 2 2 η + hη + − ηxx . ηt = (10) 2 h 2 3 3 3 ρg x This model describes a unidirectional motion of two-dimensional (n = 1) surface wave under the Boussinesq scaling

2 wave amplitude a mean depth h 2  1. (11) ε= δ = wave length λ mean depth h σ 1 This is known as the long wave approximation. In the critical case ρgh 2 = 3, the third order derivative term in the KdV equation (10) disappears and the equation falls into the inviscid Burgers’ equation. In this critical case, if we consider another long wave scaling

4 mean depth h wave amplitude a 4 1 (12) δ = ε= wave length λ mean depth h

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1 νh5 3 g 1 2 2 h5 η + hη + ηxxxx , ηxx + (13) ηt = 2 h 2 3 3 λ2 135 x

which is nowadays called the Kawahara equation. Here, we explain a fundamental property of the water wave problem. The problems (1)–(3) and (4)–(6) have trivial solutions of the forms (v, p, η) = (0, p0 − ρgxn+1 , 0) and (Φ, η) = (0, 0), respectively. In the case of a flat bottom, that is, the case b(x) ≡ 0, the linearized problem around this trivial solution has a solution of the form η(x, t) = ei(ξ·x−ωt) if and only if the wave vector ξ ∈ Rn and the angular frequency ω ∈ C satisfy the relation   σ (14) ω 2 = g|ξ| + |ξ|3 tanh(h|ξ|), ρ which is known as a dispersion relation for the linearized water wave problem. This shows that the basic equations (1)–(3) or (4)–(6) for water waves are nonlinear dispersive equations. The first term in the right-hand side of (14) represents a dispersive effect due to the gravity whereas the second term represents a dispersive effect due to the surface tension. These effects are balanced if the wave number ξc satisfies gξc = σρ ξc3 . The corresponding wave length is given by λc = 2π ξc ≈ 1.71[cm]. For long waves (λ  λc ) the effect of the gravity is dominant and that of the surface tension is negligible. The effect of the surface tension appear only for short waves (λ  λc ). Since we are mainly interested in a large scale motion in the theory of water waves, we usually neglect the surface tension and assume that σ = 0. However, there are so many literatures concerning the case σ > 0 because of mathematically rich structure of the equations in that case and also an importance to the problem of internal waves. In the rest of this article we explain the well-posedness of the initial value problem for the water wave problem (1)–(3) or equivalently (4)–(6). We also explain the validity of the approximations by the shallow water equations (7), the Green–Naghdi equations (9), the KdV equation (10), the Kawahara equation (13), and other equations, by giving error estimates between the solution of the water wave problem and that of the approximate equations. 2. Initial Value Problem The solvability of the initial value problem for the water wave problem was first given by Nalimov31 and Ovsjannikov,34 where an abstract Cauchy–

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Kowalewski theorem in a scale of Banach spaces was used. Therefore, it was assumed that the initial data are analytic. Well-posedness of the initial value problem is closely related to the solvability of the problem in Sobolev spaces. The solvability locally in time was first given by Nalimov,32 where he considered the two-dimensional water waves of infinite depth and assumed that the initial data were sufficiently small. The surface tension was neglected. He transformed the equations equivalently to equations on the water surface by using the Lagrangian coordinates, that is, he assumed that the water surface is parametrized as Γ(t) : (x1 , x2 ) = (ξ+X1 (ξ, t), X2 (ξ, t)), ξ ∈ R, where X = (X1 , X2 ) satisfies Xt (ξ, t) = v(ξ + X1 (ξ, t), X2 (ξ, t), t). Then, the equations are transformed as $ X1tt (1 + X1ξ ) + (X2tt + g)X2ξ = 0, X2t = K(X)X1t , where K(X) is a linear operator depending on X = (X1 , X2 ) the unknown water surface. The principal part of the operator K(X) is the Hilbert transform. By a reduction to a system of quasilinear equations, he found that the dynamics is essentially given by the equation utt + a(−∂ξ2 )1/2 u = f, where the function a is strictly positive. Therefore, he succeeded to show the solvability of the problem in the class C([0, T ]; H s+1/2 ) ∩ C 1 ([0, T ]; H s ) ∩ C 2 ([0, T ]; H s−1/2 ). 2

(15)

2 1/2 + X2ξ ) a

= Moreover, he showed that the function a satisfies ((1 + X1ξ )  ∂p  − ∂N Γ(t) , where N is the unit outward normal to the water surface. Therefore, the positivity of the function a is equivalent to the condition ∂p  (16) −  ≥ c > 0, ∂N Γ(t) which is nowadays known as a generalized Rayleigh–Taylor sign condition. This is an almost necessary condition for the well-posedness of the initial value problem. In the case of small solutions this condition is automatically satisfied. Later, Yosihara46,47 extended the result by Nalimov32 to the case with an almost flat bottom and to the case with surface tension. We note that in the case with the surface tension the well-posedness of the problem holds true without assuming the generalized Rayleigh–Taylor sign condition (16). Iguchi, Tanaka, and Tani21 considered an internal wave by taking into account the motion of the upper fluid, too, and showed the well-posedness

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of the problem for small initial data if we take into account the effect of the surface tension. They also showed that in the case without the surface tension the problem is ill-posed in general even if the density of the lower fluid is heavier than that of the upper fluid, in which case the linearized equations around a trivial flow is well-posed. Therefore, the ill-posedness arises as a nonlinear effect. Concerning the well-posedness for large initial data, Iguchi15 gave a positive answer for the two-dimensional water waves with the surface tension. Moreover, for the bottom topography he merely assumed its Lipschitz continuity. Wu35 considered the problem in same situation as Nalimov’s32 and showed the solvability of the problem including the case where the water surface turns over. As an important key to her result she proved that the generalized Rayleigh–Taylor sign condition always holds for any smooth nonself-intersecting surface. Wu36 also extended her result to the threedimensional case by using the Clifford analysis. We also refer to Ambrose and Masmoudi3,4 for the three-dimensional water waves with the surface tension. In view of the relation between the Eulerian and Lagrangian coordinates, it would be necessary to use the Lagrangian coordinates in order to show the well-posedness of the problem in the class (15). However, Lannes28 gave an existence theorem of the (n + 1)-dimensional water waves with finite depth by using the Euler coordinates, that is, he adopted the formulation %  2 φt + gη + 12 |∇φ|2 − 12 (1 + |∇η|2 )−1 Λ(η, b)φ + ∇η · ∇φ = 0, (17) for t > 0, ηt − Λ(η, b)φ = 0 where φ(x, t) = Φ(x, h + η(x, t), t) is the trace of the velocity potential on the water surface and Λ(η, b) is the Dirichlet-to-Neumann map to Laplace’s equation. Of course, this system is equivalent to (4)–(6) in the case σ = 0. Here, we note that the water wave problem has a conserved energy defined by

  η(x)  1 2 ρ|∇X Φ(X)| dX + ρgxn+1 dxn+1 dx ρH = Ω(t) 2 Rn 0 ρg ρ η2 . = (Λ(η, b)φ, φ) + 2 2 Moreover, the water wave problem has a Hamiltonian structure whose Hamiltonian is H and the canonical variables are η and φ. Namely, the water wave problem can be written by φt =

δH , δη

ηt = −

δH . δφ

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This fact was first found by Zakharov48 in the case of infinite depth. Explicit calculation of the above variational derivatives leads (17). Therefore, (17) is sometimes called the Zakharov formulation, although he did not use explicitly the Dirichlet-to-Neumann map. Concerning the global solvability in time, Wu37 showed an almost global existence of the solution for small data in the two-dimensional case of infinite depth. In the three-dimensional case of infinite depth, Wu38 and Germain, Masmoudi, and Shatah14 showed the global existence of the solution for small data. For large initial data, it is natural to expect that the water surface has a singularity in finite time. Recently, Castro, C´ordoba, Fefferman, Gancedo, and G´ omez-Serrano,5,6 and Coutand and Shkoller7 gave examples of solutions to the water wave problem, which have so-called splash or splat singularities in finite time. 3. Shallow Water Approximations Once we establish the solvability of the initial value problem, we can proceed to discuss the shallow water and the long wave approximations. The first step is to rewrite the equations in a non-dimensional form. In the following we adopt the formulations (4)–(6) and (17) of the problem. Let h, λ, a, and B be the mean depth of the water, the typical wave length, the amplitude of the water surface, and the maximal amplitude of the bottom variation. We introduce non-dimensional parameters δ = hλ , ε = ha , and β = B h as in (8), (11), and (12), and rescale independent and dependent variables by √ " η = a" xn+1 , t = √λgh " η , and b = B"b. x = λ" x, xn+1 = h" t, Φ = ελ ghΦ, Putting these into (4)–(6) and dropping the tilde sign in the notation, we obtain 2 Φ=0 δ 2 ΔΦ + ∂n+1

$

in

Ωε,β (t),

t > 0,

  δ 2 Φt + 12 ε|∇Φ|2 + η + ε 12 (∂n+1 Φ)2 = 0, on Γε (t), δ 2 ηt + ε∇Φ · ∇η − ∂n+1 Φ = 0 ∂n+1 Φ − βδ 2 ∇b · ∇Φ = 0

on Σβ ,

t > 0,

t > 0,

(18) (19)

(20)

where we assumed σ = 0. The water surface Γε (t) and the bottom of the water Σβ are of the forms   Γε (t) = X = (x, xn+1 ) ∈ Rn+1 ; xn+1 = 1 +εη(x, t) ,  Σβ = X = (x, xn+1 ) ∈ Rn+1 ; xn+1 = βb(x) .

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Similarly, we can rewrite (17) in a non-dimensional form as ⎧ 1 2 ⎪ ⎨ φt + η + 2 ε|∇φ|  2 − 12 εδ 2 (1 + ε2 δ 2 |∇η|2 )−1 Λ(εη, βb, δ)φ + ε∇η · ∇φ = 0, ⎪ ⎩ η − Λ(εη, βb, δ)φ = 0 for t > 0, t

(21)

where Λ(εη, βb, δ) is the Dirichlet-to-Neumann map of the scaled Laplace equation (18). More precisely, the map is defined as follows: for any φ ∈ H 1 (Rn ) the boundary value problem ⎧ 2 2 in b(x) < xn+1 < 1 + η(x), ⎨ δ ΔΦ + ∂n+1 Φ = 0 Φ=φ on xn+1 = 1 + η(x), ⎩ ∂n+1 Φ − δ 2 ∇b · ∇Φ = 0 on xn+1 = b(x) has a unique solution Φ. Using this solution we define a linear operator Λ(η, b, δ) by Λ(η, b, δ)φ = (δ −2 ∂n+1 Φ − ∇η · ∇Φ)|xn+1 =1+η(x) , which is a conormal derivative on the water surface associated to the scaled Laplace equation. Shallow water approximations corresponds to the parameter region ε = β = 1 and δ  1, which will be assumed throughout this section. 3.1. Shallow water equations As a next step we approximate the equations in (21). To this end we need to expand the operator Λ(η, b, δ) in terms of small δ:   Λ(η, b, δ)φ = −∇ · (1 + η − b)∇φ + O(δ 2 ), which can be justified in an appropriate sense. Therefore, taking the limit δ → 0 in (21) we obtain $ 0 φt + η 0 + 12 |∇φ0 |2 = 0,  ηt0 + ∇ · (1 + η 0 − b)∇φ0 = 0. These are known as the shallow water equations. In fact, putting u0 = ∇φ0 and taking the gradient of the first equation we recover the well-known form (7) of the shallow water equations. In order to justify this formal approximation of equations we need to show that the solution (η δ , φδ ) of the full water wave problem (21) converges to a solution (η 0 , u0 ) of the shallow water equations (7) in the limit δ → 0, or more strongly, η δ (t) − η 0 (t)s + ∇φδ (t) − u0 (t)s  δ 2 for 0 ≤ t ≤ T, where the time T should be independent of δ. A mathematically rigorous justification of this shallow water approximation was first given by Ovsjannikov35,36 for two-dimensional water waves under the periodic boundary condition with respect to the horizontal spatial variable, and then by

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Kano and Nishida.23 In these results an analyticity of the initial data was assumed. The justification in Sobolev spaces was first given by Li30 for two-dimensional water waves over a flat bottom. She used a conformal map to reformulate the problem, so that here result was restricted in twodimensional case. Then, the justification for the three-dimensional case in Sobolev spaces was given by Iguchi,19 and Alvarez-Samaniego and Lannes.2 In both results, non-flat bottoms were allowed. In the result,2 they used the Nash-Moser implicit function theorem so that a higher-order regularity of the initial data was imposed. 3.2. Tsunami generation In numerical computations of tsunamis due to submarine earthquakes, it is frequently used the shallow water equations (7) under the assumption that the initial displacement of the water surface is equal to the permanent shift of the seabed (bottom of the water) and that the initial velocity field is equal to zero, that is, η|t=0 = b1 − b0 and u|t=0 = 0, where b0 and b1 are bottom topographies before and after the submarine earthquake, respectively. In order to give a justification of this tsunami model, we have to consider the case where the bottom is a moving boundary, so that the function b should depend also on the time t. In this case the boundary condition on the bottom (6) should be replaced by bt + ∇b · ∇Φ − ∂n+1 Φ = 0

on Σ(t),

t > 0.

(22)

We assume that the seabed deforms only for a time interval [0, t0 ] in the dimensional variable t, so that the function b = b(x, t) which represents the bottom topography can be written in the form $ b0 (x) for τ ≤ 0, b(x, t) = β(x, t/), β(x, τ ) = (23) b1 (x) for τ ≥ 1 in the non-dimensional variables, where  is a non-dimensional parameter defined by  = λ/t√0 gh , which is the ratio of the time of the submarine earthquake to the time period of the linear shallow water wave, so that this is in general a small parameter for tsunamis. We note that in this non-dimensional time variable the bottom deforms only for the short time interval 0 ≤ t ≤ . The justification of the tsunami model was given by Iguchi20 where it was shown that under appropriate assumptions on the initial data and the bottom the solution (η δ, , φδ, ) of the full water wave problem can be approximated by the solution of the shallow water equations

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(7) with b replaced by b1 under the initial condition η = η0 + (b1 − b0 ),

u = ∇φ0

at

t = 0,

(24)

in a scaling regime δ 2    1, where (η0 , φ0 ) are initial data for the full water wave problem. More precisely, it holds that η δ, (t) − η 0 (t)s + ∇φδ, (t) − u0 (t)s   for  ≤ t ≤ T. Moreover, in the critical case δ 2    1 the initial condition (24) should be replaced by   2  1 1 βτ (·, τ ) dτ at t = 0. (25) η = η0 + (b1 − b0 ), u = ∇ φ0 + 2 0 For details we refer to Iguchi.20 It is natural to assume the condition δ 2    1 for tsunamis. However, very rarely, the condition is not satisfied, particularly, the condition on . One of such events is the Meiji-Sanriku earthquake, which occurred at June 15 in 1896. The seismic scale of this earthquake was small, but it continued for several minutes. As a result, a huge tsunami attacked the Sanriku coast line. To simulate such a tsunami, it might be better to consider the limit δ → 0 keeping  is of order one. In that case the solution of the full water wave problem can be approximated by the solution of the shallow water equations with a source term   $ ηt + ∇ · (h + η − b)u = bt , (26) ut + (u · ∇)u + g∇η = 0. In the next subsection, we explain a higher-order approximation in this scaling regime. 3.3. Green–Naghdi equations We first explain what are the Green–Naghdi equations. For simplicity, we consider a linearized problem around the trivial flow in the case of a flat bottom. Since the Dirichlet-to-Neumann map in the trivial case can be written explicitly in terms of the Fourier multipliers as Λ(0, 0, δ) = 1δ |D| tanh(δ|D|), the linearized equations for the full equations (21) have the form $ φt + η = 0, (27) ηt − 1δ |D| tanh(δ|D|)φ = 0. Letting δ → 0 we obtain linearized shallow water equations. To obtain a higher-order approximation, we use the Taylor expansion tanh x = x − 1 3 1 1 2 2 5 4 3 x + O(x ) (x → 0) and obtain δ |D| tanh(δ|D|) = −Δ − 3 δ Δ + O(δ ).

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Putting this into the linearized equations (27) and neglecting the terms of order O(δ 4 ), we obtain higher-order approximate equations $ ηt + Δφ + 13 δ 2 Δ2 φ = 0, (28) φt + η = 0. This system has a non-trivial solution of the form η(x, t) = η0 ei(ξ·x−ωt) and φ(x, t) = φ0 ei(ξ·x−ωt) if and only if the wave vector ξ ∈ Rn and the angler frequency ω ∈ C satisfy ω 2 − 1 − 13 δ 2 |ξ|2 )|ξ|2 = 0, which is the dispersion relation for (28). In the case |ξ| > δ32 , the solutions ω of this dispersion relation are purely imaginary, so that the approximate equations (28) have a solution which grows exponentially as |ξ| → ∞ for each t > 0. Therefore, the initial value problem for (28) is ill-posed, and (28) is not good approximation for the linearized equations (27). −1  φ On the other hand, introducing new variables by ψ = 1 − 13 δ 2 Δ and u = ∇ψ in (27) and neglecting the terms of order O(δ 4 ), we obtain another higher-order approximate equations $ ηt + ∇ · u = 0, (29) ut + ∇η = 13 δ 2 Δut .   The dispersion relation of these approximate equations is 1 + 13 δ 2 |ξ|2 ω 2 − |ξ|2 = 0, so that the initial value problem for (29) is well-posed. The corresponding nonlinear equations are called the Green–Naghdi equations, whose solution should approximate the solution of the full water wave problem up to O(δ 4 ). The justification was first given by Li30 for two-dimensional water waves over a flat bottom, and then by Alvarez-Samaniego and Lannes2 for three dimensional case where non-flat bottoms were allowed. In connection with a tsunami generation, the justification was given by Fujiwara and Iguchi13 for the case where the bottom is a moving boundary. 4. Long Wave Approximations In this last section we always consider the two-dimensional case, that is, the case n = 1. We also assume, for simplicity, that the bottom is flat except in section 4.3, and we take into account the effect of the surface tension. In such a case the corresponding equations to (21) are rewritten as ⎧  2 ⎪ ⎨ φt + η + 12 εφ2x − 12 εδ 2 (1 + ε2δ 2 ηx2 )−1 Λ(εη, βb, δ)φ + εηx φx (30) −μδ 4 (1 + ε2 δ 2 ηx2 )−1/2 ηx x = 0, ⎪ ⎩ η − Λ(εη, βb, δ)φ = 0 for t > 0, t

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By putting u = φx and using the above expansion, the equations in (30) can be approximated by the partial differential equations $ 2 4 δ uxxxxx = O(δ 6 + εδ 2 ), ηt + ux + ε(ηu)x + 13 δ 2 uxxx + 15 (31) 2 ut + ηx + εuux − μδ ηxxx = O(εδ 2 ). 4.1. KdV equation The KdV equation is derived from (31) in the long wave regime δ 2  ε  1, which means that an effect of the nonlinear terms is balanced with that of the third order dispersion terms, and that both effects are sufficiently small. We assume that δ 2 = ε. Letting δ 2 = ε → 0 in (31) we obtain the linear shallow water equations $ ηt + ux = 0, ut + ηx = 0. The solution of these equations is given by $ η(x, t) = α1 (x − t) + α2 (x + t), u(x, t) = α1 (x − t) − α2 (x + t). As in the case of the shallow water approximation, the solution of the full water wave problem can be approximated by the above solution up to O(δ 2 ) for some time interval 0 ≤ t ≤ T independent of the parameter ε. We proceed to give a higher-order approximation, especially, an approximation for a long time interval. Here we note that the functions α1 and α2 in the above solution represent profiles of the waves moving to the right and to the left, respectively. The profiles should change in the slow time scale τ = εt. This consideration leads the ansatz $ η(x, t) = α1 (x − t, εt) + α2 (x + t, εt) + O(δ 2 ), (32) u(x, t) = α1 (x − t, εt) − α2 (x + t, εt) + O(δ 2 ). Then, we see that α1 = α1 (x, τ ) and α2 = α2 (x, τ ) satisfy the KdV equations $ 2α1τ + 3α1 α1x + ( 13 − μ)α1xxx = 0, (33) 2α2τ − 3α2 α2x − ( 13 − μ)α2xxx = 0. In order to give a justification of the approximation (32) we have to give an existence theorem for the full water wave problem for a long time interval

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0 ≤ t ≤ T /ε, where T is a positive time independent of the parameter ε. Such a theorem was first given by Craig8 under a unidirectional motion of the wave. Schneider and Wayne37,38,40 removed the condition of the unidirectional motion, but they did not give uniform bounds of derivatives of the solution. Iguchi17 gave the uniform bounds and succeed to give a justification of the approximation (32). He also considered an effect when the bottom is not flat. Here we remark that in both of the approximations by the Green– Naghdi equations and the KdV equation we used approximate equations for water waves up to O(δ 4 ). In the former case the solution is approximated with an accuracy of O(δ 4 ) for a short time interval 0 ≤ t ≤ T , whereas in the latter case the solution is approximated with an accuracy of O(δ 2 ) for a long time interval 0 ≤ t ≤ T /ε. We also remark that the condition δ 2  ε  1 is generally matched for tsunamis. This is one of the reasons why the linear shallow water equations are used to simulate the propagation of tsunamis in the open sea. One may expect that the tsunami can be simulated by the KdV equation. However, we have to take care because the KdV dynamics appears after a very long time t = O(1/ε). 4.2. Kawahara equation In the case μ = 13 the KdV equations in (33) fall into the inviscid Burgers’ equations. In this case if the effect of the nonlinear terms is balanced with that of the higher-order dispersive term, then a higher order KdV equation with an additional fifth order derivative term called Kawahara equation describes the dynamics of the water waves for a long time interval 0 ≤ t ≤ T /ε. In fact, under the assumptions μ = 13 + νδ 2 and δ 4 = ε we can show a justification of the approximation (32) for a long time interval 0 ≤ t ≤ T /ε, where the functions α1 = α1 (x, τ ) and α2 = α2 (x, τ ) satisfy the Kawahara equations $ 1 α1xxxxx = 0, 2α1τ + 3α1 α1x − να1xxx + 45 1 α2xxxxx = 0. 2α2τ − 3α2 α2x + να2xxx − 45 For details we refer to Schneider and Wayne39,40 and Iguchi.18 4.3. Forced KdV equation In the case where the fluid has non-trivial background flow of the form v ∞ = (c, 0) like a river and the bottom is not flat, it is natural to expect

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that the background flow together with uneven bottom creates waves on the water surface. In the long wave scaling δ 2  ε  1 such waves can be described the KdV equation with a source term. We introduce other non√c which is the Froude number. dimensional parameters β = B h and γ = gh 2 2 Under the assumptions δ = ε and ε = β, the solution of the full water wave problem can be approximated as $ η(x, t) = α1 (x − (γ + 1)t, εt) + α2 (x − (γ − 1)t, εt) + O(δ 2 ), (34) u(x, t) = α1 (x − (γ + 1)t, εt) − α2 (x − (γ − 1)t, εt) + O(δ 2 ) for a long time interval 0 ≤ t ≤ T /ε, where the functions α1 = α1 (x, τ ) and α2 = α2 (x, τ ) satisfy   $ 2α1τ + 3α1 α1x +  13 − μα1xxx = γb (x + (γ + 1)τ /ε), 2α2τ − 3α2 α2x − 13 − μ α2xxx = γb (x + (γ − 1)τ /ε). In the critical case γ = 1 the second equation becomes the forced KdV equation. The justification of this approximation was given by Iguchi.16 4.4. Benjamin–Ono equation If we take into account the motion of the upper fluid, then the motion of the water surface can be described by the Benjamin–Ono equation in the scaling regime δ  ε  1. In fact, under the assumption δ = ε, μ > 0, and 0 ≤ ρ1 < ρ2 , where ρ1 and ρ2 are constant densities of upper and lower fluids, respectively, the water surface can be approximated as η(x, t) = α1 (x − t, εt) + α2 (x + t, εt) + O(ε) for a long time interval 0 ≤ t ≤ T /ε, where the functions α1 = α1 (x, τ ) and α2 = α2 (x, τ ) satisfy the Benjamin–Ono equations $ 2α1τ + 3α1 α1x + rHα1xx = 0, 2α2τ − 3α2 α2x − rHα2xx = 0, where r = ρ1 /ρ2 and H is the Hilbert transform defined by  u(y) 1 dy. Hu(x) = p.v. π x −y R The justification of this approximation was given by Ohi and Iguchi.33 See also Lannes,29 where a stability criterion for internal water waves is carefully analyzed. Finally, we mention that two time scales t and τ = εt are used in all of these long wave approximations. In the time scale t the waves of water

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surface move in the right and the left without changing their shapes. In the time scale τ = εt nonlinear and dispersion effects appear and the change of each wave profile is described by a corresponding equation.

References 1. G. B. Airy, Tides and waves, Encyclopaedia metropolitana, London, 5 (1845), 241–396. 2. B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D waterwaves and asymptotics, Invent. Math., 171 (2008), 485–541. 3. D. M. Ambrose and N. Masmoudi, Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci., 5 (2007), 391–430. 4. D. M. Ambrose and N. Masmoudi, The zero surface tension limit of threedimensional water waves, Indiana Univ. Math. J., 58 (2009), 479–521. 5. A. Castro, D. C´ ordoba, C. Fefferman, F. Gancedo, and J. G´ omez-Serrano, Splash singularity for water waves, arXiv:1106.2120. 6. A. Castro, D. C´ ordoba, C. Fefferman, F. Gancedo, and J. G´ omez-Serrano, Finite time singularities for the free boundary incompressible Euler equations, arXiv:1112.2170. 7. D. Coutand and S. Shkoller, On the finite-time splash and splat singularities for the 3-D free-surface Euler equations, arXiv:1201.4919. 8. W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Commun. Partial Differ. Equations, 10 (1985), 787.1003. 9. W. Craig and M. D. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion 19 (1994), 367–389. 10. W. Craig, P. Guyenne, D. P. Nicholls, and C. Sulem, Hamiltonian long wave expansions for water waves over a rough bottom, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 839–873. 11. W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73–83. 12. K. O. Friedrichs, On the derivation of the shallow water theory, Appendix to: “The formulation of breakers and bores” by J.J. Stoker in Comm. Pure Appl. Math., 1 (1948), 1–87. 13. H. Fujiwara and T. Iguchi, A shallow water approximation for water waves over a moving bottom, submitted. 14. P. Germain, N. Masmoudi, and J. Shatah, Global Solutions for the Gravity Water Waves Equation in Dimension 3, Ann. of Math., 175 (2012), 691–754. 15. T. Iguchi, Well-posedness of the initial value problem for capillary-gravity waves, Funkcial. Ekvac., 44 (2001), 219–241. 16. T. Iguchi, A mathematical justification of the forced Korteweg-de Vries equation for capillary-gravity waves, Kyushu J. Math., 60 (2006), 267–303. 17. T. Iguchi, A long wave approximation for capillary-gravity waves and an effect of the bottom, Comm. Partial Differential Equations, 32 (2007), 37– 85.

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18. T. Iguchi, A long wave approximation for capillary-gravity waves and the Kawahara equation, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 179–220. 19. T. Iguchi, A shallow water approximation for water waves, J. Math. Kyoto Univ., 49 (2009), 13–55. 20. T. Iguchi, A mathematical analysis of tsunami generation in shallow water due to seabed deformation, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 551–608. 21. T. Iguchi, N. Tanaka, and A. Tani, On the two-phase free boundary problem for two-dimensional water waves, Math. Ann., 309 (1997), 199–223. 22. T. Kano, Une th´eorie trois-dimensionnelle des ondes de surface de l’eau et le d´eveloppement de Friedrichs, J. Math. Kyoto Univ., 26 (1986), 101–155 and 157–175. 23. T. Kano and T. Nishida, Sur les ondes de surface de l’eau avec une justification math´ematique des ´equations des ondes en eau peu profonde, J. Math. Kyoto Univ., (1979) 19, 335–370. 24. T. Kano and T. Nishida, Water waves and Friedrichs expansion. Recent topics in nonlinear PDE, 39–57, North-Holland Math. Stud., 98, North-Holland, Amsterdam, 1984. 25. T. Kano and T. Nishida, A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves, Osaka J. Math., 23 (1986), 389–413. 26. D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422–443. 27. H. Lamb, Hydrodynamics, 6th edition, Cambridge University Press. 28. D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605–654. 29. D. Lannes, A stability criterion for two-fluid interfaces and applications, arXiv:1005.4565. 30. Y. A. Li, A shallow-water approximation to the full water wave problem, Comm. Pure Appl. Math., 59 (2006), 1225–1285. 31. V. I. Nalimov, A priori estimates of solutions of elliptic equations in the class of analytic functions, and their applications to the Cauchy-Poisson problem, Dokl. Akad. Nauk SSSR, 189 (1969), 45–48. 32. V. I. Nalimov, The Cauchy-Poisson problem, Dinamika Sploˇsn. Sredy, 18 (1974), 104–210. 33. K. Ohi and T. Iguchi, A two-phase problem for capillary-gravity waves and the Benjamin-Ono equation, Discrete Contin. Dyn. Syst., 23 (2009), 1205– 1240. 34. L. V. Ovsjannikov, A nonlinear Cauchy problem in a scale of Banach spaces, Dokl. Akad. Nauk SSSR, 200 (1971), 789–792. 35. L. V. Ovsjannikov, To the shallow water theory foundation, Arch. Mech., 26 (1974), 407–422. 36. L. V. Ovsjannikov, Cauchy problem in a scale of Banach spaces and its application to the shallow water theory justification. Applications of methods of functional analysis to problems in mechanics, 426–437. Lecture Notes in

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Math., 503. Springer, Berlin, 1976. 37. G. Schneider and C. E. Wayne, The long-wave limit for the water wave problem I. The case of zero surface tension, Comm. Pure Appl. Math., 53 (2000), 1475–1535. 38. G. Schneider and C. E. Wayne, Corrigendum: The long-wave limit for the water wave problem I. The case of zero surface tension, Comm. Pure Appl. Math., 65 (2012), 587–591. 39. G. Schneider and C. E. Wayne, Kawahara dynamics in dispersive media, Phys. D, 152/153 (2001), 384–394. 40. G. Schneider and C. E. Wayne, The rigorous approximation of longwavelength capillary-gravity waves, Arch. Ration. Mech. Anal., 162 (2002), 247–285. 41. J. J. Stoker, Water waves: the mathematical theory with application, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York. 42. S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., 130 (1997), 39–72. 43. S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445–495. 44. S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45–135. 45. S. Wu, Global well-posedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125–220. 46. H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. RIMS Kyoto Univ., 18 (1982), 49–96. 47. H. Yosihara, Capillary-gravity waves for an incompressible ideal fluid, J. Math. Kyoto Univ., 23 (1983), 649–694. 48. V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190–194.

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ON THE EXISTENCE AND NONEXISTENCE OF MAXIMIZERS ASSOCIATED WITH TRUDINGER-MOSER TYPE INEQUALITIES IN UNBOUNDED DOMAINS Michinori Ishiwata Department of Industrial System, Fukushima University, Fukushima, Japan ∗ E-mail: [email protected] It is known that the classical Trudinger-Moser inequality for a bounded domain has several extensions to the RN -case. In this note, we review some recent results on the existence and the nonexistence of maximizers for the RN -case. Keywords: Trudinger-Moser type inequality in unbounded domain, maximizing function

1. Introduction and main results This note is based on the recent work of Ishiwata12 and Ishiwata, Nakamura 1 and Wadade.13 Let N ≥ 2 and let αN := N |S N −1 | N −1 , where |S N −1 | is the surface area of the (N − 1)-dimensional unit sphere. Let Ω be a bounded domain in RN . The classical Trudinger-Moser inequality asserts that, for α ∈ (0, αN ], there exists b > 0 which depends only on N and α satisfying  N N −1 eα|u| ≤ b|Ω|, ∀u ∈ M0,N (Ω), (1) Ω

where M0,N (Ω) = {u ∈ W01,N (Ω); ∇uLN (Ω) = 1}, see Trudinger,17 Moser.21 The existence of a function which attains the equality in (1) is shown by Carleson and Chang7 if Ω is a N -dimensional ball and by Flucher1 if Ω is a two-dimensional smooth bounded domain. There are several extensions of this inequality to unbounded domains. In this note, we consider the following one. Let N ≥ 2, Ω ⊂ RN and let ΦN,α (t) := eαt −

N −2 j=0

∞ αj j αj j t = t . j! j! j=N −1

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Then we have



(TMA)Ω supu∈W 1,N (Ω), ∇u 0

LN (Ω)

=1

(TMB)Ω supu∈W 1,N (Ω), u 0

N

Ω

=1 W 1,N (Ω)

 Ω

ΦN,α (u N −1 ) < ∞ if α ∈ (0, αN ), uN LN (Ω) N

ΦN,α (u N −1 ) < ∞ if α ∈ (0, αN ].

(TMA)R2 with α sufficiently small is first appeared in Ogawa.22 Later, (TMA)RN with general N ≥ 2 is proved by Adachi and Tanaka.1 In Adachi and Tanaka,1 it is also shown that αN is the critical exponent in the sense that, for every α ≥ αN , there exists (un ) ⊂ W 1,N (RN ) satisfying ∇un LN (RN ) = 1 and N  N −1 ) RN ΦN,α (un →∞ N un LN (RN ) as n → ∞. For further extension of (1) closely related to (TMA)Ω , see e.g. Ogawa and Ozawa,23 Ozawa,26 Kozono, Sato and Wadade15 and references therein. (TMB)R2 first appears in Cao.6 Ruf 25 gave a detailed analysis of the variational problems associated with (TMB)R2 with α = 4π. The case N ≥ 3 is also treated in a recent paper Li and Ruf.20 The purpose of this note is to show the attainability of the best constants associated with (TMA)RN and (TMB)RN , respectively. Let us denote  N N −1 ) Ω ΦN,α (u cN,α (Ω) = sup ., uN u∈W01,N (Ω), ∇u LN (Ω) =1 LN (Ω)  N dN,α (Ω) = sup ΦN,α (u N −1 ). u∈W01,N (Ω), u W 1,N (Ω) =1

Ω

Our main results read as follows. Theorem 1.1 (Ishiwata, Nakamura and Wadade13 ). 1 Let N ≥ 2 and let αN = N |S N −1 | N −1 , where |S N −1 | is the surface area of the (N − 1)-dimensional unit sphere. Then cN,α (RN ) is attained for all α ∈ (0, αN ). Moreover, cN,α (Ω) is attained if and only if Ω = RN . One may regard the variational problem associated with (TMA) as a “subcritical” problem since α is below the critical exponent αN . This is not the case from the variational analysis point of view. The associated variational problem has a critical nature due to the lack of compactness caused by the scale invariance of (TMA) even if α is in a subcritical range. Indeed, assume that there exists φ ∈ W 1,N (RN ) which attains cN,α (RN ). Let

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(λn ) ⊂ R+ be a sequence consists of positive numbers and let us introduce a sequence (ϕn ) ⊂ W 1,N (RN ) by ϕn (x) := φ(λn x). Then we easily see that N   N N −1 N −1 ) ) RN ΦN,α (ϕn RN ΦN,α (φ = = cN,α . ∇ϕn LN = ∇ϕLN = 1, ϕn N φN LN (RN ) LN (RN ) Particularly, (ϕn ) is a maximizing sequence for cN,α having no convergent subsequence (in W 1,N ) if λn → ∞ or λn → 0. The similar lack of compactness arising from the scale-invariance is also observed in the variational problem associated with the higher dimensional critical Sobolev inequality u

Np N −p

≤ C,

∇up = 1,

where 1 < p < N , see e.g. Lions,18 Lions.19 Our second result is concerned with (TMB)R2 (we here consider the two-dimensional case for the simplicity): Theorem 1.2 (Ishiwata12 ). Let N = 2 and let B2 =

sup

φ =0,φ∈H 1

φ44 . ∇φ22 φ22

Then d2,α is attained for 2/B2 < α ≤ α2 = 4π. Remark 1.1. It is known that B2 > 1/(2π) and B2 is attained, see e.g. Weinstein,36 Beckner;3 thus the interval (2/B2 , α2 ] is non-empty. In Theorem 1.2, a condition on α appears in the two-dimensional case. For the bounded domain case, it is an easy exercise to show the existence of a maximizer for the varitational problem associated with (1) by using the compactness argument when α < 4π. Hence one may expect that the restriction on α appears in Theorem 1.2 is rather a technical one. Surprisingly, this is not the case and we have the following nonexistence result for small α in the two-dimsensional case, which reveals the essential difference between the problem on bounded domains and unbounded domains: Theorem 1.3 (Ishiwata12 ). Let N = 2. If α  1, then d2,α is not attained. Remark 1.2.   2 Indeed, we prove that the functional J(u) := Ω Φ2,α (u2 ) = Ω (eαu −1) possesses no critical points in {u ∈ H 1 (R2 ); uH 1 = 1} if α is sufficiently small.

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As is mentioned above, the main difference between (TMA) and (TMB) is that (TMA) is scale invariant while (TMB) is not. Reflecting this difference, the proof of Theorem 1.1 and that of Theorem 1.2 require different approaches. The crucial point of the proof of Theorem 1.1 is the renormalization of a maximizing sequence with the aid of the scale invariance. Theorem 1.1 is proved in §2. The proof of Theorem 1.2 is based on the analysis of vanishing/concentrating sequences in the spirit of concentrationcompactness principle. Since the proof of Theorem 1.2 is rather involved, we will give in this note the sketch of the proof. For the detailed proof of Theorem 1.2 and Theorem 1.3, see Ishiwata.12 For the variational problems associated with (TMA) and (TMB), it is enough to consider radially symmetric functions by virtue of symmetrization. Hence, in the following, we only consider radially symmetric, nonnegative functions. Notation  · Lp (Ω) denotes the standard Lp (Ω)-norm. We also use the abbreviation  · p,Ω and we occasionally omit the subscript Ω. The norm N N 1,N of W01,N (Ω) is defined by uN W 1,N (Ω) := ∇uLN (Ω) + uLN (Ω) . Wr denotes the set consists of radially symmetric W01,N -functions. |S N −1 | denotes the surface area of the (N − 1)-dimensional unit sphere. Let 1 αN := N |S N −1 | N −1 . The constant C may vary from line to line. We pass to subsequences freely. 2. Proof of Theorem 1.1 Let N ≥ 2 and Ω ⊂ RN . Throughout this section, we assume that 1

α < αN := N |S N −1 | N −1 .

(2)

Let  FN,α,Ω (u) :=

N

Ω

ΦN,α (u N −1 ) . uN N

We can show the following fact. See Ishiwata, Nakamura and Wadade13 for details. Lemma 2.1. Let α < αN and let (un ) be a bounded sequence in Wr1,N (Ω) satisfying ∇un N,Ω = 1 and un  u weakly in W01,N (Ω), where Wr1,N denotes the

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set consists of radially symmetric functions in W01,N . Then

  N N αN −1 αN −1 N −1 N N N −1 |un | |u| ΦN,α (un ) − → ΦN,α (u )− (N − 1)! (N − 1)! Ω Ω as n → ∞. We also note that cN,α (Ω) >

αN −1 (N − 1)!

(3)

holds. Indeed, let u ∈ W01,N (Ω) be any function satisfying ∇uN = 1. Then N ∞ αj N −1 j u N −1 N j=N j! α αN −1 N −1 j + , cN,α (Ω) ≥ FN,α,Ω (u) = > N (N − 1)! (N − 1)! uN hence (3). Proof of Theorem 1.1 Let (un ) be a maximizing sequence for cN,α (RN ), i.e., ∇un N = 1,

FN,α,RN (un ) → cN,α (RN )

as n → ∞. Without loss of generality, by virtue of the radially symmetric rearrangement, we can assume that un is a radially symmetric function. Let vn (x) := un (un N x),

x ∈ RN .

It is easy to see that ∇vn N = ∇un N = 1,

vn N = 1,

FN,α,RN (vn ) = FN,α,RN (un ) → cN,α (RN ) as n → ∞. Particularly, there exists v ∈ Wr1,N (RN ) such that vn  v weakly in W01,N ,

(4)

1 ≥ vN , ∇vN

(5)

Then, by virtue of (4), (5) and Lemma 2.1, we have cN,α (RN ) = FN,α,RN (vn ) + o(1)

 N αN −1 αN −1 N N −1 + |v| = ΦN,α (v )− (N − 1)! (N − 1)! RN

(6)

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as n → ∞. Hence (3) yields v = 0. Note that (5) and (6) imply that

 N αN −1 αN −1 + |v|N ΦN,α (v N −1 ) − cN,α (RN ) = (N − 1)! (N − 1)! RN    N αN −1 N ΦN,α (v N −1 ) − (N −1)! |v| αN −1 RN ≤ + (N − 1)! vN N = FN,α,RN (v).

(7)

We now show that ∇vN ≥ 1. By the definition of cN,α (RN ), we have cN,α (R ) ≥ FN,α,RN N

v ∇vN

N

N

j

N −1 ∞ αj v NN−1 j αN −1 + ≥ (N − 1)! j! vN N j=N

⎛ = FN,α,RN (v) + ⎝

j

N −1 ∞ αj v NN−1 j j! ∇v NN−1 j j=N −1 N N ⎛ ⎞ N −1 N N v N N 1 α N −1 +⎝ − 1⎠ N N N ! v N −1 N ∇v

∇vN N = vN N

N

⎞

N

N

N −1 α v NN−1 N ⎠ −1 . N ! vN N

N

1 N

N −1 ∇vN

This relation together with (7) gives N

N −1 ≥ 1, ∇vN

which implies 1 ≤ ∇vN ; thus ∇vN = 1 follows from (5). Consequently, by (7), we see that v is a maximizer for cN,α (RN ). Now assume that cN,α (Ω) with Ω = RN is attained by some u ∈ and we derive a contradiction. Without loss of generality, we can assume u ≥ 0 in RN . First we assume that N ≥ 3 and Let v(x) = u(uN x), where u is a zero-extension of u to RN . The scale-invariance of the problem yields cN,α (Ω) = cN,α (RN ); thus

W01,N (Ω)

vN,RN = 1,

∞ N αj j v NN−1 j = FN,α,RN (v) = cN,α (RN ). N −1 j!

(8)

j=N −1

Recall that (dFN,α,RN )w (ϕ) =

⎡

 ∞ N αj N 1 ⎣ N |w| N −1 j−2 wϕ w N (j − 1)! N − 1 w2N N j=N −1 ⎤  ∞ j N α j w NN−1 j ⎦ − N |w|N −2 wϕ N −1 j! j=N −1

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for arbitrary w, ϕ ∈ W 1,N (RN ). The Lagrange multiplier rule together with (8) and the above relation yields the existence of λ ∈ R satisfying ⎡ −ΔN v = λ ⎣

∞

j=N −1

⎤ N αj N |v| N −1 j−2 v − N |v|N −2 vcN,α (RN )⎦ (9) (j − 1)! N − 1

in RN , where ΔN v := ∇ · (|∇v|N −2 ∇v). Multiplying v to (9), integrating over RN and by using (8), we see that ⎡ ⎣ ∇vN N = λ ⎡ = λ⎣

∞ j=N −1 ∞ j=N −1

⎤ N αj N j N ⎦ v NN−1 j − N vN N cN,α (R ) N −1 (j − 1)! N − 1 ⎤ ∞ j N N αj α N j j v NN−1 j − N vN v NN−1 j ⎦ N N −1 N −1 (j − 1)! N − 1 j!



∞ N αj j N −1 j =λ v N j N −1 , N −1 j! N −1

j=N −1

j=N −1

whence λ > 0 follows. Consequently, (9) gives −ΔN v + λN cN,α (RN )|v|N −2 v = λ

∞ j=N −1

N αj N |v| N −1 j−2 v ≥ 0 (j − 1)! N − 1

in RN . This relation together with the strong maximum principle, see e.g. V´azquez [29, Theorem 5], yields v > 0 in RN . This is a contradiction since v ≡ 0 in {x ∈ RN ; uN x ∈ Ω}. This completes the proof.

3. Proof of Theorem 1.2 In this section, we assume that N = 2 and Ω = R2 unless stated. Let (un ) be a bounded sequence in H 1 satisfying un  u weakly in H 1 . We introduce several facts needed in the analysis of the compactness of maximizing sequences for d2,α in the spirit of the concentration compactness at infinity, see e.g. Ben-Naoum, Troestler and Willem,5 Bianchi, Chabrowski and Szulkin.8

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Let us introduce

 (|∇un |2 + |un |2 ),

μ0 = lim lim

R→∞ n→∞

B

R

μ∞ = lim lim

R→∞ n→∞



c BR

Φ2,α (u2n ),

ν0 = lim lim

R→∞ n→∞

BR

ν∞ = lim lim

R→∞ n→∞



c BR

Φ2,α (u2n ), 

|un | , η∞ = lim lim 2

η0 = lim lim

R→∞ n→∞

(|∇un |2 + |un |2 ),

R→∞ n→∞

BR

c BR

|un |2 ,

taking an appropriate subsequences if necessary. For a sufficiently large number R, take a function hR (r) ∈ C ∞ (R2 ) such that ⎧ hR (r) = 1 for 0 ≤ r ≤ R, ⎪ ⎪ ⎨ 0 ≤ hR (r) ≤ 1 for R ≤ r ≤ R + 1, ⎪ hR (r) = 0 for R + 1 ≤ r, ⎪ ⎩  for all r |hR (r)| ≤ 2 and introduce cut-off functions φ0R and φ∞ R by φ0R (x) := hR (|x|),

φ∞ R (x) = 1 − hR (|x|).

The following fact is proved by the same argument as in Ben-Naoum, Troestler and Willem,5 Bianchi, Chabrowski and Szulkin8 and Ishiwata.11 See for Ishiwata12 for the detailed argument. Lemma 3.1. Let u∗n,R := un φ∗R (∗ = 0, ∞). We have  (|∇u∗n,R |2 + |u∗n,R |2 ), μ∗ = lim lim R→∞ n→∞ R2  ν∗ = lim lim Φ2,α ((u∗n,R )2 ), R→∞ n→∞ R2  η∗ = lim lim |u∗n,R |2 . R→∞ n→∞

R2

The proof of Theorem 1.2 needs the study of the supremum of Φ2,α (u2n ) with a vanishing or a concentrating sequence (un ). At first we introduce the precise definition of a vanishing/concentrating sequence.



Definition 3.1.

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Let (un ) ⊂ H 1 be a sequence such that un  u weakly in H 1 . (a) It is said that (un ) is a normalized concentrating sequence ((NCS) in short) if (un ) satisfies un H 1 = 1, u = 0 and limn→∞ B c (|∇un |2 +|un |2 ) = ρ

0 for all ρ > 0. A (NCS) consisting of radially symmetric functions is called a radially symmetric normalized concentrating sequence ((RNCS) in short). (b) It is said that (un ) is a normalized vanishing sequence ((NVS) in short) if (un ) satisfies un H 1 = 1, u = 0 and ν0 = 0, where ν0 is defined in (10). A (NVS) consisting of radially symmetric functions is called a radially symmetric normalized vanishing sequence ((RNVS) in short). The notion “normalized concentrating sequence” already appears in various literatures, see e.g. Carleson and Chang,7 de Figueiredo, do O and Ruf 10 and Ruf.25 Next we introduce obstacle values for the compactness of maximizing sequences. Definition 3.2. (a) A number

 dncl (2, α) =

Φ2,α (u2n )

sup (un ):(RNCS)

is called a normalized concentration limit. (a) A number dnvl (2, α) =



sup

Φ2,α (u2n )

(un ):(RNVS)

is called a normalized vanishing limit. In Ruf,25 a normalized concentration limit dncl (2, 4π) is called a “Carleson-Chang limit” and dncl (2, 4π) = eπ is proved. As for dnvl , we have the following estimates. See Ishiwata12 for details. Proposition 3.1. There holds dnvl (2, α) = α for α ∈ (0, 4π]. Remark 3.1. Combining (10) and Proposition 3.1, we see that dnvl (2, 4π) = 4π = 4π > eπ = dncl (2, 4π).

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From this relation, it is observed that the main obstacle to the compactness of a maximizing sequence for d2,4π is not the concentrating behavior but the vanishing behavior. Hence the exclusion of the vanishing behavior is crucial for the compactness of maximizing sequences. Next we give a lower bound for d2,α . This bound gives a crucial information for the exclusion of a concentration or a vanishing behavior of maximizing sequences, see Ishiwata.12 Proposition 3.2. It holds that d2,α supφ =0,φ∈H 1

>

α for α

φ 44 . ∇φ 22 φ 22

∈

(2/B2 , 4π], where B2

=

From Proposition 3.1 and Proposition 3.2, we see that d2,α > max(dnvl (2, α), dncl (2, α)). By using this crucial relation, we can show Theorem 1.2. See Ishiwata12 for details. 4. Proof of Theorem 1.3 In this section, we always assume that N = 2. We first introduce the Gagliardo-Nirenberg-Sobolev type inequality with the sharp asymptotics of the best constant, see e.g. Ogawa,22 Ogawa and Ozawa,23 Kozono, Ogawa and Sohr,14 Edmunds and Ilyin.8 Here we recall the argument introduced by Ogawa and Yokota.24 Let β ∈ (0, 4π). Then it follows from (TMA)R2 that  2  2j ∞ u β j u2j β k u2k u22 β ∇u 2k 2 ≤ = (e − 1)dx ≤ C , β j! ∇u2j k! ∇u2k ∇u22 2 2 k=1 where Cβ is a universal constant; thus we have Lemma 4.1. For any β ∈ (0, 4π), there exists Cβ > 0 such that u2j 2j ≤ Cβ

j! 2(j−1) ∇u2 u22 βj

holds for any j ∈ N and any u ∈ H 1 (R2 ). Now we are in the position to give the proof of Theorem 1.3. Without loss of generality, we always assume α < 2π. Let M := {u ∈ H 1 (R2 ); uH 1 (R2 ) = 1}. For any v ∈ M , we introduce a family of functions √ √ vt (x) := tv( tx),

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where t > 0 is a positive parameter. Let wt := vt /vt H 1 (R2 ) . Observe that wt is a curve in M passing through v, since wt H 1 (R2 ) = 1 and d J(wt )t=1 . By w1 = v1 /v1 H 1 = v/vH 1 = v. Now we shall compute dt p (p−2)/2 p 2 vp and ∇vt 2 = t∇v22 , we the direct calculation using vt p = t see that    ∞ j   tj−2 v2j d α 2j 2 2   J(wt ) = (−t∇v2 + (j − 1)v2 ) 2 2 j+1 dt j! (t∇v2 + v2 )  t=1 j=1 ≤ −αv22 ∇v22 + ⎡

∞ j=2

t=1

αj v2j 2j (j − 1)!

⎤ 2j ∞ j−1 v α 2j ⎦ = αv22 ∇v22 ⎣−1 + 2 ∇v2 . (j − 1)! v 2 2 j=2

(10)

By Lemma 4.1 with β = 3π and by vH 1 = 1, we have v2j j! 2j ≤ C3π , v22 ∇v22 (3π)j

j ≥ 2.

From this relation and from α/(3π) < 2/3, we see that ⎤ ⎡ ∞ j−2 α j⎦ (10) ≤ αv22 ∇v22 ⎣−1 + αC3π j (3π) j=2 < αv22 ∇v22 [−1 + αC] ,  2 j−2 C3π ∞ where C := (3π) j. Consequently, we have 2 j=2 3   d J(wt ) ≤ αv22 ∇v22 [−1 + αC] < 0 dt t=1 for α < 1/C. Hence, no v can be a critical point of J in M for α < 1/C. This completes the proof of Theorem 1.3. References 1. Adachi, S., Tanaka, K., Trudinger type inequalities in RN and their best exponents. Proc. Amer. Math. Soc. 128 (2000), no. 7, 2051 –2057. 2. Adams D. R., A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. (2) 128 (1988), 385–398. 3. Beckner, W., Estimates on Moser embedding. Potential Anal. 20 (2004), 345– 359. 4. Bianchi, G., Chabrowski, J., Szulkin, A., On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent. Nonlinear Anal. 25 (1995), no. 1, 41–59.

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5. Ben-Naoum, A. K., Troestler, C., Willem, M., Extrema problems with critical Sobolev exponents on unbounded domains. Nonlinear Anal. 26 (1996), no. 4, 823–833. 6. Cao, D. M., Nontrivial solution of semilinear elliptic equation with critical exponent in R2 . Comm. Partial Differential Equations 17 (1992), no. 3-4, 407–435. 7. Carleson, L., Chang, S.-Y. A., On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. (2) 110 (1986), no. 2, 113–127. 8. Edmunds, D. E., Ilyin, A. A., Asymptotically sharp multiplicative inequalities. Bull. London Math. Soc. 27 (1995), 71–74. 9. Flucher, M., Extremal functions for the Trudinger-Moser inequality in 2 dimensions. Comment. Math. Helv. 67 (1992), no. 3, 471–497. 10. de Figueiredo, D. G., do O, Joao M., Ruf, B., On an inequality by N. Trudinger and J. Moser and related elliptic equations. Comm. Pure Appl. Math. 55 (2002), no. 2, 135–152. 11. Ishiwata, M., Concentration compactness principle at infinity with partial symmetry and its application. Nonlinear Anal. 51 (2002), 391–407. 12. Ishiwata, M., Existence and nonexistence of maximizers for variational problems associated with Trudinger-Moser type inequalities in RN . Mathematische Annalen 351 Number 4, (2011), 781-804. 13. Ishiwata, M., Nakamura, M., Wadade, H., On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form, preprint. 14. Kozono, H., Ogawa, T., Sohr, H., Asymptotic behaviour in Lr for weak solutions of the Navier-Stokes equations in exterior domains. Manusc. Math. 74 (1992), 253–275. 15. Kozono, H., Sato, T., Wadade, H., Upper bound of the best constant of a Trudinger-Moser inequality and its application to a Gagliardo-Nirenberg inequality. Indiana Univ. Math. J. 55 (2006), no. 6, 1951–1974. 16. Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), no. 2, 109–145. 17. Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), no. 4, 223–283. 18. Lions, P.-L. The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. 19. Lions, P.-L. The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1 (1985), no. 2, 45–121. 20. Li, Y., Ruf, B., A sharp Trudinger-Moser type inequality for unbounded domains in Rn . Indiana Univ. Math. J. 57 (2008), no. 1, 451–480. 21. Moser, J., A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/71), 1077–1092. 22. Ogawa, T., A proof of Trudinger’s inequality and its application to nonlinear Schrodinger equations. Nonlinear Anal. 14 (1990), no. 9, 765–769. 23. Ogawa, T., Ozawa, T., Trudinger type inequalities and uniqueness of weak

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24.

25. 26. 27. 28. 29. 30.

solutions for the nonlinear Schr¨ odinger mixed problem, J. Math. Anal. Appl. 155 (1991), no. 2, 531–540. Ogawa, T., Yokota, T., Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain. Comm. Math. Phys. 245 (2004), 105-121. Ruf, B., A sharp Trudinger-Moser type inequality for unbounded domains in R2 . J. Funct. Anal. 219 (2005), no. 2, 340–367. Ozawa, T., On critical cases of Sobolevfs inequalities, J. Funct. Anal. 127 (1995), 259–269. Smets, D., A concentration-compactness lemma with applications to singular eigenvalue problems. J. Funct. Anal. 167 (1999), no. 2, 463–480. Trudinger, N. S., On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 1967 473–483. Vazquez, J. L., A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12 (1984), no. 3, 191–202. Weinstein, M. I., Nonlinear Schrodinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1982/83), no. 4, 567–576.

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Computer-assisted Uniqueness Proof for Stokes’ Wave of Extreme Form K. Kobayashi Graduate School of Commerce and Management, Hitotsubashi University 2-1 Naka, Kunitachi, Tokyo, 186-8601 Japan E-mail: [email protected] In this paper, we present an outline of the computer-assisted proof for the global uniqueness of Stokes’ wave of extreme form. Stokes’ wave of extreme form is a water wave which forms a corner of 120 degrees at the crest, and is considered to be the limit of the positive solution of Nekrasov’s equation which expresses periodic gravity waves of permanent form on the free surface. The numerical verification method plays an important role in the proof. As for the global uniqueness of Stokes’ wave of extreme form, it is not only a longtime open problem, but also it is related to an important conjecture which is called Stokes’ conjecture. Keywords: Stokes’ wave of extreme form, Nekrasov’s equation, Numerical verification method, gravity wave, Stokes’ conjecture.

1. Introduction We are concerned in this paper with Stokes’ wave of extreme form (see, e.g., Okamoto and Sh¯oji11 ), which is a positive solution of a nonlinear integral equation for the unknown θ : (0, π] → R and which is written: ⎧  π sin θ(t) ⎪ ⎪ θ(s) = K(s, t)  t dt, ⎪ ⎪ ⎪ sin θ(w)dw 0 ⎨ 0 π (1) s ∈ (0, π), 0 < θ(s) < ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ θ(π) = 0, where K(s, t) =

  ∞   sin s+t 1 2 sin ks sin kt 2  = log  s−t  . 3π k 3π sin 2 k=1

Stokes’ wave of extreme form is derived from the following equation by

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Nekrasov:10

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩



π

θ(s) =

K(s, t) 0

sin θ(t) dt, t μ−1 + 0 sin θ(w)dw

π s ∈ (0, π), 2 θ(0) = θ(π) = 0,

0 < θ(s)
3 (there is no positive solution when μ ≤ 3) was first proved by Keady and Norbury.4 When μ → ∞, the speed at the crest tends to zero and the crest becomes sharper. Keady and Norbury guaranteed all waves having one crest and one trough except for the case of μ = ∞. The waves of extreme form satisfy (1) and the existence of a solution to (1) was proved by Toland.16 He proved that as μ → ∞, the solution of (2) have a convergent subsequence and that the limit function satisfies (1). As for the extreme wave, Stokes15 recognized the following two propositions which are nowadays called Stokes’ conjectures: (a) the crest forms a corner of angle 2π/3, that is, lims↓0 θ(s) = π/6. (b) the wave profile between two consecutive crests is downwards convex, that is, θ (s) < 0 for s ∈ (0, π). Conjecture (a) was proved independently and simultaneously by Amick et al.1 and by Plotnikov.12 A simplified proof of conjecture (a) which also applies to a more general problem was recently given by Varvaruca.19 Toland and Plotnikov18 proved that there exists a wave which satisfies θ (s) < 0. However, this does not settle conjecture (b) completely. Because of the lack of the proof of the uniqueness, it is not excluded that there is a positive solution of (1) which does not satisfy θ (s) < 0. For readers who are interested in these topics, the author recommends Toland17 which provide a good survey on the analytical results of Stokes waves. Although the existence of solutions for Nekrasov’s equation and Stokes’ wave of extreme form are proved, despite the effort of many mathematicians for many decades, the global uniqueness had been remained open. Only the local uniqueness, i.e., the uniqueness in a certain neighborhood of the solution, have proved by Fraenkel.2 On the other hand, we had succeeded to prove the global uniqueness of a positive solution for Nekrasov’s equation when μ ≤ 170 (see Kobayashi5). Using a similar technique as in Kobayashi,5 we present a method to prove the global uniqueness of Stokes’ wave of extreme form. In the two proofs, we employ the numerical verification method (see, for instance, Rump,13 Nakao and Yamamoto9 ) to obtain

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rigorous mathematical results by numerical computations. This article is a simplified version of our previous paper6 and is organized as follows: In section 2, we explain the idea of the proof of uniqueness and the numerical results are shown in section 3. Influence of discretization and rounding error in section 3 are both rigorously evaluated. Consequently, though we use numerical computation, mathematically rigorous proof can be obtained by our method. We omit the details of the discretization and implementation. Refer to Kobayashi6 for the complete and detailed proof. In this paper, θ always denotes Stokes’ wave of extreme form, namely, a solution of (1), and s is assumed to run in 0 < s ≤ π. Also, 1A denotes the function which takes the value 1 if condition A holds, and which is 0 otherwise.

2. The idea behind the proof of uniqueness In this section, we first derive a formula regarding the bounds of θ. Secondly, we present the specific lower bound of θ. Finally, we explain how to prove the uniqueness. First of all, we prove the following theorem, Theorem 2.1. are given as

Assume that, for θ(s), an upper bound and a lower bound

0 ≤ θ(s) ≤ θ(s) ≤ θ(s) ≤ π/2, then, it holds that     J θ, θ (s) ≤ θ(s) ≤ J θ, θ (s), where 



1 J φ, ψ (s) = 6π

Proof.

⎛

 0

⎞



π−|s+t−π| sin φ(w)dw t |s−t| cot · log ⎝1 +  min(s,|s−t|)  2 sin ψ(w)dw + |s−t|

π

0

min(s,|s−t|)

⎠ dt. sin φ(w)dw

Note first that

1 t+s t−s 1 sin s ∂ K(s, t) = cot − cot = · . ∂t 6π 2 2 3π cos t − cos s

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With this and (1), we have  t

 π d θ(s) = K(s, t) log sin θ(w)dw dt dt 0 0 1 = 3π



π

sin s · log cos s − cos t

0

t

sin θ(w)dw  0s 0 sin θ(w)dw

 dt.

Then, using that w ≥ s implies t ≥ s, 1 θ(s) ≤ 3π

1 = 6π

1 = 6π

1 = 6π

1 = 6π



π 0



π

⎛  |t|

t−s cot · log ⎝ 2 −π



s+π s−π



π 0

1 = 6π

≤

1 6π



π 0

⎛t log ⎝



0



s

0

0

⎠ dt

sin θ(w)dw



⎞ ⎠ du



sin 1w 0, h (t) = −μ ux (t, h(t)), ⎪ ⎪ ⎪ ⎪ −h0 ≤ x ≤ h0 , u(0, x) = u0 (x), ⎪ ⎪ ⎩ −g(0) = h(0) = h0 > 0, where a, b, d and μ are positive constants, x = g(t) and x = h(t) are the moving boundaries to be determined together with u(t, x), and μ−1 represents the pressure level on the free boundary. The initial function u0 belongs to X (h0 ) for some h0 > 0, where X (h0 ) := φ ∈ C 2 ([−h0 , h0 ]) : φ(−h0 ) = φ(h0 ) = 0, φ (−h0 ) > 0, . (4) φ (h0 ) < 0, φ(x) > 0 in (−h0 , h0 ) . For any given h0 > 0 and u0 ∈ X (h0 ), by a classical solution of (3) on the time-interval [0, T ] we mean a triple (u(t, x), g(t), h(t)) belonging to C 1,2 (GT ) × C 1 ([0, T ]) × C 1 ([0, T ]), such that all the identities in (3) are satisfied pointwisely, where   GT := (t, x) : t ∈ (0, T ], x ∈ [g(t), h(t)] . Denote g∞ := lim g(t), h∞ := lim h(t) t→∞

t→∞

when they exist. Du and Lin7 obtained the following result. Theorem 2.1. Let (u, g, h) be a solution of (3). Then either (i) spreading: (g∞ , h∞ ) = R1 and a lim u(t, x) = locally uniformly in R1 , t→∞ b

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or (ii) vanishing: (g∞ , h∞ ) is a finite interval and lim

max

t→∞ g(t)≤x≤h(t)

u(t, x) = 0.

/ / Moreover, the spreading occurs if h0 ≥ π2 ad . When h0 < π2 ad , there exits μ∗ > 0 depending on u0 such that u vanishes when 0 < μ ≤ μ∗ and u spreads when μ > μ∗ .

Y. Du and Z. Guo10 studied the problem in higher space dimensions and obtained similar results. The above theorem is an essential progress in this field, it implies that for small h0 and small μ, the species can not spread. In other words, the hair-trigger effect observed in Aronson and Weinberger3,4 is not true for problem (3). However, it yet does not clearly answer our question proposed in section 1 – does a solution u converge to 0 for small u0 L∞ (R) or small mes[supp(u0 )] ? Y. Du and B. Lou8 gave a rather complete answer to this question. They considered problem (3) with the equation replaced by a more general one ut = uxx + f (u),

g(t) < x < h(t), t > 0,

(5)

where f satisfies f (u) is C 1 and f (0) = 0.

(6)

Among others, they focused on two types of nonlinearities: (fM ) monostable case,

(fB ) bistable case.

In the monostable case (fM ), f ∈ C 1 and it satisfies f (0) = f (1) = 0,

f  (0) > 0,

f  (1) < 0,

(1−u)f (u) > 0 for u > 0, u = 1.

Clearly f (u) = u(1 − u) belongs to the type of (fM ). (fB ), f ∈ C 1 and it satisfies ⎧ ⎨ < 0 in f (0) = f (θ) = f (1) = 0, f (u) > 0 in ⎩ < 0 in

In the bistable case (0, θ), (θ, 1), (1, ∞)

for some θ ∈ (0, 1), f  (0) < 0, f  (1) < 0 and  1 f (s)ds > 0. 0

A typical bistable f (u) is u(u − θ)(1 − u) with θ ∈ (0, 12 ).

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Du and Lou studied the spreading and vanishing dichotomy/trichotomy according to the size of the initial data. They obtained the following two theorems. Theorem 2.2.8 Let (u, g, h) be a solution of problem (3) with the equation replaced by (5), and let f be of (fM ) type. Then either (i) spreading: (g∞ , h∞ ) = R1 and lim u(t, x) = 1 locally uniformly in R1 ,

t→∞

or (ii) vanishing: (g∞ , h∞ ) is a finite interval with length no bigger than 0 π/ f  (0) and lim

max

t→∞ g(t)≤x≤h(t)

u(t, x) = 0.

Moreover, if u0 = σφ with φ ∈ X (h0 ), then there exists σ ∗ = σ ∗ (h0 , φ) ∈ [0, ∞] such that vanishing happens when 0 < σ ≤ σ ∗ , and spreading happens when σ > σ∗ . In addition, 0 ⎧ if h0 ≥ π/(20f  (0)), ⎨= 0 σ ∗ ∈ (0, ∞] if h0 < π/(20f  (0)), ⎩ ∈ (0, ∞) if h0 < π/(2 f  (0)) and if f is globally Lipschitz. Theorem 2.3.8 Let (u, g, h) be a solution of problem (3) with the equation replaced by (5), and let f be of (fB ) type. Then either (i) spreading: (g∞ , h∞ ) = R1 and lim u(t, x) = 1 locally uniformly in R1 ,

t→∞

or (ii) vanishing: (g∞ , h∞ ) is a finite interval and lim

max

t→∞ g(t)≤x≤h(t)

u(t, x) = 0,

or (iii) transition: (g∞ , h∞ ) = R1 and for some a ∈ [−h0 , h0 ], lim u(t, x) = v∞ (x − a) locally uniformly in R1 ,

t→∞

where v∞ is the unique positive solution to v  + f (v) = 0 (x ∈ R1 ), v  (0) = 0, v(−∞) = v(+∞) = 0. Moreover, if u0 = σφ for some φ ∈ X (h0 ), then there exists σ ∗ = σ ∗ (h0 , φ) ∈ (0, ∞] such that vanishing happens when 0 < σ < σ ∗ , spreading

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happens when σ > σ ∗ , and transition happens when σ = σ ∗ . In addition, there exists ZB > 0 such that σ ∗ < ∞ if h0 ≥ ZB , or if h0 < ZB and f is globally Lipschitz. Some problems in applied fields can be reduced to outer problems (or, in one space dimension case, problems in the half line). X. Liu and B. Lou9 recently studied equation (5) for such problems with free boundaries. More precisely, they considered the following problem ⎧ ut = uxx + f (u), 0 < x < h(t), t > 0, ⎪ ⎪ ⎨ u(t, h(t)) = 0, t > 0, (7) ⎪ t > 0, h (t) = −μ ux(t, h(t)), ⎪ ⎩ h(0) = h0 , u(0, x) = u0 (x), 0 ≤ x ≤ h0 , with a Dirichlet condition at x = 0: u(t, 0) = 0,

t > 0,

(8)

or, a Robin condition at x = 0: αu(t, 0) + (α − 1)ux (t, 0) = 0,

t > 0,

(9)

where α ∈ (0, 1) is a constant. (Note that the problem with homogeneous Neumann condition at x = 0 is equivalent to a similar problem as (3) by reflection). Since the problem with condition (8) has similar results as that with condition (9), we only state the result for the former. The stationary problem for (7) with (8) is written as $ vxx + f (v) = 0, x ∈ (0, h∞ ), (10) v(0) = 0. When f is of (fM ) type, the bounded nonnegative solutions of (10) can be classified into the following categories: (1) zero; (2) solutions with compact supports; (3) a function v ∗ (x) defined on [0, ∞) satisfying v ∗ (0) = 0,

(v ∗ ) (x) > 0 (x ≥ 0) and

v ∗ (x) → 1 (as x → ∞).

Liu and Lou9 obtained the following result. Theorem 2.4. Assume that f is of (fM ) type and (u, h) is a solution of (7)-(8). Then either

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(i) spreading: h∞ = ∞ and lim u(t, x) = v ∗ (x) locally uniformly on [0, ∞),

t→∞

or (ii) vanishing: h∞ < ∞ and lim

max u(t, x) = 0.

t→∞ 0≤x≤h(t)

Moreover, if u0 = σφ with φ ∈ Y (h0 ) = φ ∈ C 2 ([0, h0 ])

  φ(0) = φ(h0 ) = 0, φ (0) > 0, .   φ (h0 ) < 0, φ(x) > 0 in (0, h0 ) ,

then there exists σ ∗ = σ ∗ (h0 , φ) ∈ [0, ∞] such that vanishing happens when 0 < σ ≤ σ ∗ , and spreading happens when σ > σ ∗ . In addition, 0 $ =0 if h0 ≥ π/0f  (0), ∗ σ ∈ (0, ∞] if h0 < π/ f  (0). Recently, Y. Kaneko and Y. Yamada11 also studied problem (7)-(8) with logistic type of nonlinearity. They gave spreading and vanishing dichotomy. But their results do not include the sharp threshold of the initial data for the spreading and vanishing dichotomy. 2.2. Key lemmas of the proof The proof of Theorem 2.2 and 2.3 is based on the following general convergence results: Lemma 2.5. Suppose that (6) holds and (u, g, h) is a time global solution of (3) with the equation replaced by (5), and u(t, x) is bounded, namely u(t, x) ≤ C for all t > 0, x ∈ [g(t), h(t)] and some C > 0. Then limt→∞ u(t, x) exists, and the limit is a nonnegative solution of vxx + f (v) = 0, x ∈ (g∞ , h∞ ).

(11)

Moreover, either (g∞ , h∞ ) is a finite interval and v ≡ 0, or (g∞ , h∞ ) = R1 and v is a constant or a symmetrically decreasing solution of (11). Similarly, the proof of Theorem 2.4 is based on the following general convergence result: Lemma 2.6. Suppose that (6) holds and (u, h) is a solution of (7)-(8) that is defined for all t > 0, and u(t, x) is bounded, namely u(t, x) ≤ C for all t > 0, x ∈ [0, h(t)] and some C > 0.

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Then limt→∞ u(t, x) exists, and the limit is a nonnegative solution v(x) of (10). Moreover, either (0, h∞ ) is a finite interval and v ≡ 0, or h∞ = ∞ and the limit is v ∗ (x). 3. Asymptotic spreading speed When spreading happens, that is, the solution u converges to a nonnegative stationary solution, the asymptotic spreading speed is an important factor to be understood in applied field. For the problem (3) with the equation replaced by (5), when spreading happens, this asymptotic spreading speed is determined by the following problem $ qzz − cqz + f (q) = 0 for z ∈ (0, ∞), (12) q(0) = 0, μqz (0) = c, q(∞) = 1, q(z) > 0 for z > 0. If f is of (fM ), or (fB ) type, then for each μ > 0, (12) has a unique solution (c, q) = (c∗ , q ∗ ). We call q ∗ a “semi-wave” with speed c∗ . For the problem (3) with the equation replaced by (5), Du-Lin7 and Du-Lou8 obtained the following result on spreading speed: Theorem 3.1. Assume that f is of (fM ), or (fB ) type, and spreading happens. Let c∗ be given as the above. Then h(t) −g(t) = lim = c∗ , t→∞ t t and for any small ε > 0 and any δ ∈ (0, −f  (1)), there exist positive constants M and T0 such that lim

t→∞

max

|x|≤(c∗ −ε)t

|u(t, x) − 1| ≤ M e−δt for all t ≥ T0 .

References 1. R. Fisher, Ann. Eugenics 7, 335 (1937). 2. I. P. A.N. Kolmogorov and N. Piskunov, Bull. Moscow Univ., Math. Mech. 1, 1 (1937). 3. D. Aronson and H. Weinberger, Nonlinear diffusion in population genetics, conbustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, , Partial Differential Equations and Related Topics Vol. Lecture Notes in Math. 446 (Springer, Berlin, 1975) pp. 5–49. 4. D. Aronson and H. Weinberger, Adv. in Math. 30, 33 (1978). 5. S. Allen and J. Cahn, Acta Metallurgica 27, 1084 (1979). 6. P. Fife and J. McLeod, Arch. Ration. Mech. Anal. 65, 335 (1977). 7. Y. Du and Z. Lin, SIAM J. Math. Anal. 42, 377 (2010).

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8. Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, preprint. 9. X. Liu and B. Lou, Spreading and vanishing in nonlinear diffusion problems in half line with free boundaries, preprint. 10. Y. Du and Z. Guo, The stefan problem for the fisher-kpp equation, preprint. 11. Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, preprint.

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RECENT PROGRESS ON OBSERVABILITY FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS Qi L¨ u∗ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, China ∗ E-mail: [email protected] Zhongqi Yin College of Mathematics, Sichuan Normal University Chengdu, China, 610068 E-mail: [email protected] We survey some recent progress in the observability estimate for stochastic partial differential equation, including the stochastic parabolic equation, the stochastic hyperbolic equation and the stochastic Schr¨ odinger equation. All the observability estimates are obtained by global Carleman estimate. Keywords: Observability estimate; global Carleman estimate; stochastic partial differential equations.

1. Introduction Let T > 0, G ⊂ Rn (n ∈ N) be a given bounded domain with a C 2 boundary 



Γ. Put Q = (0, T ) × G, Σ = (0, T ) × Γ. Let (Ω, F , {Ft }t≥0 , P ) be a complete filtered probability space on which a 1-dimensional standard Brownian motion {B(t)}t≥0 is defined. Let H be a Banach space. We denote by L2F (0, T ; H) the Banach space consisting of all H-valued {Ft }t≥0 -adapted processes X(·) such that E(|X(·)|2L2 (0,T ;H) ) < ∞; by L∞ F (0, T ; H) the Banach space consisting of all H-valued {Ft }t≥0 adapted bounded processes; and by L2F (Ω; C([0, T ]; H)) the Fr´echet space consisting of all H-valued {Ft }t≥0 -adapted continuous processes X(·) such that E(|X(·)|2C([0,T ];H) ) < ∞; (similarly, one can define L2F (Ω; C k ([0, T ]; H)) for any positive integer k), all of these spaces are endowed with the canonical norms respectively.

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This paper is devoted to surveying some recent results for the observability estimates for stochastic partial differential equations(SPDEs for short). We only focus on the three typical SPDEs, that is, the stochastic parabolic equation, the stochastic hyperbolic equation and the stochastic Schr¨odinger equation. Observability estimate plays an important role in many application problems, such as controllability (see [3,13,24] for example), optimal control (see [26] for example), inverse problems (see [8] for example) and so on. There are a great many studies on observability estimate for deterministic partial differential equations (PDEs for short)(see [2,3,6,13,22,24] and the rich references cited therein). It would be natural to extend the observability estimate to the stochastic case. However, the study of such kind of problems is highly undeveloped. To our best knowledge, [1,15,16,21,23] are the only published papers concerning these problems. Lots of things should be done, and some of which seem to be very difficult. People have introduced several powerful methods to establish the observability estimate for PDEs. For example, for the parabolic equations, we have the spectral method (see Ref. 11 for example) and the Carleman estimates (see Ref. 6 for example); for the hyperbolic equations, we have the spectral method (see Ref. 20 for example), the multiplier method (see Ref. 13 for example), the microlocal analysis method (see Ref. 2 for example), and the Carleman estimates (see Ref. 22 for example); for Schr¨ odinger equations, we have the multiplier method (see Ref. 19 for example), the microlocal analysis method (see Ref. 10 for example), and the Carleman estimates(see Ref. 9 for example). One will meet substantially new difficulties in the study of observability estimate for SPDEs. For instance, unlike the PDEs, the solution of a SPDE is usually non-differentiable with respect to the variable with noise (say, the time variable considered in this paper). Also, the usual compact embedding result does not remain true for the solution spaces related to SPDEs. These new phenomenons lead that some useful methods for establishing observability estimate for PDEs cannot be applied to SPDEs. Especially, it seems that the multiplier method cannot be used to establish observability for stochastic hyperbolic equations or stochastic Schr¨odinger equations. Until now, the most useful tool for studying the observability for SPDEs is the global Carleman estimate. Carleman estimate is simply a weighted energy method. There are two crucial steps in Carleman estimate. The first one is to obtain a suitable point-wise estimate related to the principal operator of the equation. The second one is to choose a proper weight

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function. This method has many advantages. For example, it is robust for a class of lower order terms, and it can give an explicit estimate on the observability constant with respect to suitable Sobolev space norms of the coefficients in the equations, which is crucial for some control problems. The rest of this paper is organized as follows. Section 2 is devoted to introducing the observability estimate for stochastic parabolic equations. In Section 3, we review the observability estimate for stochastic hyperbolic equations. Section 4 is addressed to an introduction of the observability estimate for stochastic Schr¨odinger equations. As we have mentioned before, all the observability estimates presented in this paper are obtained by some global Carleman estimate and the crucial steps for the global Carleman estimate is to establish suitable point-wise estimate and to choose proper weight function, hence we organize the rest three sections in the following way: We first present the equation. Next we introduce the result of observability estimate. Further, we give the pointwise estimate and the weight function. For the detailed proofs, please find them in the references. 2. Observability estimate for stochastic parabolic equation Throughout this section, we make the following assumptions on the coefficients aij : Ω × [0, T ] × G → lRn×n (i, j = 1, 2, · · · , n): (H1) aij ∈ L2F (Ω; C 1 ([0, T ]; W 2,∞ (G))) and aij = aji ; (H2) For any δ > 0, there is ρ > 0 such that |aij (ω, t, x1 ) − a (ω, t, x2 )| ≤ δ almost surely for any t ∈ [0, T ] and x1 , x2 ∈ G which satisfy the following inequality: |x1 − x2 | ≤ ρ; ij



(H3) There is some constant s0 > 0 such that aij (ω, t, x)ξ i ξ j ≥ s0 |ξ|2 ,

(ω, t, x, ξ) ≡ (ω, t, x, ξ 1 , · · · , ξ n ) ∈ Ω×Q×lRn .

i,j

Here and henceforth, we denote

n i,j=1

(1) simply by . For simplicity, we i,j

will use the notation yi ≡ yi (x) = ∂y(x)/∂xi , where xi is the i-th coordinate of a generic point x = (x1 , · · · , xn ) in lRn . In a similar manner, we use the notations zi , vi , etc. for the partial derivatives of z and v with respect to xi . Also, we denote the scalar product in lRn by  ·, · , and use C to denote a generic positive constant depending only on G, G0 and (aij )n×n , which may change from line to line.

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Consider the following stochastic parabolic equation: ⎧ ⎪ dz − (aij zi )j dt = [ a, ∇z  +bz]dt + czdB(t) in Q, ⎪ ⎪ ⎪ ⎨ i,j z=0 ⎪ ⎪ ⎪ ⎪ ⎩ z(0) = z0

on Σ,

(2)

in G,

Here ∗

n ∞ ∞ n ∞ 1,∞ (G)). a ∈ L∞ F (0, T ; L (G; lR )), b ∈ LF (0, T ; L (G)), and c ∈ LF (0, T ; W

We begin with the following notion. Definition 2.1. We call z ∈ L2F (Ω; C([0, T ]; L2 (G))) ∩ L2F (0, T ; H 1 (G)) a solution of equation (2) if the following hold: 1. z(0) = z0 in G, P-a.s., 2. For any t ∈ [0, T ] and any η ∈ H01 (G), it holds   z(t, x)η(x)dx − z(0, x)η(x)dx G

 t -

= G

0

G

−



aij (s, x)zi (s, x)ηj (x)

i,j

. +[ a(s, x), ∇z(s, x)  +b(s, x)z(s, x)]η(x) dxds  t + c(s, x)z(s, x)η(x)dxdB(s), P − a.s. 0

(3)

G

We refer to Ref. 7 for the well-posedness of equation (2). We have the following observability estimate for equation (2). Theorem 2.1. [21, Theorem 2.3] Let assumptions (H1), (H2) and (H3) be satisfied. Then there is a constant C > 0 such that all solutions z of equation (2) satisfy that |z(T )|L2 (Ω,FT ,P ;L2 (G)) ≤ CeC[T

−4

(1+r12 )+T r12 ]

|z(·)|L2F (0,T ;L2 (G0 ))

(4)

with 

n ∞ r1 = |a|L∞ + |b|L∞ n∗ (G)) + |c|L∞ (0,T ;W 1,∞ (G)) . F (0,T ;L (G;lR )) F F (0,T ;L

A similar result was established in Ref. 1 if (aij )1≤i,j≤n is the identity matrix. The key tool for proving Theorem 2.1 is the following point-wise estimate.

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For any nonnegative and nonzero function ψ ∈ C 3 (G) , any k ≥ 2 , and any (large) parameters λ > 1 and μ > 1, put  = λα,

α(t, x) =

eμψ(x) − e2μ|ψ|C(G) , tk (T − t)k

ϕ(t, x) =

eμψ(x) . − t)k

tk (T

(5)

Theorem 2.2. [21, Theorem 4.1] Let u be a C 2 (G)-valued semimartingale. Set θ = e , v = θu, Ψ = 2 aij ij . (6) i,j

Then for any x ∈ G and ω ∈ Ω (a.s. dP ), it follows 

T

θ −

2



0

! ! (aij vi )j + Av du − (aij ui )j dt

i,j



T

+2 0







(aij vi dv)j dt +

T

0

  θ2 aij i (du)2 dt

0

i,j

+2

i,j T



     2aij ai j i vi vj  − aij ai j i vi vj  + Ψaij vi v

i ,j 

i,j

−aij ≥2

 i,j



−

0





−

T

θ2 A − 0

i,j

A=−

T

(aij vi )j + Av2 dt

0

aij dui duj −

where

j

Bv 2 dt + 

(7)

 ! Ai + Ψ2i v 2 dt



0





T

cij vi vj dt +

θ2 0



T

T

j

i,j

i,j



aij i j + (aij i )j

! (du)2 ,

i,j

! aij i j − (aij i )j − Ψ,

i,j



B = 2 AΨ −

! (Aaij i )j − At − (aij Ψj )i − 2t , i,j



cij =

i ,j 

i,j

! aij     2aij (ai j i )j  − (aij ai j i )j  − t + Ψbij . 2

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Moreover, for λ and μ large enough, it holds A = −λ2 μ2 ϕ2 bij ψi ψj + λϕO(μ2 ), i,j

B ≥ 2s20 λ3 μ4 ϕ3 |∇ψ|4 + λ3 ϕ3 O(μ3 ) + λ2 ϕ2 O(μ4 ) + λϕO(μ4 ) −1



−1

+λ2 ϕ2+2k O(e4μ|ψ|C(G) ) + λ2 ϕ2+k O(μ2 ) + λϕ1+k

−1

O(μ2 ),

(8)

cij vi vj ≥ [s20 λμ2 ϕ|∇ψ|2 + λϕO(μ)]|∇v|2 .

i,j

Once we have Theorem 2.2, we just need to choose a ψ ∈ C 4 (G) such that ψ > 0 in G and ψ = 0 on ∂G and |∇ψ(x)| > 0 for all x ∈ G \ G1 in inequality (7) and let u = z. Then Theorem 2.1 follows from inequality (7) and the standard energy estimate for equation (2). 3. Observability estimate for stochastic hyperbolic equations Throughout this section, we make the following assumptions on the coefficients bij ∈ C 1 (G): (H4). bij = bji (i, j = 1, 2, · · · , n); (H5). For some constant s0 > 0,  bij ξ i ξ j ≥ s0 |ξ|2 , ∀(x, ξ) = (x, ξ 1 , · · · , ξ n ) ∈ G × Rn . (9) i,j

Let us consider the following stochastic hyperbolic equation: ! ⎧ ij ⎪ dz − (b z ) dt = b z +b ,∇z+b z +f dt+(b4 z +g)dB(t) in Q, t i j 1 t 2 3 ⎪ ⎪ ⎪ ⎨ i,j (10) ⎪ z=0 on Σ, ⎪ ⎪ ⎪ ⎩ z(0) = z0 , zt (0) = z1 in G. Here (z0 , z1 ) ∈ L2 (Ω, F0 , P ; H01 (G) × L2 (G)), ∞ b1 ∈ L∞ F (0, T ; L (G)),

∞ n b2 ∈ L∞ F (0, T ; L (G; R )),

p b3 ∈ L ∞ F (0, T ; L (G))(p ≥ n),

∞ b4 ∈ L∞ F (0, T ; L (G)),

(11)

and f ∈ L2F (0, T ; L2(G)),

g ∈ L2F (0, T ; L2 (G)).

We first give the definition of the solution to equation (10).

(12)

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Definition 3.1. z ∈ L2F (Ω; C([0, T ]; H01 (G))) ∩ L2F (Ω; C 1 ([0, T ]; L2(G))) is called a solution of equation (10) if the following hold: 1. z(0) = z0 in G, P-a.s., and zt (0) = z1 in G, P-a.s. 2. For any t ∈ [0, T ] and any η ∈ H01 (G), it holds that   zt (t, x)η(x)dx − zt (0, x)η(x)dx G

 t -

= 0

G

−

G



bij (x)zi (s, x)ηj (x) + [b1 (s, x)zt (s, x) + b2 (s, x) · ∇z(s, x)

i,j

. +b3 (s, x)z(s, x) + f (s, x)]η(x) dxds  t + (b4 (s, x)z(s, x) + g(s, x))η(x)dxdB(s), P-a.s. 0

G

(13) For the well-posedness of equation (10), please see.23 For the further purpose, next we introduce two conditions. Condition 3.1. There exists a positive function d(·) ∈ C 2 (G) satisfying the following: 1. For some constant μ0 > 0, it holds !. -   i j  2bij (bi j di )j  − bij di ξ i ξ j ≥ μ0 bij ξ i ξ j , j b i,j i,j i ,j  (14) ∀(x, ξ 1 , · · · , ξ n ) ∈ G × Rn . 2. There is no critical point of d(·) in G, i.e., min |∇d(x)| > 0.

(15)

x∈G

Remark 3.1. If (bij )1≤i,j≤n is the identity matrix, then d(x) = |x − x0 |2 satisfies Condition 3.1, where x0 is any point which belongs to Rn \ G. Remark 3.2. Condition 3.1 was first given in5 for the purpose of obtaining an internal observability estimate for hyperbolic equations. In that paper, the authors also gave some explanation of Condition 3.1 and some interesting nontrivial examples satisfying it. It is easy to check that if d(·) satisfies Condition 3.1, then for any given constants a ≥ 1 and b ∈ R, the function d˜ = ad + b still satisfies Condition 3.1 with μ0 replaced by aμ0 . Therefore we may choose d, μ0 , c0 > 0, c1 > 0 and T to guarantee that one of the following conditions holds:

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Condition 3.2. 1 ij   b (x)di (x)dj (x) ≥ R12 = max d(x) ≥ R02 = min d(x), 1. 4 i,j x∈G x∈G

∀x ∈ G. (16)

2.



T > T0 = 2 inf{R1 : d(·) satisfies (16) }.  2R 2 1

3. 4.

T

< c1
0

Throughout this section, we use C to denote a generic positive constant depending on G, T , Γ0 , bij (i, j = 1, · · · , n), δ, d, c0 and c1 (unless otherwise stated), which may change from line to line. Now we give the observability estimate. Here and henceforth, we denote by ν = (ν1 , · · · , νn ) the unit outer normal vector of Γ. Put  .   Γ 0 = x ∈ Γ bij di (x)ν j (x) > 0 (17) i,j

and

⎧  ⎨ r2 = |b2 |L∞ (0,T ;L∞ (G;Rn )) + |(b1 , b4 )|L∞ (0,T ;(L∞ (G))2 , F F ⎩



(18)

p r3 = |b3 |L∞ . F (0,T ;L (G))

The first result is for the boundary observability estimate for equation (10). Theorem 3.1. Let Condition 3.1 and Condition 3.2 be satisfied. For any solution of equation (10), we have that |(z0 , z1 )|L2 (Ω,F0 ,P ;H01 (G)×L2 (G)) 1   3/2−n/p C(r22 +r3 +1)  ∂z  ≤ Ce + |f |L2F (0,T ;L2 (G))   2 ∂ν LF (0,T ;L2 (Γ0 ))  +|g|L2F (0,T ;L2 (G)) .

(19)

An observability for the final state was established in Ref. 23 if (bij )1≤i,j≤n is identity matrix. A key tool for obtaining Theorem 3.1 is the following result. Theorem 3.2. [23, Theorem 4.1] Let pij ∈ C 1 ((0, T ) × Rn ) satisfy pij = pji ,

i, j, = 1, 2, · · · , n,

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102 2 l, Ψ ∈ C 2 ((0, T ) × Rn ). Assume that u is an Hloc (Rn )-valued {Ft }t≥0 2 n adapted process such that ut is an L (R )-valued semimartingale. Set θ = el and v = θu. Then, for a.e. x ∈ Rn and P-a.s. ω ∈ Ω,   ! pij li vj + Ψv dut − (pij ui )j dt θ − 2lt vt + 2

+

 i,j

i,j

i,j

ij i j 

2p p

ij i j 

li vi vj  − p p

 li vi vj  − 2pij lt vi vt + pij li vt2

i j 

 Ψi  ij 2 ! +Ψpij vi v − Ali + p v dt (20) 2 j  Ψt  2 ! pij lt vi vj − 2 pij li vj vt + lt vt2 − Ψvt v + Alt + v +d 2 i,j i,j ! = ltt + (pij li )j − Ψ vt2 − 2 [(pij lj )t + pij ltj ]vi vt

+



i,j

i,j

 !      (p lt )t + 2pij (pi j li )j  − (pij pi j li )j  + Ψpij vi vj

i,j

+Bv 2 +

ij



i ,j 

− 2lt vt + 2



pij li vj + Ψv

2 .

dt + θ2 lt (dut )2 ,

i,j

where (dut )2 denotes the quadratic variation process of ut , A and B are stated as follows: ⎧  ij ⎪ ⎪ A = (lt2 − ltt ) − (pij li lj − pij ⎪ j li − p lij ) − Ψ, ⎪ ⎨ i,j (21) ! 1 ⎪  ij ij ⎪ ⎪ = AΨ + (Al ) − (Ap l ) + Ψ − (p Ψ ) . B t t i j tt i j ⎪ ⎩ 2 i,j

i,j

Once Theorem 3.2 is obtained, in order to prove Theorem 3.1, we choose l = d(x) − c2 (t − T2 )2 . 4. Observability estimate for stochastic Schr¨ odinger equations Let us consider the following stochastic Schr¨odinger equation: ⎧ ⎪ ⎨ idz + Δzdt = (a1 · ∇z + a2 z + f )dt + (a3 z + g)dB in Q, z=0 on Σ, ⎪ ⎩ in G z(0) = z0

(22)

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with initial datum y0 ∈ L2 (Ω, F0 , P ; H01 (G)), % 1,∞ 1,∞ ia1 ∈ L∞ (G; Rn )), a2 ∈ L∞ (G)), F (0, T ; W0 F (0, T ; W 1,∞ a3 ∈ L ∞ (G)), F (0, T ; W

and f ∈ L2F (0, T ; H01(G)),

g ∈ L2F (0, T ; H 1 (G)).

(23)

We first recall the definition of the solution to equation (22). Definition 4.1. We call z ∈ L2F (Ω; C([0, T ]; H01 (G))) a solution of equation (2) if the following hold: 1. z(0) = z0 in G, P-a.s.; 2. For any t ∈ [0, T ] and any η ∈ H01 (G), it holds that   iz(t, x)η(x)dx − iz(0, x)η(x)dx G G  t !   ∇z(s, x) · ∇η(x) + a1 · ∇z + a2 z + f η(x) dxds = G 0  t (a3 z + g)η(x)dxdB, P-a.s. + G

0

We refer to Chapter 6] for the well-posedness of equation (2).  [4,  n Let x0 ∈ lR \ G and Γ0 be given by    Γ0 = x ∈ Γ : (x − x0 ) · ν(x) > 0 .

(24)

Put 

r4 = |a1 |2L∞ (0,T ;W 1,∞ (G;Rn )) + |a2 |2L∞ (0,T ;W 1,∞ (G)) F

F

0

+|a3 |2L∞ (0,T ;W 1,∞ (G)) F

(25)

+ 1.

In this section, we denote by C a generic positive constant depending only on T , G and x0 , which may change from line to line. The observability estimate for equation (22) is as follows. Theorem 4.1. For any solution of equation (22), it holds that |z0 |L2 (Ω,F0 ,P ;H01 (G))  ∂z     ≤ exp(Cr4 )   2 + |f |L2F (0,T ;H01 (G)) + |g|L2F (0,T ;H 1 (G)) . ∂ν LF (0,T ;L2 (Γ0 )) (26)

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Let β(t, x) ∈ C 2 (lR1+m ; lR), and let bjk (t, x) ∈ C 1,2 (lR1+m ; lR) satisfy j, k = 1, 2, · · · , n.

bjk = bkj ,

(27)

We define a (formal) second order stochastic partial differential operator P as 

Pz = iβ(t, x)dz +

m

(bjk (t, x)zj )k dt,

i=

√

−1.

(28)

j,k=1

As before, the key instrument for obtaining Theorem 4.1 is the following identity concerning P. Theorem 4.2. Let , Ψ ∈ C 2 (lR1+m ; lR). Assume that z is an 2 (Rn , C)-valued {Ft }t≥0 -adapted process. Put θ = e , v = θz. Then Hloc for a.e. x ∈ Rn and P-a.s. ω ∈ Ω, it holds that

θ(PzI1 + PzI1 ) + dM + divV = 2|I1 | dt + 2

m

cjk (vk v j + v k vj )dt + D|v|2 dt

j,k=1

+i

m

! (βbjk j )t + bjk (βt )j (v k v − vk v)dt

j,k=1

+i βΨ +

m

! (βbjk j )k (vdv − vdv)

j,k=1 2

(29)

+(β t )dvdv + i

m

βbjk j (dvdv k − dvk dv),

j,k=1

where ⎧ m ⎪  ⎪ ⎪ I = −iβ v − 2 bjk j vk + Ψv, 1 t ⎪ ⎪ ⎨ j,k=1 m m ⎪ ⎪  jk ⎪ ⎪ = b   − (bjk j )k − Ψ, A j k ⎪ ⎩ j,k=1

j,k=1

(30)

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⎧ m ⎪  2 2 ⎪ ⎪ M = β  |v| + iβ bjk j (v k v − vk v), t ⎪ ⎪ ⎪ ⎪ j,k=1 ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ V = [V 1 , · · · , V k , · · · , V m ], ⎪ ⎪ ⎪ ⎪ m ⎪ ! ⎪ ⎪ k ⎨ bjk j (vdv − vdv) + bjk t (vj v − v j v)dt V = −iβ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

j=1

−Ψ

m

bjk (vj v + v j v)dt +

j=1

+

m

(31)

bjk (2Aj + Ψj )|v|2 dt

j=1



m



2bjk bj



k

   − bjk bj k j (vj  v k + v j  vk )dt,

j,j  ,k =1

and ⎧ m !     ⎪ jk  ⎪ ⎪ c = 2(bj k j  )k bjk − (bjk bj k j  )k − bjk Ψ, ⎪ ⎪ ⎨ j  ,k =1 m m ! ⎪ ⎪  ⎪ 2 jk jk ⎪ = (β  ) + (b Ψ ) + 2 (b  A) + AΨ . D ⎪ t t k j j k ⎩ j,k=1

(32)

j,k=1

In the above lemma, the weight function θ and the function  are chosen in the following manner. Let ψ(x) = |x − x0 |2 + τ,

(33)

where τ is a positive constant such that ψ ≥ 56 |ψ|L∞ (G) . Let s > 0 and λ > 0. Put =s

e4λψ − e5λ|ψ|L∞ (G) , t2 (T − t)2

θ = e .

(34)

5. Some open problems There are a great many unsolved problems for observability estimate for SPDEs. We only list a few problems which are important in our opinion. 1. Compared with the observability estimate for SPDEs, it is more challenging to establish the observability estimate for backward SPDEs with one observation. For example, let {Ft }t≥0 be the natural filtration generat-

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ed by {B(t)}t≥0 and consider the following backward parabolic equation: ⎧ dz + Δzdt = (az + bZ)dt + ZdB in Q, ⎪ ⎪ ⎨ z=0 on Σ, (35) ⎪ ⎪ ⎩ z(T ) = zT in G, ∞ where zT ∈ L2 (Ω, FT , P ; L2 (G)) and a, b ∈ L∞ F (0, T ; L (G)). We are looking forward to the following inequality:

|z(0)|L2 (Ω,FT ,P ;L2 (G)) ≤ C|z|L2F (0,T ;L2 (G0 )) . As far as we know, Ref. 15 is the only published paper concerning this problem, in which the observability estimate for a very special backward stochastic parabolic equation was obtained. 2. Both inequality (19) and inequality (26) are boundary observability estimate. A natural question is to establish the internal observability estimate for equation (10) and (22). For example, for the solution of equation (10) with initial datum (z0 , z1 ) ∈ L2 (Ω, F0 , P ; L2 (G) × H −1 (G)), we expect the following inequality: |(z0 , z1 )|L2 (Ω,F0 ,P ;L2 (G)×H −1 (G)) 1   3/2−n/p 2 +1)   ≤ CeC(r1 +r2 z 2 2 

LF (0,T ;L (Oδ (Γ0 )))

+ |f |L2F (0,T ;L2 (G))

(36)

+|g|L2F (0,T ;L2 (G)) . However, to our best knowledge, people do not know how to achieve this result now. 3. It is well known that a sharp sufficient condition for establishing observability estimate for deterministic hyperbolic equations and deterministic Schr¨odinger equations with time invariant lower order terms is that the triple (G, Γ0 , T )((G, Oδ , T )) satisfies the geometric optic condition introduced in Ref. 2. It would be quite interesting and challenging to extend this result to the stochastic setting. However, in order to achieve this task, it seems that one needs the theory of propagation of singularity for SPDEs, which is completely undeveloped until now. Acknowledgement This work is partially supported by the NSF of China under grant 11101070 and the ERC Advanced Grant FP7-246775 NUMERIWAVES, the Grant PI2010-04 of the Basque Government, the ESF Research Networking Programme OPTPDE and Grant MTM2008-03541 of the MICINN, Spain.

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References 1. V. Barbu, A. R˘ ascanu and G. Tessitore, Carleman estimate and controllability of linear stochastic heat equatons, Appl. Math. Optim., 47(2003), 97–120. 2. C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilizion of waves from the boundary, SIAM J. Control Optim., 30(1992), 1024–1065. 3. J.-M. Coron, Control and Nonlinearity, Math. Surveys Monogr., vol 136, Amer. Math. Soc., Providence, RI 2007. 4. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. 5. X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007), 1578–1614. 6. A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, Seoul, 1996. 7. N. V. Krylov, A W2n -theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Related Fields, 98 (1994), 389–421. 8. V. Isakov, Inverse Problems for Partial Differential Equations, SpringerVerlag, Berlin, 1998. 9. I. Lasiecka, R. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schr¨ odinger equations via pointwise Carleman estimates: Part I. H 1 -estimates, J. Inv. Ill-posed Problems, 11(2004), 43–123. 10. G. Lebeau, Contrˆ ole de l´equation de Schr¨ odinger, J. Math. Pures Appl., 71(1992), 267–291. 11. G. Lebeau and L. Robbiano, Contrˆ ole exact de l’´equation de la chaleur, Comm. Partial Differential Equations, 20(1995), 335–356. 12. X. Li and J. Yong, Optimal control theory for infinite-dimensional systems, Systems & Control: Foundations & Applications, Birkh¨ auser Boston, Inc., Boston, MA, (1995). 13. J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30(1988), 1–68. 14. Q. L¨ u, Control and Observation of Stochastic Partial Differential Equations, Ph D Thesis, Sichuan University, 2010. 15. Q. L¨ u, Some results on the controllability of forward stochastic heat equations with control on the drift, J. Funct. Anal., 260(2011), 832–851. 16. Q. L¨ u, Observability estimate for stochastic Schr¨ odinger equations, C. R. Acad. Sci. Paris, Ser I, 348(2010), 1159–1162. 17. Q. L¨ u, Observability estimate for stochastic Schr¨ odinger equations, submitted. 18. Q. L¨ u, Boundary and internal observability estimate for stochastic hyperbolic equations, submitted. 19. E. Machtyngier, Exact controllability for the Schr¨ odinger equation, SIAM J. Control Optim., 32(1994), 24–34. 20. D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open problems, SIAM Rev., 20(1978), 639–739.

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21. S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim., 48(2009), 2191–2216. 22. X. Zhang, Explicit observability estimate for the wave equation with lower order terms by means of Carleman inqualities, SIAM J. Control Optim., 39(2000), 812–834. 23. X. Zhang, Carleman and observability estimates for stochastic wave equations, SIAM J. Math. Anal., 40(2008), 851–868. 24. E. Zuazua, Some problems and results on the controllability of partial differential equations, in Progress in Mathematics, Vol: 169, Birkh¨ auser Verlag, Basel/Switzerland, 1998, 276–311.

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The Nonlinear “Hot Spots” Conjecture in Balls of S2 and H2 Yasuhito Miyamoto Department of Mathematics, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo, 152-8551, Japan E-mail: [email protected] Let B be a ball in a two dimensional hemisphere or in a two dimensional hyperbolic space. Let f ∈ C 2 . We prove the following nonlinear “hot spots” conjecture on B: If the nonconstant solution u of the Neumann problem ∆LB u + f (u) = 0 in B,

∂ν u = 0 on ∂B

has a critical point in B\∂B, then the Morse index of u is 2 or larger, where ∆LB denotes the Laplace-Beltrami operator on B and the Morse index is the number of the strictly positive eigenvalues. Keywords: Nonlinear hot spots conjecture; Maximum point; Neumann problem.

1. Introduction and Main results Yanagida24 posed the following conjecture: Conjecture 1.1 (The nonlinear “hot spots” conjecture). Let Ω ⊂ Rn be a convex domain in Rn , and let f be a function of class C 2 . If the nonconstant solution u of the Neumann problem ∆u + f (u) = 0 in Ω,

∂ν u = 0 on ∂Ω

(1)

has a critical point inside Ω, then the Morse index of u is 2 or larger. Here P ∆ = nj=1 ∂ 2 /∂x2j , ∂ν denotes the outer unit normal derivative on ∂Ω, and the Morse index is the number of the strictly positive eigenvalues. He pointed out that Conjecture 1.1 is a nonlinear version of the following “hot spots” conjecture of J. Rauch:23 Conjecture 1.2 (“hot spots” conjecture). If the domain is a convex set in Rn , then every second eigenfunction of the Neumann Laplacian attains its maximum only on the boundary.

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Conjecture 1.2 immediately follows from the contrapositive of Conjecture 1.1 when f (u) = μ1 (Ω)u. Here μ1 (Ω) is the second Neumann eigenvalue on Ω. When the domain is not a convex set in Rn , there are counterexamples5,7,8 to Conjecture 1.2. These domains have a hole(s). The counterexample without a hole is not known. There are partial positive answers1,4,13,14,19,20,22 to Conjecture 1.2. However, Conjecture 1.2 remains open for a general convex domain even in R2 . Conjecture 1.1 also remains open for a general convex domain even in R2 . However, there are partial positive answers. Conjecture 1.1 holds for a disk17 and a rectangle.18 Let Sn and Hn denote an n-dimensional sphere and hyperbolic space, n denote an open hemisphere of Sn . In this article we respectively. Let D+ consider the Neumann problem in a ball of S2 or H2 . Specifically, we consider ΔLB u + f (u) = 0 in B,

∂ν u = 0 on ∂B,

(2)

where B is a ball in S2 or H2 and ΔLB denotes the Laplace-Beltrami operator on B. The main theorem of this article is a partial positive answer to Conjecture 1.1 for (2). Theorem 1.1. The following (i) and (ii) hold: 2 . If the (i) Suppose that the domain B is a ball strictly included in D+ nonconstant solution of (2) has an interior critical point, then the Morse index of the solution is 2 or larger. (ii) Suppose that the domain B is a ball in H2 . If the nonconstant solution of (2) has an interior critical point, then the Morse index of the solution is 2 or larger. In the case of a convex domain in Rn the Morse index of every nonconstant solution of (1) is 1 or larger.9,16 We have a similar result: Theorem 1.2. The following (i) and (ii) hold: 2 , then the Morse index (i) If the domain B is a ball strictly included in D+ of every nonconstant solution of (2) is 1 or larger. (ii) If the domain B is a ball in H2 , then the Morse index of every nonconstant solution of (2) is 1 or larger. The elliptic problems on Sn and Hn have attracted much attention for these two decades. See Refs. 2,3,6,15,21 for results of the Dirichlet problems. In particular, if the domain is a ball in Hn or if the domain is a ball strictly n , then the moving plane method is applicable and every included in D+

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positive solution is radial.15,21 However, if the domain is a ball containing a closed hemisphere, then there is a nonradial positive solution.3 Therefore, we are interested in the question whether Conjecture 1.1 holds for a ball containing a closed hemisphere. Using Sturm’s comparison theorem, one can easily show that Conjecture 1.2 holds for all balls in Sn . This fact suggests the following: Conjecture 1.3. Conjecture 1.1 holds for all balls in Sn . When the domain is not in Sn or Hn , there is a surface such that Conjecture 1.2 does not hold for a simply connected domain on the surface.10 Let us explain the strategy of the proof. We essentially follow the proof of the disk case.17 However, the test functions should be modified. We use the nodal curve of test functions which relates the shape of the solution with the Morse index. In Section 2 we collect known results about the nodal curves. Let DR := {(x, y) ∈ R2 ; x2 + y 2 < R2 }. When the domain is a disk DR in R2 , ∂x and ∂y commute with Δ. Differentiating (1) with respect linear equation Δux + f  (u)ux = 0. Let to x, wesee that ux satisfies the  H[φ] := DR −|∇φ|2 + f  (u)φ2 dx. Then we have    H[ux ] = −|∇ux |2 + f  (u)u2x dx D  R  = (Δux + f  (u)ux ) ux dx − ux ∂ν ux dσ DR ∂DR  =− ux ∂ν ux dσ. ∂DR

   Moreover, H[ux ] + H[uy ] = − DR ∂ν u2x + u2y /2dσ. Casten-Holland9 and    Matano16 showed that Ω ∂ν u2x + u2y dσ ≤ 0 if the domain Ω is convex. Since DR is convex, we have H[ux ] + H[uy ] ≥ 0. This inequality plays a crucial role in the disk case. 2 . The stereoWe consider the case where the domain is a ball in D+ 2 into the disk DR centered at the graphic projection maps a ball in D+ origin with radius R (0 < R < 1) and ΔLB to p−2 Δ, respectively. Here

p=

2 , 1 + r2

r=

0

x2 + y 2 .

(3)

∂ν u = 0 on ∂DR .

(4)

Then (2) becomes p−2 Δu + f (u) = 0 in DR ,

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Let u be a solution of (4). The associated eigenvalue problem is p−2 Δφ + f  (u)φ = μφ in DR ,

∂ν φ = 0 on ∂DR .

(5)

Now we define test functions. We set 1 + x2 − y 2 1 − x2 + y 2 ∂x + xy∂y and ∂Y := xy∂x + ∂y . 2 2 We use uX := ∂X u and uY := ∂Y u as test functions. Then we see by −2 with p−2 direct calculation that ∂X and  ∂Y commute  Δ. Thus p ΔuX +  2  2 2 f (u)uX = 0. Let Hp [φ] := DR −|∇φ| + f (u)φ p dx. We have    Hp [uX ] = −|∇uX |2 + f  (u)u2X p2 dx D  R    = ΔuX + f  (u)uX p2 uX dx − uX ∂ν uX dσ DR ∂DR  =− uX ∂ν uX dσ. ∂X :=

∂DR

Moreover,





Hp [uX ] + Hp [uY ] = −

∂ν ∂DR

u2X + u2Y 2

dσ.

(6)

In Lemma 3.1 we show that Hp [uX ] + Hp [uY ] > 0 if 0 < R < 1 and if u is not constant. The restriction 0 < R < 1 comes from Lemma 3.1. We use uθ := −yux + xuy in addition to uX and uY . Then uθ commutes with p−2 Δ and ∂ν uθ = 0 on ∂DR . Therefore, p−2 Δuθ + f  (u)uθ = 0 and 0 is an eigenvalue of (5) provided that u is nonradial. Using uX , uY , and uθ , we construct a test function v, which is defined by (8), in the proof of Theorem 1.1. Analyzing the nodal curves of v, we show that the second eigenvalue is positive and obtain Theorem 1.1 (i). When the domain is in H2 , the problem (2) on an arbitrary ball in H2 becomes q −2 Δu + f (u) = 0 in DR ,

∂ν u = 0 on ∂DR ,

(7)

where 0 < R < 1 and q := 2/(1 − r2 ). We use the test functions 1 − x2 + y 2 1 + x2 − y 2 ux − xyuy and uη := −xyux + uy . 2 2 Then the proof of this case is similar to that of the previous case. In Section 3.2 we mention different points. This article consists of three sections. In Section 2 we recall known results about the nodal curve. In Section 3 we prove Theorems 1.1 and 1.2. uξ :=

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2. Preliminaries The nodal curve plays an important role in the proof. Proposition 2.1. Let Ω be a planar connected domain with smooth boundary. Let V (x, y) ∈ C 0,γ (Ω) (0 < γ < 1), and let u(x, y) be a function such that Δu + V u = 0 in Ω. Then u ∈ C 2 (Ω). Furthermore, u has the following properties: (i) If u(x0 , y0 ) = 0 and (x0 , y0 ) ∈ Ω, then either  u ≡ 0 in Ω or there is m ∈ m/2 N such that u(x, y) = Hm (x − x0 , y − y0) + o (x − x0 )2 + (y − y0 )2 , where Hm is a real valued, non-zero, harmonic polynomial of degree m and {u = 0} has exactly 2m branches at (x0 , y0 ). (ii) If u(x0 , y0 ) = 0 and (x0 , y0 ) ∈ ∂Ω and u satisfies the Neumann (resp. Dirichlet) boundary condition, then either u ≡ 0 in Ω or u(x, y) = C0 rm cos(mθ) + o(rm ) for some C0 ∈ R\{0}, where (r, θ) is a poler coordinate of (x, y) around (x0 , y0 ) and the angle θ is chosen so that the tangent to the boundary at (x0 , y0 ) is given by the equation sin θ = 0 (resp. cos(mθ) = 0). (i) is well-known.12 The second case of (ii) is taken from Proposition 4.1 of Ref. 11. The first case of (ii), which is a strong unique continuation property at a boundary point, can be proved by straightening the boundary, extending the solution across the boundary as an even function, and applying the strong unique continuation theorem at an interior point. By Proposition 2.1 (i) we have Corollary 2.1. Let u be as in Proposition 2.1, and let (x0 , y0 ) be an interior point of Ω. If (x0 , y0 ) is a degenerate zero, i.e., u = ux = uy = 0 at (x0 , y0 ), then the following (i) or (ii) holds: (i) {u = 0} has at least four branches emanating from (x0 , y0 ). (ii) u ≡ 0 in Ω. Proof. Since (x0 , y0 ) is a degenerate zero of u, the order of the degeneracy is 2 or larger. Hence m ≥ 2, where m is defined in Proposition 2.1 (i). We obtain the conclusion. Let u be a solution of (4). We denote the cardinal number of the zeros of uθ on ∂DR by Z[uθ ]. Specifically, Z[uθ ] := {θ ∈ [0, 2π); uθ (R cos θ, R sin θ) = 0}, where S is the cardinal number of the set S. We will see in the next section that Z[uθ ] relates the shape of u and the Morse index of u.

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3. Proofs 3.1. Proofs of Theorems 1.1 (i) and 1.2 (i) 0  We define u, v := DR u(x)v(x)p2 dx, u2 := u, u, and H 1 (DR ) := . / 2 + u2 p2 ) dx < ∞ . We denote the eigenu(x); uH 1 (DR ) := (|∇u| DR

pair of (5) by {(μj , φj )}∞ j=0 . In particular, (μ0 , φ0 ) and (μ1 , φ1 ) are the first and second eigenpairs, respectively.

Lemma 3.1. Let u be a nonconstant solution to (4). Then the following (i) or (ii) holds: (i) Hp [uX ] > 0 or Hp [uY ] > 0. (ii) The Morse index of u is 2 or larger. Proof. We use the polar coordinate (x, y) = (r cos θ, r sin θ). By direct calculation we have that u2X + u2Y = (1 + r2 )2 u2r /4 + (1 − r2 )2 u2θ /4. Then   R4 − 1 2 u . ∂r u2X + u2Y r=R = 2R3 θ

By (6) we have 

H[uX ] + H[uY ] = ∂DR

1 − R4 2 u dσ ≥ 0, 4R3 θ

where we use 0 < R < 1. If H[uX ] + H[uY ] > 0, then (i) holds. Otherwise, H[uX ] = H[uY ] = 0 and uθ ≡ 0 on ∂DR . Since ∂ν uθ = 0 on ∂DR and p−2 Δuθ + f  (u)uθ = 0, each point on ∂DR is a degenerate zero of uθ . Hence we apply Proposition 2.1 (ii). If the order of the degeneracy of the zero of uθ on ∂DR is finite, then the zero of uθ on ∂DR should be isolated in ∂DR , which is a contradiction. By the first case of Proposition 2.1 (ii) we see that uθ ≡ 0 in DR . Hence u is radial. In this case it is well-known that the first eigenfunction φ0 is also radial. Since uX = (1 + r2 ) cos θur /2, uX ≡ 0 in DR . Since uX = (1 + r2 )ux /2, uX (−x, y) = −uX (x, y), hence uX is odd in x. In particular, uX , φ0  = 0. From a variational characterization of the second eigenvalue we have μ1 =

sup ψ∈H 1 (DR )\{0} ψ∈span{φ0 }⊥

Hp [ψ] 2 ψ2

≥

Hp [uX ] 2

uX 2

= 0,

where span{φ0 }⊥ := {u(x); u, φ0  = 0}. By contradiction we show that μ1 > 0. Suppose the contrary, i.e., μ1 = 0. Then uX is the second eigenfunction, hence uX satisfies the Neumann boundary condition on ∂DR . We

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have 1 + R2 cos θurr (R) = 0, 2 which indicates that urr = 0 at r = R. Since p−2 urr + f (u) = 0 at r = R, u(R) is a zero of f . Because the constant function u(R) is the unique solution of the problem ∂r uX |r=R =

p−2 (urr + ur /r) + f (u) = 0,

ur (R) = 0,

and u(R) is a zero of f ,

we see that u(r) ≡ u(R) (0 ≤ r ≤ R). Thus u is constant, which is a contradiction. Since μ1 > 0, (ii) holds. Since μ0 =

sup ψ∈H 1 (DR )\{0}

Hp [ψ] 2

ψ2

,

by Lemma 3.1 we obtain Theorem 1.2 (i). In particular, every stable steady state of the associated parabolic equation is constant. By the proof of Lemma 3.1 we see that if the second eigenvalue μ1 associated with a radial solution u is nonpositive, then u is constant. Thus μ1 > 0 provided that u is a nonconstant radial solution. We obtain Corollary 3.1. Let u be a nonconstant solution to (4). If u is radial, then the Morse index of u is 2 or larger. In the proof of Lemma 3.1 we see that each zero of uθ on ∂DR is isolated if uθ is nonradial. It is clear that the converse is also true. Thus we have Corollary 3.2. The following (i) and (ii) hold: (i) u is a nonradial solution of (4) if and only if Z[uθ ] is a finite integer greater than 1. (ii) u is a radial solution of (4) if and only if Z[uθ ] = ℵ1 . The next lemma indicates that the shape of the solution on the boundary is related with the Morse index. Lemma 3.2. Let u be a nonconstant solution. If Z[uθ ] ≥ 3, then the Morse index of u is 2 or larger. Proof. If Z[uθ ] is infinite, then by Corollary 3.2 we see that u is radial. Hence the conclusion follows from Corollary 3.1. We consider the case where u is nonradial. Then uθ ≡ 0 and p−2 Δuθ + f  (u)uθ = 0. Hence uθ is an eigenfunction corresponding to the zero eigenvalue of (5). It follows from

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Proposition 2.1 (ii) that for each isolated zero of uθ on ∂DR there is a branch of {uθ = 0} hitting the zero. Because Z[uθ ] ≥ 3 and each branch hits ∂DR or meets another branch, there are at least three nodal domains of uθ . Then 0 is not the first or second eigenvalue, since it is well-known that the numbers of the nodal domains of the first and second eigenfunctions are 1 and 2, respectively. Thus μk = 0 for some k ≥ 2. We have 0 = μk < μ1 . We obtain the conclusion. Proof of Theorem 1.1 (i). Let u be a nonconstant solution to (4). If u is radial, then it follows from Corollary 3.1 that the Morse index is 2 or larger. Hereafter we assume that u is nonradial. We suppose that (x0 , y0 ) ∈ DR is a critical point of u. Then ux = uy = 0 at (x0 , y0 ). We define v by v := uθ + a0 uX + b0 uY , where a0 :=

2y0 −2x0 , b0 := , and r0 = 1 − r02 1 − r02

(8) /

x20 + y02 .

We define α and β by 1 + x2 − y 2 1 − x2 + y 2 + b0 xy and β := x + a0 xy + b0 . 2 2 By direct calculation we see that v = αux + βuy and that α = β = 0 at (x0 , y0 ). Since α := −y + a0

v = αux + βuy = 0 at (x0 , y0 ), vx = αx ux + αuxx + βx uy + βuyx = 0 at (x0 , y0 ), and vy = αy ux + αuxy + βy uy + βuyy = 0 at (x0 , y0 ), (x0 , y0 ) is a degenerate zero of v. Because of Corollary 2.1, there are two possibilities: Case 1: {v = 0} has at least four branches emanating from (x0 , y0 ) and Case 2: v ≡ 0 on DR . Case 1: We divide this case into the following two cases: Case 1a: {v = 0} has a closed loop in DR and Case 1b: {v = 0} does not have a loop. Case 1a: We denote the open set enclosed by the loop of {v = 0} by ω(⊂ DR ). Without loss of generality we can assume that v does not change sign in ω. We define w by % v in ω, w := 0 in DR \ω.

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Because of Lemma 3.1, we assume that Hp [uX ] > 0. (The case Hp [uY ] > 0 can be similarly treated.) We can assume that φ0 > 0. Since w does not change sign, w, φ0  = 0. Hence we can define z by z := uX + cw, where c = − uX , φ0  / w, φ0 . Note that z, φ0  = 0. We have    −|∇(uX + cw)|2 f  (u)(uX + cw)2 p2 dx Hp [z] = D  R      = −|∇uX |2 + f  (u)u2X p2 dx + c2 −|∇w|2 + f  (u)w2 p2 dx DR DR     + 2c −∇uX · ∇w + f (u)uX wp2 dx DR    = Hp [uX ] + c2 −|∇w|2 + f  (u)w2 p2 dx ω    −∇uX · ∇v + f  (u)uX wp2 dx, + 2c ω

where we use w = |∇w| = 0 in DR \ω. We have    −|∇w|2 + f  (u)w2 p2 dx ω     w Δw + f  (u)wp2 dx − = ω

 ω



w∂ν wdσ = 0 and

∂ω

 −∇uX · ∇w + f  (u)uX wp2 dx      2 = w ΔuX + f (u)uX p dx − ω

w∂ν wdσ = 0,

∂ω f  (u)uX p2

where we use Δw + f  (u)wp2 = 0 in ω, ΔuX + = 0 in ω, and w = 0 on ∂ω. Thus we have Hp [z] > 0. It follows from a variational characterization of the second eigenvalue that Hp [ψ] Hp [z] sup ≥ > 0. μ1 = 2 ψ∈H 1 (DR )\{0} ψ z22 2 ψ∈span{φ0 }⊥

This inequality indicates that the Morse index is 2 or larger. Case 1b: Since {v = 0} does not have a loop, each branch emanating from (x0 , y0 ) hits ∂DR and ∂DR ∩ {v = 0} has at least four points. Let (x1 , y1 ) ∈ ∂DR ∩ {v = 0}. We will show that ux = uy = 0 at (x1 , y1 ). Since ∂ν u = 0 at (x1 , y1 ), xux + yuy = 0 at (x1 , y1 ). Since v = 0 at (x1 , y1 ), αux + βuy = 0 at (x1 , y1 ). Thus we have



x1 y1 ux (x1 , y1 ) 0 = . α(x1 , y1 ) β(x1 , y1 ) uy (x1 , y1 ) 0

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We have

(1 − R2 )(x0 x1 + y0 y1 ) x1 y1 det = R2 + α(x1 , y1 ) β(x1 , y1 ) 1 − r02 ≥

(1 − R2 )R2 (1 − R2 )(x0 x1 + y0 y1 ) 1 − R2 2 + ≥ (R − r0 R) > 0, 1 − r02 1 − r02 1 − r02

where we use x0 x1 + y0 y1 ≥ −r0 R. Since the matrix has the inverse, ux = uy = 0 at (x1 , y1 ). Hence uθ = 0 on ∂DR ∩ {v = 0}. Since ∂DR ∩ {v = 0} has at least four points, Z[uθ ] ≥ 4. By Lemma 3.2 we see that the Morse index is 2 or larger. Case 2: We already show that if (x1 , y1 ) ∈ ∂DR ∩ {v = 0}, then uθ = 0 at (x1 , y1 ). Since v ≡ 0 in DR , v ≡ 0 on ∂DR and uθ ≡ 0 on ∂DR . In the proof of Lemma 3.1 we showed that u becomes radial. We obtain a contradiction, since we assume that u is nonradial. We have verified all the cases. The proof is complete. 3.2. Proofs of Theorems 1.1 (ii) and 1.2 (ii) The proof of Theorem 1.1 (ii) is almost the same as that of Theorem 1.1 (i). However, we have to modify several points. In particular, we use uξ and uη instead of uX and uY . We use v := uθ + a0 uξ + b0 uη = αux + βuy , where / 2y0 −2x0 , b := , r := x20 + y02 , a0 := 0 0 1 + r02 1 + r02 1 − x2 + y 2 1 + x2 − y 2 − b0 xy, and β := −x − a0 xy + b0 . α := −y + a0 2 2    Let Hq [φ] := DR −|∇φ|2 + f  (u)φ2 q 2 dx. Then we have 

Hq [uξ ] + Hq [uη ] = ∂DR



det

x1 y1 α(x1 , y1 ) β(x1 , y1 )

1 − R4 2 u dσ ≥ 0 and 4R3 θ

= R2 −

1 + R2 (x0 x1 + y0 y1 ) 1 + r02

1 + R2 r0 R 1 + r02 (R − r0 )(1 − r0 R)R = > 0, 1 + r02

≥ R2 −

where (x1 , y1 ) ∈ ∂DR . The details of the proof are left to the reader.

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Acknowledgment The author would like to thank the organizers for the kind invitation to the conference held at Chern Institute of Mathematics, Nankai University. This work was partially supported by Grant-in-Aid for Young Scientists (B) (Subject No. 21740116). References 1. R. Atar and K. Burdzy, On Neumann eigenfunctions in lip domains, J. Amer. Math. Soc. 17 (2004), 243–265. 2. C. Bandle, A. Brillard, and M. Flucher, Green’s function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature, Trans. Amer. Math. Soc. 350 (1998), 1103–1128. 3. C. Bandle and J. Wei, Non-radial clustered spike solutions for semilinear elliptic problems on Sn , J. Anal. Math. 102 (2007), 181–208. 4. R. Ba˜ nuelos and K. Burdzy, On the “ hot spots” conjecture of J. Rauch, J. Funct. Anal. 164 (1999), 1–33. 5. R. Bass and K. Burdzy, Fiber Brownian motion and the “ hot spots” problem, Duke Math. J. 105 (2000), 25–58. 6. H. Brezis and L. Peletier, Elliptic equations with critical exponent on spherical caps of S3 , J. Anal. Math. 98 (2006), 279–316. 7. K. Burdzy, The hot spots problem in planar domains with one hole, Duke Math. J. 129 (2005), 481–502. 8. K. Burdzy and W. Werner, A counterexample to the ”hot spots” conjecture, Ann. of Math. 149 (1999), 309–317. 9. R. Casten and C. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations 27 (1978), 266–273. 10. P. Freitas, Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces, Indiana Univ. Math. J. 51 (2002), 305–316. 11. B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and M. P. Owen, Nodal sets for groundstates of Schr¨ odinger operators with zero magnetic field in non-simply connected domains, Comm. Math. Phys. 202 (1999), 629–649. 12. P. Hartman and A. Wintner, On the local behavior of solutions of nonparabolic partial differential equations, Amer. J. Math. 75 (1953), 449–476. 13. D. Jerison and N. Nadirashvili, The “ hot spots” conjecture for domains with two axes of symmetry, J. Amer. Math. Soc. 13 (2000), 741–772. 14. B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, 1150. Springer-Verlag, Berlin, 1985. iv+136 pp. 15. S. Kumaresan and J. Prajapat, Serrin’s result for hyperbolic space and sphere, Duke Math. J. 91 (1998), 17–28. 16. H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci. 15 (1979), 401–454. 17. Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball II, J. Differential Equations 239 (2007), 61–71.

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18. Y. Miyamoto, On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains, Quart. Appl. Math. 65 (2007), 357– 374. 19. Y. Miyamoto, The “ hot spots” conjecture for a certain class of planar convex domains, J. Math. Phys. 50 (2009), 103530, 7 pp. 20. Y. Miyamoto, A planar convex domain with many isolated “ hot spots” on the boundary, preprint. 21. P. Padilla, Symmetry properties of positive solutions of elliptic equations on symmetric domains, Appl. Anal. 64 (1997), 153–169. 22. M. Pascu, Scaling coupling of reflecting Brownian motions and the hot spots problem, Trans. Amer. Math. Soc. 354 (2002), 4681–4702. 23. J. Rauch, Five problems: an introduction to the qualitative theory of partial differential equations, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), pp. 355–369. Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975. 24. E. Yanagida, Private communication (2006).

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Mean-Field Models Describing Micro Phase Separation in the Two-Dimensional Case Barbara Niethammer Oxford Centre for Nonlinear PDE Mathematical Institute, University of Oxford 24-29 St Giles’, Oxford, OX1 3LB, United Kingdom Yoshihito Oshita Department of Mathematics, Okayama University Tsushimanaka 3-1-1, Okayama, 700-8530, Japan PRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama, Japan E-mail: [email protected] We study the free boundary problem describing the micro phase separation of diblock copolymer melts in the regime that one component has small volume fraction ρ such that the micro phase separation results in an ensemble of small disks of one component in the two dimensional case. Starting from the free boundary problem restricted to disks we rigorously derive the heterogeneous mean-field equations on a time scale of the order of R3 ln(1/ρ), where R is the mean radius of disks. On this time scale, the evolution is dominated by coarsening and stabilization of the radii of the disks, whereas migration of disks becomes only relevant on a larger time scale.

1. Introduction Diblock copolymer molecules consist of subchains of two different type of monomers, say A- and B-monomers. The different type of subchains tend to segregate, and hence the phase separation take place. However since the subchains are chemically bonded, the two subchains mix on a macroscopic scale, while on a molecular scale, A- and B-subchains still segregate and the micro-domains are formed. This is called micro phase separation. For more physical background on this phenomenon we refer to.2,9 In the strong segregation regime, energetically favorable configurations have been characterized in the Ohta–Kawasaki theory22 by minimizers of

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an energy functional, which is in the two dimensional case of the form    γ (−Δ)−1/2 (χE − ρ)2 dx . (1) J(E) = H1 (∂E) + 2 (0,L)2 Here (0, L)2 ⊂ R2 is the domain covered by the copolymers, E ⊂ [0, L)2 denotes the region covered by, say, A-monomers, ρ = |E| L2 ∈ (0, 1) the average density, γ ∈ R+ = (0, ∞) is a parameter related to the polymerization index, χE is the characteristic function of E, and H1 denotes one dimensional Hausdorff measure. The first term in the energy prefers large blocks of monomers, the second favors a very fine mixture. Competition between these terms leads to minimizers of E which represent micro phase separation. Starting with the pioneering work,19 where the Ohta–Kawasaki theory is formulated on a bounded domain as a singularly perturbed problem and the limiting sharp interface problem is identified, there has been a large body of analytical work. Minimizers of the energy functionals have been characterized in,1,3,4,25 the existence/stability of stationary solutions has been investigated in20,21,24,26 and a time dependent model has been considered in.8,10 The mean field models in the three dimensional case have been considered in.6,11,15 We consider the gradient flow of the energy, which is a standard way to set up a model for the evolution of the copolymer configuration that decreases energy and preserves the volume fraction. Then the evolution equation becomes the following extension of the Mullins–Sekerka evolution for phase separation in binary alloys.12 The normal velocity v of the interface ∂E = ∂E(t) satisfies v = [∇w · n]

on ∂E,

(2)

where [∇w · n] denotes the jump of the normal component of the gradient of the potential across the interface. Here n denotes the outer normal to E and [f ] = lim f (x) − lim f (x). x∈E / x→∂E

x∈E x→∂E

The chemical potential w is for each time determined via −Δw = 0 −1

w = κ + γ(−Δ)

(χE − ρ)

in (0, L)2 \∂E,

(3)

on ∂E,

(4)

where κ is the curvature of ∂E. We are interested in the case that the volume of E(t) is preserved in time and can thus impose Neumann or

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periodic boundary conditions for w on ∂(0, L)2 . In what follows we will consider a periodic setting and hence always require that the potential w is (0, L)2 -periodic. For local well-posedness of this evolution see.7 The evolution defined by (2)-(4) has a formal interpretation as a gradient flow of the energy (1) on a Riemannian manifold. Indeed, consider the manifold of subsets of the 2-dimensional flat torus T of length L with fixed volume, that is M := {E ⊂ T ; |E| = L2 ρ}, whose tangent space TE M at an element E ∈ M is described by all kinematically admissible normal velocities of ∂E, that is, $ 1  v dS = 0 . TE M = v : ∂E → R ; ∂E

The Riemannian structure is given by the following metric tensor on the tangent space:  1 2 ∇w1 · ∇w2 dx, (5) gE (v , v ) = T

where w : T → R (α = 1, 2) solves α

in T \∂E,

−Δwα = 0 [∇w · n] = v α

α

on ∂E

(6)

for v α ∈ TE M (α = 1, 2). The gradient flow of the energy (1) is now the dynamical system where at each time the velocity is the element of the tangent space in the direction of steepest descent of the energy. In other words, v is such that gE(t) (v, v˜) = −DJ(E(t)), v˜

(7)

for all v˜ ∈ TE(t) M. Choosing v˜ = v we immediately obtain the energy estimate associated with each gradient flow, which is  T gE(t) (v, v) dt + J(E(T )) = J(E(0)) for all T > 0. (8) 0

In what follows we consider the micro phase separation in the two dimensional case in the regime where the fraction of A-monomers is much smaller than the one of B-monomers. In this case A-phase consists of an ensemble of many small approximately circular particles. We reduce the evolution to the gradient flow on circular particles. For that purpose we define the submanifold N ⊂ M of all sets E  which are the union of disjoint balls E = i BRi (Xi ), where the centers {Xi }i and the radii {Ri }i are variables. Hence N can be identified with an open subspace of the hypersurface {Y = {Ri , Xi }i ; (Ri , Xi ) ∈

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 R+ × T, π i Ri2 = L2 ρ} in R3N , where N is the number and i = 1, . . . , N an enumeration of the particles with centers in the torus T. Since the nordXi i n on ∂BRi (Xi ), the tangent space mal velocity v satisfies v = dR dt + dt ·  can be identified with the hyperplane . ∂ ∂ (Vi +ξi · ) ; (Vi , ξi ) ∈ R×R2 , Ri Vi = 0 ⊂ R3N , TY N = Z = ∂Ri ∂Xi i i

such that Vi describes the rate of change of the radius of particle i and ξi the rate of change of its center. We use the abbreviation Z = {Vi , ξi }i for  ∂ ∂ + ξi · ∂X ). Z = i (Vi ∂R i i The metric tensor is then given by  ∇w1 · ∇w2 dx, gY (Z1 , Z2 ) = T

where the function w : T → R solves α

−Δwα = 0 [∇w · n] = α

Viα

in T \ ∪i ∂BRi (Xi ), +

ξiα

· n

on ∂BRi (Xi ).

(9)

for Zα = {Viα , ξiα }i ∈ TY N , α = 1, 2. For the following it will be convenient to split the metric tensor into the radial and shift part respectively. For any {Vi , ξi }i , we write w =u+φ where u and φ are harmonic in- and outside the particles and where [∇u · n] = Vi

on ∂BRi (Xi ),

[∇φ · n] = ξi · n

on ∂BRi (Xi ) .

(10)

We consider the energy J(Y) = Jsurf (Y) + γJnl (Y), where  Jsurf (Y) = 2π Ri and Jnl (Y) = |∇μ|2 dx T

i

with μ : T → R solving −Δμ = χ∪BRi − ρ. We obtain the differentials of ˜ = {V˜i , ξ˜i }i as the energies in the direction of a tangent vector Z ˜ = 2π DJsurf (Y), Z V˜i i

and ˜ = −2 DJnl (Y), Z

 T

∇μ · ∇w ˜ dx = 2

 i

∂BRi (Xi )

  μ V˜i + ξ˜i · n dS .

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˜ The integration Here w ˜ : T → R is a function wα satisfying (9) for Zα = Z. by parts yields  ˜ =− w(V˜i + ξ˜i · n) dS. gY (Z, Z) ∂BRi (Xi )

i

From now on we consider an arrangement of particles as described above which evolves according to the gradient flow equation. This means that for d Y(t) that any t ≥ 0, it holds for Z(t) = dt ˜ = −DJ(Y), Z, ˜ gY (Z, Z) that is,  i

w(V˜i + ξ˜i ·n) dS = 2π



∂BRi (Xi )

V˜i + 2γ

i

 i

  μ V˜i + ξ˜i ·n dS

∂BRi (Xi )

(11) ˜ is an arbitrary element of the tangent space we ˜ ∈ TY N . Since Z for all Z conclude from (11) that w satisfies    1 w − R1i − 2γμ dS = λ(t) (12) 2πRi ∂BRi (Xi ) and





 w − 2γμ n dS = 0

(13)

∂BRi (Xi )

for all i such that Ri > 0, with a Lagrange parameter λ(t) that ensures volume conservation. Equations (12) and (13) are the analogue of (4) in the restricted setting. Our aim is to identify the evolution in the limit of vanishing volume fraction of particles. More precisely, we consider a sequence of systems characterized by the parameter

 d  −1/2 (14) ε := ln R in the limit ε → 0. Here d is defined by 1 = L2 d2

(15)

i

and R by R2

i

1=

i

Ri (0)2 .

(16)

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Then d12 denotes the number density of particles, and π R2 the average volume of particles. Here and throughout this paper we use the abbreviation   i = i:Ri >0 . Our main result informally says that when L ∼ Lsc , with L2sc := d2 ln(d/R) ∼ d2 ln(1/ρ),

(17)

on the time scale of order R3 ln(1/ρ), the number density of particles with radius r and center x, denoted by ν = ν(t, r, x) (suitably normalized), satisfies 1   ∂t ν + ∂r 2 rψ − 1 − γr3 ν = 0 , (18) r where ψ = ψ(t, x) satisfies for each t that  ∞

 ∞   ∞ 1 γ ν dr = 2π r2 ν dr dy in T ν dr + 2 −Δψ + 2πψ r L T 0 0 0 (19) in the limit ε → 0. Here γ is also suitably normalized. We remark that on the other hand, in the case that L  Lsc , that is, in the very dilute case, one obtains a homogeneous version where ψ is constant in space, and is replaced by λ(t). More precisely that the number density of particles with radius r, denoted by ν(t, r) (suitably normalized), satisfies ∂t ν + ∂r

1   3 λr − 1 − γr ν =0 r2

(20)

with ∞

λ(t) =

0

1 rν

∞ dr + γ 0 r2 ν dr ∞ . 0 ν dr

(21)

2. The Result In this section, we will introduce suitably rescaled variables, state the precise assumption on our initial particle arrangement, and present the statement of our main result. We assume from now on that L = Lsc for the ease of presentation, and we will rescale the spatial variables by Lsc such that Lsc = L = 1 and hence

d = ε, R = ε exp(−1/ε2 ) =: αε .

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ˆ w, Notice that ρ = πα2ε ε−2 and ln(1/ρ) ∼ ε−2 . We introduce Rˆi , tˆ, Vˆi , ξ, ˆ γˆ and μ ˆ via Ri (t) = αε Rˆi (tˆ),

t = α3ε ln(1/ρ)tˆ,

w(t, x) = α−1 ˆ tˆ, x), ε w(

1 ε2 ε ξi (t) = 2 ξˆi (tˆ), Vˆi (tˆ) ∼ 2 Vˆi (tˆ), ln(1/ρ) αε αε 1 ε2 α2 ˆ γ= 3 γˆ ∼ 3 γˆ , μ(t, x) = 2ε μ ˆ (t, x). αε ln(1/ρ) αε ε

Vi (t) =

α2ε

From now on we only deal with the rescaled quantities and drop the hats in the notation. We denote the joint distribution of particle centers and radii at a given time t by νtε ∈ (Cp0 )∗ , which is given by  ζ dνtε = ε2 ζ (Ri (t), Xi (t)) for ζ ∈ Cp0 , (22) i

where Cp0 stands for the space of continuous functions on R+ × T with compact support contained in R+ × T. Here T denotes the unit flat torus, and R+ = (0, ∞). Note that since ζ(r, x) = 0 for r = 0, particles which have vanished do not enter the distribution. The natural space for νtε and its limit νt is the space (Cp0 )∗ of Borel measures on R+ × T. We define the energy in rescaled variables  Jε ({Ri , Xi }i ) := 2π ε2 Ri + γ |∇με |2 dx, i

T

 2 where με = με (t, x) solves −Δμε = αε 2 χEε − π, T με dx = 0 for Eε := ε ∪i Bαε Ri (Xi ) We are now going to make the assumptions on our initial particle arrangement precise. Notice first, that in view of (15) and (16) we have   ε 2 dν0 = ε =1 and r2 dν0ε = ε2 Ri2 (0) = 1 . (23) i

i

It follows immediately, that  r dν0ε = ε2 Ri (0) ≤ 1 ,

(24)

i

that is the surface energy of the initial particle arrangement is finite. Furthermore it is natural to assume that initially the nonlocal energy is uniformly bounded in ε, that is  |∇με (0, x)|2 dx ≤ C , (25) T

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where C is independent of ε.  We will see later that the nonlocal energy controls i ε2 Ri4 . Hence,  2 4 finiteness of the nonlocal energy initially also implies i ε Ri (0) ≤ C. For our analysis we need a little more than this. We need a certain tightness assumption which ensures, that not too much mass is contained in very large particles as ε → 0. More precisely, we assume that ε2 Ri4 (0) → 0 as M → ∞. (26) sup ε

Ri ≥M

Finally, we assume that initially particles are well separated in the sense that we assume that there is σ > 0, such that . B2σε (Xi (0)) are disjoint . (27) i

In accordance with the in (22) we will use in the following the   ∞ notation  abbreviation ζ dνt := 0 T ζ(r, x) dνt (r, x) for νt ∈ (Cp0 )∗ . Otherwise the domain of integration is specified. The natural space for potentials of diffusion fields is H 1 (T). Furthermore ◦ we will denote by H 1 (T) the subspace of H 1 (T) of functions with mean value zero. We can now state our main result which informally says that νtε converges as ε → 0 to a weak solution of (18)-(19). Theorem 2.1. Let T > 0 be given and assume that the assumptions in Section 2 are satisfied. Then there exists a subsequence, again denoted by ε → 0, a weakly continuous map [0, T ] t → νt ∈ (Cp0 )∗ , and a measurable map (0, T ) t → ψ(t) ∈ H 1 (T), such that   ζ dνtε → ζ dνt uniformly in t ∈ [0, T ] for all ζ ∈ Cp0 , 

r2 dνt = 1 for all t ∈ [0, T ], (18) and (19) hold in the following weak sense

  d 1 3 ζ dνt = ∂r ζ 2 r ψ(t, x) − 1 − γr dνt (28) dt r

distributionally on (0, T ) for all ζ ∈ Cp0 with ∂r ζ ∈ Cp0 . Here

  1 ∇ψ(t, x) · ∇ζ − 2πγζ dx + 2π ζ ψ(t, x) − r dνt = 0 T

for all ζ ∈ H 1 (T) and almost all t ∈ (0, T ). Furthermore

  r4 γ lim Jε ({Ri , Xi }i ) = 2π r+γ dνt + |∇K|2 dx ε→0 4 4 T

(29)

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uniformly in t ∈ [0, T ]. Here K(t, ·) ∈ H 1 (T) satisfies for each t 

  ∇K · ∇ζ dx = 2π r2 ζ dνt − ζ dx T

(30)

T

for all ζ ∈ H 1 (T). The proof of Theorem 2.1 goes similarly to the approach for the three dimensional case in.15 However in contrast to the three dimensional case we need to estimate 1/Ri term in proving that the tightness property is preserved in time since the Lagrange multiplier diverges when particles disappear. Indeed we need to show that the tightness property (26) is preserved in time so that no mass is lost at infinity in the limit ε → 0. Lemma 2.1 (tightness). For any t > 0 we have ε2 Ri4 (t) → 0 as M → ∞ uniformly in ε. Ri ≥M

This lemma is crucial to our proof, but the proof is much more difficult than the three dimensional case. In fact, the main idea of the proof is to show by asymptotics that, at least in some average sense, Vi satisfies approximately Ri2 Vi ∼ uRi − 1 − γRi3 where u is the Lagrange multiplier that ensures the volume conservation. In contrast to the three dimensional case, the Lagrange multiplier may diverge. However we can still show the following a-priori estimate, and thus, at least on average, Vi ≤ 0, if Ri is sufficiently large, and no mass can escape to infinity as ε → 0, from which one deduces Lemma 2.1. Lemma 2.2. For any T > 0, there exist constants CT > 0 and ε1 > 0 such that

1/2  T  1 1 ε dν dt ≤ CT (31) 1 − ε2 ln r r2 t 0 for all ε ∈ (0, ε1 ]. Here we omit the proof for the sake of brevity. Acknowledgments Y.O. was supported by the Japan Science and Technology Agency.

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References 1. G. Alberti, R. Choksi and F. Otto . Uniform energy distribution for an isoperimetric problem with long-range interactions. J. Amer. Math. Soc. 222, 569–605 (2009). 2. F.S. Bates and G.H. Fredrickson. Block Copolymers - Designer Soft Materials. Physics Today, 52-2, 32–38 (1999). 3. X. Chen and Y. Oshita. Periodicity and uniqueness of global minimizers of an energy functional containing a long-range interaction. SIAM J. Math. Anal. 37, 1299–1332 (2005). 4. X. Chen and Y. Oshita. An application of the modular function in nonlocal variational problems. Arch. Ration. Mech. Anal. 186-1, 109–132 (2007). 5. R. Choksi and M. A. Peletier. Small volume fraction limit of the diblock copolymer problem I: sharp interface functional, Preprint (2009). 6. K. Glasner and R. Choksi. Coarsening and Self-organization in Dilute Diblock Copolymer Melts and Mixtures, Physica D, 238, 1241–1255 (2009). 7. J. Escher and Y. Nishiura, Smooth unique solutions for a modified MullinsSekerka model arising in diblock copolymer melts, Hokkaido Math. J., 31-1, 137–149 (2002). 8. P. Fife and D. Hilhorst. The Nishiura-Ohnishi Free Boundary Problem in the 1D case. SIAM J. Math. Anal. 33, 589–606 (2001). 9. I.W. Hamley. The Physics of Block Copolymers. Oxford Science Publications, (1998). 10. M. Henry, D. Hilhorst and Y. Nishiura. Singular limit of a second order nonlocal parabolic equation of conservative type arising in the micro-phase separation of diblock copolymers. Hokkaido Math. J. 32, 561–622 (2003). 11. M. Helmers, B. Niethammer and X. Ren. Evolution in off-critical diblockcopolymer melts. Networks Heterogeneous Media, 3-3, 615–632, (2008). 12. N. Q. Le. On the convergence of the Ohta-Kawasaki evolution equation to motion by Nishiura-Ohnishi law, Preprint, (2009). 13. L. Modica. The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98, 357–383, (1987). 14. B. Niethammer. The LSW model for Ostwald ripening with kinetic undercooling. Proc. Roy. Soc. Edinb., 130A:1337–1361, (2000). 15. B. Niethammer and Y. Oshita. A rigorous derivation of mean-field models for diblock copolymer melts. Calc. Var. and PDE, 39, 273–305, (2010). 16. B. Niethammer and F. Otto. Ostwald Ripening: The screening length revisited. Calc. Var. and PDE, 13-1, 33–68, (2001). 17. B. Niethammer and F. Otto. Domain Coarsening in Thin Films. Comm. Pure Appl. Math., 54-3, 361–384, (2001). 18. B. Niethammer and J. J. L. Vel´ azquez. Well-posedness for an inhomogeneous LSW-model in unbounded domains. Math. Annalen, 328 3, 481-501, (2004). 19. Y. Nishiura and I. Ohnishi. Some Aspects of the Micro-phase Separation in Diblock Copolymers. Physica D, 84, 31–39 (1995). 20. Y. Nishiura and H. Suzuki. Higher dimensional SLEP equation and applications to morphological stability in polymer problems. SIAM J. Math. Anal. 36-3 , 916–966 (2004/05).

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21. I. Ohnishi, Y. Nishiura, M. Imai, and Y. Matsushita. Analytical Solutions Describing the Phase Separation Driven by a Free Energy Functional Containing a Long-range Interaction Term. CHAOS, 9-2, 329–341 (1999). 22. T. Ohta and K. Kawasaki. Equilibrium Morphology of Block Copolymer Melts, Macromolecules 19, 2621–2632 (1986). 23. X. Ren and J. Wei, On the multiplicity of solutions of two nonlocal variational problems, SIAM J. Math. Anal., 31 (2000), 909–924. 24. X. Ren and J. Wei J. Concentrically layered energy equilibria of the di-block copolymer problem. European J. Appl. Math. 13, 479–496 (2002). 25. X. Ren and J. Wei J. On energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5, 193–238 (2003). 26. X. Ren and J. Wei J. On the spectra of three-dimensional lamellar solutions of the Diblock copolymer problem. SIAM J. Math. Anal. 35, 1–32 (2003).

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Global Existence of Classical Solutions to Partially Dissipative Quasilinear Hyperbolic Systems Peng Qu∗ and Cunming Liu School of Mathematical Sciences, Fudan University, Shanghai, 200433, China ∗ E-mail: qupeng [email protected] For 1-D quasilinear hyperbolic systems, the strict dissipation or the weak linear degeneracy can prevent the classical solution from blowing up at least for small initial data. More precisely, if all the inhomogeneous sources are strictly dissipative, or all the characteristics are weakly linearly degenerate and the system is homogeneous, then the Cauchy problem with small and decaying initial data admits a unique global classical solution. In this paper, under some suitable hypotheses on the interaction, new kinds of formulas of wave decomposition are developed to show the global existence of classical solutions to a class of combined systems, in which a part of equations possesses the strict dissipation and the others are weakly linearly degenerate. Keywords: Global classical solution; Quasilinear hyperbolic system; Weak linear degeneracy; Partial dissipation.

1. Introduction We study the classical solution to the Cauchy problem of the following quasilinear hyperbolic system ⎧ ∂u ⎨ ∂u + A(u) = F (u), x ∈ R, t ≥ 0, (1) ∂t ∂x ⎩ (2) t = 0 : u = u0 (x), x ∈ R, where u = (u1 , . . . , un )T is the unknown vector function of (t, x), and A(u) is a given n × n C 3 matrix with n distinct real eigenvalues for small |u|: λ1 (u) < · · · < λn (u),

(3)

which leads to a complete set of left (resp. right) eigenvectors lk (u) = (lk1 (u), . . . , lkn (u)) (k = 1, . . . , n) (resp. rk (u) = (r1k (u), . . . , rnk (u))T (k = 1, . . . , n)) and the C 3 regularity of λk (u), lk (u) and rk (u) (k = 1, . . . , n). Without loss of generality, we assume that lk (u) rk¯ (u) ≡ δkk¯ ,

¯ ≤ n, ∀ |u| small, ∀ 1 ≤ k, k

(4)

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where δkk¯ is the Kronecker’s symbol. Let L(u) = (lkk¯ (u)) and R(u) = (rkk¯ (u)) be the matrices composed by the left and right eigenvectors, respectively. We have L(u)R(u) ≡ I. F (u) is a given C 3 inhomogeneous term with F (0) = 0.

(5)

Assume that the initial data u0 (x) are suitably smooth. In this paper, the classical solution to Cauchy problem (1)–(2) means its C 1 and H 2 solution, whose local existence and uniqueness has been well discussed (for instance, see Ref. 9). Generically speaking, the C 1 solution would form singularity in a finite time even for small and smooth initial data (see Ref. 6), but this can be prevented by suitable dissipative terms. There are mainly two kinds of results on the dissipative systems. The first kind could be called as strict dissipation that all the characteristics are strongly involved in the dissipation. For this kind of systems, Refs. 4,6 and 7 apply the method of characteristics to prove the global existence of C 1 classical solution to the Cauchy problem for strictly dissipative quasilinear hyperbolic systems with small C 1 initial data. The second kind of dissipation could be called as nonstrict dissipation that all the characteristics are involved in the dissipation, some strongly, some weakly. Ref. 7 discusses a kind of nonstrictly dissipative system with the method of characteristics. Ref. 3 applies the energy method to the hyperbolic conservation laws and gets the global H 2 classical solution for small H 2 initial data under Shizuta–Kawashima condition and the dissipative entropy condition. Then Ref. 17 gives a different proof with slightly different hypotheses. The corresponding case of several space variables and the asymptotic behavior are studied by Ref. 13 and Refs. 1,2, respectively. In this kind of system, Shizuta–Kawashima condition ∇F (0)rk (0) = 0,

∀1≤k≤n

(6)

plays an important role. It guarantees that if there is a dissipation for the whole system, then this dissipation could affect all the characteristics. For the homogeneous quasilinear hyperbolic systems with weakly linearly degenerate characteristics, Refs. 6,8,10,16 and 18 give a series of results on global C 1 classical solutions for small and decaying initial data. Besides, Refs. 14 and 15 discuss the global classical solution to 1-D gas dynamics in thermal nonequilibrium. This system is partially dissipative in which all the characteristics, except a linearly degenerate one, are involved in the dissipation by Shizuta–Kawashima condition. By all results mentioned above, it is natural to raise the following conjecture that for a quasilinear hyperbolic system, if a part of the system is involved in the dissipation, while the other part possesses weakly linearly degenerate characteristics, moreover, some suitable conditions are imposed for interactions between these two parts, then the corresponding Cauchy problem should admit a global classical solution for small and decaying smooth initial data. The earliest version of this conjecture occurs in Refs. 13 and 11. Ref. 11 also gives a series of discussions on the systems of this kind.

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We use the index set P ⊆ N = {1, . . . , n}

(7)

to denote the characteristics with strict dissipation, and set the corresponding strictly dissipative condition as −Gii (0) >



|Gpi (0)|,

∀i ∈ P,

(8)

p∈P p =i

or its equivalent forma −Gii (0) >



|Gip (0)|,

∀ i ∈ P,

(9)

p∈P p =i

or the positive definite form −



ξi Gip (0)ξp ≥ σ0

i,p∈P



ξi2 ,

∀ ξi ∈ R,

(10)

i∈P

where G(u) = L(u) ∇F (u) R(u).

(11)

Q = N \ P,

(12)

For the index set no dissipative property is required, while, the corresponding characteristics are assumed to be weakly linearly degenerate (WLD), i.e., λj (u(j) (s)) ≡ λj (0),

∀|s| small, ∀j ∈ Q,

(13)

here and hereafter, u = u(k) (s) denotes the kth characteristic trajectory passing through u = 0: du(k) (s) = rk (u(k) (s)), ds

u(k) (0) = 0

(k ∈ N ).

(14)

Remark 1.1. The WLD is weaker than the linear degeneracy (LD) in the sense of P. D. Lax that ∇λj (u)rj (u) ≡ 0. For more information about the WLD, we refer to Ref. 6. Remark 1.2. In this paper, the indices are used as follows: i, p ∈ P; j, q ∈ Q and k, r, l ∈ N . a The conditions (8) and (9) are equivalent under a transformation of eigenvectors, see Refs. 7 and 6 for more details.

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Finally, the interaction between the inhomogeneous sources F (u) and the nondissipative waves (corresponding to Q) is assumed to be weak, i.e., F (u(j) (s)) ≡ 0,

∀ |s| small, ∀ j ∈ Q.

(15)

Under these assumptions, we have Theorem 1.1. Under hypotheses (3), (5), (8), (13) and (15), there exists a positive number θ0 > 0 so small that for any given θ (0 ≤ θ ≤ θ0 ), Cauchy problem (1)–(2) with initial data u0 (x) satisfying θ = u0 W 1,1 (R) + u0 C 1 (R)

(16)

1

admits a unique global C classical solution u = u(t, x) on t ≥ 0. Theorem 1.2. assume that

Under hypotheses (3), (5), (9), (13) and (15), if we furthermore

lj (u(i) (s))F (u(i) (s)) ≡ 0,

∀ |s| small, ∀ i ∈ P, ∀ j ∈ Q,

(17)

then there exists a positive number θ0 > 0 so small that for any given θ (0 ≤ θ ≤ θ0 ), Cauchy problem (1)–(2) with initial data u0 (x) satisfying  

θ = sup (1 + |x|)1+μ |u0 (x)| + |u0 (x)| (18) x∈R

admits a unique global C 1 classical solution u = u(t, x) on t ≥ 0. Moreover, for this classical solution there exists a funstion T ∈ C 1 (Rn ; Rn ) satisfying det(∇u T )(0) = 0 such that we have a pointwise decay estimate ⎧ ⎨  ∂u 1+μ |Ti (u)| + li (u) (t, x) sup (1 + t) ∂x x∈R ⎩ i∈P ⎫  ⎬  ∂u + (1 + |mj |)1+μ |Tj (u)| + lj (u) (t, x) ≤ Cθ, (19) ⎭ ∂x j∈Q

where mj (t, x) = x − λj (0)t

(j ∈ Q),

(20)

and C is a constant independent of t and θ, but possibly depending on θ0 . Theorem 1.3. Under hypotheses (5), (10), (13) and (15), if we furthermore assume there is only one index in Q, namely, Q = {q0 },

(21)

then there exists a positive number θ0 > 0 so small that for any given θ (0 ≤ θ ≤ θ0 ), Cauchy problem (1)–(2) with initial data u0 (x) satisfying θ = u0 H 2 (R) admits a unique global H 2 classical solution u = u(t, x) on t ≥ 0.

(22)

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2. Sketch of the proof 2.1. Normalized coordinates As in Refs. 6,8 and 10, we introduce normalized coordinates. It is a kind of change of variables in u-space, that all the characteristic trajectories passing through u = 0 are changed into corresponding straight lines in a neighborhood of u = 0 in this new coordinates. So roughly speaking, conditions (13), (15) and (17) can be changed into λj (uj ej ) ≡ λj (0), F (uj ej ) ≡ 0, lj (ui ei )F (ui ei ) ≡ 0,

∀ |uj | small, ∀j ∈ Q,

(23)

∀ |uj | small, ∀ j ∈ Q.

(24)

∀ |ui | small, ∀ i ∈ P, ∀ j ∈ Q.

(25)

And roughly speaking, the other assumptions in Theorem 1.1–Theorem 1.3 keep in a similar form.b Then we only need to prove the following equivalent forms of Theorem 1.1–Theorem 1.3 in normalized coordinates. Theorem 2.1. In normalized coordinates, under hypotheses (3), (5), (8) and (23)–(24), there exists a positive number θ0 > 0 so small that for any given θ (0 ≤ θ ≤ θ0 ), Cauchy problem (1)–(2) with initial data u0 (x) satisfying (16) admits a unique global C 1 classical solution u = u(t, x) on t ≥ 0. Theorem 2.2. In normalized coordinates, under hypotheses (3), (5), (9) and (23)–(24), if we furthermore assume (25), then there exists a positive number θ0 > 0 so small that for any given θ (0 ≤ θ ≤ θ0 ), Cauchy problem (1)–(2) with initial data u0 (x) satisfying (18) admits a unique global C 1 classical solution u = u(t, x) on t ≥ 0. Moreover, for this classical solution there exists a funstion T ∈ C 1 (Rn ; Rn ) satisfying det(∇u T )(0) = 0 such that we have a pointwise decay estimate (19), where C is a constant independent of t and θ, but possibly depending on θ0 . Theorem 2.3. In normalized coordinates, under hypotheses (5), (10) and (23)– (24), if we furthermore assume (21), then there exists a positive number θ0 > 0 so small that for any given θ (0 ≤ θ ≤ θ0 ), Cauchy problem (1)–(2) with initial data u0 (x) satisfying (22) admits a unique global H 2 classical solution u = u(t, x) on t ≥ 0. 2.2. Formulas of wave decomposition As in Refs. 5, 6, 8 and 10, we introduce the formulas of wave decomposition. Setting ∂u wk = lk (u) (k ∈ N ). (26) ∂x The corresponding formulas of wave decomposition are  ∂u ∂uk Bkr (u)wr + Fk (u) (k ∈ N ), (27) + λk (u) k = ∂t ∂x r∈N

b See

Ref. 8 for details.

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  ∂uk ∂ Bkr (u)wr + Ξkr (u)uk wr + Fk (u) + (λ (u)uk ) = ∂t ∂x k r∈N

r∈N

(k ∈ N ),   ∂wk ∂ Γkrl (u)wr wl + Kkr (u)wr + (λ (u)wk ) = ∂t ∂x k

(28)

(k ∈ N ),   ∂wk ∂wk Γkrl (u)wr wl − Ξkr (u)wk wr + λk (u) = ∂t ∂x

(29)

r,l∈N l =r

r∈N

r,l∈N l =r

r∈N



+

(k ∈ N ),

Kkr (u)wr

(30)

r∈N

where Bkr (u) = (λk (u) − λr (u))rkr (u)

(k, r ∈ N ),

(31)

Ξkr (u) = ∇u λk (u) · rr (u)

(k, r ∈ N ).

(32)

Γkrl (u) = (λr (u) − λl (u))lk (u)∇u rl (u)rr (u)

(k, r, l ∈ N ),

(33)

Kkr (u) = Gkr (u) − lk (u)∇u rr (u)F (u)

(k, r ∈ N ).

(34)

Then by applying Hadamard’s formula and Taylor expansion, we can use the special structure of the system to reduce these formulas into   ∂ui ∂ ψirl (u)ul wr + Ξir (u)ui wr + (λ (u)ui ) = ∂t ∂x i r,l∈N l =r

+

r∈N



Φirl (u)ur ul +



Gip (0)up

p∈P

(r,l)∈S

(i ∈ P),   ∂uj ∂ ψjrl (u)ul wr + Ξjr (u)uj wr + (λ (u)uj ) = ∂t ∂x j r,l∈N l =r

r∈N r =j

  + uj ηjr (u) ur wj + r∈N r =j

+



Gjp (0)up

 ∂ui ∂u + λi (u) i = ψirl (u)ul wr + ∂t ∂x r,l∈N l =r



p∈P

Φjrl (u)ur ul

(r,l)∈S

p∈P

+



(35)

Gip (0)up

(j ∈ Q), 

(36)

Φirl (u)ur ul

(r,l)∈S

(i ∈ P),

(37)

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 ∂uj ∂uj ψjrl (u)ul wr + + λj (u) = ∂t ∂x r,l∈N l =r



+



Φjrl (u)ur ul

(r,l)∈S

(j ∈ Q),

Gjp (0)up

(38)

p∈P

 ∂ ∂wi Γirl (u)wr wl + + (λi (u)wi ) = ∂t ∂x r,l∈N l =r



+



ξirl (u)ul wr

(r,l)∈S

(i ∈ P),

Gip (0)wp

(39)

p∈P

 ∂wj ∂ + (λj (u)wj ) = Γjrl (u)wr wl + ∂t ∂x r,l∈N l =r



+



ξjrl (u)ul wr

(r,l)∈S

(j ∈ Q),

Gjp (0)wp

(40)

p∈P

  ∂wi ∂w Γirl (u)wr wl − Ξir (u)wi wr + λi (u) i = ∂t ∂x r,l∈N l =r

+

r∈N



ξirl (u)ul wr +

(i ∈ P),  Γjrl (u)wr wl − Ξjr (u)wj wr

 r,l∈N l =r

−

r∈N r =j

  wj ηjr (u) ur wj +

r∈N r =j

+

Gip (0)wp

p∈P

(r,l)∈S

∂wj ∂wj + λj (u) = ∂t ∂x







(41)

ξjrl (u)ul wr

(r,l)∈S

Gjp (0)wp

(j ∈ Q),

(42)

p∈P

where def.

S = N

2

\ {(j, ¯j) ∈ Q | j = ¯j}

= {(j, k) ∈ Q × N | k = j} ∪ {(i, k) ∈ P × N }. In order to prove Theorem 2.2 we need to add some weights into formulas of wave decomposition to indicate the decay rate of the corresponding solutions. Besides mj (t, x) defined in (20), we need m0 (t, x) = t, m−1 (t, x) = x − αt,

(43) (44)

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where α is a suitably selected real constant. Then for β ∈ (0, 1), we have ∂  ∂ (1 + β|mj |)1+μ uj + λj (u) (1 + β|mj |)1+μ uj ∂t ∂x  = (1 + μ)(1 + β|mj |)μ βsgn(mj ) ϕjr (u)ur uj + (1 + β|mj |)1+μ ·

 

r∈N \{j}

ψjrl (u)ul wr +

r,l∈N l =r



Φjrl (u)ur ul +



Gjp (0)up

p∈P

(r,l)∈S

(j ∈ Q),

(45)

∂ ∂  (1 + β|mj |)1+μ wj + λj (u) (1 + β|mj |)1+μ wj ∂t ∂x  ϕjr (u)ur wj + (1 + β|mj |)1+μ = (1 + μ)(1 + β|mj |)μ βsgn(mj ) ·

r∈N \{j}

 



Γjrl (u)wr wl −

Ξjr (u)wj wr −

r∈N \{j}

r,l∈N l =r

+





(wj ηjr (u))ur wj

r∈N \{j}

ξjrl (u)ul wr +



Gjp (0)wp

(j ∈ Q),

(46)

p∈P

(r,l)∈S

∂ ∂  (1 + β|m0 |)1+μ ui + λi (u) (1 + β|m0 |)1+μ ui ∂t ∂x = (1 + μ)(1 + β|m0 |)μ βui + (1 + β|m0 |)1+μ

    · ψirl (u)ul wr + Φirl (u)ur ul + Gip (0)up r,l∈N l =r

p∈P

(r,l)∈S

(i ∈ P), ∂ ∂  (1 + β|m0 |)1+μ wi + λi (u) (1 + β|m0 |)1+μ wi ∂t ∂x = (1 + μ)(1 + β|m0 |)μ βwi + (1 + β|m0 |)1+μ     · Γirl (u)wr wl − Ξir (u)wi wr + r,l∈N l =r

r∈N

(47)

ξirl (u)ul wr

(r,l)∈S

+

 p∈P

Gip (0)wp

(i ∈ P),

(48)

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∂ ∂  (1 + β|m−1 |)1+μ ui + λi (u) (1 + β|m−1 |)1+μ ui ∂t ∂x    = (1 + μ)(1 + β|m−1 |)μ βsgn(m−1 ) (λi (0) − α)ui + ϕir (u)ur ui + (1 + β|m−1 |)

1+μ

r∈N

 



ψirl (u)ul wr +

r,l∈N l =r

Φirl (u)ur ul

(r,l)∈S

+



(i ∈ P),

Gip (0)up

(49)

p∈P

∂ ∂  (1 + β|m−1 |)1+μ wi + λi (u) (1 + β|m−1 |)1+μ wi ∂t ∂x  = (1 + μ)(1 + β|m−1 |)μ βsgn(m−1 ) (λi (0) − α)wi + + (1 + β|m−1 |)

1+μ



 ϕir (u)ur wi

r∈N

 

Γirl (u)wr wl −

r,l∈N l =r

+





Ξir (u)wi wr

r∈N

ξirl (u)ul wr +



Gip (0)wp

(i ∈ P).

(50)

p∈P

(r,l)∈S

In order to prove Theorem 2.3, we need to compute the formulas of wave decomposition for higher order derivatives of the solution. Set ∂uk ∂x ∂wk zk = ∂x

yk =

We have yk = yi =

 p∈P



(k ∈ N ),

(51)

(k ∈ N ).

(52)

∀k∈N,

rkr (u)wr ,

r∈N

rip (u)wp +



φiqr (u)ur wq ,

∀i∈P

q∈Q r∈N r =q

and ∂ ∂zi + (λi (u)zi ) ∂t ∂x

    ∂Γirl (u)  wr wl Γirl (u)(zr wl + wr zl ) + yh = ∂uh r,l∈N l =r

−

h∈N

    ∂Ξ (u)  Ξir (u)(zi wr + wi zr ) + yh ir wi wr ∂uh

r∈N

h∈N

(53) (54)

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+

ξirl (u)(yl wr + ul zr ) +

 

yh

h∈N

(r,l)∈S



+

∂ξirl (u)  ul wr ∂uh

Gip (0)zp

(i ∈ P),

(55)

p∈P

∂zj ∂ + (λ (u)zj ) ∂t ∂x j

    ∂Γjrl (u)  Γjrl (u)(zr wl + wr zl ) + yh wr wl = ∂uh r,l∈N l =r

h∈N





−

Ξjr (u)(zj wr + wj zr ) +

r∈N \{j}

 

yh

h∈N



   ∂ηjr (u)  wj ηjr (u) (yr wj + 2ur zj ) + wj yh ur wj ∂uh h∈N r∈N \{j} 

   ∂ξjrl (u)  ξjrl (u)(yl wr + ul zr ) + yh + ul wr ∂uh h∈N (r,l)∈S  + Gjp (0)zp (j ∈ Q), (56) 

−



∂Ξjr (u)  wj wr ∂uh

p∈P

∂z ∂zi + λi (u) i ∂t ∂x

    ∂Γ (u)  = Γirl (u)(zr wl + wr zl ) + yh irl wr wl ∂uh r,l∈N l =r

−

h∈N

 

Ξir (u)(2zi wr + wi zr ) +

 

r∈N

 

+

h∈N

ξirl (u)(yl wr + ul zr ) +

  h∈N

(r,l)∈S

+



∂Ξ (u)  yh ir wi wr ∂uh yh

∂ξirl (u)  ul wr ∂uh

Gip (0)zp

(i ∈ P),

p∈P

∂zj ∂zj + λj (u) ∂t ∂x

    ∂Γjrl (u)  Γjrl (u)(zr wl + wr zl ) + yh wr wl = ∂uh r,l∈N l =r

−

 r∈N \{j}

h∈N



  ∂Ξjr (u)  Ξjr (u)(2zj wr + wj zr ) + yh wj wr ∂uh h∈N

(57)

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  ∂ηjr (u)  wj ηjr (u) (yr wj + 3ur zj ) + wj yh ur wj ∂uh h∈N r∈N \{j}

    ∂ξjrl (u)  ξjrl (u)(yl wr + ul zr ) + yh + ul wr ∂uh h∈N (r,l)∈S  Gjp (0)zp (j ∈ Q). (58) + −



 

p∈P

2.3. A priori estimation We use three different methods of a priori estimation in the proofs of Theorem 2.1–Theorem 2.3. • For the proof of Theorem 2.1, we mainly use the continuous Glimm functional methods introduced in Ref. 16. • For the proof of Theorem 2.2, we mainly develop the method of point wise decay estimate introduced in Ref. 18. • For the proof of Theorem 2.3, we mainly use the energy method. We omit all the details here. 3. Further discussions First, differentiating hypothesis (15) with respect to s, we have ∇F (u(j) (s))rj (u(j) (s)) ≡ 0,

∀|s| small, ∀j ∈ Q,

which shows that, generically speaking, the system discussed in this paper does not satisfy the Shizuta–Kawashima condition given in Ref. 12 which plays an important role in Refs. 1–3 and 13. We can also show the necessity of this condition in our theorems, by considering the finite time singularity of the classical solution to the following Cauchy problem ⎧ u = u21 u22 , ⎪ ⎪ ⎪ 1t ⎪ ⎪ ⎪ ⎨ u2t + u2x = −u2 + 1 um , 2 1  1 ⎪ ⎪ ⎪ εe |x|2 −1 , |x| ≤ 1, ⎪ ⎪ u2 = 0, ⎪ t = 0 : u1 = ⎩ 0, |x| ≥ 1, where m is a positive integer and ε > 0 is a constant suitably small. At last, we would like to compare our theorems with other known results on this subject. Comparing with Refs. 4, 6 and 7, we do not suppose that all the characteristics are involved in the strict dissipation, while, comparing with Refs. 2, 3, 11 and 17, we do not require the hypotheses on the structure of conservation laws or on the strictly convex entropy, moreover, generically speaking, our systems do not satisfy the Shizuta–Kawashima condition. On the other hand, in the special case that P = ∅ and Q = N , Theorem 1.2 provides a similar pointwise estimate as in Ref. 18, but under a little bit weaker hypothesis on the inhomogeneous term F (u).

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Acknowledgement The authors would like to thank Prof. Ta-Tsien Li for his patient guidance, precious suggestions and productive discussions. The first author is partially supported by the Fudan University Creative Student Cultivation Program in Key Disciplinary Areas (No. EHH1411208). References 1. K. Beauchard and E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems, Archive for Rational Mechanics and Analysis, 199 (2011), 177-227. 2. S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Communications on Pure and Applied Mathematics, 60 (2007), 1559-1622. 3. B. Hanouzet and R. Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Archive for Rational Mechanics and Analysis, 169 (2003), 89-117. 4. L. Hsiao and T. T. Li, Global smooth solution of Cauchy problems for a class of quasilinear hyperbolic systems, Chinese Annals of Mathematics, 4B (1983), 109-115. 5. F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Communications on Pure and Applied Mathematics, 27 (1974), 377405. 6. T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson/ John Wiley, 1994. 7. T. T. Li and T. Qin, Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms, Chinese Annals of Mathematics, 6B (1985), 199-210. 8. T. T. Li and L. B. Wang, Global Propagation of Regular Nonlinear Hyperbolic Waves, Progress in Nonlinear Differential Equations and Their Applications 76, Birkh¨ auser, 2009. 9. T. T. Li and W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, 1985. 10. T. T. Li, Y. Zhou and D. X. Kong, Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Analysis, Theory, Methods & Applications, 28 (1997), 1299-1332. 11. C. Mascia and R. Natalini, On relaxation hyperbolic systems violating the Shizuta–Kawashima condition, Archive for Rational Mechanics and Analysis, 195 (2010), 729-762. 12. Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to discrete Boltzmann equation, Hokkaido Mathematical Journal, 14 (1985), 249-275. 13. W. A. Yong, Entropy and global existence for hyperbolic balance laws, Arcchive for Rational Mechanics and Analysis, 172 (2004), 247-266. 14. Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Archive for Rational Mechanics and Analysis, 150

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(1999), 225-279. 15. Y. Zeng, Gas flows with several thermal nonequilibrium modes, Archive for Rational Mechanics and Analysis, 196 (2010), 191-225. 16. Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Annals of Mathematics, 25B (2004), 37-56. 17. Y. Zhou, Global classical solutions for partially dissipative quasilinear hyperbolic systems, Chinese Annals of Mathematics, 32B (2011), 771-780. 18. Y. Zhou, Point-wise decay estimate for the global classical solutions to quasilinear hyperbolic systems, Mathematical Methods in the Applied Sciences, 32 (2009), 1669-1680.

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Time averaged properties along unstable periodic orbits of the Kuramoto-Sivashinsky equation Yoshitaka Saiki∗ Department of Mathematics, Hokkaido University, Sapporo 060-0810, JAPAN ∗ E-mail: [email protected] Michio Yamada Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, JAPAN E-mail:[email protected] It has been reported in some dynamical systems in fluid dynamics that only a few UPOs (unstable periodic orbits) with low periods can give good approximations to mean properties of turbulent (chaotic) solutions. By employing the Kuramoto-Sivashinsky equation we compare time averaged properties of a set of UPOs embedded in a chaotic attractor and those of a set of segments of chaotic orbits, and report that the distributions of the time average of a dynamical variable along UPOs with lower and higher periods are similar to each other, and the variance of the distribution is small, in contrast with that along chaotic segments. The result is similar to those for low dimensional ordinary differential equations including Lorenz system, R¨ ossler system and economic system. Keywords: Chaotic Dynamical Systems; Unstable Periodic Orbits; Numerical Computation.

1. Introduction Periodic orbits are important invariant sets in dynamical systems. Especially, UPOs (unstable periodic orbits) are densely embedded in a chaotic invariant set.4 Therefore, UPOs have played an important role in analyzing chaotic dynamical systems. For example, various statistical values in chaotic systems have been estimated with weighted averages of statistical values of many UPOs.3,5,8,14 However, it is usually difficult to apply such methods in studying high dimensional chaotic systems, because detecting many UPOs from high dimensional systems is tough.6 Recently, in some turbulence systems in fluid dynamics, it has been shown that even only a few UPOs with relatively low periods can capture mean properties of chaotic motions.9,10,22 For the turbulent Couette flow of rather low Reynolds number in the full Navier-Stokes system, Kawahara and Kida10 ob-

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tained a remarkable agreement of a velocity correlation and an averaged velocity profile along a single UPO with those along a chaotic orbit in phase space of a turbulent Couette flow. Later van Veen et al.22 performed a numerical study of an isotropic Navier-Stokes turbulence with high symmetry, and found that among several UPOs there is an UPO with relatively low period where the energy dissipation rate appears to converge to a nonzero value in the limit of large Reynolds number as assumed in the Kolmogorov similarity theory. This suggests that the UPO corresponds to the isotropic turbulence of fluid motion, although the Reynolds number is not large enough to discuss the detailed properties of the fully developed turbulence because of computational difficulties. As for the universal statistical properties of fluid turbulence at high Reynolds numbers, employing the GOY shell model, Kato and Yamada9 found a single UPO which gives a fairly good approximation to the scaling exponents of structure functions of velocity, which implies that the intermittency in the model turbulence can be interpreted as a property of a single UPO, rather than a statistical contribution of complex orbits. In the above studies, only a few UPOs with relatively low periods work well to capture some mean properties of a chaotic solution. However, the chaotic attractor includes an infinite number of UPOs, and it may appear that an UPO with longer period gives a better approximation to the statistical properties of chaotic solutions, as a set of long UPOs and a set of chaotic orbits are intuitively taken to have similar statistical properties. So we may have a question why in the above systems even a small number of UPOs with rather low periods can give a remarkably good approximation to the chaotic mean values. Some studies have concerned with this problem.7,11,20,21 Kawasaki and Sasa11 studied a simple model of chaotic dynamical systems with a large degree of freedom, and found that there is an ensemble of UPOs with the special property that the expectation values of macroscopic quantities can be calculated by using a single UPO sampled from the ensemble. Hunt and Ott7 studied an optimal periodic orbit which yields the optimal (extreme) value of a time average of a given smooth performance function of dynamical variables. They obtained an implication that the optimal periodic orbit is typically a periodic orbit of low period, although they do not consider the relation of averaged statistical properties along UPOs and chaotic orbits. On the other hand, Yang et al.23 reported that the optimal UPO can be a periodic orbit of high period when the system is near a crisis. In a study on UPOs of low dimensional map systems by Saiki and Yamada,20 it is reported that UPOs with low periods are not effective to approximate time averaged properties of chaotic orbits. Recently Saiki and Yamada21 employed chaotic systems described by low dimensional ODEs and investigate the relation between the average of a dynamical quantity along an UPO and that along a chaotic orbit, especially with an attention focused on the dependence of the variance of the averaged value on the period of the UPOs. At a first glance, it may appear that if we take all the UPOs with the period around T , for example, and take the averages of a dynamical quantity along these UPOs, the variance of the averages would decrease as T increases, because an extremely long orbit would cover most part of the chaotic attractor,

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capturing possible dynamical states on the attractor. The aim was to see whether this intuitive discussion holds for chaotic systems of ordinary differential equations simple enough to obtain a large number of UPOs by available numerical computation with double accuracy. For this purpose we took three chaotic systems; Lorenz system, R¨ ossler model and a business cycle model. A set of UPOs in each model were obtained numerically, and it was found that for every chaotic system the distributions of a time average of a dynamical variable along UPOs with lower and higher periods are similar to each other and the variance of the distribution is small, in contrast with that along chaotic segments. Here, in this paper, we study Kuramoto-Sivashinsky equation12 as an example of a partial differential equation system and examine time averaged properties along UPOs and along segments of chaotic orbits. 2. Numerical method for detecting UPOs There are various numerical methods for detecting UPOs.13,15,16 We employ the Newton-Raphson-Mees method15 in which the period of the UPO is regarded as a variable to be found in the numerical calculation. We do not make use of the fixed Poincar´e section for the numerical scheme to be applicable for a large number of UPOs irrespective of whether the UPO intersects the Poincar´e section or not. We briefly review the method below. Let us consider a real valued n dimensional ordinary differential equation system dx = F (x), x ∈ Rn . (1) dt We set the orbit passing through x (∈ Rn ) at t = 0 as {φt (x)}t∈R . A periodic orbit is determined by the zeros (x, t) = (X, T ) (T > 0) of H(x, t) := φt (x) − x,

(2)

where X is a point on the periodic orbit and T is the period. The numerical algorithm is as follows. Linearizing y := H(x, t), we obtain Δy = Dx H(x, t)Δx + Dt H(x, t)Δt = {Φt (x) − I}Δx + F (φt (x))Δt,

(3) (4)

where Dx H and Dt H indicate the gradient of H at x and t, respectively. Φt (x) is the n × n matrix, and the gradient of φt (x) at x. I is the n × n unit matrix. Δx and Δt are determined to satisfy H(x, t) + Δy = 0 as the usual Newton-Raphson method, namely {Φt (x) − I}Δx + F (φt (x))Δt = −H(x, t).

(5)

Since this has n constraints with n + 1 unknowns, Δx and Δt are not uniquely decided. Then a constraint is added, that is, Δx should be orthogonal to the orbit, i.e., < F (x), Δx >= 0. (6)

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Thus, under the initial condition (x, t) = (X (i) , T (i) ), we solve the equation about ΔX (i) and ΔT (i) ,    

ΔX (i) ΦT (i) (X (i) ) − I F (φT (i) (X (i) )) X (i) − φT (i) (X (i) ) , (7) = (i) t (i) 0 F (X ) 0 ΔT and renew the point as (X (i+1) , T (i+1) ) = (X (i) + ΔX(i) , T (i) + ΔT (i) ),

(8)

where i is the iteration count. We can detect periodic orbits by iterating the algorithm by Eqs. (7) and (8) from an initial guess (X (0) , T (0) ). We define two sorts of errors appearing in the detection algorithm presented above. One of them is a practical error errprac which is the distance between the initial point X (i) and the terminal point φT (i) (X (i) ). The other error denoted by errcor is the absolute value of the correction vector (ΔX (i) , ΔT (i) ). errprac := |H(X (i) , T (i) )| = |φT (i) (X (i) ) − X (i) |, errcor := |(ΔX

(i)

, ΔT

(i)

)|.

(9) (10)

We consider an unstable periodic orbit to be numerically detected if both of the errors are sufficiently small at the i-th iteration, and X (i) and T (i) are then considered to be a point and the period of the unstable periodic orbit. Remark that even if the above conditions are satisfied, when |X (i) | 1, we must confirm that another relative error at the i-th iteration errrel := |errprac /X (i) |

(11)

is also sufficiently small. 3. Time averaged properties along UPOs UPOs in the Kuramoto-Sivashinsky equation have already been studied by several researches.2,13,17–19,24 Christensen et al.2 showed that cycle expansion theory works in the system with a periodic boundary condition in some set of parameter values. Zoldi and Greenside24 investigated UPOs of the Kuramoto-Sivashinsky equation with a rigid boundary condition, which shows spatio temporal chaotic behaviors. In this paper, we study Kuramoto-Sivashinsky equation with a periodic boundary condition with the same setting as that studied in the previous studies.2,19 The original system ut = −(u2 )x − uxx − νuxxxx

(12)

is written in the Fourier space as b˙k = (k2 − νk4 )bk − ik

∞  m=−∞

bm bk−m

(13)

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with

∞ 

u(x, t) =

bk (t)eikx ,

(14)

k=−∞

where the coefficients bk are in general complex variables. Here, we simplify the system by assuming that bk are pure imaginary, bk = −iak /2, where ak are real and obtain the evolution equation2 k 2

a˙k = (k2 − νk4 )ak −

∞ 

am ak−m .

(15)

m=−∞

We reduce this system to 16 dimensional ODEs and fix ν as 0.02991. The system generates two chaotic attractors which are symmetric to each other. Here we focus our attention to the distribution of time averaged values of a dynamical variable along UPOs of the Kuramoto-Sivashinsky system. We found more than 650 UPOs of the periods from 0.87072 through 12.30608, corresponding from 1 through 14 Poincar´e map periods (PERIODs), respectively. Detected UPOs are classified into three types. UPOs embedded in a chaotic attractor are classified into the first type. UPOs outside a chaotic attractor which mediate an attractor merging crisis at a different parameter value are classified into the second type. Before the merging crisis the stable manifold of the UPO forms the basin boundary of two chaotic attractors and the orbit becomes embedded in the big attractor after the merging crisis.18 Other existing UPOs which are outside a chaotic attractor are classified into the third type.

0.75 0.5

a2

0.25 0 -0.25 -0.5 -0.75 -2 -1.5 -1 -0.5

0 a1

1

1.5

2

0.75

0.5

0.5

0.25

0.25 a2

a2

0.75

0.5

0 -0.25

0 -0.25

-0.5

-0.5

-0.75

-0.75 -2 -1.5 -1 -0.5

0 a1

0.5

1

1.5

2

-2 -1.5 -1 -0.5

0 a1

0.5

1

1.5

2

Fig. 1. Projections of a chaotic attractor (a) and UPOs ((b) T =0.870729, (c) T =6.172071) onto a1 -a2 plane. These UPOs are embedded in the chaotic attractor.

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In this paper we investigate time-averaged properties of UPOs of the first type which are embedded in a chaotic attractor. 576 UPOs among all detected UPOs are classified into the first type. In Fig.1 two examples of the (a1 , a2 ) projections of UPOs of the first type ((b) T = 0.870729, (c) T = 6.172071) are shown in contrast with that of (a) a chaotic attractor. We define the Poincar´e map by the Poincar´e section a1 = 0 with da1 /dt > 0. In our numerical calculation, we identified most of UPOs with PERIOD (period of the Poincar´e map) less than or equal to 12.

Number of UPOs

1000 100 10 1 0.1 2

4

6

8

10

12

N Fig. 2. Number of detected UPOs with PERIOD N of the first type of the KuramotoSivashinsky system which are embedded in a chaotic attractor in comparison with 0.4 · 1.6N , representing the exponential growth of the number of UPOs.

One of the most important indices representing the complexity of a dynamical system is the topological entropy,1 which is estimated by the exponential growth rate of the number of periodic orbits; htop = lim sup log(#{PERIOD-N UPOs})/N, N →∞

(16)

which in this case is estimated to be log(1.6) (Fig. 2). We should remark a clear linear dependence of log{#UPO} on N which suggests that the number of UPOs with PERIOD N detected in our computation is sufficient to study statistical properties of UPOs. We now calculate the time average of a2  T a2  ≡ a2 /T dt (17) t=0

along each UPO with period T . a2 s along UPOs take similar but different values around the average value of a2 s along chaotic segments (-0.06477) (Fig. 3). Then we would like to see the distributions of a2  by classifying UPOs by

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-0.062

-0.063 -0.064 -0.065 -0.066 0 1 2 3 4 5 6 7 8 9 10 11 T T Fig. 3. Time averages a2 s (a2  ≡ t=0 a2 /T dt) along UPOs with period T . UPOs are classified by the PERIODs (period of the Poincar´e map), which are distinguished by the symbols.

0.8 0.7 0.6 Rate

0.5 0.4

UPO(7) UPO(8) UPO(9) UPO(10) UPO(11) UPO(12)

0.3 0.2 0.1 0 -0.07

-0.065

-0.06

T Fig. 4. Distribution of time averages a2 s (a2  ≡ t=0 a2 /T dt) along UPOs with PERIOD N (= 7, · · · , 12), which are similar to each other.

the PERIODs. Fig.4 shows the distributions of a2 s along UPOs for PERIOD N (= 7, 8, · · · , 12), which are normalized so that the sum over all UPOs is unity for each N . We can find that the distributions have similar shapes, indicating that even longer UPO is not necessarily suitable for evaluation of a2 averaged along a long chaotic orbit. This may be contrary to the expectation that an UPO with longer period would give better approximations to statistical properties of chaotic orbits. The property can be seen clearly by comparing standard deviations of a2 

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0.1

UPO Chaos 0.02 N-1.05

STD

0.01

0.001

1e-04 0

2

4

6 N

8

10

12

Fig. 5. Standard deviation of the distribution of a2 s along UPOs with PERIOD N (+), and that along 105 chaotic segments with the time lengths T (= 0.8774 · N )(×), with 0.02N −1.05 (line) for comparison.

0.8 0.7

UPO(10) Chaos(10)

0.6 Rate

0.5 0.4 0.3 0.2 0.1 0 -0.08

-0.07

-0.06

-0.05

Fig. 6. Distributions of a2 s along UPOs with PERIOD 10 (a2  = −0.06442) (average period=8.7625) in comparison with that along 105 chaotic segments with the time-length T (= 0.8774 · 10) (a2  = −0.06477). Time averages of a2  along UPOs are localized around the mean value along a chaotic orbit.

of UPOs and segments of chaotic orbits with similar time lengths. Actually in Fig. 5 the standard deviations of the density distribution of a2 s along UPOs with PERIOD N are nearly constant as N increases. The figure also shows the standard deviations of a2 s along segments of chaotic orbits with time length T = N ·

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0.8774, where 0.8774 stands for the corresponding recurrent time to the Poincar´e section. We see that as N increases, the latter standard deviation decreases nearly as N −1.05 . The difference between the distributions of time averages along UPOs and along segments of chaotic orbits is clearly observed in Fig. 6. 4. Summary We have investigated time averages of dynamical variables along UPOs in the Kuramoto-Sivashinsky equation. By using more than 650 UPOs detected from the system we found that time averaged properties along the set of UPOs and a set of segments of chaotic orbits with finite lengths are significantly different from each other. In addition, a longer UPO is not necessarily advantageous compared to a shorter UPO to estimate mean properties of the chaotic state as long as our numerical computations are concerned. The result is similar to those obtained for the case of ODEs (the Lorenz system, the R¨ ossler system and a 6-dimensional business cycle model).21 It implies that we can employ a short UPO for the estimation of the mean properties of the chaotic state without significant reduction of plausibility. It has been found in some fluid dynamical systems that only a few UPO with low periods give fairly good approximations to some statistical properties.9,10 Our result about the Kuramoto-Sivashinsky equation suggests that the estimation by using a short UPO is as reliable (or unreliable) as that by using a long UPO. 5. Acknowledgments This work is supported partly by Grant-in-Aid for Young Scientists (B)(23740065), Collaborative Research Program for Young Scientists of ACCMS and IIMC, Kyoto University, and Hokkaido University. References 1. R. Bowen, Topological entropy and AxiomA, Global Analysis (Proc. Sympos. Pure Math., AMS, Providence, RI) 14, 1970 23. 2. F. Christiansen, P. Cvitanovi´c and V. Putkaradze, Nonlinearity 10, 55 (1997). 3. P. Cvitanovi´c, Phys. Rev. Lett. 61, 2729 (1988). 4. R. Devaney, An introduction to Chaotic Dynamical Systems (Addison-Wesley, 1986). 5. C. Grebogi, E. Ott, J. A. Yorke, Phys. Rev. A 37, 1711 (1988). 6. A. E. Hramov and A. A. Koronovskii, Europhys. Lett. 80, 10001 (2007). 7. B. Hunt and E. Ott, Phys. Rev. Lett. 76, 2254 (1996): B. Hunt and E. Ott, Phys. Rev. E 54, 328 (1996). 8. T. Kai and K. Tomita, Prog. Theor. Phys. 64, 1532 (1980). 9. S. Kato and M. Yamada, Phys. Rev. E 68, 25302 (2003). 10. G. Kawahara and S. Kida, J. Fluid Mech. 449, 291 (2001). 11. M. Kawasaki and S. Sasa, Phys. Rev. E 72, 37202 (2005). 12. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence Springer Series in Synergetics, Vol. 19 (Springer-Verlag, Berlin, 1984).

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13. Y. Lan and P. Cvitanovi´c, Phys. Rev. E 69, 016217 (2004): Y. Lan and P. Cvitanovi´c, Phys. Rev. E 78, 026208 (2008). 14. Y.-C. Lai, Y. Nagai and C. Grebogi, Phys. Rev. Lett. 79, 649 (1997): Y.-C. Lai, Phys. Rev. E 56, 6531 (1997). 15. T. S. Parker, L. O. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, 1989). 16. A. Pyragas, Phys. Lett. A 170 421 (1992). 17. E. L. Rempel, A. C.-L. Chian, E. E. N. Macau and R. R. Rosa, Chaos 14, 545 (2004). 18. E. L. Rempel, A. C.-L. Chian , A. J. Preto and S. Stephany, Nonl. Proc. in Geophys. 11, 691 (2004). 19. E. L. Rempel and A. C.-L. Chian, Phys. Rev. E 71, 016203 (2005). 20. Y. Saiki and M. Yamada, Nonl. Proc. in Geophys. 15, 675 (2008). 21. Y. Saiki and M.Yamada, Phys. Rev. E 79, 015201 (2009): Y. Saiki and M. Yamada, Phys. Rev. E 81, 018202 (2010): M. A. Zaks and D. S. Goldobin, Phys. Rev. E 81, 018201 (2010). 22. L. van Veen , S. Kida and G. Kawahara, Fluid Dyn. Res. 38, 19 (2006). 23. T.-H. Yang, B. Hunt and E. Ott, Phys. Rev. E 62, 1950 (2000). 24. S. M. Zoldi and H. S. Greenside, Phys. Rev. E 57, 2511 (1998).

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Anomalous enstrophy dissipation via the self-similar triple collapse of the Euler-α point vortices Takashi Sakajo Department of mathematics, Hokkaido University, Sapporo, Hokkaido 060-0810, JAPAN E-mail: [email protected]

Keywords: Euler-α equations, α-point vortices, dissipative weak solutions, selfsimilar collapse, enstrophy dissipation

1. Introduction One of the characteristic properties of 2D turbulence is the emergence of the inertial range in the energy density spectrum of the flow corresponding to the forward cascade of enstrophy, which is the L2 norm of the vorticity, for sufficiently small viscosity. It is pointed out that the emergence of the enstrophy cascade is due to the enstrophy dissipation in the inviscid limit of the incompressible flow field. However, the enstrophy dissipation never occurs for smooth solutions of the 2D Euler equations, since the enstrophy is an invariant quantity for the 2D Euler flow. Hence, it is necessary to consider non-smooth solutions of the 2D Euler equations to obtain the enstrophy dissipation, which is referred to as the anomalous enstrophy dissipation. In order to obtain such non-smooth solutions of the 2D Euler with the anomalous enstrophy dissipation, we consider weak solutions that develop a singularity. Let us first remark that not all weak solutions of the Euler equations dissipate the enstrophy. The enstrophy is conserved by weak solutions for the bounded initial vorticity data.22 Eyink6 showed that weak solutions for the initial data of the vorticity in Lp (R2 ) ∩ L1 (R2 ), for which the existence of a global weak solution has been established by Diperna and Majda,4 could not dissipate the enstrophy in a weak sense. This means that we need to deal with the initial vorticity field in weaker function spaces such as the Radon measure on R2 in order to investigate how non-smooth 2D Euler flows dissipate the enstrophy. In the present article, we consider the initial vorticity field consisting of discrete δ functions that belongs to the space of Radon measure on R2 , which is called the point-vortex initial data. Naively speaking, it is impossible to construct a weak solution of the 2D Euler equations for the point-vortex initial data, since the velocity field induced by the point vortices does not belong to L2loc (R2 ),

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for which a weak solution of the Euler equations exists.3 In order to avoid this drawback, we consider the Euler-α equations for the incompressible velocity field u(t, x) in the two-dimensional space x ∈ R2 and at time t ∈ R, which are given by (1 − α2 Δ)∂t u + u · ∇(1 − α2 Δ)u − α2 (∇u)T · Δu = −∇p, ∇ · u = 0,

u(x, 0) = u0 (x).

The Euler-α equations are one of the regularized Euler equations, since they are derived by taking the spatial average of the Euler flow below the small scale α.14 The Euler-α equations is originally derived by Holm et al.7,8 as a physically relevant model of 3D turbulence and they have been investigated by many researchers. The 2D Euler-α equations and its viscous regularization the NavierStokes-α equations are also regarded as a physically relevant models of turbulent flows, since numerical investigation of the Navier-Stokes-α equations by Lunasin et al.13 show that the inertial ranges appear in the energy density spectrum corresponding to the backward energy cascade and the forward enstrophy cascade for scales larger than the filtering level α. Mathematically, Oliver and Shkoller17 have established the global existence of a unique weak solution of the 2D Euler-α equations for the initial vorticity data in the space of Radon measure on R2 in the sense of distribution. Hence, one can obtain a global weak solution of the 2D Euler-α equations for the point-vortex initial data, which is described as the motion of many point vortices, called the α-point vortex (αPV) system. Thus, by taking the α → 0 limit of the global weak solution, we are able to construct a weak solution of the 2D Euler equations in terms of the vortex dynamics. Let us now derive the equation of α-point vortex system from the Euler-α equations. Introducing the scalar α-vorticity q = (1 − α2 Δ)∇⊥ · u and taking the curl of the Euler-α equations, we have the equations for q as follows. qt + (u · ∇)q = 0,

u = K α ∗ q,

q(x, 0) = (1 − α2 Δ)∇⊥ · u0 (x).

(1)

1 ∇⊥ log |x| and Gα (x) is the Green The kernel is given by K α (x) = Gα (x) ∗ 2π function associated with the Helmholtz operator (1 − α2 Δ), i.e.,

 |x| 1 α , K0 G (x) = − 2πα2 α

in which K0 (x) is a modified Bessel function of the second kind.20 In the meantime, it has been shown that there exist two kinds of singular selfsimilar evolutions of three point vortices for a certain special condition.1,10,16,18 One is the self-similar triple collapse, in which three point vortices collide selfsimilarly at a point in finite time, and the other is called the self-similar expansion, in which three point vortices emerge from a point and then expand self-similarly to the infinity for t → ∞. Thus, if the same special condition is considered in the αPV system, we expect that the evolution of the three α-point vortices converges to a singular evolution that is close to the singular self-similar orbits in the limit of α → 0. Indeed, Sakajo19 has confirmed that an evolution of the three α-point vortices converges to the self-similar triple collapse for t < 0 and the self-similar

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triple expansion for t > 0, with which the temporal variations of the energy dissipation rate and the enstrophy along the singular orbit are investigated. As a result, it is shown that the energy dissipation rate is always zero and the enstrophy dissipation occurs via the self-similar triple collapse in the sense of distribution. Moreover, it is numerically shown that the convergence to the singular self-similar orbit with the enstrophy dissipation is still observed even if the special condition is perturbed. Thus the anomalous enstrophy dissipation via the self-similar triple collapse is a robust mechanism in the sense that it is observed for a certain range of the parameter region. In §2 and §3.1, we summarize the results presented in the paper by Sakajo,19 which is referred to as S hereafter. In §3.3, we also show some new additional materials to complement the results on the robustness of the anomalous enstrophy dissipation via the self-similar triple collapse given in S. 2. The α-point vortex system Suppose that the α-vorticity field q(t, x) is represented by N discrete δ-functions, say α-point vortices, with the strength Γm whose singularities are located at xm (t) = (xm (t), ym (t)) ∈ R2 for m = 1, . . . , N . q(t, x) =

N 

Γm δ(x − xm (t)).

(2)

m=1

The explicit representation of the kernel K α (x) in (1) is provided by Holm et al.9 as follows.  

 |x| 1 . K α (x) = − ∇⊥ log |x| + K0 2π α2 Substituting (2) into u = K α ∗ q and evaluating the velocity field at x = xm (t), we derive the equations for the α-point vortices.

 N 1  ym − yn dxm lmn =− , Γn B K 2 dt 2π α lmn

(3)

 N dym 1  xm − xn lmn Γn B = , K 2 dt 2π α lmn

(4)

n =m

n =m

where lmn = |xm − xn | and BK (x) = 1 − xK1 (x). The function K1 (x) is a modified Bessel function of the second kind.20 We call the differential equations (3) and (4) the α-point-vortex (αPV) system. Since BK (x) → 1 due to K1 (x) → e−x as x → ∞, the αPV system is reduced to the point-vortex system for α → 0.18 On the other hand, we have BK (x) → 0 for x → 0, since K1 (x) ∼ 1/x as x → 0. Thus the velocity field vanishes when the distance between two α-point vortices tends to zero, which means that the velocity field is regularized thanks to the function BK (lmn /α). Following the same idea by Novikov,15 who provided us with an explicit representation of the kinetic energy along the evolution of the point vortices, we

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obtain the kinetic energy E (α) (t) and the enstrophy Z (α) (t) that vary with the evolution of α-point vortices,

    N N lmn (t) lmn (t) lmn (t) 1   + , E (α) (t) = − Γm Γn log lmn (t) + K0 K1 2π α 2α α m=1 n=m+1

Z

(α)

N 1  (t) = 2 4πα

N 

m=1 n=m+1

lmn (t) K1 α



lmn (t) α

.

whose derivation can be found in S.19 In order to investigate the αPV system and take the limit of α → 0, we introduce the scaled variables X m (t) = (Xm (t), Ym (t)) and Lmn (t) = |X m (t) − X n (t)| as follows. Xm (t) =

1 xm (α2 t), α

Ym (t) =

1 ym (α2 t), α

Lmn (t) =

1 lmn (α2 t). α

(5)

Then, we have the equations for Xm (t) and Ym (t): N dXm 1  Ym − Yn =− Γn BK (Lmn ), dt 2π L2mn n =m

N 1  Xm − Xn dYm = Γn BK (Lmn ). dt 2π L2mn n =m

(6) The canonical system is written as a Hamiltonian dynamical system. dXm dYm = {Xm , H} , = {Ym , H} , dt dt in which the Poisson bracket between two functions f and g is defined by 

N  ∂g ∂f 1 ∂f ∂g {f, g} = − , Γm ∂xm ∂ym ∂xm ∂ym m=1

and the Hamiltonian H = H0 (t) + H1 (t) is described by H0 (t) = −

N 1  2π

N 

N 1  2π

N 

Γm Γn log Lmn (t),

(7)

Γm Γn K0 (Lmn (t)) .

(8)

m=1 n=m+1

H1 (t) = −

m=1 n=m+1

The solution of the αPV system is recovered from that of the canonical system with the following relations. xm (t) = αXm (t/α2 ),

ym (t) = αYm (t/α2 ),

lmn (t) = αLmn (t/α2 ),

(9)

which will be used to take the α → 0 limit of the motion of α-point vortices. Let us define the linear impulse L ∈ C and the angular momentum I ∈ R by L = Q + iP =

N  m=1

Γm (xm + iym ),

I=

N  m=1

2 Γm (x2m + ym ).

(10)

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Since we have {P, H} = {Q, H} = {I, H} = 0, they are invariant quantities. N Moreover, we can easily obtain {P 2 + Q2 , H} = 0, {P, Q} = m=1 Γm ≡ Γ, {Q, I} = P and {P, I} = Q. Hence, since the invariant quantities P 2 + Q2 , I, H are in involution, the canonical system for N ≤ 3 is integrable for all values of the vortex strengths. For N = 4, it is integrable only if the total amount of the vortex strengths Γ is zero. The integrability for the αPV system is the same as that for the point vortex system.18 3. The three α-vortex problem 3.1. Self-similar triple collapse with enstrophy dissipation We consider the canonical equations (6) for N = 3 with the following conditions. 1 1 1 + + = 0, Γ1 Γ2 Γ3

(11)

M = Γ1 Γ2 L212 + Γ2 Γ3 L223 + Γ3 Γ1 L231 = 0.

(12)

Let us remark here that M is a conserved quantity due to M = ΓI − Q2 − P 2 and these conditions are necessary to construct the self-similar triple collapse and expansion in the point vortex system.10,18 The canonical equations (6) for N = 3 are rewritten in terms of the distances Lmn between the three α-point vortices as follows.   2 1 1 d 2 B (L ) − B (L ) , (13) L12 = Γ3 A 23 31 K K dt π L223 L231   d 2 2 1 1 B (L ) − B (L ) , (14) L23 = Γ1 A 31 12 K K dt π L231 L212   d 2 2 1 1 B (L ) − B (L ) , (15) L31 = Γ2 A 12 23 K K dt π L212 L223 where A represents the signed area of the triangle formed by the three α-point vortices. This is computed from the distances with Heron’s formula, 1 (L12 + L23 + L31 ), 2

(16)

 1 1  2 2 2 ≥ 0. 2 L12 L23 + L223 L231 + L231 L212 − L412 − L423 − L431 4

(17)

A2 = r(r − L12 )(r − L23 )(r − L31 ),

r=

and its magnitude |A| is given by |A| =

The evolution of the signed area is obtained by differentiating (16) and substituting (13) – (15) into it.   dA 1  −2 −2 Γ1 L−1 = 23 L31 BK (L31 ) − L12 BK (L12 ) dt 4π    (r − L31 )(r − L12 ) · (r − L23 )(r − L31 )(r − L12 ) + r

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  r  −2 −2 Γ1 L−1 23 L31 BK (L31 ) − L12 BK (L12 ) (r − L31 )(r − L12 ),(18) 2π  in which denotes the summation over cyclic permutation of indices. The equations (13) – (15) and (18) define a four-dimensional dynamical system. Owing to the invariance of M , we can further reduce the dynamical system to a twodimensional one for the variables (L23 , L31 ). Since the Hamiltonian H is also invariant, we can describe the evolution of the three α-point vortices by plotting the contour lines of the Hamiltonian in the two-dimensional phase space. In what follows, we summarize some properties of the dynamical system, whose proofs can also be found in S.19 First, the condition |A| ≥ 0 should be satisfied so that the three α-point vortices form the triangle, which is described in terms of (L23 , L31 ) as follows. −

Proposition 3.1. Let L223 /L231 = k. Then there exist two reals k1 and k2 with 0 < k1 < 1 < k2 such that |A| ≥ 0 is equivalent to k1 ≤ k ≤ k2 . Proposition 3.1 shows that the contour lines of the Hamiltonian are plotted in the wedge-shaped region k1 ≤ L223 /L231 ≤ k2 in the (L23 , L31 )-plane, which we call the physical region. Next, the equilibria of the equations (13)–(15) and (18) are given as follows. Proposition 3.2. The equilibria for the canonical system (13)–(15) and (18) with the conditions (11) and (12) are equilateral triangles. Following the book of Newton,18 we introduce the following variables. b1 =

L223 , Γ1

b2 =

L231 , Γ2

b3 =

L212 . Γ3

Then the condition (12) is equivalent to b1 + b2 + b3 = 0. On use of (11), the Hamiltonian H = H0 + H1 is described in terms of the new variables.   Γ1 Γ2 Γ3 Γ1 b1 1 Γ2 b2 1 log + log , 4π Γ1 Γ3 b3 Γ2 Γ3 b3             Γ1 Γ2 Γ3 2  2 K0 Γ1 b1 − K0 Γ3 b3 + K0 Γ2 b2 − K0 Γ3 b3 . H1 = − 4π Γ1 Γ2

H0 = −

Figure 1(a) and (b) show the contour plots of the Hamiltonian for (Γ1 , Γ2 ) = (1, 1) and (3, 1) respectively. The contour line connecting the origin is the straight L2

1 b1 = Γ = 1 corresponding to the stationary similar equilateral triangles line L23 2 Γ2 b2 31 given in Proposition 3.2. The other contour lines start from the boundary of the physical region and extend to the infinity. Let us remark that a point in the physical region represents two configurations of the three α-point vortices corresponding to (L23 , L31 , L12 , A) and (L23 , L31 , L12 , −A), in which the shapes of the triangle are the same but their signs of the area are opposite. This means that a point moving along a contour line of the Hamiltonian in the physical region reach its boundary and then the point moves back in the opposite direction on the same contour line. Thus if we choose the collinear configuration as the initial

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(b) b2

(a) b2

0

b1

0

b1

Fig. 1. Contour plots of the Hamiltonian in the (b1 , b2 ) phase space. The vortex strengths are (a)(Γ1 , Γ2 ) = (1, 1) and (b)(Γ1 , Γ2 ) = (3, 1).

data, which is always possible without loss of generality owing to the structure of the contour lines of the Hamiltonian, the evolution of the three α-point vortices is symmetric with respect to time, whose exact statement is given as follows. Proposition 3.3. For the canonical system (6) for N = 3 with (11) and (12), if A(0) = 0 then we have Lmn (t) = Lmn (−t) and A(t) = −A(−t) for t ≥ 0. Since the contour lines are unbounded, we need to observe the behaviour of the contour lines of the Hamiltonian as b1 → ∞ in order to understand the asymptotic behaviour of the evolution for t → ∞. Proposition 3.4. As b1 → ∞, the contour line of the Hamiltonian (8) asymptotically approaches a straight line bb12 = const. in the physical region. This proposition shows that the contour line tends to a straight line bb12 = (const.) as b1 , b2 → ∞, which means that the configuration of the three α-point vortices converges asymptotically to a self-similar triangle for t → ±∞. According to  |t| for Kimura,10 the distances between the three point vortices behave like the self-similar evolution of the point vortices. Besides, the canonical system gets closer to the equation of the point vortices as t → ∞, since BK (Lmn ) → 1 as Lmn → ∞. Accordingly, the distance Lmn (t) behaves asymptotically as  (19) Lmn (t) −→ C∞ |t| for t → ±∞, for a certain constant C∞ > 0. Figure 2(a) and (b) show the evolutions of the distances L12 (t), L23 (t) and L31 (t) for (Γ1 , Γ2 ) = (1, 1) and (3, 1) respectively. The initial configuration is collinear with M = 0 and L12 (0) + L23 (0) = L31 (0) = 1.

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They clearly support the asymptotic behaviour of the distances (19). Then, owing to (9), we have the convergence of the α-point vortices for t = 0 as follows. 

lmn (t) = αLmn



t α2

−→ C∞ α

 |t| = C∞ |t|, α

α → 0.

Consequently, in the limit of α → 0, the evolution of the three α-point vortices converges to the self-similar collapse for t < 0 and to the self-similar expansion for t > 0. 3

10

3

2

1

10

100

0

10

1

10

2

3

10 10 Time

10

4

5

10

L12 L23 L31 t-0.5

1

10

0

10 10

10-1 -1 10

(b)

10 Distances

102 Distances

10

L12 L23 L31 t-0.5

(a)

-1

10-2 10-3 10-2 10-1 100 101 102 103 104 105 Time

Fig. 2. Log-log Plot of the distances L12 (t), L23 (t) and L31 (t). The vortex strengths are given by (a)(Γ1 , Γ2 ) = (1, 1) and (b)(Γ1 , Γ2 ) = (3, 1) respectively.

Now, we investigate how the energy dissipation rate and the enstrophy evolve along the singular self-similar orbit. First, on use of (9) and (11), the energy E (α) (τ ) is represented by E (α) (τ ) = H −

3 3  τ    τ  1   Γm Γn Lmn K L . 1 mn 4π m=1 n=m+1 α2 α2 (α)

By differentiating it, we have the energy dissipation rate, denoted by DE (τ ), as follows. (α)

3 3  τ    τ  dLmn  τ  1   Γm Γn Lmn K0 Lmn 2 2 2 4πα m=1 n=m+1 dτ α α α2 1  τ  ≡ 2F , α α2

DE (τ ) =

in which the function F (τ ) is defined by F (τ ) =

3 3 1   dLmn Γm Γn (τ )Lmn (τ )K0 (Lmn (τ )). 4π m=1 n=m+1 dτ

(20)

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The function F (τ ) is odd, i.e., F (τ ) = −F (−τ ), due to Proposition 3.3. See also Figure 3(a). Hence, for any compactly supported function ϕ(τ ),  ∞    ∞ 1  τ  (α) DE , ϕ = F ϕ(τ )dτ = F (s)ϕ(α2 s)ds 2 α2 −∞ α −∞  ∞ F (s)ds = 0, −→ ϕ(0) −∞

as α → 0. Hence we have the following theorem. Theorem 3.1. For the singular self-similar orbit defined as the α → 0 limit of (α) the three α point vortices, the energy dissipation rate DE converges to 0 in the sense of distribution, namely (α)

lim DE = 0

α→0

in

D .

Owing to (5) and (9), the enstrophy variation is rewritten as 3 3  τ    τ  1   Γm Γn Lmn K1 Lmn 2 2 4πα m=1 n=m+1 α α2   1 τ ≡ − 2 HK . α α2 Note that the function HK (τ ) is even due to Proposition 3.3. In order to observe the function HK (τ ), we solve the canonical equations (6) numerically for the collinear initial data M = 0 and L12 (0) + L23 (0) = L31 (0) = 1. Figure 3(b) shows the plots of the function HK (τ ) along the evolution of the three α-point vortices for Γ1 = 1, 3, 6, 9 and Γ2 = 1, which indicate that HK (τ ) is positive, rapidly decreasing as t → ±∞ and thus it is integrable on R. Hence, we have the convergence of the enstrophy variation as α tends to zero in the sense of distribution. Namely, for any compactly supported smooth function ϕ(τ ), we have  ∞    τ  1 Z (α) , ϕ = − HK ϕ(τ )dτ 2 α2 −∞ α  ∞ HK (s)ϕ(α2 s)ds −→ −z0 ϕ(0), =−

Z (α) (τ ) =

−∞

as α → 0, where



∞

z0 =

HK (s)ds > 0. −∞

Thus, the enstrophy variation tends to Dirac’s δ measure with a negative mass −z0 , which is restated in terms of the total variation of the enstrophy,  T Z (α) (τ )dτ −→ −z0 H(T ), −∞

in which H represents the Heaviside function. The total variation of the enstrophy is zero until the collapsing time T = 0 and it abruptly drops to −z0 beyond the critical time, which means that the enstrophy dissipation occurs via the selfsimilar triple collapse in the α → 0 limit of the αPV system.

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0.012

Γ1=1 Γ1=3 Γ1=6 Γ1=9

(b)

0.01 HK(τ)

F(τ)

0.014

Γ1=1 Γ1=3 Γ1=6 Γ1=9

(a)

0.008 0.006 0.004 0.002

0

0 -3000 -2000 -1000

1000 2000 3000

Time

0

1000 2000 3000

Time

Fig. 3. Plots of (a) the function F (τ ) and (b) the function HK (τ ). They are computed from the numerical solution for Γ1 = 1, 3, 6, 9 and Γ2 = 1.

3.2. Robustness of the enstrophy dissipation via the self-similar triple collapse In order to discuss a continuity of the convergence to the self-similar triple collapse with the enstrophy dissipation under the perturbation of the conditions (11) and (12), we investigate the evolution of the three α-point vortices subject to 1 1 1 + + = 0, Γ1 Γ2 Γ3

M ≡ Γ1 Γ2 L12 + Γ2 Γ3 L23 + Γ3 Γ1 L31 = εM = 0,

(21)

in which εM is a non-zero small constant. We here observe the contour plot of the Hamiltonian with the help of the standard analysis used for the integrable point vortex system in order to see the evolution from a global point of view. With the same analytical technique in the book of Newton,18 we introduce the

3

b3

b2 √3/2

-√3/2

b1

Fig. 4.

Trilinear coordinates

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new trilinear coordinates as follows. b1 =

L223 ,  Γ1 M

b2 =

L231 ,  Γ2 M

b3 =

L212 ,  Γ3 M

M = in which M . Then the second condition in (21) is equivalent to b1 + 3Γ1 Γ2 Γ3 b2 + b3 = 3. With using the trilinear coordinate system, a configuration of the three α-point vortices is represented by a point (x0 , y0 ) in the two-dimensional (x, y)-phase √ space, whose distances from the three sides of the equilateral triangle with length 3 are given by b1 , b2 and b3 as we see in Figure 4. The point (x0 , y0 ) are computed explicitly from (b1 , b2 ) as follows. 

√ 1 2 3 − √ b 1 − √ b2 , b1 . (x0 , y0 ) = 3 3

Now we assume that Γ1 ≥ Γ2 > 0 without loss of generality. Then Γ3 should be negative due to (21). The configuration of the three α-point vortices satisfies |A| ≥ 0. In terms of the trilinear coordinate, it is equivalent to (Γ1 b1 )2 + (Γ2 b2 )2 + (Γ3 b3 )2 ≤ 2(Γ1 Γ2 b1 b2 + Γ2 Γ3 b2 b3 + Γ3 Γ1 b3 b1 ), which we call the physical region again. A collinear configuration corresponds to a point at the boundary |A| = 0, which is a hyperbola. We plot the contour lines of the Hamiltonian of the canonical system, which is represented in terms of the trilinear coordinates with (21) by   Γ1 Γ2 Γ3 1 Γ1 b 1 1 Γ2 b 2 H0 = − log + log , 4π Γ1 Γ3 b 3 Γ2 Γ3 b 3  





 Γ1 Γ2 Γ3 2 b1 + 2 K0 b2 + 2 K0 b3 H1 = − K0 Γ1 M Γ2 M Γ3 M . 4π Γ1 Γ2 Γ3 Figure 5(a) and (b) show the contour plots of the Hamiltonian for εM = 0.03 when the vortex strengths are given by (Γ1 , Γ2 ) = (1, 1) and (3, 1) respectively. The contour lines of the Hamiltonian drawn in the middle of the phase space are unbounded. Thus we observe the behaviour of the contour lines as b1 → −∞, since b1 , b2 < 0 and b3 > 0 due to εM > 0. Suppose first that b2 converges to a finite constant b, then b3 → ∞ due to b1 + b2 + b3 = 3. While the second term H1 remains bounded, the first term H0 , which is denoted by b1 and b2 by   Γ1 Γ 2 Γ 3 1 Γ 1 + Γ2 (Γ1 + Γ2 )b2 /b1 1 H0 = − log log , + 4π Γ1 Γ2 (1 + b2 /b1 − 3/b1 ) Γ2 Γ1 (1 + b2 /b1 − 3/b1 ) due to 1/Γ1 + 1/Γ2 + 1/Γ3 = 0 and b1 + b2 + b3 = 3, diverges as b1 → −∞ and b2 → b. This contradicts to the conservation of the Hamiltonian. Next, suppose that b2 /b1 → ∞ as b1 → −∞, then we have the divergence of the first term in a similar manner. Thus the ratio b2 /b1 converges to a non-zero constant. Since Γ2 b2 Γ1 b1

=

L2 31 L2 23

= const., the evolution acquires the self-similarity, which means that  the distances between the three α-point vortices behave asymptotically as |t|

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(a)

(b)

y

y

x

x

Fig. 5. Contour plots of the Hamiltonian in the trilinear phase space for εM = 0.03. The vortex strengths are (a)(Γ1 , Γ2 ) = (1, 1) and (b)(Γ1 , Γ2 ) = (3, 1).

(a)

(b)

y

y

x

x

Fig. 6. Contour plots of the Hamiltonian in the (b1 , b2 ) phase space for εM = −0.03. The vortex strengths are (a)(Γ1 , Γ2 ) = (1, 1) and (b)(Γ1 , Γ2 ) = (3, 1).

for t → ±∞ in a similar way as in §3.1. Thus we also obtain the convergence to the singular self-similar orbit. Figure 6(a) and (b) show the contour plots of the Hamiltonian for εM = −0.03 when the vortex strengths are given by (Γ1 , Γ2 ) = (1, 1) and (3, 1) respectively. In this case, the contour lines are bounded and connect two points at the boundary of the physical region. As explained in § 3.1, the contour lines in the trilinear phase space can be viewed as double sided, the three α-point vortices evolve periodically along the contour line and thus the evolution of the thee α-point vortices is periodic and the distances between them are bounded for all time, which yields that there exists a constant CL such that |lmn (t)| = αLmn αt2 ≤ CL |α| → 0

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as α tends to zero. Therefore three α-point vortices converge to a point in the limit of α → 0, which is not the convergence to the self-similar orbit.

3

2

Distances

10

(a)

L12(t) L23(t) L31(t) -0.5 t

1

10

0

10

10-1 10-2 -3 -2 -1 0 1 2 3 4 5 10 10 10 10 10 10 10 10 10 Time

1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -600

L12(t) L23(t) L31(t)

(b)

Distances

10

-400

-200

0

200

400

600

Time

Fig. 7. (a) Log-log plot of the evolutions of the distances L12 (t), L23 (t) and L31 (t) for εM = 0.03. (b) The evolutions of the distances for εM = −0.03. The vortex strength is Γ1 = 3 and Γ2 = 1 and the initial configurations corresponds to the red circle in the contour lines of Hamiltonian in Figure 5(b) and in Figure 6(b) respectively.

Numerical evolutions of the three α-point vortices with the strengths Γ1 = Γ2 = 1 for an isosceles initial data satisfying M = εM and L23 (0) = L31 = 1 were presented in S,19 which shows that the evolution for εM = 0.03 converges to the singular self-similar evolution with the enstrophy dissipation and the periodic orbit for εM = −0.03 tends to a point as α → 0. These evolutions correspond to the unbounded contour line drawn as the gray lines in the middle of the physical region for εM = 0.03 in Figure 5(a) and for εM = −0.03 in Figure 6(a) respectively. Let us finally show some additional numerical examples for the other vortex strengths, Γ1 = 3 and Γ2 = 1. The evolutions of the distances L12 (t), L23 (t) and L31 (t) for εM = 0.03 and εM = −0.03 are shown in Figure 7(a) and (b) respectively. We consider the isosceles initial configuration with M = εM and L23 (0) = L31 (0) = 1, which are drawn as the filled circles in the contour plots of the Hamiltonian in Figure 5(b) and in Figure 6(b) respectively. These figures  also show that the distances behave asymptotically as |t| for εM = 0.03 and they are periodic for εM = −0.03, which support that the convergence to the selfsimilar triple collapse occurs for εM = 0.03 and that to a point for εM = −0.03 again. Let us note that we can still observe the convergence to the singular selfsimilar orbit for εM > 0 when we consider the same isosceles initial configuration. Hence, we plot the functions F (τ ) and HK (τ ) along the numerical evolutions of the three α-point vortices for εM = 0.03, 0.06 and 0.09 in Figure 8(a) and (b) respectively, which indicate that the energy dissipation rate converges to zero and the enstrophy dissipation does to the δ-measure with a negative mass in the sense of distribution in a similar manner as we discussed in §3.1.

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0.0016

εM=0.03 εM=0.06 εM=0.09

(a)

2e-06

0.0014 0.0012

1e-06

0.001 HK(τ)

F(τ)

εM=0.03 εM=0.06 εM=0.09

(b)

0

0.0008

-1e-06

0.0006

-2e-06

0.0004

-3e-06

0.0002

-4e-06 -4000 -2000

0 Time

2000

4000

0 -4000 -2000

0

2000

4000

Time

Fig. 8. Plots of (a) the function F (τ ) and (b) the function HK (τ ). They are computed from the numerical solution for εM = 0.03, 0.06 and 0.09. The vortex strength is Γ1 = 3 and Γ1 = 1 and the initial isosceles configuration is considered.

References 1. Aref, H. 1979 Motion of three vortices. Phys. Fluids 22(3), 393–400. 2. Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids Supppl. II 12, 233–239. 3. Delroit, J. -M. 1991 Existence de nappe de tourbillion en dimension deux. J. Amer. Math. Soc. 4, 553-586. 4. DiPerna, R. J. & Majda, A. J. 1987 Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. 40, 301–345. 5. Duchon, J. & Robert, R. 2000 Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13, 249–255. 6. Eyink, G. L. 2001 Dissipation in turbulence solutions of 2D Euler equations. Nonlinearity 14, 787–802. 7. Holm, D. D., Marsden, J. E. and Ratiu T. S. 1998 Euler-Poincar´e models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 80(19), 4173–4176. 8. Holm, D. D. 2002 Variational principles for Lagrangian-averaged fluid dynamics. J. Phys. A 35, 679–688. 9. Holm, D. D. Nitsche, M. & Putkaradze V. 2006 Euler-alpha and vortex blob regularization of vortex filament and vortex sheet motion. J. Fluid Mech. 555, 149–176. 10. Kimura, Y. 1987 Similarity solution of two-dimensional point vortices. J. Phys. Soc. Japan 56, 2024-2030. 11. Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423. 12. Leith, C. E. 1968 Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11, 671–672. 13. Lunasin, E., Kurien, S., Taylor, M. A. and Titi, E. S. 2007 A study of the Navier-Stokes-α model for two-dimensional turbulence. J. Turbulence 8, 1–21. (doi:10.1080/14685240701439403) 14. Marsden, J. E. & Shkoller, S. 2003 The anisotropic Lagrangian averaged

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Euler and Navier-Stokes equations. Arch. Ration. Mech. Anal. 166, 27–46. 15. Novikov, E. A. 1976 Dynamics and statistics of a system of vortices. Sov. Phys. JETP 41, 937–943. 16. Novikov, E. A. & Sedov, Y. B. 1979 Vortex collapse. Sov. Phys. JETP 50(2), 297–301. 17. Oliver, S. & Shkoller, S. 2001 The vortex blob method as a second-grade non-Newtonian fluid. Comm. Partial Diff. Eq. 26, 295–314. 18. Newton, P. K. 2001 The N -vortex problem, Analytical techniques. SpringerVerlag. 19. Sakajo, T. 2012 Instantaneous energy and enstrophy variations in Euleralpha point vortices via the triple collapse. J. Fluid Mech, accepted. 20. Watson, G. N. 2008 A treatise on the theory of Bessel functions. Merchant Books. 21. Wolibner, W. 1933 Un theor´eme sur l’existence du mouvement plan d’un fluide parfait, homoge´ene, incompressible, pendant un temps infiniment long. Math. Z. 37, 698–726. 22. Yudovich, V. I. 1963 Nonstationary motion of an ideal incompressible liquid. USSR Comp. Math. Phys. 3, 1407–1456.

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Action Minimizing Periodic Solutions in the N -Body Problem M. Shibayama Mathematical Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, JAPAN E-mail: [email protected] http://www.sigmath.es.osaka-u.ac.jp/˜shibayama/ Using the variational method Chenciner and Montgomery proved the existence of a new periodic solution of figure-eight shape to the planar three-body problem. Since then, number of periodic solutions have been found as minimizers of variational formulation of the n-body problem in various different settings. In this paper we survey the theory and present some numerical algorithm for finding periodic solutions by based on the theory. Keywords: N -body problem, variational method, choreography

1. Introduction This paper is concerned with the Newtonian N -body problem which is given by the following set of ODEs: q¨i = −



mj

j=i

qi − qj , |qi − qj |3

q1 , . . . , qN ∈ V,

(1)

where mj > 0 and V = R2 or R3 . Using the variational method Chenciner and Montgomery1 proved the existence of a new periodic solution of figure-eight shape to the planar three-body problem (Fig. 1). As an important property, three

0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

−1

−0.8

−0.6

Fig. 1.

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure-eight solution

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particles chase each other along one curve. Sim´ o named a periodic solution, whose particles chase each other along one curve, a choreography and numerically found many choreographies.5,6 In this paper we survey the theory and then present a numerical algorithm for finding choreographies. Next section we formulate a variational structure with a symmetry for the n-body problem, and introduce some known results. In Section 3, those results will be used to show the existence of the figure-eight solution. In Section 4, we consider a simple and interesting symmetry “choreographic symmetry”. In Section 5, we describe a numerical algorithm for obtaining choreographies. In the last section, we give the numerical results. 2. Variational formulation and symmetric constraint The N -body problem is equivalent to the variational problem with respect to the action functional  T L(q, q)dt ˙ A(q) = 0

where the function L is the Lagrangian L(q, q) ˙ = Let X be defined by

N  mi mj 1 mk |q˙k |2 + . 2 k=1 |qi − qj | i