The Large Flux Problem to the Navier-Stokes Equations: Global Strong Solutions in Cylindrical Domains [1st ed. 2019] 978-3-030-32329-5, 978-3-030-32330-1

Table of contents :
Front Matter ....Pages i-vi
Introduction (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 1-9
Notation and Auxiliary Results (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 11-30
Energy Estimate: Global Weak Solutions (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 31-57
Local Estimates for Regular Solutions (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 59-82
Global Estimates for Solutions to Problem on (v, p) (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 83-93
Global Estimates for Solutions to Problem on (h, q) (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 95-97
Estimates for ht (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 99-105
Auxiliary Results: Estimates for (v, p) (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 107-115
Auxiliary Results: Estimates for (h, q) (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 117-128
The Neumann Problem (3.6) in L2-Weighted Spaces (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 129-141
The Neumann Problem (3.6) in Lp-Weighted Spaces (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 143-155
Existence of Solutions (v, p) and (h, q) (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 157-168
Back Matter ....Pages 169-179

Citation preview

Lecture Notes in Mathematical Fluid Mechanics

Joanna Rencławowicz Wojciech M. Zaja¸czkowski

The Large Flux Problem to the Navier-Stokes Equations Global Strong Solutions in Cylindrical Domains

Advances in Mathematical Fluid Mechanics Lecture Notes in Mathematical Fluid Mechanics

Editor-in-Chief Giovanni P Galdi University of Pittsburgh, Pittsburgh, PA, USA

Series Editors Didier Bresch Universit´e Savoie-Mont Blanc, Le Bourget du Lac, France Volker John Weierstrass Institute, Berlin, Germany Matthias Hieber Technische Universit¨at Darmstadt, Darmstadt, Germany Igor Kukavica University of Southern California, Los Angles, CA, USA James Robinson University of Warwick, Coventry, UK Yoshihiro Shibata Waseda University, Tokyo, Japan

Lecture Notes in Mathematical Fluid Mechanics as a subseries of “Advances in Mathematical Fluid Mechanics” is a forum for the publication of high quality monothematic work as well lectures on a new field or presentations of a new angle on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations and other significant viscous and inviscid fluid models. In particular, mathematical aspects of computational methods and of applications to science and engineering are welcome as an important part of the theory as well as works in related areas of mathematics that have a direct bearing on fluid mechanics.

More information about this subseries at http://www.springer.com/series/15480

Joanna Rencławowicz Wojciech M. Zajaczkowski ˛

The Large Flux Problem to the Navier-Stokes Equations Global Strong Solutions in Cylindrical Domains

Joanna Rencławowicz Institute of Mathematics Polish Academy of Sciences Warsaw, Poland

Wojciech M. Zajaczkowski ˛ Institute of Mathematics Polish Academy of Sciences Warsaw, Poland Institute of Mathematics and Cryptology Military University of Technology Warsaw, Poland

ISSN 2297-0320 ISSN 2297-0339 (electronic) Advances in Mathematical Fluid Mechanics ISSN 2510-1374 ISSN 2510-1382 (electronic) Lecture Notes in Mathematical Fluid Mechanics ISBN 978-3-030-32329-5 ISBN 978-3-030-32330-1 (eBook) https://doi.org/10.1007/978-3-030-32330-1 Mathematics Subject Classification: 35Q30, 76D03, 76D05, 35A01, 35B65, 35B45, 35D30, 35D35, 35G61 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkh¨auser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Notation and Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Spaces and Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Auxiliary Problems, Results, and Green Functions . . . . . . . . . . 2.3 Interpolation, Imbeddings, Trace Theorems, and the Korn Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 17

3

Energy Estimate: Global Weak Solutions . . . . . . . . . . . . . . . . . . . . . 3.1 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Estimates for vt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 34 47

4

Local Estimates for Regular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 A Priori Estimates for Function h = v,x3 . . . . . . . . . . . . . . . . . . . . . 4.2 A Priori Estimates for Vorticity Component χ . . . . . . . . . . . . . . . 4.3 Relating v and h: rot–div System . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 64 72 75

5

Global Estimates for Solutions to Problem on (v, p) . . . . . . .

83

6

Global Estimates for Solutions to Problem on (h, q) . . . . . . .

95

7

Estimates for ht . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

8

Auxiliary Results: Estimates for (v, p). . . . . . . . . . . . . . . . . . . . . . . . . 107

9

Auxiliary Results: Estimates for (h, q) . . . . . . . . . . . . . . . . . . . . . . . . 117

10

The Neumann Problem (3.6) in L2 -Weighted Spaces . . . . . . 129

11

The Neumann Problem (3.6) in Lp -Weighted Spaces . . . . . . 143

12

Existence of Solutions (v, p) and (h, q) . . . . . . . . . . . . . . . . . . . . . . . . 157 12.1 Existence of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 12.2 Existence of Regular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

24

v

vi

Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Chapter 1

Introduction

Abstract This chapter is an introduction, where we describe the problem and define some key parameters to formulate main theorems: Theorems 1.1 and 1.2. We consider incompressible Navier-Stokes equations in cylindrical domain Ω with large inflow and outflow on the bottom and the top of the cylinder and the slip boundary conditions on the lateral part of the boundary. In order to prove the global existence of (v, p), where v is a velocity and p a pressure, with arbitrary large flux d, we show first the existence of weak solution, next we find conditions guaranteeing regularity of weak solution for large time T , and finally we achieve the existence of global regular solutions. In Theorem 1.1 we conclude that for sufficiently small parameter Λ2 (T ), which depends on data: norms of flux derivatives tangent to the bottom and the top of the cylinder, the initial condition h(0) = v,x3 (0), the derivative with respect to x3 of force and with the estimate for weak solutions denoted with A, we have v, h ∈ W22,1 (Ω T ), ∇p, ∇q = ∇p,x3 ∈ L2 (Ω T ). In Theorem 1.2, for sufficiently large time T and from decay property of the equations we prove the existence of solutions in the interval [kT, (k + 1)T ], k ∈ N0 , therefore we can extend solutions step by step.

In this book, the motion of incompressible fluid described by the NavierStokes equations with large inflow and outflow is considered. The aim is to prove the existence of global regular solutions with arbitrary large flux. The domain is a straight non-axially symmetric cylinder with arbitrary crosssection. We assume the slip boundary conditions on the lateral part of the boundary, whereas on the bottom and the top of the cylinder there are some inflow and outflow fluxes. There is no restriction on the magnitude of the flux and we admit arbitrary large L2 norm of initial velocity, but some homogeneity of the flux is necessary. Moreover, the initial velocity does not change too much along the axis of cylinder and the inflow does not change much along the directions perpendicular to this axis either with respect to time.

© Springer Nature Switzerland AG 2019 J. Renclawowicz, W. M. Zajączkowski, The Large Flux Problem to the

Navier-Stokes Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-32330-1 1

1

2

1

Introduction

Fig. 1.1 Domain Ω

Physically, the problem can describe a motion of blood through straight parts of arteries. It seems that slip boundary conditions are more appropriate than nonslip because the blood can slip on the boundary. The model is the first step for further analysis of inflow-outflow problems and next, flows around some obstacles, with large velocities. This can also be the starting point to describe blood motion because walls of arteries are in fact elastic, so the problem with free boundary seems to be more natural: this will be the next step. We consider the following initial boundary value problem to the NavierStokes equations in a cylindrical domain Ω with boundary ∂Ω = S = S1 ∪ S2 (Fig. 1.1) and with inflow and outflow vt + v · ∇v − div T(v, p) = f

in Ω T = Ω × (0, T ),

div v = 0

in Ω T ,

v·n ¯=0

on S1T = S1 × (0, T ),

νn ¯ · D(v) · τ¯α + γv · τ¯α = 0, α = 1, 2

on S1T ,

(1.1)

on S2T = S2 × (0, T ),

v·n ¯=d n ¯ · D(v) · τ¯α = 0, α = 1, 2 v|t=0 = v(0)

on S2T , in Ω,

where the stress tensor has the form T(v, p) = νD(v) − pI. By ν > 0 we denote the constant viscosity coefficient, γ > 0 is the slip coefficient, n ¯ is the unit outward vector normal to S, τα , α = 1, 2, are vectors tangent to S, I is the unit matrix, and D(v) is the dilatation tensor of the form

1

Introduction

3

D(v) = {vi,xj + vj,xi }i,j=1,2,3 . We set S = S1 ∪ S2 , where S1 is parallel to the x3 axis and S2 is perpendicular to it. Hence S1 = {x ∈ R3 : ϕ0 (x1 , x2 ) = c0 , −a < x3 < a}, S2 (−a) = {x ∈ R3 : ϕ0 (x1 , x2 ) < c0 , x3 = −a},

(1.2)

S2 (a) = {x ∈ R : ϕ0 (x1 , x2 ) < c0 , x3 = a}, 3

where a, c0 are given positive numbers and ϕ0 (x1 , x2 ) = c0 describes a sufficiently smooth closed curve in the plane x3 = const. To describe the inflow and outflow we define d1 = −v · n ¯ |S2 (−a) , d2 = v · n ¯ |S2 (a) ,

(1.3)

with di ≥ 0, i = 1, 2. Equation (1.1)2 implies the following compatibility condition   d1 dS2 = d2 dS2 . (1.4) S2 (−a)

S2 (a)

The goal of this book is to prove the existence of global regular solutions to problem (1.1)–(1.4) with arbitrary large flux d. In order to demonstrate such results we are going to proceed in three main steps: first, we show the existence of weak solution, next we find the conditions guaranteeing regularity of weak solution for large time, and finally we achieve the existence of global regular solutions. This book consists of 12 chapters. In Chap. 2 notation and auxiliary results are introduced. In Chap. 3 the energy type estimate for weak solutions to problem (1.1)– (1.4) is derived. The main result is formulated in Proposition 3.4. Since (1.1)– (1.4) is an inflow-outflow problem the normal component of velocity on S2 does not vanish (see (1.1)5 ). This makes the direct derivation of energy type estimate for solutions to (1.1)–(1.4) impossible. To make it possible we introduce the Hopf function (3.2) which help us to transform the problem (1.1)–(1.4) into the new one, with homogeneous Dirichlet boundary conditions. Namely, we construct some extension δ such that w = v − δ (see (3.7)) satisfies that w · n ¯ |S = 0. Since w is a solution to problem (3.8) the energy type inequality (3.10) can be derived by integration by parts. Therefore, in Lemma 3.2, we are able to obtain the energy type estimate for solutions of problem (3.8). However, to obtain an energy estimate we need some estimates in weighted Sobolev spaces for the extension δ

4

1

Introduction

(see (3.12), (3.13), and (3.14)). In fact we need estimates in these weighted spaces for solutions to the auxiliary Neumann problem for the Poisson equation (3.6). The existence of solutions to (3.6) in L2 -weighted spaces is shown in Chap. 10 and in Lp -weighted spaces in Chap. 11. The weight is equal to a power function of the distance to S2 which is not a Muckenhoupt weight. In reality, in Proposition 3.4 we prove global estimate using a step by step in time approach. It is shown that vV (kT,t) , t ∈ [kT, (k+1)T ], k ∈ N0 , where the norm V (kT, t) is defined in (2.3), is bounded by a constant depending on data but independent of k. We have to emphasize that this estimate is proved in detail and that we derive a priori estimates only, while the existence is shown briefly. The existence of weak solutions is then proved by the Faedo-Galerkin method in Chap. 12, Theorems 12.1 and 12.2 and in detail in Renclawowicz and Zaj¸aczkowski [RZ3] (see also arguments by the Leray-Schauder fixed point theorem in [RZ4, Z5]). The step by step in time approach implies less restriction on data because only finite time integrals appear in the bound. In Chap. 3 there is also derived an a priori bound for vt V (kT,t) , t ∈ [kT, (k + 1)T ]. The bound depends on v2,2,Ω×(kT,(k+1)T ) (with the notation for Sobolev spaces and norms introduced in (2.2)) which is estimated in Chap. 4. In Chap. 4 we find an estimate guaranteeing regularity of weak solutions for large finite time interval. In order to get this, we transform problem (1.1) into a series of problems. Then assuming smallness of the following data: v,x3 (0), f,x3 (0), d,x , where x = (x1 , x2 , x3 ) are the Cartesian coordinates such that x3 is along the cylinder and x = (x1 , x2 ), such estimate can be shown. Since the idea of the proof is little complicated we sketch its steps. We introduce quantities h = v,x3 , q = p,x3 , g = f,x3 which satisfy problem (4.6). In virtue of Lemmas 4.3, 4.4, and Corollary 4.5 we achieve the estimate hV (Ω t ) ≤ ϕ(D1 (t), V(t))(Λ1 (t) + |h(0)|2,Ω ),

(1.5)

where ϕ is an increasing positive function, and norms notation is given in (2.1), (2.2), and (2.3), D1 (t) = |d1 |3,6,S2t , V(t) = |∇v|3,2,Ω t (see (4.40)),  T (dx 21,S2 + dt 21,S2 + |f3 |24/3,S2 + |g|26/5,Ω )dt Λ21 (T ) = 0

+A2 sup dx 21,3/2,S2 t

 v2V (Ω t )

t

≤ ϕ(sup d1,3,S2 ) t

0

(see (4.38)), (1.6)

(|f |26/5,Ω + d21,3,S2 + dt 21,6/5,S2 )dt +|v(0)|22,Ω ≡ A2 (t)

(see (4.13)).

1

Introduction

5

Next, in order to increase regularity of the first two components of velocity, i.e., v  = (v1 , v2 ), we introduce the vorticity χ(x, t) = v2,x1 − v1,x2 satisfying problem (4.9) (see Lemma 4.2). We are allowed to use vorticity since Ω is a cylindrical domain parallel to the x3 -axis. In Corollary 4.8 we obtain the inequality (4.54) χ2V (Ω t ) ≤ c(1 + A2 )(|h3 |23,∞,Ω t + |F3 |26/5,2,Ω t +v  2W 1,1/2 (Ω t ) + v  25/6,2,∞,Ω t + |χ(0)|22,Ω ). 2

The above inequality can be attained because slip boundary conditions are used. Then, from Lemmas 4.9 and 4.10 and sufficiently small Λ2 = Λ1 +|h(0)|2,Ω (see (4.61)) we derive the inequality vW 2,1 (Ω T ) ≤ ϕ(A)H2 + ϕ(D), 5/3

where D describes all data norms, A = A(t) is the estimate for weak solution given in (4.13), and H2 = H1 + |h|10/3,Ω t , H1 = sup |h|3,Ω + h1,2,Ω t . t

Using the estimate for the Stokes system (4.63), Lemma 4.12 implies vW 2,1 (Ω t ) + |∇p|2,Ω t ≤ ϕ(D, H2 ). 2

(1.7)

Applying the Stokes system connected with problem (4.6) and smallness of Λ2 (T ) = Λ1 (T ) + |h(0)|2,Ω , Lemma 4.13 implies the estimate hWσ2,1 (Ω t ) + ∇qLσ (Ω t ) ≤ ϕ(D),

10 5 ≤σ≤ . 3 3

(1.8)

Moreover, the imbedding H2 ≤ chWσ2,1 (Ω t ) , for σ ≥

5 3

holds. Therefore, combining this with (1.7) and (1.8) we conclude vW 2,1 (Ω t ) + |∇p|2,Ω t ≤ ϕ(D). 2

The above considerations in Chap. 4 imply bounds for the quantities vW 2,1 (Ω T ) , 2

hW 2,1 (Ω T ) , 2

(1.9)

6

1

Introduction

for sufficiently small Λ2 (T ), where T can be chosen as large as we want (see definition of Λ1 in (1.6)). This means that in Chap. 4 long-time estimates are proved. Applying the Leray-Schauder fixed point theorem we prove long-time existence of regular solutions to (1.1)–(1.4) and (4.6). For application of the Leray-Schauder fixed theorem see Chap. 12 and [RZ4], [Z5]. Estimates (1.8) and (1.9) hold for any t ∈ R+ . On the other hand, D depends on time integrals of some quantities, so for large t they must decay sufficiently fast. Considering estimates (1.8) and (1.9) in time interval (0, T ) we show, in Chaps. 5–8, the existence of T sufficiently large so that (1.13) hold, i.e. v((k + 1)T )1,Ω ≤ v(kT )1,Ω , h((k + 1)T )1,Ω ≤ h(kT )1,Ω ,

for any k ∈ N0 .

These inequalities imply the existence of global regular solutions to problem (1.1) for t ∈ R+ . Hence, we have the following. Theorem 1.1 Consider problems (1.1)–(1.4) for v and (4.6) for h = v,x3 , where the domain Ω is non-axially symmetric. Let time T > 0 be given and denote q = p,x3 , F3 = f2,x1 − f1,x2 , and g = f,x3 . Assume that Λ2 (T ) = Λ1 (T ) + |h(0)|2,Ω , where Λ1 (T ) is given by (1.6), is sufficiently small. Additionally, we require that: • external forces satisfy f, g ∈ L2 (Ω T ) and F3 ∈ L6/5,2 (Ω T ), • initial velocity satisfies v(0), h(0) ∈ H 1 (Ω), 3/2,3/4

• flux satisfies d ∈ L∞ (0, T ; W31 (S2 )) ∩ W2

dt ∈

1 L2 (0, T ; W6/5 (S2 )).

3/2,3/4

(S2T ), dx ∈ W2

(S2T ),

Then there exists a solution such that v, h ∈ W22,1 (Ω T ), ∇p, ∇q ∈ L2 (Ω T ) and the following estimate holds vW 2,1 (Ω T ) + hW 2,1 (Ω T ) + |∇p|2,Ω T + |∇q|2,Ω T 2 2  ≤ ϕ |f |2,Ω T , |g|2,Ω T , |F3 |6/5,2,Ω T , v(0)1,Ω , h(0)1,Ω ,

(1.10)  1 (S )) , dx  sup d1,3,S2 , dW 3/2,3/4 (S T ) , dt L2 (0,T ;W6/5 , 3/2,3/4 2 W (S T )

t≤T

2

2

2

2

where ϕ is an increasing positive function and x = (x1 , x2 ). In Chap. 5 we prove the global existence of solutions to problem (1.1)– (1.4), (4.6) by applying the step by step in time argument. From Lemma 5.2 we have (see (5.13)) v(kT )1,Ω ≤ Q1 (T ) + exp(−νkT )v(0)1,Ω

(1.11)

1

Introduction

7

and Lemma 5.4 implies (see (5.14)) h(kT )1,Ω ≤ Q2 (T ) + exp(−νkT )h(0)1,Ω ,

(1.12)

where Q1 (T ), Q2 (T ) are equal to the first terms on the r.h.s. of (5.13) and (5.14), respectively. Inequalities (1.11) and (1.12) follow from the considerations from Chaps. 6 to 9. From (1.11), (1.12) for T sufficiently large and some v(0)1,Ω , h(0)1,Ω we obtain v((k + 1)T )1,Ω ≤ v(kT )1,Ω ,

(1.13)

h((k + 1)T )1,Ω ≤ h(kT )1,Ω . Therefore we have the following.

Theorem 1.2 Let the assumptions of Theorem 1.1 hold. Then for T sufficiently large and conditions implying (1.13) we have the existence of solutions to problems (1.1)–(1.4) and (4.6) in the interval [kT, (k + 1)T ], k ∈ N0 such that, with Ω × (kT, (k + 1)T ) = Ω kT , S2 × (kT, (k + 1)T ) = S2kT , vW 2,1 (Ω kT ) + hW 2,1 (Ω kT ) + |∇p|2,Ω kT + |∇q|2,Ω kT ≤ ϕ(D(kT, (k + 1)T )), 2

2

where  D(kT, (k + 1)T ) ≡ |f |2,Ω kT , |g|2,Ω kT , |F3 |6/5,2,Ω kT , v(0)1,Ω , h(0)1,Ω , sup kT ≤t≤(k+1)T

d(t)1,3,S2 , dW 3/2,3/4 (S kT ) , 2

2

 1 (S )) , dx  dt L2 (kT,(k+1)T ;W6/5 . 3/2,3/4 2 W (S kT ) 2

2

The aim of this monograph is to prove the existence of global regular solutions to the Navier-Stokes equations in a bounded straight cylinder with large inflow and outflow. Our approach is strictly connected with the structure of the Navier-Stokes system and is motivated by the classical paper of Ladyzhenskaya [L1]. Using ideas of [L1], W.M. Zaj¸aczkowski came to the conclusion that stability of 2d solutions was possible. Hence the global regular solutions close to 2d solutions are proved in [Z5, Z6, Z7, Z8]. To show the existence of such solutions we need geometrical restrictions such that the considered domain must be a straight cylinder. We need also the slip boundary conditions on the part of the boundary because with such conditions a problem for 2d rotation can be applied (see also [Z5, Z6]).

8

1

Introduction

The case with inflow-outflow was considered in Renclawowicz and Zaj¸aczkowski papers [RZ3, RZ4, RZ5, RZ6] and in case of Poiseuille flow in [PZ]. In [RZ3] the existence of weak solutions to problem (1.1) is proved. The proof bases on the following steps. 1. Appropriate extension of the nonhomogeneous boundary conditions on S2 . 2. A global estimate for solutions to (1.1) is proved by using some weighted estimates. 3. Once we have the estimate, the existence of weak solutions is proved by the Faedo-Galerkin method. Under the smallness conditions presented in Theorem 1.1, the long-time existence of solutions to problem (1.1) was proved in [RZ4]. In [RZ5, RZ6] the global existence of solutions to problem (1.1) formulated in Theorem 1.2 was proved. However, we can clarify some aspects of the proof presented there. The global existence of solutions to problem (1.1) without inflow and outflow was shown in [Z5, RZ7] under smallness of the following quantity Λ(T ) = |f3 |24/3,S T + |g|26/5,Ω T + |h(0)|22,Ω . 2

We mention also papers [SZ1, SZ2, SZ3] where problem (1.1) without inflow and outflow but coupled with the heat convection is considered. There is a huge literature concerning on the existence and regularity of solutions to the Navier-Stokes equations, see [S, Le, G2]. In [S], Sohr developed a very advanced and beautiful theory of semigroup with applications to the Navier-Stokes equations. Unfortunately, such theory does not feel the structure of the Navier-Stokes system, so in reality it can be applied to prove only either local existence or global existence with small data. In [Le], Lemari´e-Rieusset presented a lot of spaces (Besov, Lorentz, Morrey and Sobolev) and a lot of different techniques and approaches applied to the Navier-Stokes equations. He also mentioned the idea of semigroup. Nevertheless, the chapter corresponding to the global existence of regular solutions is very short (see Ch. 33). The book is indeed a vast and deep review of different techniques, spaces, and methods which can be used in the problems of Navier-Stokes equations but it is not adequate in our case. The new book of Lemarie-Rieusset, [Le2], is some extension of the previous one. The author recalls properties of mild solutions to Navier-Stokes equations in H s , s ≥ 1/2, Lp , p ≥ 3 and Besov, Morrey spaces. Moreover, the author describes the regularity criterion of Caffarelli-Kohn-Nirenberg and L3 -criterion of Escaurioza-Seregin-Sverak on suitable weak solutions. However, this book does not contain any results on the existence of global regular solutions close to either axisymmetric or two-dimensional solutions. The new book of Robinson et al. [RRS] is focused on the regularity problem for solutions to the Navier-Stokes equations. The authors distinguish the

1

Introduction

9

following topics: the existence of global-in-time weak solutions of Leray-Hopf problem, the local regularity theory originated from the Serrin criterion, and partial regularity developed by Scheffer, Caffarelli, Kohn, and Nirenberg. The book does not contain any results on the existence of global special regular solutions (to compare, see [Z6, Z11, Z12]). Since the existence of regular weak solutions to the Navier-Stokes equations is still an open problem we can distinguish three kinds of possibilities of showing the existence of global regular non-small solutions. 1. The existence of global regular solutions close to the two-dimensional solutions proved by Ladyzhenskaya in [L2]. Such solutions are shown in [RZ7], [Z5]. 2. In [Z11, Z12] the existence of global regular solutions is established, which are close to the axially symmetric solutions with vanishing angular coordinate of velocity. The existence of corresponding axially symmetric solutions was proved by Ladyzhenskaya [L3] (see also [UY]). 3. The existence of regular global solutions to the Navier-Stokes equations describing fast rotating fluids (see papers of Babin et al.: [BMN1, BMN2, BMN3], Mahalov and Nicolaenko [MN], and Farwig et al.: [FST]). To prove the existence of global regular solutions we need a method that is not sensitive to nonlinearity, and therefore the most appropriate approach is the energy method and L2 -type Sobolev spaces. Although it seems oldfashioned, it is the proper one. Motivated by the results of Chap. 8 from the book of Galdi [G2], where the inflow-outflow problem for stationary Navier-Stokes is considered, we started to consider the nonstationary problem in the cylinder. We have to mention that Takeshita counter example, see [T], does not work in our case.

Chapter 2

Notation and Auxiliary Results

Abstract In this chapter we introduce the basic notation for classical norms (Lebesque spaces Lp and Sobolev spaces Wps and H s ) and define some more complicated spaces and norms: • anisotropic Sobolev spaces on Ω and QT for Ω, Q ⊂ R3 : Wpk1 ,p2 (Ω), l,l/2

• • • • •

Wr (QT ), l,l/2 anisotropic Sobolev spaces with mixed norms: Wp1 ,p2 (Ω T ), energy type spaces: V s (Ω T ), V (t1 , t2 ), l,l/2 Sobolev-Slobodetskii spaces: Wr (S T ), l,l0 Besov and Nikolskii spaces: Br,θ (G), Hrl,l0 (G), l weighted spaces: Lkp,μ (Ω), Hμk (Ω ), Vp,β (Q).

We consider some auxiliary problems in cylindrical domains (Poisson equation and stationary Stokes system) and calculate corresponding Green functions. We also construct partition of unity and collect some facts useful in the next chapters: trace and extensions theorems, imbeddings and interpolation inequalities, and Korn inequality.

2.1

Spaces and Basic Notation

First we introduce the simplified notation for standard Lebesgue spaces Lp and Sobolev spaces Wps . Definition 2.1 (Lebesque and Sobolev spaces) (see [AF]) We set the following notation for Lebesque spaces uLp (Q) = |u|p,Q ,

uLp (Qt ) = |u|p,Qt ,

uLq (0,t;Lp (Q)) = |u|p,q,Qt ,

© Springer Nature Switzerland AG 2019 J. Renclawowicz, W. M. Zajączkowski, The Large Flux Problem to the

Navier-Stokes Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-32330-1 2

(2.1)

11

12

2

Notation and Auxiliary Results

and Sobolev spaces uH s (Q) = us,Q ,

where H s (Q) = W2s (Q),

(2.2)

uWps (Q) = us,p,Q ,

where Q is a bounded set in Rn (in particular, Q = Ω or Q = S ≡ ∂Ω) or Q = Rn , Qt = Q × (0, t), p, q ∈ [1, ∞), s ∈ R. 

In definitions below we use the notation: Dxα = ∂xα11 ∂xα22 ∂xα33 , Dxα = ∂xα11 ∂xα22 , where α = (α1 , α2 ) and α = (α1 , α2 , α3 ) are multiindices such that αi ∈ N0 and |α | = α1 + α2 , |α| = α1 + α2 + α3 . We also set uLq (0,t;Wpk (Ω)) = uk,p,q,Ω t , uLp (0,t;Wpk (Ω)) = uk,p,Ω t . Definition 2.2 (Weighted spaces) Let η(x3 ) = mini=1,2 dist(x3 , S2 (ai )). Then we introduce weighted spaces by uLkp,μ (Ω) =

   |α|=k

1/p |Dxα u|p η pμ dx

< ∞,

μ ∈ R, p ∈ (1, ∞),

Ω

and Lp,μ (Ω) = L0p,μ (Ω). In Chap. 10, in Definition 10.1 we also define Hμk (Ω ) for Ω = {x ∈ Ω : 0 < x3 < }. l In Chap. 11, we also use more specific weighted spaces: Vp,β (Q) for Q ⊂ R3+ is a set of functions with the finite norm uVp,β l (Q) =

 |α|≤l

p(β+|α|−l)

|Dxα u|p x3

dx dx3

1/p ,

p ∈ [1, ∞),

Q

where x = (x1 , x2 ), β ∈ R, α = (α1 , α2 , α3 ) is a multiindex. Moreover, 0 l Vp,β (Q) = Lp,β (Q), V2,β (Q) = Hβl (Q),    α p p(β+|α|−l) = ess sup u| x |D . uV∞,β l (Q) x 3 |α|≤l

x∈Q

Definition 2.3 (Anisotropic Sobolev spaces on Ω) Introduce the anisotropic Sobolev space  uWpk ,p 1

2

(Ω)

a

= −a



 S2 |α |≤k



|Dxα u|p1 dx1 dx2

p2 /p1

1/p2 dx3

< ∞,

2.1 Spaces and Basic Notation

13

where p1 , p2 ∈ [1, ∞], k ∈ N0 = N ∪ {0}, x = (x1 , x2 ). For k = 0 we have uWp0

1 ,p2

(Ω)

= uLp1 ,p2 (Ω) = |u|p1 ,p2 ,Ω .

For p1 = ∞ or p2 = ∞ we have, respectively  uWpk ,∞ (Ω) = ess 1

x3 ∈(−a,a)

2

(Ω)



a

= ess sup

x ∈S2

uW∞,∞ k (Ω) = ess sup

x∈Ω



1/p1



|Dxα u|p1 dx1 dx2

< ∞,

S2 |α |≤k

 uW∞,p k



sup

−a |α |≤k

1/p2



|Dxα u|p2 dx3

< ∞,



|Dxα u| < ∞.

|α |≤k

l,l/2

Definition 2.4 (Anisotropic Sobolev spaces on QT ) By Wr (QT ), Q ∈ R3 , where r ∈ [1, ∞] and l is even, we denote the anisotropic Sobolev space with the following finite norm ⎛ uWrl,l/2 (QT ) = ⎝



0

α+2α0 ≤l

uW∞ = ess l,l/2 (QT )

⎞1/r

 t

|Dxα ∂tα0 u|r dxdt⎠



sup

,

r ∈ [1, ∞)

Q

(x,t)∈QT α+2α ≤l 0

|Dxα ∂tα0 u|.

We will apply also the following notation  uWp2,1 (Ω t ) =

(|∇ u| + |∂t u| + |u| )dxdt 2

p

p

p



1/p = u(2),p,Ω t .

Ωt

Definition 2.5 (Anisotropic Sobolev spaces with mixed norm) Introl,l/2 duce the anisotropic Sobolev space with the mixed norm: Wp1 ,p2 (Ω T ), l-even, p1 , p2 ∈ [1, ∞], where uWpl,l/2 = T ,p (Ω ) 1

2

 |α|≤l

Dxα uLp2 (0,T ;Lp1 (Ω)) +



∂tα0 uLp2 (0,T ;Lp1 (Ω))

α0 ≤l/2

+uLp2 (0,T ;Lp1 (Ω)) , where α = (α1 , α2 , α3 ), |α| = α1 + α2 + α3 , αi ∈ N0 , i = 1, 2, 3. To describe the energy type estimates for solutions to the Navier-Stokes equations we need the following spaces characteristic for parabolic equations.

14

2

Notation and Auxiliary Results

Definition 2.6 (Energy Type spaces) V s (Ω T ) = {u : ess sup u(t)s,Ω + ∇us,2,Ω T < ∞}, t≤T

s ∈ N0 .

For s = 0, we have V (Ω T ) = V 0 (Ω T ) and uV (Ω T ) = ess sup |u(t)|2,Ω + |∇u|2,Ω T . t≤T

In the step by step in time approach we need spaces V (Ω×(t1 , t2 )) = V (t1 , t2 ), t1 < t2 , with the finite form  uV (t1 ,t2 ) = ess

sup |u(t)|2,Ω +

t1 ≤t≤t2

t2

|∇u(t



)|22,Ω dt

1/2 .

(2.3)

t1

In particular, for t1 = 0, t2 = t, we denote sometimes V (0, t) = V (Ω × (0, t)) = V (Ω t ). To describe traces of u on the boundary S of Ω we introduce the Sobolevl,l/2 Slobodetskii spaces Wr (S T ), where l is non-integer. The norm of this space is the following (see Ladyzhenskaya et al. [LSU, Ch. 2, Sect. 3], Il’in and Solonnikov [IS], and Solonnikov [S2]). Definition 2.7 (Sobolev-Slobodetskii spaces) For r ∈ [1, ∞) and nonl,l/2 integer l we define space Wr (S T ) with the finite norm uWrl,l/2 (S T ) = ⎛ +⎝





+⎝

 0 li − ki > 0, r ∈ [1, ∞] and we set l0 , li , ∈ R+ , where l = (l1 , l2 , l3 ) is l,l0 the multiindex. Then the Besov space Br,θ (G) is a set of function with finite norm uB l,l0 (G) = uLr (G) + r,θ

3  i=1

⎛ +⎝



h0

0

Δ

m0

⎛ ⎝



h0 0

ki i Δm i (h, Ω)∂xi uLr (G) hli −ki

(h, (0, T ))∂tk0 uLr (G) hl0 −k0

θ

θ

⎞1/θ dh ⎠ h

⎞1/θ dh ⎠ h

,

r ∈ [1, ∞),

where h0 is an arbitrary parameter, G ⊂ Ω × (0, T ), and θ ∈ [1, ∞). For r = ∞ we define uB l,l0 (G) = ess sup |u(x, t)| ∞,θ

⎛  3  ⎝ + i=1

⎛ +⎝



h0 0

(x,t)∈G

⎞1/θ  θ mi ki  h0  ess sup (x,t)∈G |Δi (h, Ω)∂xi u|  dh  ⎠     h hli −ki 0

⎞1/θ  θ k0 m0  ess sup  dh |Δ (h, (0, T ))∂ u| (x,t)∈G t   ⎠ .     h hl0 −k0

The spaces are equivalent for any mi , ki satisfying mi + ki > li > ki > 0. l,l0 Golovkin in [G] proved that the norms of spaces Wτl,l0 (Ω T ) and Br,θ (Ω T ) are equivalent. l,l0 For θ = ∞, Br,∞ (G) are denoted by Hrl,l0 (G) and called the Nikolskii spaces. Definition 2.9 (Nikolskii spaces) Let i = 1, 2, 3, mi , m0 , ki , k0 ⊂ N, mi > li − ki > 0, r ∈ [1, ∞] and we set l0 , li , ∈ R+ . Then the Nikolskii space Hrl,l0 (G) is a set of function with finite norm

16

2

uHrl,l0 (G) = uLr (G) + + sup 0 23 because for μ = 23 the first integral on the r.h.s. of the above inequalities is not integrable. In view of the definition of the weighted spaces used in (3.12), the weight in (3.12) is x2μ 3 , 3μ > 2. Therefore, the weight is not the Muckenhoupt weight, so estimate in (3.12) of the singular operator is proved in Chaps. 10 and 11 (see also [RZ1, RZ2]).

3.1 Weak Solutions

37

Then ˜ 3,S + μ+1/3 sup |d˜,x |3,S ]w2 ≡ E 2 . |I2 | ≤ c[εμ−2/3 sup |d| 1,Ω 2 3 2 x3

x3

Now, we consider the term    w · ∇δ · wdx = w · ∇b · wdx + w · ∇∇ϕ · wdx ≡ I3 + I4 . Ω

Ω

(3.14)

Ω

For I4 , we have             |I4 | =  div (w · ∇ϕw)dx − w · ∇w · ∇ϕdx =  w · ∇w · ∇ϕdx ≤ E 2 Ω

Ω

Ω

because the second term in I4 is equal to I2 and the first vanishes in view of (3.8)3 . 2 On the other hand, using b = α¯ e3 = i=1 d˜i ηi e¯3 , we find a bound for I3 :  2   |I3 | =  ˜

S2 (ai ,)

i=1

   2 =  i=1

≤

˜2 (ai ,) S

  ˜ w · ∇(di ηi )w3 dx   (w · ∇d˜i ηi w3 + w · ∇ηi d˜i w3 )dx

2   i=1

˜2 (ai ,) S

|w · ∇d˜i ηi | |w3 |dx

    w3  ˜  supp ηi dσi dx1 dx2 +ε w d 3 i   ˜2 (ai ,) σi S ≤c



2 

(3.15)

|w|6,S˜2 (ai ,) |w3 |3,S˜2 (ai ,) |∇d˜i |2,S˜2 (ai ,)

i=1

+ cε

2 

 |w3 |6,S˜2 (ai ,) |d˜i |3,S˜2 (ai ,)

i=1

≤ c1/6

2 

 S2 (ai )

|w|26,S˜2 (ai ,) |∇d˜i |2,S˜2 (ai ,)

i=1

+ cε

2 

|w|6,S˜2 (ai ,) |∇w3 |2,S˜2 (ai ,) |d˜i |3,S˜2 (ai ,)

i=1

˜ 1,3,Ω . ≤ c(1/6 + ε)w21,Ω d



dx1 dx2 r

 2 1/2  w3  dσi   σi

38

3

Energy Estimate: Global Weak Solutions

Thus, we can summarize the estimates for I1 –I4 to conclude that the nonlinear term in (3.10) is bounded by     ˜ 3,S  (w · ∇δ · w + δ · ∇w · w)dx ≤ c(εμ−2/3 sup |d| 2   x Ω

μ+1/3

+

3 (3.16) 1/6 ˜ ˜ ˜ 1,Ω )w2 . sup |d,x3 |3,S2 + ( + ε)d1,3,Ω + 1/6 d 1,Ω

x3

Next, we examine the second term on the r.h.s. of (3.10), 2 

2 

|δ · τ¯α |22,S1 ≤

α=1

(|b · τ¯α |22,S1 + |∇ϕ · τ¯α |22,S1 )

α=1

≤ |α|22,S1 + c∇ϕ21,3/2,Ω ≤

2 

|di |22,S1 + c|div b|23/2,Ω

i=1

˜2 ≤ cd 1,3/2,Ω + c

2 

|∇(d˜i ηi )|23/2,Ω

i=1

˜2 ≤ cd 1,3/2,Ω + c

2 

(|∇d˜i ηi |23/2,Ω + |d˜i ∇ηi |23/2,Ω )

i=1

˜2 ≤ cd 1,3/2,Ω + c

2 

|d˜i ∇ηi |23/2,Ω .

i=1

We estimate the last expression in more detail 2 

|d˜i ∇ηi |23/2,Ω ≤ ε2

 −a+   dx3

i=1

S2 (a1 )

−a+r

 +

S2 (a2 )

 ≤ε



dx3 a−

2

sup |d˜1 |23/2,S2 (a1 ) x3

  −a+ 4/3    1 3/2   dx3  a + x3  −a+r

 + sup |d˜2 |23/2,S2 (a2 ) x3

 ˜2 ≤ cε2 sup |d| 3/2,S2 x3

 r

dy y 3/2

4/3

     d˜2 3/2 4/3  dx  a − x3 



a−r

    d˜1 3/2 4/3  dx  a + x3  

a−r a−

  4/3   1 3/2   dx3  a − x3 

−1/2 ˜2 ≤ cε2 sup |d| − −1/2 ]4/3 3/2,S2 [r x3

3.1 Weak Solutions

˜2 ≤ cε2 sup |d| 3/2,S2 x3

39



1

 exp

2/3

1 2ε



  4/3 2 ε2 ˜2 ≤ c 2/3 exp −1 sup |d| 3/2,S2 . 3ε x3 

Combining the inequalities above, we infer 2 

ε2 ˜2 |δ · τ¯α |22,S1 ≤ cd + c exp 1,3/2,Ω 2/3 α=1





2 3ε

˜2 sup |d| 3/2,S2 .

(3.17)

x3

We also estimate the term |D(δ)|22,Ω ≤ |D(b)|22,Ω + |D(∇ϕ)|22,Ω ≤

2 

(|∇d˜i ηi |22,Ω + |d˜i ∇ηi |22,Ω ) + |∇2 ϕ|22,Ω

i=1

≤c

2  (|∇d˜i ηi |22,Ω + |d˜i ∇ηi |22,Ω ) i=1

≤c

2 

 d˜i 1,2,Ω + cε2

i=1



dx3 a−

≤c

2  

2  

2   i=1

≤c

S2 (a1 )

   ˜ 2   d2  dx  a − x3  

d˜i 21,2,Ω + ε2 sup |d˜i |22,S2 

d˜i 21,2,Ω + ε2 sup |d˜i |22,S2 x3

i=1

≤c

S2 (a2 )



dx3

x3

i=1

≤c



a−r

+ ε2 c

−a+r

   d˜1 2  dx  a + x3 



−a+

1 d˜i 21,2,Ω + ε2 sup |d˜i |22,S2  x3

 r

(3.18) dy y2



1 1 − r  



   1 exp −1 ε

2  ε2 [d˜i 21,2,Ω + e1/ε sup |d˜i |22,S2 ].  x3 i=1

Analyzing the last integral on the r.h.s. of (3.10) we have  (f −δt −δ·∇δ)·wdx ≤ Ω

 

ε1 |w|26,Ω +c(1/ε1 )(|f |26/5,Ω +|δt |26/5,Ω )+

Ω

  δ·∇δ·wdx.

40

3

Energy Estimate: Global Weak Solutions

We estimate |δt |6/5,Ω = |bt + ∇ϕt |6/5,Ω ≤ |d˜t |6/5,Ω + |div bt |6/5,Ω ≤ |d˜t |6/5,Ω + |∇d˜t |6/5,Ω + |d˜t ∇η|6/5,Ω ≤ d˜t 1,6/5,Ω   5/6 dx3 + ε sup |d˜t |6/5,S2 6/5 x3 r x3 ε ≤ d˜t 1,6/5,Ω + c 1/6 e1/6ε sup |d˜t |6/5,S2 ,  x3

(3.19)

where we used that 

 r

5/6

dx3 6/5

x3

r 5/6   5/6 5/6 1 1  1 5 5/6 = 5 1/5  =5 − 1/5 = 1/6 (e1/5ε −1)5/6 . 1/5 r   x3 

Finally, we examine      δ · ∇δ · wdx ≤ |∇δ|2,Ω |δ|3,Ω |w|6,Ω ≤ ε2 |w|26,Ω + c(1/ε2 )δ41,2,Ω  Ω (3.20) ε4 2/ε 2 4 4 ˜ ˜ ≤ ε2 |w|6,Ω + c(1/ε2 )(d1,2,Ω + 2 e sup |d|2,Ω ).  x3 Employing the above estimates in (3.10) yields 2  1 d |w|22,Ω + νw21,Ω + γ |w · τ¯α |22,S1 2 dt α=1

˜ 3,S + μ+1/3 sup |d˜,x |3,S ≤ cw21,Ω [εμ−2/3 sup |d| 2 3 2 x3

x3

˜ 1,3,Ω ] + (1/6 + ε)d

(3.21)

˜ 4 + d ˜ 2 + d˜t 2 + c[|f |26/5,Ω + d 1,Ω 1,Ω 1,6/5,Ω +

4 2 ε2 1/ε ˜ 2 + ε e2/ε sup |d| ˜ 4 + ε e2/3ε sup |d| ˜2 e sup |d| 2,S2 2,S2 3/2,S2 2 2/3    x3 x3 x3

+

ε2 1/3ε e sup |d˜t |26/5,S2 ]. 1/3 t

Introduce the anisotropic Sobolev spaces  uWp1

1 ,p2

(Ω)

a

= −a



 S2 |α|≤1

p2 /p1 |Dxα u|p1 dx1 dx2

1/p2 dx3

,

3.1 Weak Solutions

41

  uLp1 ,p2 (Ω) =

p2 /p1 |u|p1 dx1 dx2

1/p2 dx3

,

S2

where p1 , p2 ∈ [1, ∞]. Therefore, the norms of d˜ under the first square bracket on the r.h.s. of (3.21) are bounded by ˜ W 1 (Ω) , d 3,∞ and the first three norms of d˜ under the second square bracket are bounded by the quantities ˜ 1,Ω , d

d˜t 1,6/5,Ω .

Then, we express (3.21) in the form 2  1 d 2 2 |w| + νw1,Ω + γ |w · τ¯α |22,S1 2 dt 2,Ω α=1

˜ W 1 (Ω) ] ≤ cw21,Ω [(εμ−2/3 + μ+1/3 + 1/6 + ε)d 3,∞  ˜ 2 + d ˜ 4 + d˜t 2 + c |f |2 + d 6/5,Ω

1,Ω

1,Ω

1,6/5,Ω

(3.22)



 ε2 1/ε ε2 ε4 2/ε ˜ 4 ˜2 e + 2/3 e2/3ε d dL2,∞ (Ω) L2,∞ (Ω) + 2 e     ε2 + 1/3 e1/3ε d˜t 2L6/5,∞ (Ω) .  +

We set μ > 23 and  < 1. Then μ+1/3 < 1/6 . Therefore, the expression under the first square bracket in (3.22) is bounded by ˜ W 1 (Ω) . (ε + 1/6 )d 3,∞ Setting ˜ W 1 (Ω) ≤ c(ε + 1/6 )d 3,∞

ν 2

(3.23)

42

3

Energy Estimate: Global Weak Solutions

we derive from (3.22) the inequality 2  d |w|22,Ω + νw21,Ω + 2γ |w · τ¯α |22,S1 dt α=1

˜ 2 + d ˜ 4 + d˜t 2 ≤ c[|f |26/5,Ω + d 1,Ω 1,Ω 1,6/5,Ω ]  2  2 ε 1/ε ε ˜2 e + 2/3 e2/3ε d +c L2,∞ (Ω)  

(3.24)

 ε4 2/ε ˜ 4 ε2 1/3 ˜ 2 + 2 e dL2,∞ (Ω) + 1/3 e dt L6/5,∞,Ω .  

From (3.23) we calculate ε=



ν ˜ W 1 (Ω) 4cd 3,∞

,

=

ν ˜ W 1 (Ω) 4cd 3,∞

6 .

(3.25)

Then (3.24) takes the form 2  d |w|22,Ω + νw21,Ω + 2γ |w · τ¯α |22,S1 dt α=1

˜ 2 + d ˜ 4 + d˜t 2 ≤ c[|f |26/5,Ω + d 1,Ω 1,Ω 1,6/5,Ω ]    (3.26) 1 ˜2 2 4 ˜ ˜ dW 1 (Ω) + c (1 + dW 1 (Ω) )dW 1 (Ω) exp 3,∞ 3,∞ 3,∞ c      1 ˜ 1 ˜ 10 2 ˜ ˜ 1 1 dW3,∞ (Ω) + exp dW3,∞ (Ω) dt L6/5,∞,Ω , + dW 1 (Ω) exp 3,∞ c c ˜ W 1 (Ω) , ε = c , where we used the following calculations. Let a = d 3,∞ a " c #6 ρ= . Then the first term under the second square bracket in the r.h.s. a of (3.24) equals  $ c %2 a/c a $ c %6 e a

$ c %2  2 a + $ a %4 e 3 c a2 ≤ c(a2 + 1)a4 ea/c , c a

the second $ c %4 a

$ c %12 e a

2a/c 2

a = ca8 e2a/c a2 ,

3.1 Weak Solutions

43

and the third $ c %2 a

$ c %2 e

a/3c

d˜t 2L6/5,∞,Ω ≤ cea/c d˜t 2L6/5,∞,Ω .

a

To simplify notation we write (3.26) in the form d ˜ W 1 (Ω) )· |w|2 + νw21,Ω ≤ c[|f |26/5,Ω + ϕ(d 3,∞ dt 2,Ω ˜2 1 + d˜t 2 )] ≡ F 2 (t), · (d

(3.27)

1,6/5,Ω

W3,∞ (Ω)

where ϕ is an increasing positive function which form follows from the form of the r.h.s. of (3.26) and the following estimates uL6/5,∞ (Ω) ≤ cu1,6/5,Ω , 1 u1,Ω ≤ cuW3,∞ (Ω) .

To obtain a global estimate we write (3.27) in the form d (|w|22,Ω eνt ) ≤ F 2 (t)eνt . dt

(3.28)

Integrating (3.28) with respect to time from t = kT , to t ∈ (kT, (k + 1)T ], k ∈ N0 , we obtain |w(t)|22,Ω

≤e

−νt



t



F 2 (t )eνt dt + exp(−ν(t − kT ))|w(kT )|22,Ω .

(3.29)

kT

Setting t = (k + 1)T we get 

(k+1)T

|w((k + 1)T )|22,Ω ≤

F 2 (t)dt + exp(−νT )|w(kT )|22,Ω .

(3.30)

kT

Let  A21 (T )

(k+1)T

F 2 (t)dt.

= sup k∈N0

(3.31)

kT

Then (3.30), by iteration, implies the estimate |w(kT )|22,Ω ≤

A21 (T ) + exp(−νkT )|w(0)|22,Ω . 1 − exp(−νT )

(3.32)

44

3

Energy Estimate: Global Weak Solutions

Next, we integrate (3.27) with respect to time from t t ∈ (kT,(k+1)T]. Using (3.32) yields  w2V (Ω×(kT,t)) ≤ A21 (T ) + = A21 (T )

t

≤

=

kT to

F 2 (t )dt + |w(kT )|22,Ω

kT

A21 (T ) + exp(−νkT )|w(0)|22,Ω 1 − exp(−νT )

(3.33)

2 − exp(−νT ) + exp(−νkT )|w(0)|22,Ω ≤ A22 (T ), 1 − exp(−νT )

where A22 (T ) = A21 (T )

2 − exp(−νT ) + |w(0)|22,Ω . 1 − exp(−νT )

(3.34)

Having estimate (3.33) for w and using transformation (3.7) we derive v2V (Ω×(kT,t)) ≤ w2V (Ω×(kT,t)) + δ2V (Ω×(kT,t)) .

(3.35)

We restrict our considerations to the time interval (0, T ) only because estimates in any interval [kT, (k + 1)T ] can be performed in the same way. The last term in the r.h.s. of (3.35) is bounded by δ2V (Ω t ) ≤ b2V (Ω t ) + ∇ϕ2V (Ω t ) ≤ |b|2,∞,Ω t + b21,2,2,Ω t +|∇ϕ|22,∞,Ω t + ∇ϕ21,2,2,Ω t ≡ I. Now, we estimate the particular terms in I. The first term |b|22,∞,Ω t ≤ |α|22,∞,Ω t ≤

2 

˜ 2,∞,Ω t . |d˜i |22,∞,Ω t = |d|

i=1

The second term 

t

b21,2,2,Ω t =

(|∇b|22,Ω + |b|22,Ω )dt ≤

0

2   i=1

+

2  t  i=1

0

|d˜i ∇ηi |22,Ω dt +

t

|∇d˜i ηi |22,Ω dt

0 2   i=1

t 0

|d˜i |22,Ω dt

(3.36)

3.1 Weak Solutions

45

 ≤

˜2 cd 1,2,2,Ω t

+ε

−a+ 

t

2

  dx 



dx3 0

S2 (a1 )

−a+r

2 d˜1  a + x3 

2  d˜2  a − x3  0 a− S2 (a2 )   t  −a+   1 2 2 2 2 ˜ ˜   ≤ cd1,2,2,Ω t + ε sup |d1 |2,S2 (a1 )  a + x3  dx3 x3 0 −a+r    a−r   1 2   dx3 dt + sup |d˜2 |22,S2 (a2 )  a − x3  x3 a−  t   dy 2 2  ˜2 ˜ ≤ cd + cε sup | d| dt 2,S2 1,2,2,Ω t 2 x 3 0 r y    t 1 2 2 2  1 ˜ ˜ − = cd1,2,2,Ω t + cε sup |d|2,S2 dt = r  0 x3 

+

t

dt



  dx 



a−r

dx3

    ˜ 2 dt 1 exp 1 − 1 sup |d| 2,S2  ε 0 x3    t 1 ε2 ˜ 2 dt ≡ I1 . + c exp sup |d| 2,S2   0 x3 

t

2 ˜2 = cd 1,2,2,Ω t + cε

˜ 1,2,2,Ω t ≤ cd Employing (3.25) gives

 ˜2 ˜ 1 (Ω) exp I1 ≤ cd 1,2,2,Ω t + c sup dW3,∞

˜ W 1 (Ω)  supt d 3,∞ c

t



t

· 0

(3.37)

˜ 2 dt ≡ J. sup |d| 2,S2 x3

Next, we consider the third term in I (see (3.36)) |∇ϕ|22,∞,Ω t = sup |∇ϕ(t )|22,Ω . t

Multiplying (3.6)1 by ϕ, integrating by parts, and using (3.6)2 give 



|∇ϕ|22,Ω = Ω

=

 div (bϕ)dx −

div b ϕdx = Ω



2   i=1

b · ∇ϕdx Ω

(3.38)

di ϕdS2 − S2 (ai )

b · ∇ϕdx. Ω

46

3

Energy Estimate: Global Weak Solutions

Applying the H¨ older and Young inequalities to the r.h.s. of (3.38) and using the Poincar´e inequality to the l.h.s. of (3.38) we derive ϕ21,Ω ≤ ε1 |∇ϕ|22,Ω +

2 c 2 c  ˜ 2 |b|2,Ω + ε2 |ϕ|22,S2 + |di |2,S2 (ai ) . ε1 ε2 i=1

Hence, for sufficiently small ε1 and ε2 we have ϕ21,Ω ≤ c|b|22,Ω + c|d|22,S2 ≤ c|d|2L2,∞ (Ω) . Then it follows ϕ1,2,∞,Ω t ≤ c sup |d(t )|L2,∞ (Ω) . t ≤t

(3.39)

Finally, we estimate the last term in I (see (3.36))  ∇ϕ21,2,2,Ω t

t

=

(|∇2 ϕ|22,Ω + |∇ϕ|22,Ω )dt ≡ I2 .

0

In view of solvability of problem (3.6) we have 

t

I2 ≤ c 0

|d(t )|2L2,∞ (Ω) dt + c



t

|div b(t )|22,Ω dt ≡ I3 .

0

The second term in I3 is bounded by c

 t 2

(|∇d˜i |22,Ω + |d˜i ∇ηi |22,Ω )dt ≤ cJ,

0 i=1

where J is introduced in (3.37). Summarizing, we obtain the estimate ˜2 ˜2 δ2V (Ω t ) ≤ c|d| 2,∞,Ω t + cd1,2,2,Ω t   t 1 4 ˜ ˜ ˜ 2 dt 1 sup dW3,∞ +c sup dW 1 (Ω) exp sup |d| (Ω) 2,S2 3,∞ c t t 0 x3

(3.40)

+c sup d(t )2L2,∞ (Ω) . t ≤t

A similar estimate holds for any time interval [kT, (k + 1)T ], k ∈ N0 . The above considerations imply the following result. Lemma 3.2 Let (v, p) be a solution to problem (1.1). Let w be defined by (3.8). Then

3.2 Estimates for vt

47

(1) w2V (Ω×(kT,t)) ≤ cA21 (T ) + exp(−νkT )|w(0)|22,Ω , where T > 0, t ∈ [kT, (k + 1)T ], k ∈ N0 , and  A21 (T )

(k+1)T

= sup k∈N0

kT

(|f (t)|26/5,Ω + d(t)21,3,S2

+dt (t)26/5,S2 )dt ϕ(sup d(t)1,3,S2 ) < ∞, t

where ϕ is an increasing positive function. Next (2) v2V (Ω×(kT,t)) ≤ w2V (Ω×(kT,t)) + ϕ(sup d(t)1,3,S2 ) · t

·d 1 ,2,S2 ×(kT,t) ≡ 2

2

w2V (Ω×(kT,t))

+ F12 (t),

where d ∈ L∞ (kT, (k + 1)T ; W31 (S2 )) ∩ L2 (kT, (k + 1)T ; H 1/2 (S2 )) for any k ∈ N0 . Proof Property (1) follows from (3.33) and property (2) from (3.35) and (3.40). This ends the proof.  

3.2

Estimates for vt

Consider the problem vtt − div T(vt , pt ) = −v · ∇vt − vt · ∇v + ft

in Ω T ,

div vt = 0

in Ω T ,

vt · n ¯=0

on S1T ,

νn ¯ · D(vt ) · τ¯α + γvt · τ¯α = 0, α = 1, 2,

on S1T ,

vt · n ¯ = dt

on S2T ,

n ¯ · D(vt ) · τ¯α = 0, α = 1, 2

on S2T ,

vt |t=0 = vt (0)

in Ω.

(3.41)

48

3

Energy Estimate: Global Weak Solutions

Now, instead of (3.4)–(3.7) we have αt =

2 

d˜i,t ηi ,

bt = αt e¯3

(3.42)

i=1

u t = v t − bt ,

div ut = −α,x3 t , Δϕt = −div bt

n ¯ · ∇ϕt = 0  ϕt dx = 0,

ut · n ¯ |S = 0,

in Ω, on S,

(3.43)

Ω

wt = ut − ∇ϕt = vt − (bt + ∇ϕt ) = vt − δt .

(3.44)

Consequently, (wt , pt ) is a solution to the problem wtt + w · ∇wt + w · ∇δt + δ · ∇wt + wt · ∇w + δt · ∇w +wt · ∇δ − div T(wt , pt ) = −δtt − δ · ∇δt − δt · ∇δ +νdiv D(δt ) + ft ≡ F1 (δ, δt , t)

in Ω T ,

div wt = 0

in Ω T ,

¯=0 wt · n

on S T , (3.45)

νn ¯ · D(wt ) · τ¯α + γwt · τ¯α = −ν n ¯ · D(δt ) · τ¯α − γδt · τ¯α α = 1, 2

on S1T ,

n ¯ · D(wt ) · τ¯α = −¯ n · D(δt ) · τ¯α , α = 1, 2

on S2T ,

wt |t=0 = vt (0) − δt (0) ≡ wt (0)

in Ω.

Multiply (3.45) by wt and integrate over Ω. Then we derive 1 d |wt |22,Ω + 2 dt

 (w · ∇wt + wt · ∇w + w · ∇δt + δ · ∇wt + δt · ∇w Ω



+ wt · ∇δ) · wt dx −

div T(wt + δt , pt ) · wt dx

(3.46)

Ω



(ft − δtt − δ · ∇δt − δt · ∇δ) · wt dx.

= Ω

In view of the boundary conditions the third integral on the l.h.s. of (3.46) assumes the form

3.2 Estimates for vt

49







Dij (wt + δt )wtj,xi dx − ν

ν Ω

div (pt wt )dx ≡ I,

∂xj [Dij (wt + δt )wti ] + Ω

Ω

where the first integral equals ν |Dij (wt )|22,Ω + ν 2

 Dij (δt )wtj,xi dx, Ω

the second has the form  (|wtτα |2 + wtτα δtτα )dS1 ,

γ S1

and the third vanishes in view of the Green theorem and (3.45)3 . Applying the Korn inequality (see Lemma 2.22) we obtain the inequality 2  1 d |wt |22,Ω + νwt 21,Ω + γ |wt · τ¯α |22,S1 2 dt α=1  ≤ − (w · ∇wt + wt · ∇w + w · ∇δt + δ · ∇wt Ω

+δt · ∇w + wt · ∇δ) · wt dx +

c|D(δt )|22,Ω

+c

2 

(3.47) |δt ·

τ¯α |22,S1

α=1



(ft − δtt − δ · ∇δt − δt · ∇δ) · wt dx.

+ Ω

In view of (3.45)2,3 the first integral in the first term on the r.h.s. of (3.47) vanishes. The second can be written in the form  wt · ∇wt · wdx Ω

and estimated by ε1 |∇wt |22,Ω + c/ε1 |w|2∞,Ω |wt |22,Ω . Now, we shall estimate the other terms from the first integral on the r.h.s. of (3.47). The third term has the form 





w · ∇δt · wt dx = Ω

w · ∇bt · wt dx + Ω

w · ∇∇ϕt wt dx ≡ J1 + J2 . Ω

50

3

Energy Estimate: Global Weak Solutions

For J2 we have      |J2 | =  w · ∇ϕt ∇wt dx ≤ |∇wt |2,Ω |w|6,Ω |∇ϕt |3,Ω . Ω

In view of (3.12) and (3.13) we obtain the estimate |J2 | ≤ c(εμ−2/3 sup |d˜,t |3,S2 + μ+1/3 sup |d˜,tx3 |3,S2 )w1,Ω wt 1,Ω . x3

x3

Next we consider J1 . Repeating the estimate of I3 in (3.15), we have |J1 | ≤ c(1/6 + ε)d˜t 1,3,Ω w1,Ω wt 1,Ω . The fourth integral in the first term on the r.h.s. of (3.47) equals 





δ · ∇wt · wt dx =

b · ∇wt · wt dx +

Ω

Ω

∇ϕ · ∇wt wt dx ≡ J3 + J4 . Ω

Repeating estimate of I1 from (3.11) we have ˜ 1,Ω wt 2 , |J3 | ≤ c1/6 d 1,Ω and using estimate of I2 yields ˜ 3,S + μ+1/3 sup |d˜,x |3,S )wt 2 . |J4 | ≤ c(εμ−2/3 sup |d| 1,Ω 2 3 2 x3

x3

Continuing, the fifth integral in the first term on the r.h.s. of (3.47) has the form    δt · ∇w · wt dx = bt · ∇wwt dx + ∇ϕt · ∇w · ∇wt dx ≡ J5 + J6 . Ω

Ω

Ω

Repeating the estimates of terms I1 and I2 from (3.11) we have |J5 | ≤ c1/6 d˜t 1,Ω w1,Ω wt 1,Ω and |J6 | ≤ c[εμ−2/3 sup |d˜t |3,S2 + μ+1/3 sup |d˜tx3 |3,S2 ]w1,Ω wt 1,Ω . x3

x3

Finally, the last integral from the first term on the r.h.s. of (3.47) takes the form

3.2 Estimates for vt

51







wt · ∇δ · wt dx = Ω

wt · ∇b · wt dx + Ω

wt · ∇∇ϕ · wt dx ≡ J7 + J8 . Ω

Repeating the estimates of I3 and I4 from (3.14) we derive ˜ 1,3,Ω wt 2 |J7 | ≤ c(1/6 + ε)d 1,Ω and ˜ 3,S + μ+1/3 sup |d˜,x |3,S ]wt 2 . |J8 | ≤ c[εμ−2/3 sup |d| 1,Ω 2 3 2 x3

x3

Summarizing the above estimates gives      (w · ∇wt + wt · ∇w + w · ∇δt + δ · ∇wt + δt · ∇w + wt · ∇δ) · wt dx   Ω

˜ 1,3,Ω wt 2 ≤ c(1/6 + ε)d˜t 1,3,Ω w1,Ω wt 1,Ω + c(1/6 + ε)d 1,Ω ˜ 1,Ω wt 2 + c1/6 d˜t 1,Ω w1,Ω wt 1,Ω + c1/6 d 1,Ω

(3.48)

+ c(εμ−2/3 sup |d˜,t |3,S2 + μ+1/3 sup |d˜,tx3 |3,S2 )w1,Ω wt 1,Ω x3

x3

˜ 3,S + μ+1/3 sup |d˜,x |3,S )wt 2 . + c(εμ−2/3 sup |d| 1,Ω 2 3 2 x3

x3

Next, we examine the third term on the r.h.s. of (3.47). Repeating the considerations leading to (3.17), we infer 2  α=1

|δt ·

τ¯α |22,S1

ε2 ≤ cd˜t 21,3/2,Ω + c 2/3 exp 



2 3ε

 sup |d˜t |23/2,S2 .

(3.49)

x3

In virtue of estimate (3.18) we have the following estimate for the second term on the r.h.s. of (3.47) |D(δt )|22,Ω ≤ c

2 

[|d˜i,t 21,2,Ω +

i=1

ε2 1/ε e sup |d˜i,t |22,S2 ].  x3

(3.50)

Analyzing the last integral on the r.h.s. of (3.47) one gets      (ft − δtt − δ · ∇δt − δt · ∇δ) · wt dx   Ω

≤ ε2 |wt |26,Ω + c/ε2 (|ft |26/5,Ω + |δtt |26/5,Ω )      +  (δ · ∇δt + δt · ∇δ) · wt dx. Ω

(3.51)

52

3

Energy Estimate: Global Weak Solutions

Repeating estimate (3.19) in this case we have ε |δtt |6/5,Ω ≤ d˜tt 1,6/5,Ω + c 1/6 e1/6ε sup |d˜tt |6/5,S2 .  x3

(3.52)

Finally, in view of (3.20), we obtain      (δ · ∇δt + δt · ∇δ) · wt dx   Ω

≤ |∇δt |2,Ω |δ|3,Ω |wt |6,Ω + |∇δ|2,Ω |δt |3,Ω |wt |6,Ω ≤ ε3 |wt |26,Ω + c(1/ε3 )δt 21,2,Ω δ21,2,Ω   ε2 1/ε ˜2 ˜2 e ≤ ε3 |wt |26,Ω + c(1/ε3 ) d + sup | d| 1,2,Ω 2,S2  x3   ε2 · d˜t 21,2,Ω + e1/ε sup |d˜t |22,S2 .  x3

(3.53)

Employing the above estimates in (3.47) yields 2  1 d |wt |22,Ω + νwt 21,Ω + γ |wt · τ¯α |22,S1 ≤ c|w|2∞,Ω |wt |22,Ω 2 dt α=1

+ c[(1/6 + ε)d˜t 1,3,Ω w1,Ω wt 1,Ω ˜ 1,3,Ω wt 2 + (1/6 + ε)d 1,Ω ˜ 1,Ω wt 2 + c1/6 d˜t 1,Ω w1,Ω wt 1,Ω + 1/6 d 1,Ω + (εμ−2/3 sup |d˜t |3,S2 + μ+1/3 sup |d˜,tx3 |3,S2 )w1,Ω wt 1,Ω x3

x3

˜ 3,S + μ+1/3 sup |d˜,x |3,S )wt 2 ] + (εμ−2/3 sup |d| 1,Ω 2 3 2 x3

x3

ε + c[|ft |26/5,Ω + d˜tt 21,6/5,Ω + 1/6 e1/6ε sup |d˜tt |26/5,S2  x3 ε2 + d˜t 21,3/2,Ω + 2/3 e2/3ε sup |d˜t |23/2,S2 + d˜,t 21,2,Ω  x3   2 ε 1/ε ε2 1/ε 2 2 2 ˜ ˜ ˜ + e sup |d,t |2,S2 + d1,2,Ω + e sup |d|2,S2   x3 x3   2 ε · d˜t 21,2,Ω + e1/ε sup |d˜t |22,S2 ].  x3

(3.54)

3.2 Estimates for vt

53

We estimate the first square bracket on the r.h.s. of (3.54) by ˜ W 1 (Ω) wt 2 c(εμ−2/3 + μ+1/3 + 1/6 + ε)[d 1,Ω 3,∞ 1 +d˜t W3,∞ (w) w1,Ω wt 1,Ω ].

The terms under the second square bracket on the r.h.s. of (3.54) are estimated by  +c

˜4 ˜ 4 c[|ft |26/5,Ω + d˜tt 21,6/5,Ω + d˜t 21,2,Ω + d 1,2,Ω + dt 1,2,Ω ] ε 1/6

e1/6ε sup |d˜tt |26/5,S2 + x3

ε2 2/3ε ε2 e sup |d˜t |22,S2 + e1/ε sup |d˜t |22,S2 2/3   x3 x3  4 ε4 ˜ 4 + ε e2/ε sup |d˜t |4 + 2 e2/ε sup |d| 2,S2 2,S2 .  2 x3 x3

Using that ε small and  < 1 inequality (3.54) simplifies to 2  1 d |wt |22,Ω + νwt 21,Ω + γ |wt · τ¯α |22,S1 ≤ c|w|2∞,Ω |wt |22,Ω 2 dt α=1

˜ W 1 (Ω) wt 2 + c(εμ−2/3 + μ+1/3 + 1/6 + ε)[d 1,Ω 3,∞ 1 + d˜t W3,∞ (Ω) w1,Ω wt 1,Ω ]

˜4 ˜ 4 + c[|ft |26/5,Ω + d˜tt 21,6/5,Ω + d˜t 21,2,Ω + d 1,2,Ω + dt 1,2,Ω ]  ε ε2 + c 1/6 e1/6ε sup |d˜tt |26/5,S2 + e1/ε sup |d˜t |22,S2   x3 x3  4 4 ε ˜ 4 + ε e2/ε sup |d˜t |4 + 2 e2/ε sup |d| 2,S2 2,S2 .  2 x3 x3

(3.55)

Now, we simplify (3.55). Assume that μ > 23 and  will be chosen small. Then the second term on the r.h.s. of (3.55) is bounded by ˜ W 1 (Ω) + d˜t W 1 (Ω) )(wt 2 + w2 ). c(1/6 + ε)(d 1,Ω 1,Ω 3,∞ 3,∞ Next, we assume that ˜ W 1 (w) + d˜t W 1 (Ω) ) ≤ ν , c(1/6 + ε)(d 3,∞ 3,∞ k0 where k0 will be chosen sufficiently large. Introducing the notation

(3.56)

54

3

Energy Estimate: Global Weak Solutions

˜ W 1 (Ω) + d˜t W 1 (Ω) , α = d 3,∞ 3,∞ we obtain from (3.56) that 1/6 ≤

c , α

ε≤

c . α

(3.57)

To examine the last term on the r.h.s. of (3.55), we calculate ε 1/6

e1/6ε ≤ cecα ,

ε2 1/ε (c/α)2 cα e ≤ e = cα4 ecα  (c/α)6 ε4 2/ε (c/α)4 cα e ≤ e = cα8 ecα . 2  (c/α)12 In view of the above considerations we obtain from (3.55) the inequality 2  d |wt |22,Ω + νwt 21,Ω + γ |wt · τ¯α |22,S1 dt α=1

≤ c|w|2∞,Ω |wt |22,Ω +

ν w21,Ω k

˜4 + c[|f |26/5,Ω + |ft |26/5,Ω + d˜tt 21,6/5,Ω + d˜t 21,2,Ω + d 1,2,Ω

(3.58)

˜ W 1 (Ω) , d˜t W 1 (Ω) ) · [|d˜tt |2 + d˜t 41,2,Ω ] + cϕ(d 6/5,∞,Ω 3,∞ 3,∞ ˜4 ˜ 4 ˜ 1 (Ω) + d˜t 2 + |d˜t |22,∞,Ω + |d| 2,∞,Ω + |dt |2,∞,Ω + dW3,∞ 1,6/5,Ω ] ≡ c|w|2∞,Ω |wt |22,Ω +

ν w21,Ω + Γ 2 (t). k

From (3.27) and (3.58) we have d (|w|22,Ω + |wt |22,Ω ) + ν(w21,Ω + wt 21,Ω ) dt

(3.59)

≤ c|w|2∞,Ω |wt |22,Ω + F 2 (t) + Γ 2 (t). From (3.59) it follows d (|w|22,Ω + |wt |22,Ω ) + ν(|w|22,Ω + |wt |22,Ω ) dt ≤ c|w|2∞,Ω (|w|22,Ω + |wt |22,Ω ) + F 2 (t) + Γ 2 (t).

(3.60)

3.2 Estimates for vt

55

Hence, one obtains     t d |w(t )|2∞,Ω dt (|w|22,Ω + |wt |22,Ω ) exp νt − dt kT    t ≤ [F 2 (t) + Γ 2 (t)] exp νt − |w(t )|2∞,Ω dt .

(3.61)

kT

Integrating with respect to time from kT to t ∈ (kT, (k + 1)T ], k ∈ N0 , we derive  t  2 2  2  |w(t)|2,Ω + |wt (t)|2,Ω ≤ exp |w(t )|∞,Ω dt 

t

·





kT







t

(F (t ) + Γ (t ))dt + exp − ν(t − kT ) + 2

2

|w(t

kT

kT

·(|w(kT )|22,Ω



)|2∞ dt



(3.62)

+ |wt (kT )|22,Ω ).

Setting t = (k + 1)T , introducing the notation  B12 (T )

(k+1)T

= sup exp k∈N0

kT

|w(t)|2∞,Ω dt

  ·

(k+1)T

(F 2 (t) + Γ 2 (t))dt, kT

(3.63) and assuming that the following restriction holds, with A2 defined in (3.34) 

(k+1)T kT

|w(t)|2∞,Ω dt ≤

1/2 cT 1/4 A2 (T )



3/4

(k+1)T

|wxx (t)|22,Ω dt (3.64)

kT

ν ≤ T 2

for any k ∈ N0 ,

we obtain from (3.62) the inequality |w((k + 1)T )|22,Ω + |wt ((k + 1)T )|22,Ω ≤ B12 (T )   ν + exp − T (|w(kT )|22,Ω + |wt (kT )|22,Ω ). 2

(3.65)

By iteration, we have |w(kT )|22,Ω + |wt (kT )|22,Ω ≤ +

exp(−νkT )(|w(0)|22,Ω

B12 (T ) $ % 1 − exp − ν2 T

+

|wt (0)|22,Ω ).

(3.66)

56

3

Energy Estimate: Global Weak Solutions

In view of (3.63) and (3.64) we obtain |w(t)|22,Ω + |wt (t)|22,Ω

" ν # ≤ B12 (T ) + exp − T (|w(kT )|22,Ω + |wt (kT )|22,Ω ). 2

(3.67)

Using (3.66) in (3.67) we obtain for t ∈ (kT, (k + 1)T ] the inequality |w(t)|22,Ω + |wt (t)|22,Ω ≤ +

B12 (T ) $ % 1 − exp − ν2 T

exp(−νkT )(|w(0)|22,Ω

+

(3.68)

|wt (0)|22,Ω ).

Integrating (3.59) with respect to time from t = kT to t ∈ (kT, (k + 1)T ] and using (3.68), we obtain w2V (Ω×(kT,t)) + wt 2V (Ω×(kT,t))    (k+1)T B12 2 $ % ≤ 1+ |w(t)|∞,Ω dt 1 − exp − ν2 T kT   + exp(−νkT )(|w(0)|22,Ω + |wt (0)|22,Ω ) + c

(3.69) (k+1)T

(F 2 (t) + Γ 2 (t)dt). kT

The above considerations imply the following result. Lemma 3.3 Assume that F, Γ ∈ L2 (kT, (k+1)T ) for any k ∈ N0 , where F is defined in (3.27) and Γ in (3.58). Assume that w ∈ L2 (kT, (k + 1)T ; L∞ (Ω)) and restriction (3.64) holds, with initial data such that w(0), wt (0) ∈ L2 (Ω). Then estimates (3.68) and (3.69) hold for any k ∈ N0 , where B1 (T ) is defined in (3.63). Having estimate (3.69) for w and wt we can derive an estimate for v and vt . Proposition 3.4 Let the assumptions of Lemmas 3.2 and 3.3 hold. Then vV (Ω×(kT,t)) ≤ cwV (Ω×(kT,t)) + F12 (kT, t),

(3.70)

where F1 (kT, t) is defined in the property (2) of Lemma 3.2 as a quantity ϕ(supt d(t)1,3,S2 ) · d21 ,2,S2 ×(kT,t) . Next 2

vt V (kT,t) ≤ wt V (kT,t) + δt V (kT,t) ,

(3.71)

3.2 Estimates for vt

57

where δt V (kT,t) ≤ ϕ(sup d(t)1,3,S2 , sup dt (t)1,3,S2 ) t

t

·(d1/2,2,S2 ×(kT,t) + dt 1/2,2,S2 ×(kT,t) ) ≡ Γ12 (kT, t). Proof From the definition of δt we have δt 2V (Ω×(kT,t)) ≤ bt 2V (Ω×(kT,t)) + ∇ϕt 2V (Ω×(kT,t)) ≤ |bt |22,∞,Ω×(kT,t) + bt 21,2,2,Ω×(kT,t) + |∇ϕt |22,∞,Ω×(kT,t) +∇ϕt 21,2,2,Ω×(kT,t) ≡ J. Now we estimate the particular terms in J. Repeating the considerations leading to estimate I and (3.57) we obtain that J ≤ Γ12 . This concludes the proof.  

Chapter 4

Local Estimates for Regular Solutions

Abstract The aim of this chapter is to show the estimate vW 2,1 (Ω t ) ≤ M, 2

where M depends on some norms of data but does not depend on time t ∈ (0, T ] explicitly. The dependence on time is only through integrals with respect to time of data functions f, d and their time and space derivatives. To prove the above inequality, we need smallness of the following quantity 

t

Λ2 (t) = 0

(dx 2H 1 (S2 ) + dt 2H 1 (S2 ) + |f3 |2L4/3 (S2 ) + |g|2L6/5 (Ω) ) dt +A sup dx 2W 1 t

3/2 (S2 )

+ |h(0)|2L2 (Ω) ,

where h = v,x3 , g = f,x3 , and A estimates the weak solution (see Chap. 3). To achieve this, we consider the problems for variables h, q = p,x3 and χ = v2,x1 − v1,x2 . Then for sufficiently small Λ2 (t) we can apply some fixed point argument and use energy estimates for solutions to problems on h, q and χ to obtain the desired bound on v by M.

The main parts of this monograph are this chapter and Chap. 5. In this chapter the finite time existence of solutions to problem (1.1) is established. The existence time is inversely proportional to the quantity Λ2 . In this chapter we are not interested in the length of T but we perform all estimates in such a way that in fact there is no restriction on magnitude of T . Here we derive an a priori estimate and with this we are able to prove the existence of regular solutions in Chap. 12, Sect. 12.2 by the method of the Leray-Schauder theorem. On the other hand, to achieve the estimate in this chapter we need the estimate and existence of weak solutions (see Chap. 3). To infer the estimate for weak solutions in Chap. 3, we had to apply some delicate considerations implied by the fact that the inflow-outflow problem © Springer Nature Switzerland AG 2019 J. Renclawowicz, W. M. Zajączkowski, The Large Flux Problem to the

Navier-Stokes Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-32330-1 4

59

60

4

Local Estimates for Regular Solutions

was analyzed. Then, the corresponding result was shown in Lemma 3.2, in Chap. 3. The first result in this chapter is the energy type inequality for hV (Ω T ) (see (4.42)). To prove this inequality we need the existence and estimates for Green functions to problem (4.19) and (4.20) which are proved in Lemmas 2.11 and 2.12. Since Ω is the cylindrical domain parallel to the x3 -axis we can introduce vorticity χ(x, t) which helps us to increase the regularity of the first two components of velocity, i.e., v  = (v1 , v2 ) (see Lemma 4.9 and inequality (4.56)). We next use the following estimate for the Stokes system (see [Z8], Theorem A), for σ ∈ (1, ∞). vWσ2,1 (Ω T ) +∇pLσ (Ω T ) ≤ c(|f¯|σ,Ω T +dWσ2−1/σ,1−1/2σ (S T ) + v(0)Wσ2−2/σ (Ω) ),

(4.1)

2

where the Stokes system has the form v,t − div T(v, p) = −v  · ∇v − v3 h + f ≡ f¯

in Ω T ,

div v = 0

in Ω T ,

v·n ¯ = 0, ν n ¯ · D(v) · τ¯α + γv · τ¯α = 0, α, 1, 2

on S1T ,

v·n ¯ = d, n ¯ · D(v) · τ¯α = 0, α = 1, 2

on S2T ,

v|t=0 = v(0)

in Ω.

Combining this with inequality (4.56) we conclude (see (4.62)) vW 2,1 (Ω T ) ≤ ϕ(A)H2 + ϕ(D),

(4.2)

5/3

where D describes all data norms, A is the estimate for the weak solution given in (4.13), and H2 = H1 + |h|10/3,Ω t H1 = sup |h|3,Ω + h1,2,Ω t . t

For solutions to problem (4.6) we derive the estimate hV (Ω t ) ≤ ϕ(D1 , V)Λ2 (t), where D1 (t) = |d1 |3,6,S t , V(t) = |∇v|3,2,Ω t . Then for the problem (4.9) for two-dimensional vorticity we obtain inequality (4.54)

4

Local Estimates for Regular Solutions

61

χ2V (Ω t ) ≤ c(1 + A2 )(|h3 |23,∞,Ω t + |F3 |26/5,2,Ω t +v  2W 1,1/2 (Ω t ) + v  25/6,2,∞,Ω t + |χ(0)|22,Ω ). 2

Next the rot-div problem (4.57) implies the inequality (4.60) |v  2V 1 (Ω t ) ≤ ϕ(D1 , V)Λ22 (t) +c(1 + A2 )(|h3 |23,∞,Ω t + h3 2V (Ω t ) + |F3 |26/5,2,Ω t +|χ(0)|22,Ω + v  2W 1,1/2 (Ω t ) + v  25/6,2,∞,Ω t ). 2

Consider the Stokes problem (4.63) where the nonlinear term v · ∇v is expressed in the form v  · ∇ v + v3 · h and estimated by (4.64), (4.65) as follows |v · ∇v|5/3,Ω T ≤ |v  · ∇v|5/3,Ω T + |v3 h|5/3,Ω T ≤ A(v  V 1 (Ω T ) + |h|10/3,Ω T ). Then, for sufficiently small Λ2 we obtain the estimate (see (4.62) and Lemma 4.10) vW 2,1 (Ω T ) ≤ M, 5/3

where M depends on H2 and data. Using this for the Stokes problem (4.63) we increase regularity of v to obtain (see (4.73)) vW 2,1 (Ω T ) ≤ Φ(H2 , D),

(4.3)

2

where D contains norms of data functions. Using for imbedding

5 3

≤ σ ≤

10 3

the

H2 ≤ chWσ2,1 (Ω T ) and applying (4.3) and (4.1) for the problem (4.6) we attain, for sufficiently small parameter Λ2 , the estimate (4.75) of the form: hWσ2,1 (Ω T ) ≤ cD9 (t),

10 5 ≤σ≤ , 3 3

(4.4)

where D9 is defined by (4.77). This yields (4.4) and the estimate for v ∈ W22,1 (Ω T ) (see Theorem 4.14) for sufficiently small Λ2 (t) and finite time

62

4

Local Estimates for Regular Solutions

t. Let us note that although T in this theorem can be chosen arbitrarily large, on the other hand we require smallness of Λ2 (T ), where  Λ22 (T )

T

= 0

(dx 21,S2 + dt 21,S2 + |f3 |24/3,S2 + |g|26/5,Ω )dt +A sup dx 21,3/2,S2 + |h(0)|22,Ω . t

If we look closely at this formula, we observe that for large T the necessity of smallness implies stronger decay restrictions in norms integrated in time T . Therefore, we need some balance between small Λ2 (T ) and possibly large, but not too large T, in order to satisfy the assumptions for global existence. This is the topic of Chap. 5. Proceeding into details, to obtain a higher regularity for solutions to problem (1.1) we introduce the quantities h = v,x3 ,

q = p,x3 ,

g = f,x3 .

(4.5)

Lemma 4.1 Functions h, q defined in (4.5) are solutions to the system: ht − div T(h, q) = −v · ∇h − h · ∇v + g

in Ω T ,

div h = 0

in Ω T ,

h·n ¯=0

on S1T ,

νn ¯ · D(h) · τ¯α + γh · τ¯α = 0, α = 1, 2,

on S1T ,

hi = −d,xi , i = 1, 2

on S2T ,

h3,x3 = Δ d

on S2T ,

h|t=0 = h(0)

(4.6)

in Ω,

where Δ = ∂x21 + ∂x22 . Proof Equations (4.6)1,2,3,4,7 follow directly from the corresponding equations in (1.1) by differentiation with respect to x3 , because S1 is parallel to the x3 -axis. To show (4.6)5,6 we recall that v3 |S2 = d,

(vi,x3 + v3,xi )|S2 = 0,

i = 1, 2.

(4.7)

Hence, vi,x3 = −d,xi , i = 1, 2 and (4.6)5 holds. From (1.1)2 we have that v3,x3 x3 |S2 = −(v1,x1 x3 + v2,x2 x3 )|S2 = d,x1 x1 + d,x2 x2 = Δ d and (4.6)6 follows. This ends the proof.  

4

Local Estimates for Regular Solutions

63

Lemma 4.2 The function χ = v2,x1 − v1,x2

(4.8)

is a solution to the problem χt + v · ∇χ − h3 χ + h2 w,x1 − h1 w,x2 − νΔχ = F3

in Ω T ,

χ = −vi (ni,xj τ1j + τ1i,xj nj ) + γνvj τ1j + v · τ¯1 (τ12,x1 − τ11,x2 ) ≡ χ∗

on S1T ,

(4.9)

on S2T ,

χ,x3 = 0 χ|t=0 = χ(0)

in Ω,

where F3 = f2,x1 − f1,x2 , w = v3 and the tangent and normal vectors have the form (ϕ0,x , ϕ0,x2 , 0) n ¯ |S 1 = ! 1 , ϕ20,x1 + ϕ20,x2 n ¯ |S2 (ai ) = (−1)i e¯3 , τ¯1 |S2 = (1, 0, 0) ≡ e¯1 ,

(−ϕ0,x2 , ϕ0,x1 , 0) τ¯1 |S1 = ! , ϕ20,x1 + ϕ20,x2

τ¯2 |S1 = (0, 0, 1) ≡ e¯3 ,

i = 1, 2, a1 = −a, a2 = a,

(4.10)

τ¯2 |S2 = (0, 1, 0) ≡ e¯2 .

Proof Differentiating the first equation of (1.1)1 with respect to x2 , the second equation of (1.1)1 with respect to x1 and subtracting yields (4.9)1 . To show (4.9)2 we extend vectors τ¯1 , n ¯ into a neighborhood of S1 . In this neighborhood v  = (v1 , v2 ) can be expressed in the form ¯n ¯. v  = v · τ¯1 τ¯1 + v · n Then χ|S1 = [(v · τ¯1 τ12 + v · nn2 ),x1 − (v · τ¯1 τ11 + v · n ¯ n1 ),x2 ]|S1 = [−¯ n · ∇(v · τ¯1 ) + v · τ¯1 (τ12,x1 − τ11,x2 )]|S1 ,

(4.11)

where we used that v · n ¯ |S1 = 0 and n)−n1 ∂x2 (v·¯ n) = (n2 ∂x1 −n1 ∂x2 )(v·¯ n) = −(τ11 ∂x1 +τ12 ∂x2 )(¯ v ·¯ n) = 0. n2 ∂x1 (v·¯ Utilizing (1.1)3 in (1.1)4 for α = 1 yields νn ¯ · ∇(v · τ¯1 ) − νvi (ni,xj τ1j + τ1i,xj nj ) + γv · τ¯1 = 0.

(4.12)

64

4

Local Estimates for Regular Solutions

Exploiting (4.12) in (4.11) yields (4.9)2 . By the definition of χ and (4.6)5 we have χ,x3 |S2 = (v2,x1 x3 − v1,x2 x3 )|S2 = −(d,x1 x2 − d,x2 x1 )|S2 = 0.  

This ends the proof. From Chap. 3 we have  v2V (Ω t ) ≤ ϕ(sup d1,3,S2 ) t

+

dt 21,6/5,S2 )dt

t 0

+

(|f |26/5,Ω + d21,3,S2 |v(0)|22,Ω

(4.13)

≡ A (t), 2

where ϕ is an increasing positive function.

4.1

A Priori Estimates for Function h = v,x3

To obtain the energy estimate for solutions to problem (4.6), we have to make the Dirichlet boundary condition on S2T homogeneous. For this purpose, we ˜ such that are looking for a function h ˜=0 div h

in Ω,

˜=0 h

on S1 ,

˜ i = −d,x , i = 1, 2, h i

on S2 ,

˜3 = 0 h

on S2 .

(4.14)

Lemma 4.3 Assume that d = (d1 , d2 ), d,x ∈ Wσ1 (S2 ), d,x t ∈ Lσ (S2 ), σ ∈ (1, ∞). Then there exists a solution to problem (4.14) such that ˜ ∈ W 1 (Ω), h ˜ ,t ∈ Lσ (Ω), and h σ

˜ 1,σ,Ω ≤ cd,x 1,σ,S , h 2 ˜ ,t |σ,Ω ≤ c|d,x t |σ,S . |h 2

(4.15)

Proof First we define the functions ¯ i = −(¯ h η1 d1,xi + η¯2 d2,xi ), i = 1, 2, ¯ 3 = 0, h

(4.16)

4.1 A Priori Estimates for Function h = v,x3

65

where η¯i = η¯i (x3 ), i = 1, 2, are smooth cut-off functions such that η¯1 = 1, η¯2 = 0 near S2 (−a) and η¯1 = 0, η¯2 = 1 near S2 (a). S2 (ai , ), i = 1, 2, are introduced above (3.4). We have the compatibility condition 2 

ni |S1 dα,xi = 0, α = 1, 2,

(4.17)

i=1

because n ¯ |S1 does not depend on x3 , so ¯ |S 1 = 0 ⇒ h · n ¯ |S1 = 0 ⇒ h|S2 · n ¯ |S1 = 0, v·n ¯ |S1 = 0 ⇒ v,x3 · n ¯ is a solution to and (4.16) follows from the restriction n3 |S1 = 0. Hence, h the problem ¯ = −(¯ div h η1 Δ d1 + η¯2 Δ d2 )

in Ω,

¯ · τ¯1 = −(¯ η1 d˜1,xj + η¯2 d˜2,xj )τ1j h

on S1 ,

¯ · τ¯2 = 0 h

on S1 ,

¯·n h ¯=0

on S1 ,

¯ i = −dj,x , i = 1, 2, h i

(4.18)

on S2 (aj ), j = 1, 2,

¯3 = 0 h

on S2 ,

where the tangent and normal vectors are introduced in (4.10). ˜ satisfying (4.14) we define function φ such that To construct function h Δφ = −(¯ η1 Δ d1 + η¯2 Δ d2 )

in Ω,

n ¯ · ∇φ = 0

on S,

(4.19)

and functions λ and σ such that − Δλ + ∇σ = 0

in Ω,

div λ = 0

in Ω,

¯ · τ¯β , β = 1, 2, λ · τ¯β = −¯ τβ · ∇φ + h

on S1 ,

λ·n ¯=0

on S1 ,

λj = −∇j φ, j = 1, 2

on S2 ,

λ3 = 0

on S2 .

(4.20)

66

4

Local Estimates for Regular Solutions

Then, in view of (4.18)–(4.20), the function ˜=h ¯ − (λ + ∇φ) h

(4.21)

is a solution to problem (4.14). For solutions to (4.16) we have ¯ 1,σ,Ω ≤ cd,x 1,σ,S , h 2

¯ ,t |σ,Ω ≤ c|d,x t |σ,S , |h 2

(4.22)

where σ ∈ [1, ∞]. There exists a Green function for (4.19) (see Lemma 2.11) such that φ(x, t) =

  2

G(x, y)∂yi (¯ η1 d1,yi + η¯2 d2,yi )dy

Ω i=1

=−

  2

(4.23) ∇yi G(x, y)(¯ η1 d1,yi + η¯2 d2,yi )dy,

Ω i=1

where the compatibility condition (4.17) is used. Then  ∇x φ(x, t) =

∇x ∇yi G(x, y)(¯ η1 d1,yi + η¯2 d2,yi )dy Ω

and by the properties of the singular integrals (see the book of Stein [St]) we have (see the remark at the end of the proof of Lemma 2.11, below (2.11). ∇φ1,σ,Ω ≤ cd,x 1,σ,S2 ,

|∇φ,t |σΩ ≤ c|d,x t |σ,S2 ,

(4.24)

where σ ∈ (1, ∞). Utilizing the existence of the Green function to problem (4.20) (see Lemma 2.12) we have  λi (x, t) = S1

∂Giα ¯ · τ¯α )dS1 + (−¯ τα · ∇φ + h ∂nS1

 S2

∂Gij (−∇j φ)dS2 , ∂nS2

where ∂nSi means the normal exterior derivative to Si , i = 1, 2. Therefore, the estimates hold (see [St]) ¯ 1,σ,Ω ) ≤ cd,x 1,σ,S , λ1,σ,Ω ≤ c(∇φ1,σ,Ω + h 2 ¯ ,t |σ,Ω ) ≤ c|d,x t |σ,S , |λ,t |σ,Ω ≤ (|∇φt |σ,Ω + |h 2

(4.25)

where σ ∈ (1, ∞). From (4.22), (4.24), (4.25), and (4.21) estimate (4.15) holds. This concludes the proof.  

4.1 A Priori Estimates for Function h = v,x3

67

Let us introduce the new function ˜ k = h − h.

(4.26)

Then, k is a solution to the problem ˜ ,t + g ≡ G k,t − div T(h, q) = −v · ∇h − h · ∇v − h

in Ω T ,

div k = 0

in Ω T ,

n ¯ · k = 0, ν n ¯ · D(h) · τ¯α + γh · τ¯α = 0, α = 1, 2,

on S1T , (4.27)

ki = 0, i = 1, 2, h3,x3 = Δ d

on S2T ,

˜ k|t=0 = h(0) − h(0) ≡ k(0)

in Ω,

where g = f,x3 , Δ = ∂x21 + ∂x22 , and v is a solution to problem (1.1). Introducing function δ from (3.7) we see that function w defined by (3.7) also is a solution to problem (3.8). Using that v =w+δ

(4.28)

problem (4.27) takes the form ˜ k,t − div T(h, q) = −w · ∇k − k · ∇v − δ · ∇k − v · ∇h ˜ · ∇v − h ˜ ,t + g ≡ G −h

in Ω T ,

div k = 0

in Ω T ,

n ¯ · k = 0, ν n ¯ · D(h) · τ¯α + γh · τ¯α = 0, α = 1, 2

on S1T ,

ki = 0, i = 1, 2, h3,x3 = Δ d

on S2T ,

k|t=0 = k(0)

(4.29)

in Ω.

Projecting div k on S2 , we see that div k|S2 = k3,x3 |S2 = 0.

(4.30)

Lemma 4.4 Assume that the following quantities D1 (t) = |d1 |3,6,S2t ,



t

Λ21 (t) = 0

V(t) = |∇v|3,2,Ω t ,

(dx 21,2,S2 + d,t 21,S2 + |f3 |24/3,S2 + |g|26/5,Ω )dt +A2 sup d,x 21,3/2,S2 t

68

4

Local Estimates for Regular Solutions

are finite, where A = A(t) is introduced in (4.13). Then, any solution k to problem (4.29) satisfies kV (Ω t ) ≤ ϕ(D1 , V)(Λ1 (t) + |k(0)|2,Ω ),

(4.31)

where ϕ is an increasing positive function which is described precisely by the r.h.s. of (4.41). Proof We shall obtain the energy type estimate for solutions to problem (4.29). Multiplying (4.29)1 by k and integrating over Ω yield   1 d 2 |k| − div T(h, q) · kdx = − w · ∇k · kdx 2 dt 2,Ω Ω Ω    ˜ · kdx − k · ∇v · kdx − δ · ∇k · kdx − v · ∇h 

Ω

Ω



 ˜ · ∇v · kdx − h

− Ω

Ω

˜ ,t · kdx + h Ω

(4.32)



g · kdx ≡ Ω

G · kdx. Ω

Now, we examine the particular terms in (4.32). Integrating by parts, the second term on the l.h.s. takes the form    ν − n ¯ ·T(h, q)·kdS1 − n ¯ ·T(h, q)·kdS2 + D(h)·D(k)dx ≡ I1 +I2 +I3 , 2 Ω S1 S2 where 



I1 = −

n ¯ · T(h, q) · τ¯α k · τ¯α dS1 = γ S1

 = γ|k · τ¯α |22,S1 + γ 



I2 = −

T33 (h, q)k3 dS2 = − S2

 S2

˜ · τ¯α k · τ¯α dS1 , h S1

(2νh3,x3 − q)k3 dS2 S2



˜ 3,x − q)k3 dS2 = −2ν (2ν h 3

=−

h · τ¯α k · τ¯α dS1 S1



˜ 3,x k3 dS2 + h 3 S2

qk3 dS2 . S2

To examine the last integral, we use the third component of (1.1)1 projected on S2 : ˜ 3,x − f3 = −q. dt + v  · ∇ d + dh3 − νΔ d − ν h 3 Using this relation in the last term of I2 , we obtain

(4.33)

4.1 A Priori Estimates for Function h = v,x3





˜ 3,x + f3 )k3 dS2 − (−d,t + νΔ d + ν h 3

qk3 dS2 = S2

69



S2

v  dx k3 dS2 S2



(4.34)

dk32 dS2 ,

+ S2

˜ 3 |S = 0 (see (4.14)) and we do not distinguish where we utilized that h 2 between dependence on S2 (−a) and S2 (a) because it does not have any influence on estimations. We estimate the first expression in (4.34) by ε1 |k3 |24,S2 + c(1/ε1 )(|d,t |24/3,S2 + |Δ d|24/3,S2 + |f3 |24/3,S2 ), ˜ ,x |S = Δ d is used, the second term by where the relation h 3 2 ε2 |k3 |24,S2 + c(1/ε2 )|v  |24,S2 |d,x |22,S2 , and the last one as follows (see the book of Besov et al. [BIN, Ch. 2, Sect. 10])  S2

d1 k32 dS2 ≤ |d1 |3,S2 |k3 |23,S2 ≤ (ε1/3 |∇k3 |22,Ω + cε−5/3 |k3 |22,Ω )|d1 |3,S2 1/3

−5/3

≤ ε3 |∇k3 |22,Ω + cε3

|d1 |63,S2 |k3 |22,Ω .

Employing the above estimates in (4.32), we obtain 1 d 2 ν γ ˜ · τ¯α |2 |k| + |D(k)|22,Ω + |k · τ¯α |22,S1 ≤ εk21,Ω + γ|h 2,S1 2 dt 2,Ω 2 2  2 2 6 2 ˜ 2 + c(1/ε)(|d,t |2 + ch 1,Ω 4/3,S2 + |Δ d|4/3,S2 + |f3 |4/3,S2 + |d1 |3,S2 |k|2,Ω )     +  G · kdx. Ω

Applying the Korn inequality (9.19) and taking ε sufficiently small, we get

+ c(|d,t |24/3,S2

d 2 |k| + νk21,Ω + γ|k · τ¯α |22,S1 ≤ c|d1 |63,S2 |k|22,Ω dt 2,Ω   (4.35)   ˜ 2 )+ . + |Δ d|24/3,S2 + |f3 |24/3,S2 + h G · kdx 1,Ω   Ω

Finally, we shall examine the last term on the r.h.s. of (4.35). To this end, we use the r.h.s. of (4.32). The first term of the r.h.s. of (4.32) vanishes. We estimate the second term by  k 2 |∇v|dx ≤ ε1 |k|26,Ω + c(1/ε1 )|∇v|23,Ω |k|22,Ω . Ω

70

4

Local Estimates for Regular Solutions

To examine the third term on the r.h.s. of (4.32), we use the definition of δ in (3.7) and express it in the form 

 b · ∇k · kdx + Ω

∇ϕ · ∇k · kdx ≡ I1 + I2 . Ω

In view of Sect. 3 and estimates for I1 and I2 as defined in (3.11), we have ˜ 1,Ω k2 |I1 | ≤ c1/6 d 1,Ω and ˜ 3,S + μ+1/3 sup |d˜,x |3,S )k2 . |I2 | ≤ c(ε2 μ−2/3 sup |d| 1,Ω 2 3 2 x3

x3

We estimate the fourth term by ˜2 ε3 |k|26,Ω + c(1/ε3 )|v|26,Ω |∇h| 3/2,Ω , the fifth term by ˜2 , ε4 |k|26,Ω + c(1/ε4 )|∇v|22,Ω |h| 3,Ω the sixth by ˜ ,t |2 ε5 |k|26,Ω + c(1/ε5 )|h 6/5,Ω , and, finally, the last one by ε6 |k|26,Ω + c(1/ε6 )|g|26/5,Ω . Employing the aforementioned consideration in (4.35) and assuming that , ε1 − ε6 are sufficiently small, we obtain d 2 |k| + νk21,Ω + γ|k · τ¯α |22,S1 ≤ c(|d1 |63,S2 + |∇v|23,Ω )|k|22,Ω dt 2,Ω (4.36) ˜ 2 + c(|d,t |2 + |Δ d|2 + |f3 |2 + h 4/3,S2

4/3,S2

1,Ω

4/3,S2

2 ˜ 2 ˜ 2 + v21,Ω h 1,3/2,Ω + |h,t |6/5,Ω + |g|6/5,Ω ).

From (4.36) it follows d (|k|22,Ω exp[νt − c(|d1 |63,6,S2t + |∇v|23,2,Ω t )]) dt ˜ 2 + v2 h ˜ 2 + |f3 |2 + h

≤ c(|d,t |24/3,S2 + |Δ d|24/3,S2

4/3,S2

1,Ω

1,Ω

1,3/2,Ω

4.1 A Priori Estimates for Function h = v,x3

71

2 6 2 ˜ ,t |2 +|h 6/5,Ω + |g|6/5,Ω ) exp[νt − c(|d1 |3,6,S2t + |∇v|3,2,Ω t )].

Integrating the inequality with respect to time yields |k(t)|22,Ω ≤ c exp c(|d1 |63,6,S2t + |∇v|23,2,Ω t )Λ21 (t) + exp[−νt + c(|d1 |63,6,S2t + |∇v|23,2,Ω t )]|k(0)|22,Ω ,

(4.37)

where ˜ 2 Λ21 (t) = |d,t |24/3,2,S t + |Δ d|24/3,2,S t + |f3 |24/3,2,S t + h 1,2,Ω t 2

2

2

2 ˜ 2 ˜ 2 +A h 1,3/2,∞,Ω t + |h,t |6/5,2,Ω t + |g|6/5,2,Ω t 2

and estimate (4.13) was used. In view of (4.15) we simplify Λ1 to the form 

t

Λ21 (t) = 0

(d,x 21,2,S2 + d,t 21,S2 + |f3 |24/3,S2 + |g|26/5,Ω )dt

(4.38)

+ A2 sup d,x 21,3/2,S2 . t

Integrating (4.36) with respect to time and applying (4.37) we derive k2V (Ω t ) ≤ c(|d1 |63,6,S2t + |∇v|23,2,Ω t )[exp(|d1 |63,6,S2t + |∇v|23,2,Ω t )Λ21 (t) + exp(−νt + c(|d1 |63,6,S2t

(4.39)

+ |∇v|23,2,Ω t ))|k(0)|22,Ω ] + Λ21 (t) + |k(0)|22,Ω . Introducing the notation D1 (t) = |d1 |3,6,S2t ,

V(t) = |∇v|3,2,Ω t ,

(4.40)

we can express (4.39) in the form k2V (Ω t ) ≤ c(D16 + V 2 ) exp(D16 + V 2 )Λ21 (t)

(4.41) + c(D16 + V 2 ) exp[−νt + c(D16 + V 2 )]|k(0)|22,Ω + Λ21 (t) + |k(0)|22,Ω .

Inequality (4.41) implies (4.31). This concludes the proof. Corollary 4.5 Since ˜ V (Ω t ) ≤ c(|d,x |2,∞,S t + d,x 1,2,S t ), h 2 2 ˜ |h(0)| 2,Ω ≤ c|d,x (0)|2,S2 ≤ c|d,x |2,∞,S2t ,

 

72

4

Local Estimates for Regular Solutions

we obtain from (4.31) that hV (Ω t ) ≤ ϕ(D1 , V)(Λ1 (t) + |h(0)|2,Ω ),

(4.42)

where Λ1 (t) is defined by (4.38).

4.2

A Priori Estimates for Vorticity Component χ

Now, we need to examine solutions to problem (4.9) to derive the energy type estimate. For this we need homogeneous Dirichlet boundary conditions on S1 . For this purpose we introduce the functions χ ˜ as a solution to the problem χ ˜,t − νΔχ ˜=0

in Ω T ,

χ ˜ = χ∗

on S1T ,

χ ˜,x3 = 0

on S2T ,

χ| ˜ t=0 = 0

(4.43)

in Ω,

where χ∗ is described in (4.9)2 . To show the existence of solutions to (4.43) we need the following compatibility conditions χ∗,x3 = 0

on S¯1 ∩ S¯2 .

(4.44)

To satisfy (4.44)1 we differentiate χ∗ with respect to x3 . It is possible because S1 is the part of the boundary of cylinder Ω which is parallel to the x3 -axis. Moreover, vectors n ¯ |S1 and τ¯1 |S1 do not depend on x3 . Therefore, we need to differentiate the components of velocity only. In χ∗ only two-components of velocity v1 and v2 appear. Differentiating them with respect to x3 , projecting on S2 , and using (4.6)5 , we obtain the compatibility condition (4.44)1 in the form χ∗,x3 |S¯1 ∩S¯2 = −

2   i,j=1

d,xi (ni,xj τ1j + τ1i,xj nj ) + 

γ d,x τ1j ν j (4.45)

+ d,xi τ1i (τ12,x1 − τ11,x2 ) = 0. Then, we can introduce the new function χ = χ − χ, ˜ which is a solution to the problem

4.2 A Priori Estimates for Vorticity Component χ

73

χ,t + v · ∇χ − h3 χ + h2 v3,x1 − h1 v3,x2 − νΔχ = F3 − v · ∇χ ˜ + h3 χ ˜

in Ω T ,

χ = 0

on S1T , (4.46)

χ,x3 = 0

on S2T ,

χ |t=0 = χ(0)

in Ω.

Lemma 4.6 Assume that estimate (4.13) holds. Assume that h3 ∈ L∞ (0, t; L3 (Ω)), F3 ∈ L2 (0, t; L6/5 (Ω)), χ(0) ∈ L2 (Ω), and χ ˜ ∈ L2 (0, t; W21 (Ω)),

χ ˜ ∈ L∞ (0, t; L3 (Ω)).

(4.47)

Then solutions to problem (4.9) satisfy the inequality χ2V (Ω t ) ≤ cA2 (|h3 |23,∞,Ω t + sup |χ| ˜ 23,Ω ) t

+

c(|F3 |26/5,2,Ω t

(4.48)

+ χ ˜ 2V (Ω t ) + |χ(0)|22,Ω ).

Proof First, we examine problem (4.46). Multiply (4.46)1 by χ , integrate over Ω, and use boundary conditions. Then, we have    d 2 |χ |2,Ω + ν|∇χ |22,Ω = h3 χ 2 dx − (h2 v3,x1 − h1 v3,x2 )χ dx dt Ω Ω (4.49)       + F3 χ dx − v · ∇χχ ˜ dx + h3 χχ ˜ dx. Ω

Ω

Ω

We estimate the first and last terms together. We have     h3 χ 2 dx + h3 χ χdx ˜ = h3 χ χdx, Ω

Ω

Ω

which is bounded by ε1 |χ|26,Ω + c(1/ε1 )|h3 |23,Ω |χ|22,Ω . The second term from the r.h.s. of (4.49) is bounded by ε2 |χ |26,Ω + c(1/ε2 )|h|23,Ω |v3,x |22,Ω , and the third one by ε3 |χ |26,Ω + c(1/ε3 )|F3 |26/5,Ω .

74

4

Local Estimates for Regular Solutions

Integrating by parts in the fourth term yields     v · ∇(χχ ˜  )dx − v · ∇χ χdx ˜ = dχχ ˜  dS2 − v · ∇χ χdx ˜ ≡ I1 + I2 , Ω

Ω

S2

Ω

where |I1 | ≤ ε4 |χ |24,S2 + c(1/ε4 )|d|22,S2 |χ| ˜ 24,S2 and ˜ 23,Ω . |I2 | ≤ ε5 |∇χ |22,Ω + c(1/ε5 )|v|26,Ω |χ| Using the above estimates in (4.49) and assuming that ε1 − ε5 are sufficiently small we derive the inequality d 2 |χ |2,Ω + νχ 21,Ω ≤ c(|h3 |23,Ω |χ|22,Ω + |h3 |23,Ω |v3,x |22,Ω dt

(4.50)

+ |d|22,S2 |χ| ˜ 24,S2 + |v|26,Ω |χ| ˜ 23,Ω + |F3 |26/5,Ω ). Integrating (4.50) with respect to time and utilizing (4.13) imply    t χ 2V (Ω t ) ≤ cA2 sup |h3 |23,Ω + sup |χ| ˜ 23,Ω |d|22,∞,S2t |χ| ˜ 24,S2 dt +

t

t

c|F3 |26/5,2,Ω t

|χ(0)|22,Ω .

+

0

(4.51)

˜ and χV (Ω t ) ≤ χ V (Ω t ) + χ ˜ V (Ω t ) we obtain from (4.51) Since χ = χ + χ the inequality (4.48). This concludes the proof.   Since Ω is bounded, restrictions (4.47) are equivalent to the following χ ˜ ∈ L2 (0, t; W21 (Ω)) ∩ L∞ (0, t; L3 (Ω)),

(4.52)

where χ ˜ is a solution to (4.43). Lemma 4.7 (see [Z2, Lemma 6.1], [NZ1, Theorem 1.4]) For solutions to problem (4.43) we have (see Lemma 2.20) |χ| ˜ 3,∞,Ω t ≤ |χ∗ |3,∞,S1t ≤ cv  5/6,2,∞,Ω t , |χ| ˜ 2,∞,Ω t ≤ |χ∗ |2,∞,S1t ≤ cv  1/2,2,∞,Ω t ,

(4.53)





χ ˜ 1,2,Ω t ≤ cχ∗ W 1/2,1/4 (S t ) ≤ cv W 1/2,1/4 (S t ) ≤ cv W 1,1/2 (Ω t ) . 2

1

2

1

2

4.3 Relating v and h: rot–div System

75

Corollary 4.8 In view of (4.53) inequality (4.48) takes the form χ2V (Ω t ) ≤ c(1 + A2 )(|h3 |23,∞,Ω t + |F3 |26/5,2,Ω t + v  2W 1,1/2 (Ω t ) + v  25/6,2,∞,Ω t + |χ(0)|22,Ω ).

(4.54)

2

4.3

Relating v and h: rot–div System

Lemma 4.9 Assume that estimate (4.13) for the weak solution holds (the quantity A was introduced there). We set H12 (t) = sup |h|23,Ω + h21,2,Ω t .

(4.55)

t

We require that H1 (t) and D1 (t), V(t), Λ1 (t) defined in the assumptions of Lemma 4.4 are finite. Assume also that χ(0) ∈ L2 (Ω), h(0) ∈ L2 (Ω), F3 ∈ L2 (0, t; L6/5 (Ω)), and v  ∈ L2 (Ω; H 1/2 (0, t)). Then the following inequality holds v  2V 1 (Ω t ) ≤ ϕ(D1 , V)(Λ21 (t) + |h(0)|22,Ω ) + c(1 + A2 )(H12 + A2 + v  2L2 (Ω;H 1/2 (0,t))

(4.56)

+ |F3 |26/5,2,Ω t + |χ(0)|22,Ω ). Proof Let Ω  be the cross-section of Ω with the plane perpendicular to the x3 -axis and passing through the point x3 ∈ (−a, a). Let S1 be the crosssection of S1 with the same plane. Then S1 is the boundary of Ω  . Therefore, the elliptic system (rot, div) reduces to the problem v1,x2 − v2,x1 = χ

in Ω  ,

v1,x1 + v2,x2 = −h3

in Ω  ,

v · n ¯ = 0

on S1 ,

(4.57)

where x3 is treated as a parameter. For solution to (4.57) we have sup v  L2 (−a,a;H 1 (Ω  )) + v  L2 (0,t;L2 (−a,a);H 2 (Ω  )) t

≤ c(χV (Ω t ) + h3 V (Ω t ) ).

(4.58)

76

4

Local Estimates for Regular Solutions

Moreover, (4.42) implies h V (Ω t ) ≤ cϕ(D1 , V)(Λ1 (t) + |h(0)|2,Ω ).

(4.59)

Inequalities (4.58) and (4.59) imply v  2V 1 (Ω t ) ≤ ϕ(D1 , V)(Λ21 (t) + |h(0)|22,Ω ) + c(1 + A2 )(|h3 |23,∞,Ω t + h3 2V (Ω t ) + |F3 |26/5,2,Ω t (4.60) + |χ(0)|22,Ω + v  2W 1,1/2 (Ω t ) + v  25/6,2,∞,Ω t ). 2

Using that v  W 1,1/2 (Ω t ) = v  L2 (0,t;H 1 (Ω)) + v  L2 (Ω;H 1/2 (0,t)) 2

≤ A + v  L2 (Ω;H 1/2 (0,t)) and the interpolation v  5/6,2,∞,Ω t ≤ εv  1,2,∞,Ω t + c(1/ε)|v  |2,∞,Ω t , where the second norm is bounded by A, in the r.h.s. of (4.60), implies (4.56). This concludes the proof.   Next, we shall obtain an estimate for vW 2,1 (Ω t ) in terms of some norms 5/3 of h. Lemma 4.10 Assume that D1 (t) = |d1 |3,6,S2t , 

t

Λ21 (t) = 0

(dx 21,2,S2 + d,t 21,S2 + |f3 |24/3,S2 + |g|26/5,Ω )dt +A2 sup d,x 21,3/2,S2 . t

Assume that H1 is introduced in (4.55) and H2 = H1 + |h|10/3,Ω t , Λ2 = Λ1 + |h(0)|2,Ω , D3 = A(1 + A)H2 + ϕ(A) + D2 , D2 = (1 + A)(|F3 |6/5,2,Ω t + |χ(0)|2,Ω ) + |f |5/3,Ω t +dW 7/5,7/10 (S t ) + v(0)W 4/5 (Ω) , 5/3

2

5/3

and M = cγD3 with γ > 1, where Λ2 , D2 , D3 , and H2 are finite. Let Λ2 be so small that

4.3 Relating v and h: rot–div System

77

Aϕ(D1 , M )Λ2 ≤ (γ − 1)M,

(4.61)

where ϕ already appears in (4.56). Then vW 2,1 (Ω t ) ≤ M.

(4.62)

5/3

Proof Consider the problem (1.1) written in the form of the Stokes system v,t − div T(v, p) = −v  · ∇v − v3 h + f ≡ f¯

in Ω T ,

div v = 0

in Ω T ,

v·n ¯ = 0, ν n ¯ · D(v) · τ¯α + γv · τ¯α = 0, α, 1, 2

on S1T ,

v·n ¯ = d, n ¯ · D(v) · τ¯α = 0, α = 1, 2

on S2T ,

v|t=0 = v(0)

(4.63)

in Ω.

Applying Lemma 2.21, we have that v  L10 (Ω T ) ≤ cv  V 1 (Ω T ) .

(4.64)

Then |v  · ∇v|5/3,Ω T ≤ Av  V 1 (Ω T ) ,

(4.65)

|v3 h|5/3,Ω T ≤ A|h|10/3,Ω T .

In view of the aforementioned estimates and the following estimate for the Stokes system (see [Z8], Theorem A) vWσ2,1 (Ω T ) + ∇pLσ (Ω T ) ≤ c(|f¯|σ,Ω T + dWσ2−1/σ,1−1/2σ (S T ) + v(0)Wσ2−2/σ (Ω) ),

(4.66)

2

where σ ∈ (1, ∞), we obtain the following estimate vW 2,1 (Ω t ) ≤ cA(v  V 1 (Ω t ) + |h|10/3,Ω t ) 5/3

+ c(|f |5/3,Ω t + dW 7/5,7/10 (S t ) + v(0)W 4/5 (Ω) ). 5/3

2

(4.67)

5/3

Taking into account (4.56), it yields vW 2,1 (Ω t ) ≤ cA[ϕ(D1 , V)(Λ1 + |h(0)|2,Ω ) 5/3



+ (1 + A)(H1 + A + v L2 (Ω;H 1/2 (0,t)) + |F3 |6/5,2,Ω t + |χ(0)|2,Ω )] (4.68) + cA|h|10/3,Ω t + c(|f |5/3,Ω t + dW 7/5,7/10 (S t ) + v(0)4/5,5/3,Ω ). 5/3

2

78

4

Local Estimates for Regular Solutions

Using the definition of H2 , Λ2 , and D2 , inequality (4.68) can be expressed in the form vW 2,1 (Ω t ) ≤ cA[ϕ(D1 , V)Λ2 + (1 + A)H2 + ϕ(A) 5/3

+ (1 + A)v  L2 (Ω;H 1/2 (0,t)) ] + cD2 .

(4.69)

By the interpolation (see Lemma 2.24) v  L2 (Ω;H 1/2 (0,t)) ≤ εv  W 2,1 (Ω t ) + c(1/ε)|v  |2,Ω t , 5/3

we obtain from (4.69) the inequality vW 2,1 (Ω t ) ≤ cAϕ(D1 , V)Λ2 + cA(1 + A)H2 + ϕ(A) + cD2 .

(4.70)

5/3

We have V(t) = |∇v|3,2,Ω t and the imbedding (see Lemma 2.25) |∇v|3,2,Ω t ≤ cvW 2,1 (Ω t ) . 5/3

Introducing the notation vW 2,1 (Ω t ) = M (t) we can express (4.70) in the 5/3 form M ≤ cAϕ(D1 , M )Λ2 + cD3 ,

(4.71)

where D3 is introduced in the assumptions of this lemma. Assume that M = cγD3 , γ > 1, and Λ2 is so small that Aϕ(D1 , cγD3 )Λ2 ≤ (γ − 1)cD3 . This implies (4.61) and (4.62), so we end the proof.

(4.72)  

Remark 4.11 Since all quantities in (4.72) depend on time through the timeintegral norms we can satisfy (4.72) or (4.62) for any time assuming that time-integral norms in Λ2 are sufficiently small. Hence for chosen small Λ2 we can take T large assuming sufficiently fast decay in the time of integrand functions. Then the bound (4.62) will imply the existence of solutions to problem (1.1) (see [Z8]) for large time with some small data. To increase the regularity of solutions to (1.1) for given H2 (t) we need the following. Lemma 4.12 Let H2 (t), t ≤ T , be finite. Let the assumptions of Lemma 4.10 hold. Let f ∈ L2 (Ω t ), t ≤ T , v(0) ∈ H 1 (Ω). Then any solution to problem (1.1) satisfies the inequality

4.3 Relating v and h: rot–div System

79

vW 2,1 (Ω t ) + |∇p|2,Ω t ≤ ϕ(D1 , D4 H2 + D5 )(D4 H2 + D5 )Λ2 2

+ D6 H22 + D7 H2 + D8 ≡ ϕ0 (D1 , D4 , D5 , D6 , D7 , D8 , H2 ) ≡ ϕ0 (H2 ),

(4.73)

t ≤ T,

where we express ϕ0 in more explicit form using that M = D4 H2 + D5 , D4 = cγ(1 + A)A, D5 = cγ(ϕ(A) + D2 ), D6 = D42 + (1 + A)D4 , D7 = D4 (ϕ(A) + D5 + D2 )D5 + D4 D5 + (1 + A)D5 , D8 = (ϕ(A) + D5 + D2 )D5 + c(|f |2,Ω t + v(0)1,Ω + dW 2−1/2,1−1/4 (S t ) ). 2

2

Proof We have |v  · ∇v|2,Ω t ≤ |v  |10,Ω t |∇v|5/2,Ω t ≤ cv  V 1 (Ω t ) vW 2,1 (Ω t ) ≤ cv  V 1 (Ω t ) M. 5/3

Applying notation of Lemma 4.10 we obtain from (4.56) the estimate v  V 1 (Ω t ) ≤ ϕ(D1 , M )Λ2 + c(1 + A)H1 + ϕ(A) + M + D2 . Then |v  · ∇v|2,Ω t ≤ [ϕ(D1 , M )Λ2 + c(1 + A)H1 + ϕ(A) + M + D2 ]M. Next, we calculate |v3 h|2,Ω t ≤ |v3 |5,Ω t |h|10/3,Ω t ≤ cvW 2,1 (Ω t ) |h|10/3,Ω t ≤ cM H2 . 5/3

Hence, solutions to Stokes system (4.63) satisfy vW 2,1 (Ω t ) + |∇p|2,Ω t ≤ ϕ(D1 , M )M Λ2 2

+ (ϕ(A) + M + D2 )M + c(1 + A)M H2 + c(|f |2,Ω t + v(0)1,Ω + d2−1/2,2,S2t ) ≡ ϕ(D1 , D4 H2 + D5 )(D4 H2 + D5 )Λ2 + [D42 + (1 + A)D4 ]H22 + [D4 (ϕ(A) + D5 + D2 ) + D4 D5 + (1 + A)D5 ]H2 + (ϕ(A) + D5 + D2 )D5 + c(|f |2,Ω t + v(0)1,Ω + dW 2−1/2,1−1/4 (S t ) ) 2

2

≡ ϕ(D1 , D4 H2 + D5 )(D4 H2 + D5 )Λ2 + D6 H22 + D7 H2 + D8 .

(4.74)

80

4

Local Estimates for Regular Solutions

 

This implies (4.73) and concludes the proof. Here we prove a priori estimates for solutions to system (4.6).

Lemma 4.13 Let the assumptions of Lemmas 4.4 and 4.10 hold. Let 2−2/σ f ∈ L2 (Ω T ), v(0) ∈ H 1 (Ω), g ∈ Lσ (Ω T ), h(0) ∈ Wσ (Ω), and 5/3 ≤ σ ≤ 10/3. Let Λ2 (t), t ≤ T , be defined in the assumptions of Lemma 4.10. Then for sufficiently small Λ2 the following estimate holds hWσ2,1 (Ω t ) + |∇q|σ,Ω t ≤ cγD9 ,

γ > 1, t ≤ T,

(4.75)

where D9 is defined by (4.77) and Λ2 is so small that (4.80) holds. Proof For solutions to problem (4.6) we have hWσ2,1 (Ω t ) + |∇q|σ,Ω t ≤ c(|v · ∇h|σ,Ω t + |h · ∇v|σ,Ω t ) + cD9 ,

(4.76)

where σ ∈ (1, ∞) and D9 (t) =

2 

d,xi Wσ2−1/σ,1−1/2σ (S t ) + Δ dWσ1−1/σ,1/2−1/2σ (S t ) 2

i=1

2

(4.77)

+ |g|σ,Ω t + h(0)2−2/σ,σ,Ω . Now, we examine the first two norms on the r.h.s. of (4.76). By interpolation from Lemma 2.15 we have |v · ∇h|σ,Ω t ≤ |v|σλ1 ,Ω t |∇h|σλ2 ,Ω t 1 1 ≤ |v|10,Ω t (ε1−κ hWσ2,1 (Ω t ) + ε−κ |h|2,Ω t ) ≡ I1 , 1 1

where 1/λ1 + 1/λ2 = 1 and  κ1 =

   5 5 5 3 − +1 = +1 = σ σλ2 σλ1 4

because σλ1 = 10.

Hence −3/4

1/4

I1 ≤ ε2 hWσ2,1 (Ω t ) + cε2 1/4

where we used that ε2

|v|410,Ω t |h|2,Ω t ,

1/4

= ε1 |v|10,Ω t . Similarly, by Lemma 2.15, we have

|h · ∇v|σ,Ω t ≤ |h|σλ1 ,Ω t |∇v|σλ2 ,Ω t 2 2 ≤ |∇v|10/3,Ω t (ε1−κ hWσ2,1 (Ω t ) + cε−κ |h|2,Ω t ) ≡ I2 , 3 3

4.3 Relating v and h: rot–div System

81

where  κ2 =

5 5 − σ σλ1



1 5 3 = = 2 2σλ2 4

because σλ2 =

10 . 3

Hence −3/4

1/4

I2 ≤ ε4 hWσ2,1 (Ω t ) + cε4 1/4

|∇v|410/3,Ω t |h|2,Ω t ,

1/4

where ε4 = ε3 |∇v|10/3,Ω t . Employing the estimates in (4.76), using (4.73) and (4.42) in the form hV (Ω t ) ≤ ϕ(D1 , ϕ0 (H2 ))Λ2 , we obtain hWσ2,1 (Ω t ) + |∇q|σ,Ω t ≤ ϕ1 (H2 )Λ2 + cD9 .

(4.78)

Employing the imbedding (see Lemma 2.25) H2 (t) ≤ chWσ2,1 (Ω t ) ≡ H,

σ≥

5 , 3

we derive from (4.78) the inequality for H0 = H + |∇q|σ,Ω t H0 ≤ ϕ1 (D1 , · · · , D8 , H0 )Λ2 + cD9 .

(4.79)

Setting that H0 ≤ cγD9 ,

γ>1

and assuming that Λ2 is so small that ϕ(D1 , · · · , D8 , cγD9 )Λ2 ≤ c(γ − 1)D9

(4.80)  

estimate (4.75) holds. This ends the proof. Finally, we summarize the results of this section. Theorem 4.14 Consider systems (1.1)–(1.4) and (4.6). Assume that 2−2/σ

• initial data are such that v(0) ∈ H 1 (Ω), h(0) ∈ Wσ (Ω), • external forces satisfy f ∈ L2 (Ω t ), g ∈ L6/5,2 (Ω t ) ∩ Lσ (Ω t ),

F3 ∈ L6/5,2 (Ω t ), f3 ∈ L4/3,2 (S2t ),

82

4

• flux

is such that d

∈

1/2,1/4

W2

Local Estimates for Regular Solutions

(S2t ), d

dt ∈ L2 (0, t; H 1 (S2 )), σ ∈ [5/3, 10/3], dx 1−1/σ,1/2−1/2σ dx x ∈ Wσ (S2t ),

∈

2 L∞ (0, t; W3/2 (S2 )), 2−1/σ,1−1/2σ

∈ Wσ

(S2t ),

Let the following quantities be finite A2 (t) = ϕ(d1,3,∞,S2t )(|f |26/5,2,Ω t + d21,3,2,S2t + dt 21,6/5,2,S2t ) + |v(0)|22,Ω , Λ22 (t) = d,x 21,2,S2t + d,t 21,2,S2t + |f3 |24/3,2,S2t + |g|26/5,2,Ω t +A2 d,x 21,3/2,∞,S2t + |h(0)|22,Ω , D2 (t) = (1 + A(t))(|F3 |6/5,2,Ω t + |χ(0)|2,Ω ) + |f |5/3,Ω t + dW 7/5,7/10 (S t ) 5/3

2

+v(0)W 4/5 (Ω) , 5/3

D1 (t) = |d1 |3,6,S2t , D8 (t) = |f |2,Ω t + v(0)1,Ω + dW 2−1/2,1−1/4 (S t ) , 2

D9 (t) =

2 

2

d,xi Wσ2−1/σ,1−1/2σ (S t ) + Δ dWσ1−1/σ,1/2−1/2σ (S t ) 2

i=1

2

+|g|σ,Ω t + h(0)2−2/σ,σ,Ω , where 5/3 ≤ σ ≤ 10/3. Assume that quantity Λ2 is so small that inequalities (4.61) and (4.80) hold. Then the following estimates are valid hWσ2,1 (Ω t ) + |∇q|σ,Ω t ≤ cD9 ,

5/3 ≤ σ ≤ 10/3

(4.81)

vW 2,1 (Ω t ) + |∇p|2,Ω t ≤ ϕ(D1 , D2 , D9 , A) + cD8 .

(4.82)

and 2

Chapter 5

Global Estimates for Solutions to Problem on (v, p)

Abstract In this chapter we prove the global existence of solutions (v, p) and (h = v,x3 , q = p,x3 ) by applying the step by step in time argument. From Lemma 5.2 we have v(kT )H 1 (Ω) ≤ Q1 (T ) + exp(−νkT )v(0)H 1 (Ω) , and Lemma 5.4 implies h(kT )H 1 (Ω) ≤ Q2 (T ) + exp(−νkT )h(0)H 1 (Ω) , where Q1 (T ), Q2 (T ) depend on data. The above inequalities imply that the initial data for v(kT ), h(kT ) are bounded by constants independent of k, so in an interval (kT, (k + 1)T ) we have solvability of v, h in W22,1 (Ω × (kT, (k + 1)T )). Hence the norms are bounded by quantities independent of k. These inequalities follow from considerations from Chaps. 6 to 9. Smallness of Λ2 (t), where 

t

Λ2 (t) = kT

(dx 2H 1 (S2 ) + dt 2H 1 (S2 ) + |f3 |2L4/3 (S2 ) + |g|2L6/5 (Ω) )dt

+A sup dx 2W 1 t

3/2 (S2 )

+ |h(kT )|2L2 (Ω) ,

t ∈ (kT, (k + 1)T ),

is necessary to repeat the proofs of Chap. 4 in interval (kT, (k + 1)T ). In this chapter we also derive the precise criteria of smallness for parameter Λ2 (T ) in terms of data.

In this chapter we derive estimates guaranteeing the existence of global solutions. For this we have to show that v(t)1,Ω and h(t)1,Ω do not increase with time. In reality our considerations are made step by step in time so we have to show that

© Springer Nature Switzerland AG 2019 J. Renclawowicz, W. M. Zajączkowski, The Large Flux Problem to the

Navier-Stokes Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-32330-1 5

83

84

5

Global Estimates for Solutions to Problem on (v, p)

v((k + 1)T )1,Ω ≤ v(kT )1,Ω , h((k + 1)T )1,Ω ≤ h(kT )1,Ω ,

(5.1)

where k ∈ N0 and T will be chosen later. In other words we show the existence of positive constants C1 and C2 such that v(t)1,Ω ≤ C1 ,

(5.2)

h(t)1,Ω ≤ C2 .

This is an extension of results of Chap. 4 step by step in time. For this, we need T sufficiently large while Λ2 (T ) remains small enough to fulfill results of Chap. 4. In order to justify (5.1) it is necessary to show first the differential inequality (5.11) and for this we examine artificial elliptic problem (8.1), where the first equation is identity. Next, we analyze properties of such class of functions that satisfy (5.1), so we exploit (8.8) and (8.18). To show inequality (5.1)2 we have to consider problem (7.1) as well as the artificial elliptic problems (7.2) and (9.1)—where the first equations are identities. Hence we derive inequalities (7.9) and (9.37) which yield (5.34). Then from inequalities (5.13) and (5.36) we deduce (5.1) for sufficiently large T. Therefore the role of Chaps. 7–9 is auxiliary to Chap. 4 and this chapter. First we obtain an estimate for velocity in the interval [kT, (k + 1)T ] with initial data v(kT ) ∈ H 1 (Ω). Introduce the quantity V =

ν γ |D(v)|22,Ω + |v · τ¯α |22,S1 . 4 2

(5.3)

Lemma 5.1 Let (v, p) be a solution to (1.1). Let σ be defined in (3.7) and (8.11). Assume that v ∈ L∞ (Ω), dx ∈ L2 (S2 ), d ∈ H 3/2 (S2 ), dt ∈ L2 (S2 ), and δ ∈ H 1 (Ω). Then d V + νV ≤ c(|v|2∞,Ω + |dx |82,S2 )V + c(d23/2,S2 + |dt |22,S2 dt

(5.4)

+ δ21,Ω + |dx |10 2,S2 ). Proof Multiplying (1.1)1 by −div T(v, p) and integrating over Ω yield  vt · div T(v, p)dx + |div T(v, p)|22,Ω

− Ω





(5.5)

v · ∇vdiv T(v, p)dx +

= Ω

f div T(v, p)dx. Ω

5

Global Estimates for Solutions to Problem on (v, p)

85

Applying the H¨ older and Young inequalities to the r.h.s. terms of (5.5) gives  −

1 vt · div T(v, p)dx + |div T(v, p)|22,Ω ≤ |v · ∇v|22,Ω + |f |22,Ω . 2 Ω

(5.6)

The first term on the l.h.s. of (5.6) equals 

ν div (vt · T(v, p))dx + − 2 Ω

 D(vt ) · D(v)dx ≡ I1 . Ω

Expressing v in the local coordinate system in a neighborhood of the boundary ¯, v = vα τ¯α + vn n

vα = v · τ¯α ,

vn = v · n ¯,

the first term in I1 takes the form   − ni vjt Tij (v, p)dS1 − S1

α = 1, 2,

ni vjt Tij (v, p)dS2 S2

 =−

ni (vτα ,t ταj + vn,t nj )Tij (v, p)dS1 S1

 −

ni (vτα ,t ταj + vn,t nj )Tij (v, p)dS2 ≡ I2 + I3 , S2

where 



I2 = −

vτα ,t ni Tij ταj dS1 = γ S1

vτα ,t vτα dS1 = S1

γ d |v · τ¯α |22,S1 2 dt

We have to emphasize that the summation convention over repeated indices is assumed. Next,   vn,t ni nj Tij (v, p)dS2 = − d,t T33 (v, p)dS2 . I3 = − S2

S2

Using quantity (5.3) we obtain from (5.6) the inequality d 1 V + |div T(v, p)|22,Ω ≤ |v · ∇v|22,Ω + dt 2

 d,t T33 (v, p)dS2 + |f |22,Ω . S2

We estimate the first term on the r.h.s. of (5.7) by |v · ∇v|22,Ω ≤ |v|2∞,Ω |∇v|22,Ω ≡ I1 .

(5.7)

86

5

Global Estimates for Solutions to Problem on (v, p)

Applying (8.2), we have I1 ≤ c|v|2∞,Ω (|D(v)|22,Ω + |dx |22,S2 ) ≤ c|v|2∞,Ω (V + |dx |22,S2 ). In view of the H¨older and Young inequalities the second term on the r.h.s. of (5.7) is estimated by ε|T33 (v, p)|22,Ω + c(1/ε)|dt |22,S2 ≡ I2 . By the trace theorem (see Lemma 2.16) we have |T33 (v, p)|22,S2 ≤ c(v22,Ω + p21,Ω ) ≡ I3 . In view of Lemma 8.4 we have ˜ 2 + |dx |2 ) I3 ≤ c(|div T(v, p)|22,Ω + d23/2,S2 + d 1,Ω 2,S2 ≤ c(|div T(v, p)|22,Ω + d23/2,S2 ). Employing the above inequalities in (5.7) and using that ε is sufficiently small, we have d V + ν|div T(v, p)|22,Ω ≤ c|v|2∞,Ω (V + |dx |22,S2 ) dt

(5.8)

+ c(|dt |22,S2 + d23/2,S2 ). We examine the following term from the r.h.s. of (5.8), J1 ≡ |v|2∞,Ω |dx |22,S2 . By interpolation (see [BIN, Sect. 15]) we have 3/2

1/2

J1 ≤ cv2,Ω |v|2,Ω |dx |22,S2 ≤ εv22,Ω + c(1/ε)|v|22,Ω |dx |82,S2 ≡ J2 . Using (8.2) and (8.31) one obtains J2 ≤ ε(|div T(v, p)|22,Ω + d23/2,S2 ) + c(1/ε)(|D(v)|22,Ω + |dx |22,S2 )|dx |82,S2 ≤ ε(|div T(v, p)|22,Ω + d23/2,S2 ) + c(1/ε)(V + |dx |22,S2 )|dx |82,S2 .

Employing the above estimates in (5.8) and utilizing that ε is sufficiently small yield

5

Global Estimates for Solutions to Problem on (v, p)

87

d V + |div T(v, p)|22,Ω dt

(5.9)

≤ c(|v|2∞,Ω + |dx |82,S2 )V + c(d23/2,S2 + |dt |22,S2 + |dx |10 2,S2 ). Continuing, (8.8) yields V = |D(v)|22,Ω +

2 

|v · τ¯α |22,S1 ≤ cv21,Ω

α=1

(5.10)

≤ c(|div T(v, p)|22,Ω + δ21,Ω + |dx |22,S2 ). Using (5.10) in (5.8) implies d V + νV ≤ c(|v|2∞,Ω + |dx |82,S2 )V dt +

c(d23/2,S2

+

|dt |22,S2

(5.11) +

δ21,Ω

+

|dx |10 2,S2 ).  

This implies (5.4) and concludes the proof. To prove the global existence we need the following result.

Lemma 5.2 Assume that for V defined in (5.3), V (0) < ∞ and the following conditions are fulfilled: 

(k+1)T

(|v(t)|2∞,Ω + |dx (t)|82,S2 )dt ≤ ν

sup k∈N0

kT



(k+1)T

R12 (T ) = sup

k∈N0

T , 2

kT

(d(t)23/2,S2 + |dt |22,S2

(5.12)

+ δ(t)21,Ω + |dx (t)|10 2,S2 )dt < ∞,     (k+1)T  R22 (T ) = exp sup |v(t)|2∞,Ω + |dx (t)|82,S2 dt < ∞. k∈N0

kT

Then, for k ∈ N0 and t ∈ [kT, (k + 1)T ],   kT R12 (T )R22 (T ) % + exp − ν $ V (kT ) ≤ V (0), 2 1 − exp − νT 2   R12 (T )R22 (T ) (k + 1)T % + exp − ν $ V (t) ≤ V (0). 2 1 − exp − νT 2

(5.13)

88

5

Global Estimates for Solutions to Problem on (v, p)

Proof From (5.4) we have    t d  2 8  (|v(t )|∞,Ω + |dx |2,S2 )dt ) V exp(νT − dt kT ≤ c(d23/2,S2 + |dt |22,S2 + δ21,Ω + |dx |10 2,S2 )    t  2 8   (|v(t )|∞,Ω + |dx |2,S2 )dt . · exp νt −

(5.14)

kT

Integrating with respect to time from kT to t ∈ (kT, (k + 1)T ] yields 



V (t) ≤ c exp 

t

− νt +

(|v(t kT

t

· kT



)|2∞,Ω

+

|dx |82,S2 )dt



  (d(t )23/2,S2 + |dt |22,S2 + δ(t )21,Ω + |dx (t )|10 2,S2 ) exp(νt )dt (5.15)





t

+ exp − ν(t − kT ) +

(|v(t



kT

)|2∞,Ω

+ |d

x

|82,S2 )dt

 V (kT ).

Simplifying (5.15) we get 

t

V (t) ≤ c exp 

(|v(t kT

t

· kT



)|2∞,Ω

+

|dx |82,S2 )dt



 (d(t )23/2,S2 + |dt |22,S2 + δ(t )21,Ω + |dx (t )|10 2,S2 )dt





t

+ exp − ν(t − kT ) +

(|v(t kT



)|2∞,Ω

+ |d (t x



)|82,S2 )dt

(5.16)

 V (kT ).

Setting t = (k + 1)T and assuming that −νT + 2



(k+1)T kT

(|v(t)|2∞,Ω + |dx (t)|82,S2 )dt ≤ 0

(5.17)

and using notation (5.12) we derive  V ((k + 1)T ) ≤

R22 (T )R12 (T )

Hence, by iteration, it follows

+ exp

 νT − V (kT ). 2

(5.18)

5

Global Estimates for Solutions to Problem on (v, p)



R22 (T )R12 (T ) % + exp $ V (kT ) ≤ 1 − exp − νT 2

89

 kT −ν V (0). 2

(5.19)

 (k + 1)T V (0), 2

(5.20)

Employing (5.19) in (5.16) yields V (t) ≤

R22 (T )R12 (T ) % + exp $ 1 − exp − νT 2

 −ν

where t ∈ [kT, (k +1)T ], k ∈ N0 . Then (5.13) holds. This concludes the proof.   Now, we examine condition (5.12)1 . Our aim is to show that (5.12)1 does not imply any restrictions on the magnitude of d. By interpolation the first term under the integral on the l.h.s. of (5.12)1 is bounded by 

(k+1)T

c kT

3/2 1/2 v2,Ω |v|2,Ω dt

≤

≤

1/2 sup |v(t)|2,Ω t

1/2 c sup |v(t)|2,Ω T 1/4 t





(k+1)T

3/2

v(t)2,Ω dt

kT

3/4

(k+1)T

v(t)22,Ω dt

.

kT

Now our aim is to show that the quantity 

(k+1)T

v(t)22,Ω dt

I= kT

remains bounded by the same constant for T as large as we want assuming that norms of dx , dt , g, and f3 are sufficiently small (see comments on Λ2 in the Remark 4.11). Lemma 5.3 Let the assumptions of Theorem 4.14 hold. Then v(2),2,Ω t ≤ ϕ(D1 , · · · , D9 , Λ2 ),

t ≤ T,

(5.21)

where D1 , D2 , D3 , and Λ2 are defined in Lemma 4.10, D6 , D7 , and D8 in Lemma 4.12 and D9 in (4.77). Then estimate (5.21) holds for any finite time T if Λ2 is sufficiently small for all t ≤ T . Proof From (4.13) we have  v2V (Ω t ) ≤ ϕ(sup d1,3,S2 ) t

+

dt 21,6/5,S2 )dt

t 0

+

(|f |26/5,Ω + d21,3,S2 |v(0)|22,Ω

≡A . 2

(5.22)

90

5

Global Estimates for Solutions to Problem on (v, p)

Recall that D1 (t) = |d1 |3,6,S2t ,  t (dx 21,2,S2 + dt 21,S2 + |f3 |24/3,S2 + |g|26/5,Ω )dt Λ21 (t) = 0

+A2 sup dx 21,3/2,S2 , t

Λ22 (t)

=

Λ21 (t)

+ |h(0)|22,Ω ,

D2 (t) = (1 + A)(|F3 |6/5,2,Ω t + χ(0)6/5,2,Ω ) + |f |5/3,Ω t + dW 7/5,7/10 (S t ) 2

5/3

+v(0)4/5,5/3,Ω . From (4.70) and the imbedding introduced below (4.70) we obtain vW 2,1 (Ω t ) ≤ cAϕ(D1 , vW 2,1 (Ω t ) )Λ2 + D3 , 5/3

(5.23)

5/3

where H12 = sup |h|23,Ω + h21,2,Ω t , t

H22 = H12 + |h|210/3,Ω t ,

D3 = cA(1 + A)H2 + ϕ(A) + cD2 ≡ D4 H2 + D5 . Assuming that Λ2 is so small that cAϕ(D1 , γD3 )Λ2 + D3 ≤ γD3 ,

(5.24)

we obtain that vW 2,1 (Ω t ) ≤ γD3 ,

γ > 1.

(5.25)

5/3

Next, (4.74) yields v(2),2,Ω t + |∇p|2,Ω t ≤ ϕ(D1 , D4 H2 + D5 )(D4 H2 + D5 )Λ2 +D6 H22 + D7 H2 + D8 ,

(5.26)

where D6 = D42 + (1 + A)D4 ,

D7 = D4 (ϕ(A) + D2 ) + D4 D5 + (1 + A)D5 ,

D8 = (ϕ(A) + D2 + D5 )D5 + c(|f |2,Ω t + v(0)1,Ω + d(2−1/2),2,S2t ).

5

Global Estimates for Solutions to Problem on (v, p)

91

Introducing H = h2,σ,Ω t , H0 = H + |∇q|σ,Ω t we obtain (4.79), where D9 appeared in (4.78). For Λ2 so small that ϕ(D1 , · · · , D8 , γD9 )Λ2 ≤ (γ − 1)D9 ,

(5.27)

we have the bound H0 ≤ γD9 ,

γ > 1.

(5.28)

Since H2 ≤ cH0 we obtain from (5.26) the estimate v(2),2,Ω t ≤ ϕ(D1 , · · · , D9 , Λ2 ),

(5.29)

where smallness of Λ2 holds for any time if time integrals of dx , dt , f3 , and g are sufficiently small. This concludes the proof.   Next, estimates for H  (see (6.2)) and for |h3,t |2,Ω (see (7.9)) will be derived. Introduce the quantities D12 (t) = c(1 + |d|22,S2 + v22,Ω )dx 21,Ω + cdt 21,S2 , D22 (t) = c(|v|2∞,Ω + v21,3,Ω ). Then (6.7) takes the form d  H + νH  ≤ D12 + D22 H  + c|h3t |22,S2 . dt

(5.30)

To find a bound for the last term on the r.h.s. of (5.30) we need estimates from Chap. 7. Introduce the quantities D32 (t) = c(dx t 21,S2 + dt 21,S2 + |dtt |24/3,S2 + |f3t |24/3,S2 + vt 21,Ω +|gt |26/5,Ω ) + c(v21,Ω |dx t |22,S2 + vt 21,Ω |dx |22,S2 ), D42 (t) = c(|v|2∞,Ω + |∇v|23,Ω ), D52 (t) = c(|dt |22,S2 + |vt |23,Ω ). Then (7.9) yields the inequality d |ht |22,Ω + νht 21,Ω ≤ D32 (t) + D42 |ht |22,Ω + D52 h21,Ω . dt

(5.31)

92

5

Global Estimates for Solutions to Problem on (v, p)

In view of (6.8) we derive d |ht |22,Ω + νht 21,Ω ≤ D32 + D52 dx 21,S2 + D42 |ht |22,Ω + D52 H  . dt

(5.32)

Introducing the quantity H0 = H  + |ht |22,Ω

(5.33)

inequalities (5.30) and (5.32) imply d  H + νH0 ≤ D12 + D32 + D52 dx 21,S2 + (D22 + D42 + D52 )H0 dt 0 ≡

D62

+

(5.34)

D72 H0 .

Lemma 5.4 Assume that H0 (0) < ∞ and 

(k+1)T

D72 (t)dt ≤ ν

sup k∈N0

kT



(k+1)T

D62 (t)dt < ∞,

P12 (T ) = sup

k∈N0



P22 (T )

= exp

T , 2

kT





(k+1)T

D72 (t)dt

sup k∈N0

(5.35) < ∞.

kT

Then   kT P12 (T )P22 (T ) $ νT % + exp − ν H0 (0), 2 1 − exp − 2 (5.36)   (k + 1)T P12 (T )P22 (T )   $ % + exp − ν H0 (0) for t ∈ (kT, (k + 1)T ]. H0 (t) ≤ 2 1 − exp − νT 2 H0 (kT ) ≤

Proof From (5.34) we have d  [H exp(νt − dt 0



t kT

D72 (t )dt )] ≤ D62 exp(νt −



t

D72 (t )dt ).

(5.37)

kT

Integrating (5.37) with respect to time from t = kT to t ∈ (kT, (k + 1)T ] we have

5

Global Estimates for Solutions to Problem on (v, p)

 H0 (t) ≤ exp(

t

D72 (t )dt )

kT





t

D62 (t )dt

kT t

+ exp(−(t − kT ) +

93

(5.38)

D72 (t )dt )H0 (kT ).

kT

Setting t = (k + 1)T in (5.38) and applying iteration we get (5.36)1 . Next (5.36)2 follows easily. This concludes the proof.  

Chapter 6

Global Estimates for Solutions to Problem on (h, q)

Abstract In this chapter we examine the system for (h, q), where h = v,x3 , q = p,x3 in Ω kT = Ω × (kT, (k + 1)T ) for k ∈ N0 : ht − div T(h, q) = −v · ∇h − h · ∇v + g

in Ω kT ,

div h = 0

in Ω kT ,

h·n ¯=0

on S1kT ,

νn ¯ · D(h) · τ¯α + γh · τ¯α = 0, α = 1, 2

on S1kT ,

hi = −d,xi , i = 1, 2,

on S2kT ,

h3,x3 = Δ d

on S2kT ,

h|t=0 = h(kT )

in Ω.

We derive the differential inequality for H  = 14 |D(h)|2L2 (Ω) + γ2 |h · τ¯α |2L2 (S1 ) . We use some additional facts on solutions to problem for (h, q) that are established in Chap. 9: estimates for h in H 1 (Ω) and H 2 (Ω). Denoting Ω kT = Ω × (kT, (k + 1)T ) and SikT = Si × (kT, (k + 1)T ), i = 1, 2, we consider problem (4.6) in the form ht − div T(h, q) = −v · ∇h − h · ∇v + g

in Ω kT ,

div h = 0

in Ω kT ,

h·n ¯=0

on S1kT ,

νn ¯ · D(h) · τ¯α + γh · τ¯α = 0, α = 1, 2

on S1kT ,

hi = −d,xi , i = 1, 2,

on S2kT ,

h3,x3 = Δ d

on S2kT ,

h|t=0 = h(kT )

(6.1)

in Ω,

© Springer Nature Switzerland AG 2019 J. Renclawowicz, W. M. Zajączkowski, The Large Flux Problem to the

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Global Estimates for Solutions to Problem on (h, q)

where k ∈ N0 . Introduce the notation H =

1 γ |D(h)|22,Ω + |h · τ¯α |22,S1 . 4 2

(6.2)

Lemma 6.1 Assume that d ∈ L2 (S2 ), dt ∈ H 1 (S2 ), dx ∈ H 1 (S2 ), h3t ∈ L2 (S2 ), f3 ∈ L2 (Ω), g ∈ L2 (Ω), v ∈ W31 (Ω) L∞ (Ω). Then d  H + νH  ≤ c(|v|2∞,Ω + v21,3,Ω )H  dt +c(1 + |d|22,S2 + |v|2∞,Ω + v21,3,Ω )dx 21,S2

(6.3)

+c(|h3t |22,S2 + dt 21,S2 ) + c(|f3 |22,Ω + |g|22,Ω ). Proof Multiplying (6.1)1 by −div T(h, q) and integrating over Ω yield  ht div T(h, q)dx + |div T(h, q)|22,Ω

− Ω





(6.4)

(v · ∇h + h · ∇v)div T(h, q)dx +

= Ω

gdiv T(h, q)dx. Ω

The first integral on the l.h.s. takes the form 1 d γ d |D(h)|22,Ω + |h · τ¯α |22,S1 − 4 dt 2 dt

 nj Tij (h, q)hit dS2 ≡ I1 . S2

In view of boundary conditions (6.1)5,6 the last term in I1 equals 



−



T3i (h, q)hit dS2 = S2

T3α (h, q)dxα t dS2 − S2

T33 (h, q)h3t dS2 . S2

Employing the above expressions in (6.4) and applying the H¨ older and Young inequalities, we have d  H + ν|div T(h, q)|22,Ω ≤ ε(|T33 |22,S2 + |T3α |22,S2 ) dt + c(1/ε)(|h3t |22,S2 + |dx t |22,S2 ) + c(|v · ∇h|22,Ω + |h · ∇v|22,Ω + |g|22,Ω ). The first term on the r.h.s. of (6.5) is bounded by ε(h22,Ω + q21,Ω ).

(6.5)

6

Global Estimates for Solutions to Problem on (h, q)

97

To absorb it by the second term on the l.h.s. of (6.5) we recall estimate (9.37), where F = div T(h, q). Then, for sufficiently small ε, inequality (6.5) yields d  H + νh22,Ω + q21,Ω ≤ c(dx 21,S2 + |dt |22,S2 dt + v  21,Ω dx 21,S2 + |f3 |22,Ω + |v3 |22,S2 |h3 |22,Ω + |d|22,S2 dx 21,S2 ) (6.6) + c(|h3t |22,S2 + |dx t |22,S2 + |v · ∇h|22,Ω + |h · ∇v|22,Ω + |g|22,Ω ). Simplifying, we express (6.6) in the form d  H + νH  ≤ c(1 + |d|22,S2 + v21,Ω + |v|2∞,Ω + v21,3,Ω )dx 21,S2 dt + cdt 21,S2 + c(|v3 |22,S2 + |v|2∞,Ω + v21,3,Ω )H 

(6.7)

+ c|h3t |22,S2 + c|f3 |22,Ω + c|g|22,Ω , where (9.30) was used as follows h21,Ω ≤ c(EΩ (h) + dx 21,S2 ). This implies (6.3) and concludes the proof.

(6.8)  

Chapter 7

Estimates for ht

Abstract We analyze the system for ht in Ω kT = Ω ×(kT, (k +1)T ) that we obtain by differentiating problem on h = v,x3 . We need the Korn inequality and H 1 -estimates for solutions to problem on ht . To achieve this, we use an ˜ t as a solution to the problem auxiliary function h ˜t = 0 div h

in Ω,

˜t · n ¯=0 h

on S1 ,

˜ 3t = 0 ˜ it = −dtx , i = 1, 2, h h i

on S2

˜ t . Finally, in Corollary 7.4 we derive the and also the problem for kt = ht − h differential inequality of the form d H + νH ≤ αH + L, where dt 1 γ H = |D(h)|2L2 (Ω) + |h · τ¯α |2L2 (S1 ) + |ht |2L2 (Ω) , 4 2 α, L depend on data and norms of velocity v, γ is the slip coefficient, and ν is the viscosity coefficient. From (6.1) we obtain the following problem for ht , denoting Ω kT = Ω × (kT, (k + 1)T ) and SikT = Si × (kT, (k + 1)T ), i = 1, 2, htt − div T(ht , qt ) = −vt · ∇h − v · ∇ht − ht · ∇v − h · ∇vt + gt div ht = 0

in Ω kT , in Ω kT ,

(7.1)

ht · n ¯ = 0, ν n ¯ · D(ht ) · τ¯α + γht · τ¯α = 0, α = 1, 2

on S1kT ,

© Springer Nature Switzerland AG 2019 J. Renclawowicz, W. M. Zajączkowski, The Large Flux Problem to the

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7

hti = −dxi t , i = 1, 2, ht3,x3 = Δ dt

Estimates for ht

on S2kT ,

ht |t=0 = ht (kT )

in Ω.

We need the Korn inequality and H 1 -estimates for solutions to problem (7.1). For this purpose, similarly as in Chap. 9, we consider the problem div T(ht , qt ) = Ft ,

in Ω,

div ht = 0,

in Ω,

ht · n ¯ = 0, ν n ¯ · D(ht ) · τ¯α + γht · τ¯α = 0, α = 1, 2,

on S1 ,

hti = −dtxi , i = 1, 2, h3t,x3 = Δ dt ,

on S2 ,

(7.2)

which holds on solutions of (7.1) because (7.2)1 is identity. This means that Ft ≡ div T(ht , qt ). ˜ t as Next, we repeat the considerations from Chap. 9. Introduce function h a solution to the problem ˜t = 0 div h

in Ω,

˜t · n ¯=0 h

on S1 ,

˜ 3t = 0 ˜ it = −dtx , i = 1, 2, h h i

on S2 .

(7.3)

Repeating the proof of Lemma 9.1 we have the following. Lemma 7.1 Assume that dx t ∈ H 1/2+α (S2 ), α = 0, 1. Then there exists a ˜ t ∈ H 1+α (Ω) and solution to problem (7.3) such that h ˜ t 1+α,Ω ≤ cdx t 1/2+α,S . h 2

(7.4)

In order to follow (9.17) we introduce the function ˜t, kt = ht − h

(7.5)

which satisfies ˜ t) div T(kt , qt ) = Ft − div D(h

in Ω,

div kt = 0

in Ω,

kt · n ¯ = 0, ν n ¯ · D(kt ) · τ¯α + γkt · τ¯α

(7.6)

˜ t ) · τ¯α + γ h ˜ t · τ¯α ), α = 1, 2 = −(ν n ¯ · D(h

on S1 ,

˜ 3,x t kit = 0, i = 1, 2, k3t,x3 = Δ dt − h 3

on S2 .

7

Estimates for ht

101

Repeating the proof of Lemma 9.2 for the problem (7.6) we obtain the following. Lemma 7.2 Assume that EΩ (kt ) = |D(kt )|22,Ω , div kt = 0, kt · n ¯ |S1 = 0,  1 ˜ ˜ kit |S2 = 0, i = 1, 2, k3t,x3 = Δ dt − h3t,x3 ∈ L2 (S2 ), h3t ∈ H (Ω). Then ˜ 3t 2 + |Δ dt − h ˜ 3t,x |2 ). kt 21,Ω ≤ c(EΩ (kt ) + h 1,Ω 3 2,S2

(7.7)

Using (7.5) in (7.7) implies ˜ t 2 + |Δ dt − h ˜ 3t,x |2 ). ht 21,Ω ≤ c(EΩ (ht ) + h 1,Ω 3 2,S2

(7.8)

Lemma 7.3 Assume that dt ∈ H 2 (S2 ), dtt ∈ L4/3 (S2 ), dx ∈ L4 (S2 ), vt ∈ L2 (S2 ), v ∈ L∞ (Ω) ∩ W31 (Ω), f3t ∈ L4/3 (S2 ), gt ∈ L6/5 (Ω). Then for sufficiently regular solutions to (7.1) the inequality holds d |ht |22,Ω + νht 21,Ω ≤ c(dx t 21,S2 + |dtt |24/3,S2 + |f3t |24/3,S2 + dt 21,S2 dt (7.9) + |g |2 ) + c(v2 |d  |2 + v 2 |d  |2 ) t 6/5,Ω

+

c(|v|2∞,Ω

1,Ω

+

x t 2,S2

|∇v|23,Ω )|ht |22,Ω

+

t 2,S2

c(|dt |22,S2

x 4,S2

+ |vt |23,Ω )h21,Ω .

Proof Multiplying (7.1)1 by ht and integrating over Ω yield   1 d |ht |22,Ω − div T(ht , qt ) · ht dx = − vt · ∇h · ht dx 2 dt Ω Ω    − v · ∇ht · ht dx − ht · ∇v · ht dx − h · ∇vt · ht dx 

Ω

Ω

Ω

gt · ht dx.

+ Ω

The second term on the l.h.s. of (7.10) equals 

 −

Tij (ht , qt )htj,xi dx ≡ I1 .

(Tij (ht , qt )htj ),xi dx + Ω

Ω

The second term in I1 equals 1 2

 |D(ht )|2 dx, Ω

(7.10)

102

7

Estimates for ht

and integration by parts with employing the boundary conditions in the first term in I1 yields   ni Tij (ht , qt )htj dS1 − ni Tij (ht , qt )htj dS2 − 

S1

=−

S2



n ¯ · D(ht ) · τ¯α ht · τ¯α dS1 − S1



 |ht · τ¯α |2 dS1 −

= S1

T3j (ht , qt )htj dS2 S2

T33 (ht , qt )ht3 dS2 ≡ I2 . S2

To examine the second integral in I2 we use the time derivative of the third component of (1.1) projected on S2 , T33 (ht , qt ) = dtt + v  dx t + vt dx + v3 h3t + v3t h3 − f3t . Hence,      T33 (ht , qt ) · h3t dS2  ≤ ε|h3t |24,S2 + c(1/ε)(|dtt |24/3,S2 + |f3t |24/3,S2 )  S2       2 v dx t h3t dS2 + vt dx h3t dS2 + v3 h3t dS2 + v3t h3 h3t dS2 . + S2

S2

S2

S2

Next, we estimate the last four terms on the r.h.s. of the above inequality. The third term is bounded by ε|h3t |24,S2 + c(1/ε)|v  |24,S2 |dx t |22,S2 ≤ εh3t 21,Ω + c(1/ε)v  21,Ω |dx t |22,S2 , the fourth by ε|h3t |24,S2 + c(1/ε)|vt |22,S2 |dx |24,S2 ≤ εh3t 21,Ω + c(1/ε)vt 22,S2 |dx |24,S2 , the fifth by |v3 |∞,Ω |h3t |22,S2 ≤ c|v3 |∞,Ω h3t 1,Ω |h3t |2,Ω ≤ εht 21,Ω + c(1/ε)|v3 |2∞,Ω |h3t |22,Ω ,

and finally, the sixth by ε|h3t |24,S2 + c(1/ε)|v3t |22,S2 |h3 |24,S2 ≤ εh3t 21,Ω + c(1/ε)|dt |22,S2 h3 21,Ω . Employing the above estimates in (7.10), using (7.8), and assuming that ε is sufficiently small, we derive the inequality

7

Estimates for ht

103

d |ht |22,Ω + νht 21,Ω + γ dt

 ˜ t 2 |ht · τ¯α |2 dS1 ≤ c(h 1,Ω S1



2 ˜ 3t,x |2 + |dtt |2 + |Δ dt − h 3 2,S2 4/3,S2 + |f3t |4/3,S2 )

+ c(v  21,Ω |dx t |22,S2 + |vt |22,S2 |dx |24,S2 + |v|2∞,Ω |h3t |22,Ω   + |dt |22,S2 h3 21,Ω ) − vt · ∇h · ht dx − v · ∇ht · ht dx Ω



Ω



−

ht · ∇v · ht dx − Ω

(7.11)

 h · ∇vt · ht dx +

Ω

gt · ht dx. Ω

Now, we estimate the last five integrals from (7.11). The fifth term from the end of (7.11) is bounded by ε|ht |26,Ω + c(1/ε)|vt |23,Ω |∇h|22,Ω . The fourth term equals   1 1 v · ∇h2t dx = v·n ¯ h2t dS2 ≡ I. 2 Ω 2 S2 Hence, |I| ≤ |v|∞,Ω |ht |22,S2 ≤ c|v|∞,Ω ht 1,Ω |ht |2,Ω ≤ εht 21,Ω + c(1/ε)|v|2∞,Ω |ht |22,Ω . We estimate the third term from the end of (7.11) by ε|ht |26,Ω + c(1/ε)|∇v|23,Ω |ht |22,Ω . Integration by parts in the second term from the end of (7.11) yields 





h·n ¯ vt · ht dS2 + S2

h·n ¯ vt · ht dS1 − S1

h · ∇ht · vt dx,

(7.12)

Ω

where the second integral vanishes because h · n ¯ |S1 = 0 and the first integral is expressed in the form  h·n ¯ (−vαt dxα t − dt vαt,xα )dS2 = I. S2

Using fractional derivatives we have |I| ≤ εh21,Ω + c(1/ε)v21,Ω dt 22,S2 .

104

7

Estimates for ht

The third integral in (7.12) is bounded by ε|∇ht |22,Ω + c(1/ε)|vt |23,Ω |h|26,Ω . Finally, the last term on the r.h.s. of (7.11) is estimated by ε|ht |26,Ω + c(1/ε)|gt |26/5,Ω . Employing the above estimates in (7.11), utilizing (7.4), and using that ε is sufficiently small yield d |ht |22,Ω + νht 21,Ω ≤ c(dx t 21/2,S2 + |dtt |24/3,S2 + |f3t |24/3,S2 dt + dt 21,S2 + |gt |26/5,Ω ) + c(v21,Ω |dx t |22,S2 +

vt 21,Ω |dx |22,S2 )

+

|vt |23,Ω h21,Ω

+

+

c(|v|2∞,Ω |ht |22,Ω

+

(7.13)

|dt |22,S2 h21,Ω

|∇v|23,Ω |ht |22,Ω ).

Simplifying (7.13) implies (7.9). This concludes the proof.

 

Corollary 7.4 Let the assumptions of Lemma 7.3 be satisfied. Then holds the following inequality d H + νH ≤ αH + L, dt

(7.14)

where H=

1 γ |D(h)|22,Ω + |h · τ¯α |22,S1 + |ht |22,Ω , 4 2

α(t) = c(|dt |22,S2 + |v|2∞,Ω + v21,3,Ω + |vt |23,Ω ),



t 0

L(t )dt ≤ c[(1 + |d|22,∞,S2t )dx 21,2,S2t +(|v|2∞,2,Ω t + v21,3,2,Ω t + |vt |22,S2t )dx 21,2,∞,S2t +(|v|2∞,2,Ω t + v21,3,2,Ω t )dx t 21/2,2,∞,S2t +dtt 24/3,2,S2t + dx t 21/2,2,S2t + |f3 |22,Ω t +|f3t |24/3,S2t + |g|22,Ω t + |gt |22,Ω t ].

Proof Let H = H  + |ht |22,Ω ,

(7.15)

7

Estimates for ht

105

where H =

1 γ |D(h)|22,Ω + |h · τ¯α |22,S1 4 2

is defined in (6.2) Then (6.3), (7.8), (7.9), and (9.30) imply the differential inequality d H + νH ≤ c(|dt |22,S2 + |v|2∞,Ω + v21,3,Ω + |vt |23,Ω )H dt +c(1 + |dt |22,S2 + |d|22,S2 + |v|2∞,Ω + v21,3,Ω + |vt |22,S2 )dx 21,S2 +(|v|2∞,Ω

+

(7.16)

v21,3,Ω )dx t 21/2,S2

+c(|dtt |24/3,S2 + dx t 21/2,S2 + |f3 |22,Ω + |f3t |24/3,S2 + |g|22,Ω + |gt |22,Ω ). Therefore we can formulate (7.16) in the form of (7.14) and this concludes the proof.  

Chapter 8

Auxiliary Results: Estimates for (v, p)

Abstract The aim of this chapter is to derive the estimates and prove the existence for stationary Stokes system for (v, p) with the slip boundary conditions in H 2 (Ω) space for velocity v. −νdiv D(v) + ∇p = f

in Ω,

div v = 0

in Ω,

v·n ¯=0

on S1 ,

νn ¯ · D(v) · τ¯α + γv · τ¯α = 0, α = 1, 2

on S1 ,

v·n ¯=d

on S2 ,

n ¯ · D(v) · τ¯α = 0, α = 1, 2

on S2 .

The proof is divided into two steps. First using the energy inequality and the Korn inequality we show that v ∈ H 1 (Ω). Next by applying the partition of unity we increase the regularity of v showing that v ∈ H 2 (Ω).

Consider the problem − νdiv D(v) + ∇p = f

in Ω,

div v = 0

in Ω,

v·n ¯=0

on S1 ,

νn ¯ · D(v) · τ¯α + γv · τ¯α = 0, α = 1, 2

on S1 ,

v·n ¯=d

on S2 ,

n ¯ · D(v) · τ¯α = 0, α = 1, 2

on S2 ,

(8.1)

where we assume that f ≡ −νdiv D(v) + ∇p, so the first equation is identity.

© Springer Nature Switzerland AG 2019 J. Renclawowicz, W. M. Zajączkowski, The Large Flux Problem to the

Navier-Stokes Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-32330-1 8

107

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8

Auxiliary Results: Estimates for (v, p)

Lemma 8.1 Assume that EΩ (v) = |D(v)|22,Ω < ∞, and that (8.1)2 –(8.1)6 hold. Assume also that dx ∈ L2 (S2 ). Then v21,Ω ≤ c(|D(v)|22,Ω + |d,x |22,S2 ).

(8.2)

Proof For sufficiently smooth v, we examine the integral     2 |D(v)|2 dx = (vi,xj + vj,xi )2 dx = 2 vi,x dx + 2 vi,xj vj,xi dx j Ω

Ω



=2 Ω

2 vi,x dx j

+2

The boundary integral yields   ni vi,xj vj dS1 = − 

S1

ni vi,xj vj dS2 = 

ni vi,xj vj dS.

ni,xj vi vj dS1 ,

(v3,xα vα + v3,x3 v3 )dS2 



(d,xα vα − vα,xα v3 )dS2 = 2 S2

(8.3)

S

S2

=

Ω

S1



S2

Ω



n ¯ · vv3 dL,

d,xα vα dS2 + S2

L

where it is used that ∂S2 = L, where L is the edge. Since L ⊂ S1 condition (8.1)3 implies that the integral over L disappears. Then (8.3) gives |∇v|22,Ω ≤ |D(v)|22,Ω + c|d,xα vα |2,S2 + c|v · τ¯α |22,S1

(8.4)

From the Korn lemma, i.e., Lemma 2.22, for any positive constants δ, M and for non-axially symmetric domain Ω it holds |v|22,Ω ≤ δ|∇v|22,Ω + M |D(v)|22,Ω ,

(8.5)

where δ can be chosen as small as we need. By interpolation (see [BIN, Ch. 3, Sect. 15]) inequality (8.4) implies |∇v|22,Ω ≤ c(|D(v)|22,Ω + |d,x |22,S2 + |v|22,Ω ),

(8.6)

where x = (x1 , x2 ). Employing (8.5) with sufficiently small δ yields v21,Ω ≤ c(|D(v)|22,Ω + |d,x |22,S2 ). This by the density argument gives (8.2) and concludes the proof.

(8.7)  

8

Auxiliary Results: Estimates for (v, p)

109

Lemma 8.2 Consider problem (8.1). Assume that f dx ∈ L2 (S2 ), δ ∈ H 1 (Ω), where δ is defined in (8.11). Then v21,Ω + |v · τ¯α |22,S1 ≤ c(δ21,Ω + |f |26/5,Ω + |dx |22,S2 ).

∈

L6/5 (Ω),

(8.8)

Proof First we transform problem (8.1) to make the Dirichlet boundary conditions homogeneous. Introduce a function α such that α|S(−a) = d1 ,

α|S(a) = d2

and the vector b = α¯ e3 ,

e¯3 = (0, 0, 1).

We define a function u so that u = v − b which is a solution to the problem div u = −div b = −α,x3

in Ω,

u·n ¯=0

on S,

(8.9)

so the boundary condition for u is homogeneous. Since u is not divergence free we introduce a function ϕ as a solution to the Neumann problem Δϕ = −div b

in Ω,

n ¯ · ∇ϕ = 0  ϕdx = 0.

on S,

(8.10)

Ω

Introducing the new function w = u − ∇ϕ = v − (b + ∇ϕ) = v − δ

(8.11)

we see that div w = 0,

w·n ¯ |S = 0

(8.12)

because div δ = 0, δ·n ¯ |S = d. Then function w is a solution to the problem

(8.13)

110

8

Auxiliary Results: Estimates for (v, p)

− νdiv D(w + δ) + ∇p = f

in Ω,

div w = 0

in Ω,

w·n ¯=0

on S,

νn ¯ · D(w + δ) · τ¯α + γ(w + δ) · τ¯α = 0, α = 1, 2,

on S1 ,

n ¯ · D(w + δ) · τ¯α = 0, α = 1, 2

on S2 .

(8.14)

Multiplying (8.14)1 by w, integrating over Ω, and using (8.14)2 and the boundary conditions yield    ν D(w + δ) · D(w)dx + γ (w · τ¯α + δ · τ¯α )w · τ¯α dS1 = f wdx. (8.15) 2 Ω S1 Ω Hence, we have  ν|D(w)|22,Ω

+ γ|w

· τ¯α |22,S1

≤

c(|D(δ)|22,Ω

+ |δ

· τ¯α |22,S1 ) +

f · wdx.

(8.16)

f · (v − δ)dx.

(8.17)

Ω

Transformation (8.11) implies  |D(v)|22,Ω + |v · τ¯α |22,S1 ≤ c(|D(δ)|22,Ω + |δ · τ¯α |22,S1 ) +

Ω

Then Lemma 8.1 and applications of the H¨older and Young inequalities to the last term on the r.h.s. of (8.17) yield v21,Ω + |v · τ¯α |22,S1 ≤ c(|D(δ)|22,Ω + |δ · τ¯α |22,S1 +|δ|26,Ω + |d,x |22,S2 + |f |26/5,Ω ).  

This implies (8.8) and concludes the proof.

Lemma 8.3 Assume that f ∈ L2 (Ω), v ∈ H (Ω), p ∈ L2 (Ω), and d ∈ H 3/2 (S2 ). Then 1

v2,Ω + p1,Ω ≤ c(|f |2,Ω + v1,Ω + |p|2,Ω + d3/2,S2 ).

(8.18)

Proof To prove the Lemma we use the partition of unity introduced in Chap. 2. In this Lemma the (H 2 , H 1 ) regularity for (v, p) is only shown. The existence can be proved by the regularizer technique and the Fredholm theorem. Let k ∈ M and let v (k) = vζ (k) , p(k) = pζ (k) , f (k) = f ζ (k) . Then problem (8.1) for v (k) , p(k) takes the form

8

Auxiliary Results: Estimates for (v, p)

111

− νΔv (k) + ∇p(k) = f (k) − 2∇v∇ζ (k) − vΔζ (k) + p∇ζ (k) , div v (k) = v · ∇ζ (k) .

(8.19)

Hence Lemma 2.13 implies v (k) 2,Ω + p(k) 1,Ω ≤ c(|f (k) |2,Ω + v1,Ω (k) + |p|2,Ω (k) ),

(8.20)

where Ω (k) = supp ζ (k) . For k ∈ N1 we choose ζ (k) ∈ S1 and introduce the new coordinate system y = (y1 , y2 , y3 ) with origin at ξ (k) . Hence, we have the transformation y = Y (x). Assume that S1 is described locally by the equation y3 = Fk (y1 , y2 ), where ξ (k) = (0, 0, 0) in coordinates y. Introduce the new coordinates zα = yα , α = 1, 2,

z3 = y3 − F (y1 , y2 ),

and define the transformation by z = Φ(y). In these coordinates ¯k = {z : z3 = 0}. Let Ψ = Φ ◦ Y . Then ∇x = Ψx ∇z = Ψxj ∂z and S1 ∩ Ω j ˆ ∇z = Ψx |x=Ψ −1 (z) · ∇z . Let u ˆ(k) (z) = u(Ψk−1 (z)) and u(k) = u ˆ(k) (z)ζˆ(k) (z). k For k ∈ N1 and in view of the above-mentioned notation problem (8.1) takes the form ˆ 2z )v (k) + (∇z − ∇ ˆ z )p(k) − ν∇2z v (k) + ∇z p(k) = −ν(∇2z − ∇ ˆ ζˆ(k) − vˆ(k) ∇ ˆ ζˆ(k) + pˆ(k) ∇ ˆ ζˆ(k) , ˆ v (k) ∇ + f (k) − 2∇ˆ ˆ z )v (k) + vˆ(k) ∇ ˆ z ζˆ(k) , div z v (k) = (div z − div (k)

ˆ¯ )|z3 =0 , v3 |z3 =0 = (v (k) e¯3 − v (k) n (k) | ≡ ν e¯3 · D(v (k) ) · e¯α |z3 =0 − νvα,z 3 z3 =0

(8.21) (k) νv3,τα |z3 =0

(k)

ˆ¯ · D(v (k) ) · τˆ¯α − γˆ = ν e¯3 · D(v (k) ) · e¯α − νv3,τα − ν n v (k) · τˆ¯α ˆ z ζˆ(k) + vˆj ∇ ˆ z ζˆ(k) )ˆ + νn ˆ i (ˆ vi ∇ τjα , α = 1, 2, z3 = 0. j i In view of Lemma 2.13 we have v (k) 2,R3+ + p(k) 1,R3+ ≤ cλ(v (k) 2,R3+ + p(k) 1,R3+ ) + c|f (k) |2,R3+ + c(ˆ v 1,R3+ ∩supp ζ (k) + ˆ p1,R3+ ∩supp ζ (k) ),

(8.22)

where k ∈ N1 . Since S2 is flat we do not have to pass to variables z in the neighborhood of an interior point of S2 . Therefore, for k ∈ N2 , problem (8.1) can be expressed

112

8

Auxiliary Results: Estimates for (v, p)

in coordinates y with the origin at ξ (k) which are derived from coordinates x by translation only because the origin of coordinates x is located at the middle of the x3 -axis. Then problem (8.1) has the form −ν∇2y v (k) + ∇y p(k) = f (k) − 2∇v∇ζ (k) + p∇ζ (k) , div y v (k) = v · ∇ζ (k) ,

(8.23)

(k)

v3 |y3 =0 = d(k) (k) νvα,y | = −γv · τ¯α + νni (vi ∇yj ζ (k) + vj ∇yi ζ (k) )ταj , 3 y3 =0

where n ¯ |y3 =0 = e¯3 , τ¯1 |y3 =0 = e¯1 , and τ¯2 |y3 =0 = e¯2 . Moreover, the transformation between coordinates x and y is expressed by y = x + a, where a = (a1 , a2 , a3 ) is a constant vector. In view of Lemma 2.13 from Solonnikov [S1] the following estimate for solutions to (8.23) holds v (k) 2,R3+ + p(k) 1,R3+ ≤ c(|f (k) |2,Ω + v1,R3+ ∩supp ζ (k) + |p|2,R3+ ∩supp ζ (k) + d(k) 3/2,R2 ).

(8.24)

Let k ∈ N3 and ξ (k) ∈ L. Then we introduce new variables z = (z1 , z2 , z3 ) such that neighborhood Ω (k) of ξ (k) is transformed in such a way that L locally is determined by z1 = z2 = 0 so it is the z3 -axis. Then Ψk (S1 ∩ Ω (k) ) = {z : z2 = 0} and Ψk (S2 ∩ Ω (k) ) = {z : z1 = 0}. Therefore problem (8.1) in these variables takes the form ˆ 2 )v (k) + (∇z − ∇ ˆ z )p(k) − ν∇2z v (k) + ∇z p(k) = −ν(∇2z − ∇ z ˆ ζˆ(k) − vˆ(k) ∇ ˆ 2 ζˆ(k) + pˆ(k) ∇ ˆ ζˆ(k) ≡ f (k) ˆ v (k) ∇ + f (k) − 2∇ˆ 1

(8.25)

in the dihedral right angle Dπ/2 located between two planes z1 = 0 and z2 = 0, ˆ z )v (k) + vˆ(k) ∇ ˆ ζˆ(k) ≡ g (k) div z v (k) = (div z − div 1 (k)

(k)

on z1 = 0,

(k)

on z2 = 0,

ˆ¯ )|z1 =0 + d(k) ≡ d1 v1 |z1 =0 = (v (k) · e¯1 − v (k) · n (k)

in Dπ/2 ,

ˆ¯ )|z2 =0 ≡ d2 v2 |z2 =0 = (v (k) · e¯2 − v (k) · n

8

Auxiliary Results: Estimates for (v, p)

113

(k) (k) ˆ¯ · D(v (k) ) · τˆ¯α )|z1 =0 vα,z | = (¯ e1 D(v (k) ) · e¯α − v1,zα − n 1 z1 =0

ˆ z ζˆ(k) + vˆj ∇ ˆ z ζˆ(k) )ˆ +ˆ ni (ˆ vi ∇ ταj |z1 =0 ≡ h(k) α , α = 2, 3, j i (k)

(k) ˆ¯ · D(v (k) ) · τˆ¯α )] vα,z | = [−v2,zα + (¯ ez · D(v (k) ) · e¯α − n 2 z2 =0

ˆ z ζˆ(k) + vˆj ∇ ˆ z ζˆ(k) ) · τˆαj |z =0 ≡ l(k) , α = 1, 3. +ˆ ni (ˆ vi ∇ α j i 2 Hence, problem (8.25) simplifies to (k)

− ν∇2z v (k) + ∇z p(k) = f1

in Dπ/2 ,

(k)

in Dπ/2 ,

div z v (k) = g1 (k)

(k)

(k)

(k)

v1 |z1 =0 = d1 , v2 |z2 =0 = d2 ,

(8.26)

(k) | = h(k) vα,z α , α = 2, 3, 1 z1 =0 (k) | = lα(k) , α = 1, 3. vα,z 2 z2 =0 (k)

To apply Lemma 2.17 we construct functions uj , j = 1, 2, 3, in such a way that (k)

(k)

u2 |z2 =0 = d2 ,

(k) u(k) α,z2 |z2 =0 = lα , α = 1, 3.

Introducing the new functions w = v (k) − u(k) , q = p(k) , we see that they are solutions to the problem (k)

− ν∇2z w + ∇z q = f1

+ ν∇2z u(k) ≡ f0 ,

(k)

div z w = g1 − div u(k) ≡ g0 , (k)

(k)

w1 |z1 =0 = d1 − u1 |z1 =0 ≡ d0 ,

(8.27)

(k) wα,z1 |z1 =0 = h(k) α − uα,z1 |z1 =0 ≡ hα , α = 2, 3,

w2 |z2 = 0, wα,z2 |z2 =0 = 0, α = 1, 3. Next we construct a reflection w ˜ of w with respect to the plane z2 = 0 such that w ˜2 |z2 0 , w ˜α,z2 |z2 0 ,

w ˜2 |z2 >0 = w2 |z2 >0 , w ˜α,z2 |z2 >0 = wα,z2 |z2 >0 , α = 1, 3.

114

8

Auxiliary Results: Estimates for (v, p)

After the reflection the problem (8.27) takes the form ˜ + ∇z q˜ = f˜0 , − ν∇2z w div w ˜ = g˜0 , (8.28)

w ˜1 |z1 =0 = d˜0 , ˜ α , α = 1, 2. w ˜α,z1 |z1 =0 = h

Applying Lemma 2.13 to problem (8.28) and using the properties of the reflection and extension (see Lemma 2.15, 2.16) we obtain v (k) 2,Dπ/2 + p(k) 1,Dπ/2 ≤ cλ(v (k) 2,Dπ/2 + p(k) 1,Dπ/2 ) + c(|f (k) |2,Dπ/2 + d(k) 3/2,Γ1 )

(8.29)

+ c(v1,Dπ/2 ∩supp ζ (k) + |p|2,Dπ/2 ∩supp ζ (k) ), where Γ1 = Ψk (S2 ∩ supp ζ (k) ), Γ2 = Ψk (S1 ∩ supp ζ (k) ), and Γ1 ∪ Γ2 is the boundary of Dπ/2 . Passing in (8.22), (8.29) to variables x, adding them to (8.20), (8.24), and applying the properties of the partition of unity we obtain for sufficiently small λ the inequality v2,Ω + p1,Ω ≤ c(|f |2,Ω + v1,Ω + |p|2,Ω + d3/2,S2 ). This implies (8.18) and concludes the proof

(8.30)  

Lemma 8.4 Assume that f ∈ L2 (Ω), d ∈ H 3/2 (S2 ). Then the following estimate holds v2,Ω + p1,Ω ≤ c(|f |2,Ω + d3/2,S2 + |d,x |2,S2 ).

(8.31)

Proof From the form of δ in (8.13) we have δ1,Ω ≤ cd1/2,S2 ≤ d3/2,S2 .

(8.32)

Let ϕ be defined by div ϕ = p

in Ω,

ϕ|S = 0.

(8.33)

Let p ∈ L2 (Ω). Then [KP] implies the existence of ϕ ∈ H 1 (Ω) satisfying the estimate ϕ1,Ω ≤ c|p|2,Ω .

(8.34)

8

Auxiliary Results: Estimates for (v, p)

115

Multiplying (8.1)1 by ϕ and integrating over Ω yield |p|2,Ω ≤ c(|f |2,Ω + v1,Ω ).

(8.35)

Using (8.32) in (8.8) gives v1,Ω ≤ c(|f |6/5,Ω + d1/2,S2 + |d,x |2,S2 ).

(8.36)

Employing (8.35) and (8.36) in (8.18) gives (8.31). This concludes the proof.  

Chapter 9

Auxiliary Results: Estimates for (h, q)

Abstract In this chapter we attain some more refined estimates for (h, q), where h = v,x3 , q = p,x3 and in particular we obtain inequality of the form hH 2 (Ω) + qH 1 (Ω) ≤ c|div T(h, q)|L2 (Ω) + some norms of data. To get this we consider the stationary Stokes system for (h, q) such that the slip boundary conditions are determined on S1 and the Dirichlet-Neumann conditions on S2 . div T(h, q) = F, h·n ¯ = 0,

div h = 0

in Ω,

νn ¯ · D(h) · τ¯α + γh · τ¯α = 0, α = 1, 2

hi = −d,xi , i = 1, 2,



h3,x3 = Δ d

on S1 , on S2 .

First we derive H 1 -estimate for h. Since the Dirichlet-Neumann boundary conditions on S2 are assumed we need to construct some extensions to make Dirichlet boundary conditions homogeneous. Next by using the partition of unity we increase the regularity of h showing that h ∈ H 2 (Ω). We have to emphasize that the Dirichlet boundary conditions on S2 imply that the applied Korn inequality holds also for the axially symmetric domain.

Consider the elliptic problem div T(h, q) = F

in Ω,

div h = 0

in Ω,

h·n ¯=0

on S1 ,

νn ¯ · D(h) · τ¯α + γh · τ¯α = 0, α = 1, 2,

on S1 ,

hi = −d,xi , i = 1, 2

on S2 ,

h3,x3 = Δ d

on S2 .

© Springer Nature Switzerland AG 2019 J. Renclawowicz, W. M. Zajączkowski, The Large Flux Problem to the

Navier-Stokes Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-32330-1 9

(9.1)

117

118

9

Auxiliary Results: Estimates for (h, q)

In this chapter we derive some additional estimates for solutions to problem (6.1). For this purpose we consider (9.1), where (h, q) is also a solution to (6.1). This is possible because F is exactly equal to div T(h, q), so (9.1)1 is the identity. The main aim of this chapter is to get the inequality h2,Ω + q1,Ω ≤ c|div T(h, q)|2,Ω + some norms of data. We are not able to show either the Korn inequality or the energy type estimates for solutions to (9.1) because it contains nonhomogeneous Dirichlet boundary conditions on S2 . To make the conditions homogeneous we ˜ such that construct the function h ˜=0 div h

in Ω,

˜·n h ¯=0

on S1 ,

˜3 = 0 ˜ i = −d,x , i = 1, 2, h h i

on S2 .

(9.2)

Lemma 9.1 Assume that d,x ∈ H 1/2+α (S2 ), α = 0, 1. Then there exists a ˜ ∈ H 1+α (Ω) and solution to problem (9.2) such that h ˜ 1+α,Ω ≤ cd,x 1/2+α,S . h 2

(9.3)

˜ in a few steps. First we Proof We show the existence of the function h ¯ by the formula construct a function h ¯ i |S = −d,x , i = 1, 2, h 2 i ¯ 3 |S = 0. h 2

(9.4)

Let d ∈ H 1/2 (S2 ) be given. Then, by the inverse trace theorem (see Lemma 2.17), there exists an extension d˜ on Ω such that ˜ S = d, d| 2

(9.5)

˜ 1+α,Ω ≤ cd1/2+α,S . d 2

(9.6)

and

We construct such extension that supp d˜ is located in some neighborhoods ¯ is of S2 (ai ), i = 1, 2, a1 = −a, and a2 = a. Then the vector function h defined as ¯ i = −d˜,x , i = 1, 2, h ¯ 3 = 0. h i

(9.7)

9

Auxiliary Results: Estimates for (h, q)

119

Moreover, in view of the compatibility condition 2 

ni |S1 d,xi |S¯1 ∩S¯2 = 0,

(9.8)

i=1

¯ i , i = 1, 2, in such a way that we can perform a construction of h ¯ i ni |S = 0. h 1

(9.9)

¯ is a solution to the problem Then h ¯ = −Δ d˜ div h

in Ω,

¯ · τ¯α , h ¯·n ¯ · τ¯α = h ¯ = 0, α = 1, 2 h

on S1 ,

¯ 3 = 0, i = 1, 2 ¯ i = −d,x , h h i

on S2 ,

(9.10)

where Δ = ∂x21 + ∂x22 . Now we define a function φ such that ˜ Δφ = −Δ d,

n ¯ · ∇φ|S = 0.

(9.11)

The above construction is possible under the compatibility condition (9.8). Then the function ˆ=h ¯ − ∇φ h

(9.12)

is a solution to the problem ˆ=0 div h

in Ω,

¯ · τ¯α , α = 1, 2 h ˆ·n ˆ · τ¯α = −¯ τα · ∇φ + h ¯=0 h

on S1 ,

ˆ3 = 0 ˆ = −d,x − ∇i φ, i = 1, 2, h h i

on S2 .

(9.13)

˜ we introduce the functions λ and σ such that To construct the function h − Δλ + ∇σ = F

in Ω,

div λ = 0

in Ω,

¯ · τ¯α , α = 1, 2, n λ · τ¯α = −¯ τα · ∇φ + h ¯·λ=0

on S1 ,

λi = −∇i φ, i = 1, 2, λ3 = 0

on S2 .

(9.14)

Then the function ˜=h ¯ − (λ + ∇φ) h

(9.15)

120

9

Auxiliary Results: Estimates for (h, q)

is a solution to problem (9.2). Moreover (9.6) implies ¯ 2+α,Ω ≤ cd,x 3/2+α,S , h 2 ∇φ2+α,Ω ≤ cd,x 3/2+α,S2 ,

(9.16)

λ2+α,Ω ≤ cd,x 3/2+α,S2 .  

Hence, (9.3) holds. This concludes the proof. Introduce the function ˜ k = h − h.

(9.17)

Then k is a solution to the problem ˜ div T(k, q) = F − div D(h)

in Ω,

div k = 0

in Ω,

k·n ¯ = 0, ν n ¯ · D(k) · τ¯α + γk · τ¯α

(9.18)

˜ · τ¯α + γ h ˜ · τ¯α ), α = 1, 2, = −(ν n ¯ · D(h)

on S1 ,

˜ 3,x ki = 0, i = 1, 2, k3,x3 = Δ d − h 3

on S2T .

Lemma 9.2 Assume that EΩ (k) = |D(k)|22,Ω < ∞,

div k = 0,

k·n ¯ |S1 = 0,

˜ 3,x ∈ L2 (S2 ), k3,x3 |S2 = Δ d − h 3

ki |S2 = 0, i = 1, 2,

˜ 3 ∈ H 1 (Ω). h

Then ˜ 3 2 + |Δ d − h ˜ 3,x |2 ). k21,Ω ≤ c(EΩ (k) + h 1,Ω 3 2,S2

(9.19)

Proof We have     2 |D(k)|2 dx = (ki,xj + kj,xi )2 dx = 2 ki,x dx + 2 ki,xj kj,xi dx. j Ω

Ω

Ω

Ω

(9.20) Since k is divergence free we can integrate by parts in the second integral on the r.h.s. of (9.20) to obtain 

 −2

(k3,x3 k3 + k3,xi ki )dS2 ≡ I.

ki kj ni,xj dS1 + 2 S1

S2

9

Auxiliary Results: Estimates for (h, q)

121

Using the boundary conditions on S2 we have ˜ 3,x |2 , |I| ≤ c|k  |22,S1 + ε|k3 |22,S2 + c(1/ε)|Δ d − h 3 2,S2

(9.21)

where k  = (k1 , k2 ). From (9.20), (9.21), and some interpolation from Lemma 2.15 we get ˜ 3,x |2 . |∇k|22,Ω ≤ c(EΩ (k) + |k  |22,Ω ) + ε|k3 |22,S2 + c(1/ε)|Δ d − h 3 2,S2

(9.22)

To estimate the second norm on the r.h.s. of (9.22) we will show that there exist positive constants δ and M such that |k  |22,Ω ≤ δ|∇k  |22,Ω + M EΩ (k  ),

(9.23)

δ can be chosen sufficiently small. Moreover, we have ¯ |S1 = 0. k · n

(9.24)

We argue by a contradiction (see similar argument in Solonnikov and Shchadilov [SS]). Assume that such M in (9.23) does not exist. Then for  any m ∈ N there exists a sequence k (m) ∈ H 1 (Ω) such that |k



(m) 2 |2,Ω

Then for u(m) =

≥ δ|∇k



(m) 2 |2,Ω

+ mEΩ (k



(m)

) ≡ Gm (k



(m)

).



k (m) |k (m) |2,Ω

we have 

|u(m) |2,Ω = 1,

Gm (u(m) ) =

Gm (k (m) ) ≤ 1. |k  (m) |22,Ω

Therefore, we can choose from the sequence {u(m) } a subsequence {u(mk ) } which converges weakly in H 1 (Ω) and strongly in L2 (Ω) to a limit u ∈ H 1 (Ω). Moreover, EΩ (u) = 0, so u = cη, where η = (−x2 , x1 ). Since, additionally, u · n ¯ |S2 = 0 we have that u = 0. However, this is in contradiction with |u|2,Ω = lim |u(mk ) |2,Ω = 1. mk →∞

Hence (9.23) holds. Using it in (9.22) yields ˜ 3,x |2 . |∇k|22,Ω ≤ cEΩ (k) + ε|k3 |22,Ω + c(1/ε)|Δd − h 3 2,S2

(9.25)

122

9

Auxiliary Results: Estimates for (h, q)

From definition of h3 we have 



dx

h3 dx = Ω



S2





a −a

S2





d1 dx = 0,

d2 dx −

=

[v3 (x , a) − v3 (x , −a)]dx

h3 dx3 =

S2 (a)

S2 (−a)

which holds in view of the compatibility conditions. Then by the Poincar´e inequality we have |h3 |22,Ω ≤ c|∇h3 |22,Ω .

(9.26)

Since we need the inequality for k3 we calculate ˜3 − h ˜ 3 |2 ≤ c|k3 + h ˜ 3 |2 + c|h ˜ 3 |2 |k3 |22,Ω = |k3 + h 2,Ω 2,Ω 2,Ω ˜ 3 )|2 + c|h ˜ 3 |2 ≤ c|∇k3 |2 + ch ˜ 3 2 . ≤ c|∇(k3 + h 2,Ω 2,Ω 2,Ω 1,Ω

(9.27)

Using (9.27) in (9.25) and assuming that ε is sufficiently small we obtain ˜ 3 2 + |Δ d − h ˜ 3,x |2 ) ≡ cJ. |∇k|22,Ω ≤ c(EΩ (k) + h 1,Ω 3 2,S2

(9.28)

Employing (9.28) in (9.27) gives |k3 |22,Ω ≤ cJ.

(9.29)

Hence (9.23), (9.28), and (9.29) imply (9.19). This concludes the proof.

 

Employing the relation (9.17) between k and h we obtain from (9.19) the inequality ˜ 2 + |Δ d − h ˜ 3,x |2 ) h21,Ω ≤ c(EΩ (k) + h 1,Ω 3 2,S2 ˜ 2 + |Δ d − h ˜ 3,x |2 ). ≤ c(EΩ (h) + h 1,Ω 3 2,S2

(9.30)

Next we find an estimate for EΩ (k). ˜ ∈ H 1 (Ω), Lemma 9.3 (See [RZ6, Z1]) Assume that k ∈ H 1 (Ω), h ˜ 3 ∈ H 3/2 (Ω), dt ∈ L2 (S2 ), v  dx ∈ L2 (S2 ), f3 ∈ L2 (S2 ), F ∈ L2 (Ω), h &  and  S2 v3 h3 k3 dS2  < ∞. Then

9

Auxiliary Results: Estimates for (h, q)

k21,Ω

+

2 

 |k ·

τ¯α |22,S2

˜ 2 + |h ˜ 3,x |2 + |dt |2 ≤ c h 1,Ω 2,S2 3 2,S2

α=1

+ |v



dx |22,S2

h21,Ω +

2 

123

+

|f3 |22,S2

|h · τ¯α |22,S1

+

|F |22,Ω

  + 

S2

  v3 h3 k3 dS2  ,

(9.31)  2  2 ˜ 2 + h ˜ 3 2 ≤ c h 1,Ω 3/2,Ω + |dt |2,S2 + |v dx |2,S2

α=1

  + |f3 |22,S2 + |F |22,Ω + 

S2

  v3 h3 k3 dS2  .

Proof Multiplying (9.18)1 by k and integrating over Ω yield 

 div T(h, q) · kdx = Ω

F · kdx.

(9.32)

Ω

Integrating by parts in the term on the l.h.s. we obtain    div T(h, q) · kdx = n ¯ · T(h, q) · τ¯α k · τ¯α dS1 + Ω

S1

T33 (h, q)k3 dS2 S2

   ν D(h) · D(k)dx = −γ h · τ¯α k · τ¯α dS1 + T33 (h, q)k3 dS2 2 Ω S1 S2 ν ν ˜ · D(k)dx. − |D(k)|22,Ω − D(h) 2 2 −

Employing the results in (9.32) gives  ν ˜ · τ¯α k · τ¯α dS1 |D(k)|22,Ω + γ|k · τ¯α |22,S1 = −γ h 2 S1    ν ˜ · D(k)dx + − D(h) T33 (h, q) · k3 dS2 − F · kdx. 2 Ω S2 Ω

(9.33)

Applying the H¨ older and Young inequalities to the first two terms on the r.h.s. of (9.33) implies ν ε1 1 ν2 ˜ 2 |D(k)|22,Ω + γ|k · τ¯α |22,S1 ≤ |D(k)|22,Ω + |D(h)| 2,Ω 2 2 2ε1 4   ε2 1 2˜ + |k · τ¯α |22,S1 + γ |h · τ¯α |22,S1 + T33 (h, q)k3 dS2 − F · kdx. 2 2ε2 S2 Ω

124

9

Setting ε1 =

ν 2

Auxiliary Results: Estimates for (h, q)

and ε2 = γ we derive

|D(k)|22,Ω +

2 

˜ 2 + εk2 |k · τ¯α |22,S1 ≤ ch 1,Ω 1,Ω

α−1

   + c(1/ε) 

S2

(9.34)

   T33 k3 dS2  + |F |22,Ω .

The third component of the Navier-Stokes equations (1.1) projected on S2 takes the form dt + v  · dx + v3 h3 − ν(Δ d + h3,x3 ) + q = f3 . Hence T33 (h, q)|S2 = dt + v  · dx + v3 h3 − f3 . Using that ε is sufficiently small, the formula for T33 in (9.34) and (9.19) implies the inequality k21,Ω +

2 

˜ 2 + c(|dt |2 + |v  dx |2 |k · τ¯α |22,S1 ≤ ch 1,Ω 2,S2 2,S2

α=1

+

|f3 |22,S2

+

|F |22,Ω )

  + 

S2

(9.35)

  v3 h3 k3 dS2 .

In view of relation (9.17) we have h21,Ω +

2 

˜ 2 + h ˜ 3 2 |h · τ¯α |22,S1 ≤ c(h 1,Ω 3/2,Ω )

α=1

+ c(|dt |22,S2 + |v  dx |22,S2 + |f3 |22,S2

  2 + |F |2,Ω ) + 

S2

 (9.36)  v3 h3 k3 dS2 .  

Hence, (9.35) and (9.36) imply (9.31). This concludes the proof. 

Lemma 9.4 Assume that dx ∈ H (S2 ), dt ∈ L2 (S2 ), v ∈ H (Ω), f3 ∈ L2 (S2 ), F ∈ L2 (Ω), d ∈ L2 (S2 ), and h3 ∈ L2 (Ω). Then the following inequality holds 1

h2,Ω + q1,Ω ≤ c(dx 1,S2 + |dt |2,S2 + v  21,Ω dx 21,S2 + |f3 |2,S2 + |F |2,Ω + |v3 |22,S2 |h3 |22,Ω + |d|22,S2 dx 21,S2 ).

1

(9.37)

9

Auxiliary Results: Estimates for (h, q)

125

Proof To prove the lemma we use the partition of unity introduced in Chap. 2. Let k ∈ M. Then problem (9.1) restricted to neighborhoods Ω (k) = supp ζ (k) , k ∈ M, has the form νΔh − ∇q = F,

(9.38)

div h = 0.

Using notation h(k) = hζ (k) , q (k) = qζ (k) , and F (k) = F ζ (k) we transform (9.38) to the following problem for the localized functions νΔh(k) − ∇q (k) = F (k) + 2ν∇h∇ζ (k) + νhΔζ (k) − q∇ζ (k) , div h(k) = h · ∇ζ (k) .

(9.39)

From the theory of stationary Stokes system we have the inequality h(k) 2,Ω + q (k) 1,Ω ≤ c|F k |2,Ω + c(h1,Ω (k) + |q|2,Ω (k) ).

(9.40)

Let k ∈ N1 and let ξk ∈ S1 . Moreover, Ω (k) = supp ζ (k) is a neighborhood ¯ (k) ∩ S1 . We assume also that of ξk such that ξk is the middle point of Ω (k) Ω is located in a positive distance from the edges L. Introduce a new coordinate system y = (y1 , y2 , y3 ) with origin at ξ (k) and y = Yk (x), where x = (x1 , x2 , x3 ) is the global Cartesian system in Ω and Yk is a composition of a rotation and a translation. Let us assume that Ω (k) ∩ S1 is described in the form y1 = Fk (y1 , y3 ). Then we introduce new coordinates z2 = y 2 ,

z3 = y 3 ,

(9.41)

z1 = y1 − F (y2 , y3 ). The transformation (9.41) is denoted by z = Ψk (y) = Ψk (Yk (x)) ≡ Φk (x),

z = (z1 , z2 , z3 ).

In these coordinates z we have supp ζ (k) ∩ S1 ⊂ {z : z1 = 0}.

126

9

Auxiliary Results: Estimates for (h, q)

We restrict problem (9.1) to neighborhood Ω (k) , k ∈ N1 and express it in the new coordinates z. For this we need the notation u ˆ(z) = u(Φ−1 k (z)),

u(k) = u ˆ(z)ζˆ(k) (z),

n ˆ (z) = n ¯ (Φ−1 k (z)),   ˆ z = ∂zi  ∇ ∇z , ∂x x=Φ−1 (z) i

τˆα = τ¯α (Φ−1 k (z)),

(9.42)

∇zi = ∂zi .

k

In coordinates (9.41), n ¯ z = (1, 0, 0), τ¯z1 = (0, 1, 0), and τ¯zz = (0, 0, 1). Then (9.1) in Ω (k) , k ∈ N1 , and in coordinates z has the form ˆ 2z )h(k) − (∇z − ∇ ˆ z )q (k) ν∇2z h(k) − ∇z q (k) = ν(∇2z − ∇ ˆ∇ ˆ∇ ˆ zh ˆ 2z ζˆ(k) − qˆ∇ ˆ z ζˆ(k) ˆ z ζˆ(k) + ν h + F (k) + 2ν ∇ ˆ z )h(k) + h ˆ∇ ˆ z ζˆ(k) div z h(k) = (div z − div (k)

z1 > 0, z1 > 0,

(k)

ˆ (z) h1 = h1 − h(k) · n

z1 = 0,

(9.43)

(k) ˆ z (h(k) ) · τˆα (z) + n1 ∂x ˆ (z) · D ζ (k) hj τjα h(k) α,z1 = hα,z1 − n i

ˆ z h(k) + ni ∂xj ζ (k) hi τjα − τˆα · ∇ 1

z1 = 0.

Hence, (9.43) is the stationary Stokes system in the half space z1 > 0 with the Dirichlet and Neumann conditions on z1 = 0 for coordinates of h(k) . From the theory of the Stokes system and sufficiently small λ, we get ˆ ˆ (k) h(k) 2,Ωˆ (k) + q (k) 1,Ωˆ (k) ≤ c(|F (k) |2,Ωˆ (k) + h 1,Ω + |ˆ q |2,Ωˆ (k) ),

k ∈ N1 ,

(9.44)

ˆ (k) = Φk Ω (k) . where Ω Let k ∈ N2 and ξ (k) ∈ S2 . We assume that ξ (k) is the middle point of supp ζ (k) ∩ S2 . Since S2 is flat we introduce the following new coordinates z such that z3 = x3 , zi = xi + ai , and i = 1, 2, where ai are constants and describe a translation. In this case we localize problem (9.1) to the following one ˆ + νh ˆ∇ ˆ z ζˆ(k) ν∇2z h(k) − ∇z q (k) = F (k) + 2ν∇z ζˆ(k) ∇z h

z3 > 0,

ˆ · ∇z ζˆ(k) div z h(k) = h

z3 > 0,

(k)

(9.45)

= −dˆ,zi ζˆ(k)

z3 = 0,

(k) ˆ 3 ζˆ(k) + Δ dˆζˆ(k) h3,z3 = h ,z3 z

z3 = 0.

hi

9

Auxiliary Results: Estimates for (h, q)

127

For solutions to (9.45) the following estimate holds ˆ ˆ (k) + |F (k) | ˆ (k) h(k) 2,Ωˆ (k) + q (k) 1,Ωˆ (k) ≤ c(h 1,Ω 2,Ω + dˆ,z 3/2,Sˆ(k) ).

(9.46)

2

Finally, we pass to the most difficult case—neighborhoods of edges. Let k ∈ N3 , ξ (k) ∈ L, and Ω (k) = supp ζ (k) is a neighborhood of ξ (k) . We introduce new coordinates y = (y1 , y2 , y3 ) such that Ω (k) ∩ S1 = {y : y1 = Fk (y2 , y3 )}, Ω (k) ∩ S2 ⊂ {y : y3 = 0}, and L = {y : y1 = Fk (y2 , 0)}. Next we define coordinate z by the relations z3 = y3 , z1 = y1 − Fk (y2 , y3 ), and z2 = y2 . Then the localized problem (9.1) in the right dihedral angle has the form ˆ 2z )h(k) − (∇z − ∇ ˆ z )q (k) ν∇2z h(k) − ∇z q (k) = ν(∇2z − ∇ ˆ∇ ˆ∇ ˆ zh ˆ 2 ζˆ(k) − qˆ∇ ˆ z ζˆ(k) ˆ z ζˆ(k) + ν h +F (k) + 2ν ∇ z

z1 > 0, z3 > 0,

ˆ z )h(k) + h ˆ∇ ˆ z ζˆ(k) div z h(k) = (div z − div

z1 > 0, z3 > 0,

(k)

(k)

(k)

ˆ (z) ≡ G1 h1 =h1 −h(k) · n

z1 = 0 (9.47)

(k) ˆ z (h(k) ) · τˆα (z) nz · D(h(k) ) · τ¯zα −ˆ n(z)D h(k) α,z1 =−h1,zα +¯

+ni ∂xi ζ (k) hj τjα + ni ∂xj ζ (k) hi τjα ≡ G(k) α , (k)

α = 1, 2, z1 = 0,

= −dˆzi ζˆ(k) i = 1, 2,

z3 = 0,

(k) ˆ 3 ζˆ(k) + Δ dˆζˆ(k) h3,z3 = h ,z3 z

z3 = 0,

hi

where n ¯ z = (1, 0, 0), τ¯z1 = (0, 1, 0), and τ¯z2 = (0, 0, 1) on {z : z1 = 0}. (k) We extend the functions Gj , j = 1, 2, 3, on z1 > 0 and denote the ¯ (k) , j = 1, 2, 3. Introducing the new functions extended functions by G j ¯ (k) we obtain the problem (9.47) with vanishing boundary g (k) = h(k) − G conditions on z1 = 0. Next after reflection with respect to the plane z1 = 0 we obtain problem (9.47) in the half space z3 > 0. Then for λ sufficiently small, we obtain the inequality for solutions to problem (9.47) ˆ ˆ (k) + |Fˆ (k) | ˆ (k) h(k) 2,Ω + q (k) 1,Ω ≤ c(h 1,Ω 2,Ω + |ˆ q |2,Ωˆ (k) ),

k ∈ N3 .

(9.48)

We pass to variables x in (9.44), (9.46), and (9.48). Next adding the transformed inequalities (9.44), (9.46) and (9.48) to (9.40), summing up over all neighborhoods of the partition of unity, we derive the inequality

128

9

Auxiliary Results: Estimates for (h, q)

h2,Ω + q1,Ω ≤ c(h1,Ω + |q|2,Ω + |F |2,Ω ).

(9.49)

To estimate the second norm on the r.h.s. we consider the system νΔh − ∇q = F, div h = 0.

(9.50)

Let ϕ be a function such that div ϕ = q,

ϕ|S = 0,

(9.51)

and let q ∈ L2 (Ω). Then the paper by Kapitanskii and Pileckas [KP] implies the existence of ϕ such that ϕ ∈ H 1 (Ω) and ϕ1,Ω ≤ c|q|2,Ω .

(9.52)

Multiplying (9.50) by ϕ, integrating over Ω and by parts, and using (9.51), (9.52) we have |q|2,Ω ≤ c(|F |2,Ω + h1,Ω ).

(9.53)

Employing (9.53) in (9.49) and next exploiting (9.31)2 we obtain 

˜ 1,Ω + h ˜ 3 3/2,Ω + |dt |2 + |v  dx |2 h2,Ω + q1,Ω ≤ c h 2,S2 2,S2 +

|f3 |22,S2

+

|F |22,Ω

  + 

S2

  v3 h3 k3 dS2  .

(9.54)

The last term on the r.h.s. of (9.54) is estimated by     ≤

S2

    v3 h23 dS2  + 

εh3 22,Ω

S2

  ˜ 3 |4,S ˜ 3 dS2  ≤ |d|2,S |h3 |2 + |d|2,S |h3 |4,S |h v 3 h3 h 4,S2 2 2 2 2 

˜ 3 1,Ω . + c(1/ε)|v3 |22,S2 |h3 |22,Ω + ε|h3 |24,S2 + c(1/ε)|d|2,S2 h

Moreover, the fourth term on the r.h.s. of (9.54) is estimated by |v  |24,S2 |dx |24,S2 ≤ v  21,Ω dx 21,S2 . Employing the estimates in (9.54) and assuming that ε is sufficiently small yield (9.37) and concludes the proof.  

Chapter 10

The Neumann Problem (3.6) in L2 -Weighted Spaces

Abstract In this chapter and Chap. 11 we derive weighted estimates for solutions to the Neumann problem for the Poisson equation (3.6), i.e. Δϕ = −div b

in Ω,

n ¯ · ∇ϕ = 0 on S,  ϕdx = 0. Ω

These estimates are necessary to prove global energy estimates formulated in Lemma 3.2. In this chapter we apply L2 approach to get H 2 weighted regularity and for this we need weighted energy type estimates for the Fourier transform in the directions perpendicular to the x3 -axis. The results are formulated in Lemmas 10.4–10.6. We get the estimate in a neighborhood of S2 and use a partition of unity to obtain an estimate for ϕ in whole Ω.

In this chapter and Chap. 11 we analyze the problem (3.6) to achieve a weighted estimate for solutions in a neighborhood of S2 which makes possible to prove the energy estimate established in Lemma 3.2. However, the techniques and results of these chapters are totally different. In Chap. 10 we derive the weighted estimate in the L2 -spaces using the Fourier transform in the directions perpendicular to the x3 -axis and some delicate weighted estimates possible in the L2 -approach only (see Kubica and Zaj¸aczkowski [KZ], [Z9]). However, to prove Lemma 3.2 we need Lp estimate with p ≥ 3. Therefore, in Chap. 11 we show Lp -weighted estimate using the estimate proved in Chap. 10. We apply the local regularity technique connected with the considered weights (see Maz’ya and Plamenevskii [MP]). Hence we cannot mix the above different approaches. Estimated in weighted norms can also be found in [CF].

© Springer Nature Switzerland AG 2019 J. Renclawowicz, W. M. Zajączkowski, The Large Flux Problem to the

Navier-Stokes Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-32330-1 10

129

130

10

The Neumann Problem (3.6) in L2 -Weighted Spaces

In this chapter we examine problem (3.6) in the form − Δϕ = f

in Ω,

n ¯ · ∇ϕ = 0  ϕ dx = 0,

on S,

(10.1)

Ω

where f = div b

(10.2)

and b is defined by (3.4). We assume that the considered domain has the form as in Chap. 1 (Fig. 10.1). For convenience we replace −a with 0 in this and the following chapter. The shape of Ω is appropriate for application of weighted spaces. Since the weighted spaces have the weight as a power function of the distance from either S2 (0) or S2 (a) it is convenient to examine problem (10.1) in a neighborhood of S2 (0) only, because considerations near S2 (a) are similar. To consider problem (10.1) in a neighborhood of S2 (0) we introduce a smooth cut-off function ζ = ζ(x3 ) such that ζ(x3 ) = 1 for x3 ≤ r and ζ(x3 ) = 0 for x3 ≥ , r <  < a. Let ϕ˜ = ϕζ,

f˜ = f ζ.

(10.3)

Then ϕ˜ is a solution to the problem − Δϕ˜ = f˜ − 2∇ϕ∇ζ − ϕΔζ ≡ f˜ n ¯ · ∇ϕ| ˜ S2 (0) = 0, n ¯ · ∇ϕ| ˜ S1 = 0, ϕ| ˜ x3 = = 0,

Fig. 10.1 Cylindrical domain Ω

(10.4)

10

The Neumann Problem (3.6) in L2 -Weighted Spaces

131

where we used that n ¯ |S1 does not have the third coordinate. Let ϕ(0) = ϕ|x3 =0

and

ϕ(0) ˜ = ϕ(0)ζ

(10.5)

and ˜ ϕ = ϕ˜ − ϕ(0).

(10.6)

¯ · ∇ϕ(0)|S1 = 0 because n ¯ · ∇ϕ|S1 = 0. Since Moreover, n ¯ · ∇ϕ(0)| ˜ S2 (0) = 0, n  ¯ · ∇ϕ(0)| ˜ n ¯ · ∇ζ|S1 = 0 we have that n S1 = 0. Therefore ϕ is a solution to the problem ˜ ≡ f , − Δϕ = f˜ + Δϕ(0) n ¯ · ∇ϕ |S2 (0) = 0,

(10.7)

n ¯ · ∇ϕ |S1 = 0, ϕ |x3 = = 0.

To examine the problem (10.7) in weighted spaces we introduce the following. Definition 10.1 Let Ω = {x ∈ Ω : 0 < x3 < } and Hμk (Ω ) =

   u : uHμk (Ω ) = |α|≤k

1/2

dx dx3 |Dxα u|2 x3

2(μ+|α|−k)

' . We calculate

132

10

The Neumann Problem (3.6) in L2 -Weighted Spaces







x3

ϕ(x ˜ , x3 ) − ϕ(x ˜ , s) =

ϕ˜,s (x , s )ds ,

s

older inequality where x = (x1 , x2 ). By the H¨ 

|ϕ(x ˜  , x3 ) − ϕ(x ˜  , s)|2 ≤ |x3 − s|

x3

|ϕ˜,s (x , s )|2 ds .

s

Setting s = 0 and integrating the inequality over the following set: Ω  = {x ∈ Ω : x3 = const ∈ [0, a]}, we get 

|ϕ(x ˜  , x3 ) − ϕ(x ˜  , 0)|2  dx ≤ x3

Ω



dx Ω



x3

|ϕ˜,s (x , s)|2 ds,

(10.10)

0

where the l.h.s. of (10.10) equals 

|ϕ (x , x3 )|2  dx . x3

Ω

Now, we consider the second factor on the r.h.s. of (10.9). We have for μ ∈ [0, 1) the inequalities 

|ϕ |2 x−2μ dx = 3





Ω

Ω



dx

a 0

|ϕ |2 1−2μ x dx3 x3 3

 a |ϕ | ≤ dx sup x31−2μ dx3 x3 ≤a x3 0 Ω  a   a ≤ x31−2μ dx3 dx |ϕ˜,x3 (x , x3 )|2 dx3  2

0



≤c

dx Ω



Ω



a

|ϕ˜

,x3

0



dx

+c Ω

0



(x



, x3 )|2 dx3

 ≤c

dx Ω



(10.11)



a 0

|ϕ,x3 (x , x3 )|2 dx3

a

|ϕ(0) ˜ ,x3 |2 dx3 , 0

where the last term on the r.h.s. of the above inequality equals   ˙ 2 dx3 , dx |ϕ(x , 0)|2 |ζ| c Ω

supp ζ˙

where ζ˙ = ζ,x3 and supp ζ˙ ⊂ [r, ]. Employing (10.11) in (10.9) yields 

|∇ϕ |2 dx ≤ c Ω



|f  |2 x2μ 3 dx + c Ω



dx Ω

 supp ζ˙

˙ 2 dx3 . |ϕ(x , 0)|2 |ζ| (10.12)

10

The Neumann Problem (3.6) in L2 -Weighted Spaces

133

In view of (10.9)–(10.12) and the Poincar´e inequality we derive (10.8). This concludes the proof.   Remark 10.3 Estimate (10.8) describes local properties of solutions to problem (10.1) in a neighborhood of S2 (0). We derive a similar estimate for ϕ near S2 (a). To obtain an estimate for ϕ in whole Ω we need a partition of unity of N0 Ω given as {ζ (k) (x3 )}k=1,...,N0 , k=1 ζ (k) (x3 ) = 1, where ζ (1) (x3 ) is equal to ζ(x3 ) introduced above. Similarly ζ N0 (x3 ) is equal to 1 near S2 (a). Hence, we can derive (10.8) in any supp ζ (k) , k = 2, . . . , N0 − 1, also, where the weighted spaces are not needed. Adding the estimates we obtain 

(ϕ H 1 (Ω (k) ) + ϕ L2,−μ (Ω (k) ) ) +

k=1,N0

≤c

ϕH 1 (Ω (k) )

k=2



f  L2,μ (Ω (k) ) + c

k=1,N0

+c

N 0 −1



N 0 −1

f L2 (Ω (k) )

(10.13)

k=2

ϕ(x , 0)ζ˙ (k) L2 (Ω) ,

k=1,N0

where Ω (k) = supp ζ (k) and in supp ζ (N0 ) there are introduced such local coordinates that S2 (a) is determined by x3 = 0. Having the estimate (10.13), the existence of solutions to (10.1) follows from the first Fredholm theorem. Applying local considerations, we increase the regularity of weak solutions described by Remark 10.3. The most difficult considerations are expected in neighborhoods Ω (1) and Ω (N0 ) because in these domains the technique of weighted spaces must be used. We restrict our considerations to neighborhoods of the edge L(0). Let ξ1 , . . . , ξM0 belong to L(0). Take smooth cut-off functions η (1) , . . . , η (M0 ) such that ξk is the middle point of L(0) ∩ supp η (k) . Assume that {η (k) }k=1,...,M0 is a partition of unity in a neighborhood of  M0 L(0). Hence k=1 supp η (k) cover whole L(0). Consider (10.7) in supp η (k) ⊂ (1) supp ζ . Introduce new local coordinates with origin at ξk . Denote them by y = (y1 , y2 , y3 ). They are obtained from global coordinates x = (x1 , x2 , x3 ) by a translation and a rotation. We denote the transformation by: y = Yk (x). Next we introduce new coordinates z = (z1 , z2 , z3 ) transforming neighborhood supp η (k) of point ξk into a right dihedral angle. We make such transformation that S2 (0) ∩ supp η (k) becomes a subset of {z : z3 = 0}, S1 ∩ supp η (k) becomes a subset of {z : z1 = 0}, and L(0) ∩ supp η (k) is transformed into z2 -line which is the intersection of planes z1 = 0 and z3 = 0.

134

10

The Neumann Problem (3.6) in L2 -Weighted Spaces

We denote the transformation by z = Φk (y) = Φk (Yk (x)) ≡ Ψk (x). Since the index is fixed we omit it in the next considerations. Introducing the notation ϕ (z) = ϕ (Ψ −1 (z)),

f  (z) = f  (Ψ −1 (z))

(10.14)

we express problem (10.7) in the form −∇2z ϕ = −∇2z ϕ + ∇2Ψ ϕ + f  ≡ f˜,

z3 > 0, z1 > 0,

ϕ,z3

=0

z3 = 0,

ϕ,z1

=0

z1 = 0,

(10.15)

 ∂z  where ∇Ψ = ∂x ∇ . We extend ϕ by 0 for z3 > Ψ (). Next we x=Ψ −1 (z) z make the reflection with respect to the plane z1 = 0 such that the reflected function ϕ˜ satisfies ϕ˜ (z1 , z2 , z3 ) = ϕ (z1 , z2 , z3 ), 



ϕ˜ (z1 , z2 , z3 ) = ϕ (−z1 , z2 , z3 )

z1 > 0, z1 < 0.

(10.16)

Hence, after an extension by zero, the reflected solution to (10.15) satisfies −∇2z ϕ˜ = f˜ z3 > 0, ϕ˜,z3 = 0 z3 = 0,

(10.17)

where f˜ has a compact support. For further considerations it is convenient to write (10.17) in the form −Δx u = f

x3 > 0,

u,x3 = 0

x3 = 0,

u=0

x3 = a.

Let us consider the Fourier transform   u ˜(ξ, x3 ) = e−iξ·x u(x , x3 )dx ,

(10.18)

(10.19)

R2

where ξ = (ξ1 , ξ2 ), x·ξ = x1 ξ1 +x2 ξ2 . Applying this transformation to (10.18) yields

10

The Neumann Problem (3.6) in L2 -Weighted Spaces

−

135

˜ d2 u + ξ2u ˜ = f˜ x3 > 0, dx23 (10.20)

u ˜,x3 = 0 x3 = 0, u ˜ = 0 x3 = a. We set u ˆ=u ˜−u ˜(0)ζ(x3 ),

˜(0), u ˜|x3 =0 = u

u ˆ,x3 |x3 =0 = 0,

u ˆ|x3 =a = 0,

where ζ is introduced above and supp ζ ⊂ (0, ),  < a. Then u ˆ is a solution to the problem ¨ 3 ) ≡ fˆ, ˆ = f˜ + ξ 2 u ˜(0)ζ(x3 ) − u ˜(0)ζ(x −ˆ u,x3 x3 + ξ 2 u u ˆ,x3 = 0, u ˆ = 0, Lemma 10.4 Assume that   dξ 

R2

a

x3 = 0, (10.21) x3 = a.

|fˆ|2 x2μ 3 dx3 < ∞,

0



a

2

ξ dξ R2

x3 > 0,

(10.22) |ˆ u|2 x32μ−2 dx3

< ∞.

0

Then solution u ˆ to problem (10.21) satisfies 



a

ξ 2 dξ R2

0



+



a

dξ R2

 ≤c1

(|ˆ u,x3 |2 + ξ 2 |ˆ u|2 )|x2μ 3 dx3



0 a

ξ 2 dξ R2

0

(|∂x23 u ˆ|2 x2μ ˆ|2 x32μ−2 + |ˆ u|2 x32μ−4 )dx3 (10.23) 3 + |∂x3 u 

|ˆ u|2 x32μ−2 dx3 + c



a

dξ R2

|fˆ|2 x2μ 3 dx3 .

0

¯ˆx2μ and integrating with respect to x3 over Proof Multiplying (10.21)1 by u 3 (0, a) we get 

a 0



a

¯ˆx2μ + ξ 2 |ˆ (−ˆ u,x3 x3 u u|2 )x2μ 3 3 dx3 =

¯ˆx2μ dx3 , fˆu 3

0

where v¯ is the complex conjugate to v. Integrating by parts we obtain

136



10 a

The Neumann Problem (3.6) in L2 -Weighted Spaces

 (|ˆ u,x3 |2 x2μ 3

+ξ

2

|ˆ u|2 x2μ 3 )dx3

a

= −2μ

0



a

¯ˆx2μ−1 dx3 + u ˆ,x3 u 3

0

¯ˆx2μ dx3 . fˆu 3

0

(10.24) By the H¨older and Young inequalities we estimate the first term on the r.h.s. of (10.24) by   ε1 a 4μ2 a 2 2μ−2 2 2μ |ˆ u,x3 | x3 dx3 + |ˆ u| x3 dx3 , 2 0 2ε1 0 and the second by 

ε2 2 ξ 2

a

|ˆ u|2 x2μ 3 dx3

0



1 1 + 2ε2 ξ 2

a

|fˆ|2 x2μ 3 dx3 .

0

Setting ε1 = ε2 = 1, we obtain from (10.24) the inequality 1 2





a

(|ˆ u,x3 | + ξ |ˆ u| 2

2

2

)x2μ 3 dx3

≤ 2μ

0

a

2

|ˆ u|2 x32μ−2 dx3

0

1 + 2 ξ



a

|fˆ|2 x2μ 3 dx3 .

0

We multiply this by 2ξ 2 and integrate with respect to ξ to get 



(|ˆ u,x3 | + ξ |ˆ u| 2

ξ dξ R2

0



+2





a

2



a

dξ R2

2

2

)x2μ 3 dx3

≤ 4μ

2

a

2

ξ dξ R2

|ˆ u|2 x32μ−2 dx3

0

(10.25)

|fˆ|2 x2μ 3 dx3 .

0

From (10.21) the following bound follows ∂x23 u ˆ2L2,μ (0,a) ≤ ξ 4 ˆ u2L2,μ (0,a) + fˆ2L2,μ (0,a) . Consequently, integrating with respect to ξ implies 



a

dξ  ≤

R2



a

4

ξ dξ R2

0



|ˆ u|2 x2μ 3 dx3

+



a

dξ R2

0

|∂x23 u ˆ|2 x2μ 3 dx3 (10.26) |fˆ|2 x2μ 3 dx3 .

0

On the other hand, by the Hardy inequality (see Kondratiev [K, Sect. 4]) we have   a   a 2(μ−1) 2 2 2μ dξ |∂x3 u ˆ| x3 dx3 ≥ c dξ |∂x3 u ˆ |2 x 3 dx3 R2

≥c

0



R2



a

dξ R2

0

2(μ−2) |ˆ u |2 x 3 dx3 ,

0

where we used that ∂x3 u ˆ|x3 =0 = 0, u ˆ|x3 =0 = 0.

(10.27)

10

The Neumann Problem (3.6) in L2 -Weighted Spaces

137

Then inequalities (10.25)–(10.27) imply (10.23). This concludes the proof.   We need to estimate the first term on the r.h.s. of (10.23). To this end, we introduce the sets Q1 = {(ξ, x3 ) ∈ R2 × (0, a) : |ξ|−1 x−1 3 ≤ a1 }, Q2 = {(ξ, x3 ) ∈ R2 × (0, a) : |ξ|−1 x−1 3 ≥ a2 }, Q3 = {(ξ, x3 ) ∈ R2 × (0, a) : a1 ≤ |ξ|−1 x−1 3 ≤ a2 }. Note that a1 < a2 are arbitrary, but will be chosen later. Lemma 10.5 Assume that   R2



a 0



R2

ξ 4 |ˆ u|2 x2μ 3 dξdx3 < ∞,

a

|ˆ u|2 x32μ−4 dξdx3 < ∞.

0

Then 



a

ξ 2 dξ R2

1 + 2 a2

0



 |ˆ u|2 x32μ−2 dx3 ≤ a21



R2

a

|ˆ u|2 x32μ−4 dξdx3

 R2

+

a

ξ 4 |ˆ u|2 x2μ 3 dξdx3

0



ca22−2μ

0

 R2

a

(10.28) |fˆ|2 x2μ 3 dξdx3 .

0

Proof Let us consider the expression 



a

ξ 2 dξ R2

|ˆ u|2 x32μ−2 dx3 =

0

3   i=1

ξ 2 |ˆ u|2 x32μ−2 dξdx3 ≡ Qi

where  ξ 4 |ˆ u|2 x2μ 3 dξdx3 ,

I1 ≤ a21 Q1

1 I2 ≤ 2 a2 I3 ≤



|ˆ u|2 x32μ−4 dξdx3 , Q2



a2−μ 2

|ξ|4−2μ |ˆ u|2 dξdx3 ≡ I. Q3

3  i=1

Ii ,

138

10

The Neumann Problem (3.6) in L2 -Weighted Spaces

To estimate I we recall considerations from [SZ, Sect. 4] and [Z4, (6.14)]. ¯ˆ and integrating over (0, a) we get Multiplying (10.21) by u 

a

(|ˆ u,x3 |2 + ξ 2 |ˆ u|2 )dx3 0



a

=



a

¯ fˆ · u ˆdx3 ≤

0

1/2  

a

|fˆ|2 x2μ 3 dx3

0

|ˆ u|2 x−2μ dx3 3

1/2 (10.29) .

0

Next, we recall the proof of the interpolation inequality (see [SZ, (2.3)], [Z4, Th. 2.5])  R+

|v|2 x−2μ dx3 ≤ ε2(1−μ) 3

 R+

|v,x3 |2 dx3 + cε−2μ

 R+

|v|2 dx3 .

(10.30)

To prove (10.30) we consider 

|v|2 x−2μ dx3 3

R+



r0

=

|v|2 x−2μ dx3 3



∞

+

|v|2 x−2μ dx3 ≡ I1 + I2 , 3

r0

0

where 

r0

I1 ≤ c



r0

|v,x3 |2 x32−2μ dx3 ≤ cr02−2μ

0

|v,x3 |2 dx3

0

and I2 ≤ r0−2μ



∞

|v|2 dx3 .

r0

Hence  R+

|v|2 x−2μ dx3 ≤ cr02−2μ 3



|v,x3 |2 dx3 + cr0−2μ

R+

 R+

|v|2 dx3 ,

ˆ we get so (10.30) is proved. Setting ε = |ξ|−1 and v = u  ξ 2−2μ R+

|ˆ u|2 x−2μ dx3 ≤ c 3

 R+

(|ˆ u,x3 |2 + ξ 2 |ˆ u|2 )dx3 .

(10.31)

Since u ˆ vanishes outside (0, a) we can write (10.31) in the form  ξ

a

2−2μ 0

|ˆ u|2 x−2μ dx3 3



a

≤c

(|ˆ u,x3 |2 + ξ 2 |ˆ u|2 )dx3 . 0

(10.32)

10

The Neumann Problem (3.6) in L2 -Weighted Spaces

139

From (10.29) we have 

a

(|ˆ u,x3 |2 + ξ 2 |ˆ u|2 )dx3

ξ 2−2μ 0

≤ξ



a

2−2μ

1/2  

a

|fˆ|2 x2μ 3 dx3

0

|ˆ u|2 x−2μ dx3 3

1/2

(10.33) .

0

Employing (10.32) in (10.33) yields 



a

a

(|ˆ u,x3 |2 + ξ 2 |ˆ u|2 )dx3 ≤ c

ξ 2−2μ 0

|fˆ|2 x2μ 3 dx3 .

(10.34)

0

Hence (10.34) yields estimate for I,  I ≤ ca22−2μ

R2



a

|fˆ|2 x2μ 3 dx3 dξ.

(10.35)

0

Therefore, using the bounds for I1 , I2 , and I we obtain (10.28). This concludes the proof.   Lemma 10.6 Assume that f ∈ L2,μ (R3+ ), u(0) ∈ H 2 (R2 ), u(0) = u|x3 =0 , and μ ∈ (0, 1). Then there exists a solution to (10.21) such that u ¯ = F −1 (˜ u−u ˜(0)ζ(x3 )), where F is the Fourier transform defined by (10.19), belongs to Hμ2 (R3+ ) and the estimate holds ¯ uHμ2 (R3+ ) ≤ c(f L2,μ (R3+ ) + u(0)H 2 (R2 ) ).

(10.36)

Proof Using (10.28) in the r.h.s. of (10.23) and assuming that a1 is sufficiently small and a2 is sufficiently large we derive 



(|ˆ u,x3 | + ξ |ˆ u| )dx3 + 2

ξ dξ R2

0



+  ≤c



a

2



a

dξ R2



0 a

dξ R2

2



2

a

2

ξ dξ R2

|ˆ u|2 x32μ−2 dx3

0

(|ˆ u,x3 x3 |2 x2μ u,x3 |2 x32μ−2 + |ˆ u|2 x32μ−4 )dx3 3 + |ˆ

(10.37)

|fˆ|2 x2μ 3 dx3 .

0

Employing the form of fˆ from (10.26)1 and applying the Parseval identity we derive (10.36). This concludes the proof.   Finally, we have to estimate the last norm on the r.h.s. of (10.36). Let us recall that f defined in (10.2) is introduced in (3.4), (3.6). Hence we consider the problem (10.17) formulated in the form

140

10

The Neumann Problem (3.6) in L2 -Weighted Spaces

− Δu = α,x3

x3 > 0,

u,x3 = 0

x3 = 0,

(10.38)

˜ and θ˜ is obtained from d˜ by the procedure formulated where α = θη in (10.14)–(10.16). We assume that supp θ˜ is compact. Moreover, ˜ x =0 = θ, θ| 3 ˜ and θ is an extension of d in a similar way as θ˜ is derived from d. Since d is given, we have the estimates θH s (R2 ) ≤ cdH s (S2 (0)) , ˜ H s+1/2 (R3 ) ≤ cθH s (R2 ) , θ +

(10.39)

˜ H s+1/2 (Ω) ≤ cdH s (S (0)) . d 2 Lemma 10.7 Assume that u is a solution to (10.38), u(0) = u|x3 =0 , α is described above, and d ∈ H 1 (S2 (0)). Then u(0)H 2 (S2 (0)) ≤ cdH 1 (S2 (0)) .

(10.40)

Proof Using the Neumann function, any solution to (10.38) can be expressed by 

 u(x) = R3+

 1 1 + α,y3 dy, |x − y| |x − y¯|

where y¯ = (y1 , y2 , −y3 ). Integrating by parts we obtain 

     1 1 1 1 + + ∂ y3 α dy − αdy |x − y| |x − y¯| |x − y| |x − y¯| R3+ R3+        1 x 3 − y3 1 x 3 + y3 ˜ ˜ −  + − θdy θηdy = |x − y| |x − y¯| y3 =0 |x − y|3 |x − y¯|3 R2 R3+     x 3 − y3 2 x 3 + y3 ˜  ˜ ( − θdy − = θηdy. |x − y|3 |x − y¯|3 (x − y  )2 + x23 R2 R3+ 

u(x) =

∂ y3

10

The Neumann Problem (3.6) in L2 -Weighted Spaces

141

In view of this formula we have  R2

 +4 R

|∂x2 u(0)|2 dx ≤ 4

   dx ∂x2 2

R3+

(

y3

 R2

   dx ∂x2

R2

2  1 ˜  θdy    |x − y |

2  ˜  dy3  ≤ c(θH 1 (R2 ) θdy  2

|x − y  |2 + y3

˜ H 1 (R3 ) ) ≤ cθH 1 (R2 ) , +θ + so in view of (10.39) we derive (10.40). This concludes the proof.

 

Chapter 11

The Neumann Problem (3.6) in Lp -Weighted Spaces

Abstract In this chapter we show Wp2 -weighted regularity, with p ≥ 3, for solutions to the Neumann problem for the Poisson equation (3.6), i.e. Δϕ = −div b

in Ω,

n ¯ · ∇ϕ = 0 on S,  ϕdx = 0. Ω

We also consider auxiliary problem that follows from (3.6) by reflection with respect to S1 and localization to a neighborhood of S2 . We use the H 2 weighted regularity considered in Chap. 10. The main tool in this part is the classical technique of increasing regularity through the MarcinkiewiczMikhlin type result (formulated in Proposition 11.9), connected with the partition of unity which generates the weight. By the weight we mean the power of the distance to the boundary S2 (bottom or top of the cylinder). The local estimates are possible to attain by extending the solutions on R3+ .

In Chap. 10 the local regularity of weak solutions to problem (3.6) is proved in weighted spaces Hμ2 (D), where D = {x ∈ R3+ : x ∈ R2 , 0 < x3 < a}, μ ∈ (0, 1). The result is formulated in Lemma 10.6 by Formula (10.36). We restricted our considerations to neighborhood of S2 (0) because near S2 only the weighted spaces must be used. We replaced S2 (−a) from Chap. 3 by S2 (0) for convenience. In L2 -weighted spaces the proofs from Chap. 10 depend heavily on the Fourier transform (10.19). Any element u ∈ Hμ2 , μ ∈ (0, 1) is such that u|x3 =0 = 0, u,x3 |x3 =0 = 0 but on S2 we have either inflow or outflow. Therefore, we introduce the function u ˆ defined below (10.20). Finally, the trace u|x3 =0 is estimated in (10.40).

© Springer Nature Switzerland AG 2019 J. Renclawowicz, W. M. Zajączkowski, The Large Flux Problem to the

Navier-Stokes Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-32330-1 11

143

144

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

In this chapter we consider the problem −Δu = f

in D,

u,x3 = 0

x3 = 0,

u=0

x3 = a,

(11.1)

u|x3 =0 = 0, ˆ, f = F −1 fˆ, and F is defined by (10.19). where u = F −1 u Problem (11.1) follows from the problem (3.6) by reflection with respect to S1 and localization to a neighborhood of S2 . The localization is made by using a smooth cut-off function of x3 equal 1 near S2 and vanishing near x3 = a. Therefore it is sometimes convenient to consider problem (11.1) for x3 ∈ R+ . Then we reckon that the solution to (11.1) is extended by 0 with respect to x3 . In fact this is a natural interpretation because this possibility follows from the properties of the cut-off function (in x3 ). The role of the problem (11.1) is then to help to detect such behavior of solutions to (3.6) that the energy type estimate for solutions to problem (1.1) proved in Lemma 3.2 is attainable. Therefore integrals on (0, a) and R+ are equivalent. We have to mention that the methods and tools used in this chapter had already appeared in [Z10] in a different setting. In Lemmas 10.5 and 10.6 we proved local regularity of weak solutions such that u ∈ Hμ2 (D) and uHμ2 (D) ≤ c(f L2,μ (D) + dH 1 (S2 ) ),

(11.2)

where μ ∈ (0, 1). The aim of this chapter is to show that f ∈ Lp,μ (D) implies 2 l that u ∈ Vp,μ (D), for p > 2 and Vp,β (Q), Q ⊂ R3+ , is a set of functions with the finite norm uVp,β l (Q) =

 |α|≤l

p(β+|α|−l)

|Dxα u|p x3

dx dx3

1/p ,

Q

where x = (x1 , x2 ), p ∈ [1, ∞], β ∈ R, α = (α1 , α2 , α3 ) is a multiindex. We observe that 0 (Q) = Lp,β (Q), Vp,β

l V2,β (Q) = Hβl (Q).

We recall the following local regularity result in Sobolev spaces Wp2 (see C.B. Morrey book [M]).

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

145

Lemma 11.1 (Local Regularity) Let B1 = {x ∈ R2 : |x | < r},

B2 = {x ∈ R2 : |x | < 2r},

ζi = ζi (x3 ), ζi ∈ C0∞ (R+ ), i = 1, 2, ζ1 ζ2 = ζ1 , supp ζ2 ⊂ {x3 : x3 < c1 }, and ζ1 , ζ2 are equal 1 near x3 = 0. Then for a function u ∈ Wp2 (B2 × R+ ) the following inequality holds ζ1 uWp2 (B1 ×R+ ) ≤ c(ζ2 ΔuLp (B2 ×R+ ) + ζ2 uLp1 (B2 ×R+ ) ),

(11.3)

where p1 ∈ [1, ∞]. Definition 11.2 (Partition of Unity) Consider the families {ζj }∞ j=−∞ , ∞ {σj }∞ j=−∞ , where ζj , σj ∈ C (R+ ) satisfy supp ζj ⊂ {x3 : 2j−1 < x3 < 2j+1 }, supp σj ⊂ {x3 : 2j−2 < x3 < 2j+2 }, ζj σ j = ζ j ,

|∂xα3 ζj | + |∂xα3 σj | ≤ cα 2−jα , α ∈ N0 .

Properties of these partitions make it possible to show the following statement. Lemma 11.3 Let β ∈ R. Then for any function u ∈ Wp2 (R2 × {2j−2 < x3 < 2j+2 }), the following inequality holds 2 (R2 ×R ) ≤ cσj ΔuL 2 + cσj uLp,β−2 (R2 ×R+ ) . ζj uVp,β p,β (R ×R+ ) +

Proof We define B = {x : |x | < 2}, Bμ = {x : |x | < 21+μ }, K = {x3 : 1 < x3 < 2}, Kμ = {x3 : 2μ < x3 < 2μ+1 }.

(11.4)

146

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

Applying Lemma 11.1 with p1 = p we obtain 2 

Dα (ζj u)Lp (B×K) ≤ cσj ΔuLp (B1 ×2K) + cσj uLp (B1 ×2K) ,

|α|=0

) where 2K = x3 :

1 2

* < x3 < 4 . In view of scaling x → 2μ x we have 2 

2μ(|α|−2) Dα (ζj u)Lp (Bμ ×Kμ )

|α|=0

≤ cσj ΔuLp (Bμ+1 ×2Kμ ) + c2−2μ σj uLp (Bμ+1 ×2Kμ ) , where 2Kμ = {x3 : 2μ−1 < x3 < 2μ+2 }. Now we multiply this formula by 2βμ and then raise the resulting inequality to the power p. Next, we note that  = x3 ∼ 2μ and we sum over μ to obtain 2 

β+|α|−2 Dα (ζj u)Lp (R3+ ) ≤ cβ σj ΔuLp (R3+ )

|α|=0

+cβ−2 σj uLp (R3+ ) . 2 Applying the definition of spaces Vp,β and Lp,β to this estimate yields (11.4). This concludes the proof.  

Corollary 11.4 For β ∈ R and u as in Lemma 11.3, 2 (R3 ) ≤ cΔuL 3 uVp,β + cuLp,β−2 (R3+ ) . p,β (R+ ) +

(11.5)

Proof We sum up inequalities (11.4) with respect to j to obtain (11.5). This ends the proof.   Let P (∂x , ∂x3 ) = −Δ,

P (iξ, ∂x3 ) = −∂x23 + ξ 2 .

Let A(ξ) denote the operator of the problem u = fˆ, P (iξ, ∂x3 )ˆ u ˆ,x3 |x3 =0 = 0,

u ˆ|x3 =a = 0.

(11.6)

Lemma 11.5 Ker A(ξ) = 0. Proof Take ξ = 0. Then every solution of homogenous Eq. (11.6)1 has the form

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

147

u ˆ = α sin h(|ξ|x3 ) + β cos h(|ξ|x3 ), where α, β are arbitrary parameters and (11.6)2 implies the equations for α, β [|ξ|α cos h(|ξ|x3 ) + |ξ|β sin h(|ξ|x3 )]|x3 =0 = 0, [α sin h(|ξ|x3 ) + β cos h(|ξ|x3 )]|x3 =a = 0, so α = 0, β = 0. If ξ = 0, then any solution to homogeneous (11.6)1 has the form u ˆ = αx3 + β, and now (11.6)2 gives α = 0, β = 0. This concludes the proof.

 

Corollary 11.6 There exists an inverse operator A−1 (ξ) to problem (11.6) such that u ˆ(ξ, x3 ) = A−1 (ξ)fˆ(ξ, x3 ). Corollary 11.7 From Lemma 10.6 we have for β ∈ (0, 1) −1 ˆ ˆV 0 (R ) . 2 (R ) = A(ξ) 2 (R ) ≤ cf f V2,β ˆ uV2,β + + + 2,β

(11.7)

Let fˆν = fˆζν and let u ˆν be a solution to the problem uν = fˆν , P (ξ, ∂x3 )ˆ u ˆν,x3 |x3 =0 = 0,

u ˆν |x3 =a = 0.

(11.8)

Then u ˆν = A−1 (ξ)fˆν .

(11.9)

Hence ˆμ = σμ A−1 (ξ)fˆν . σμ u The further presentation depends heavily on the considerations from Maz’ya and Plamenevskii [MP, Section 7]. Lemma 11.8 In view of Corollary 11.7 and ξ = 0 we have −ε|μ−ν|+2μ 0 (R )→V 0 (R ) ≤ c2 σμ A−1 (ξ)fˆν V2,β , + + 2,β

where β ∈ (0, 1) and ε > 0.

(11.10)

148

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

Proof In view of (11.7) and properties of the partition of unity, we have εμ εμ ˆ 2 (R ) ≤ 2 2 0 ˆν V2,β σμ u ˆν V2,β−ε σμ u (R+ ) ≤ 2 fν V2,β−ε (R+ ) + 0 (R ) . ≤ 2ε(μ−ν) fˆν V2,β +

Since −2μ 2 (R ) ≥ 2 0 (R ) , σμ u ˆν V2,β σμ u ˆν V2,β + +

we conclude ε(μ−ν)+2μ ˆ 2 (R ) ≤ 2 0 (R ) σμ u ˆν V2,β fν V2,β + + 0 (R ) . = 2−ε(ν−μ)+2μ fˆν V2,β +

(11.11)

Taking −ε instead of ε we get similarly −ε(μ−ν)+2μ ˆ 0 (R ) ≤ 2 0 (R ) . σμ u ˆν V2,β fν V2,β + +

(11.12)  

In view of (11.9), (11.11), and (11.12) we conclude the proof.

Now, we recall the following Marcinkiewicz-Mikhlin type result (see Dunford and Schwartz [DS, Part 2, Ch. 11, Theorem 28]). Proposition 11.9 (Marcinkiewicz-Mikhlin Theorem) Let Lp (Rd ; H) be the space of functions with the finite norm  f Lp (Rd ;H) =

1/p Rd

f (z, ·)pH dz

< ∞,

where H is a Hilbert space. Let M (ξ), ξ ∈ Rd , be a bounded linear operator in H. Assume that for s = 0, · · · , d, ik = il , + + + + ∂sM s+ |ξ| + (ξ)+ ≤ const . ∂ξi1 . . . ∂ξis +H→H −1 M (ξ)Fz→ξ is a continuous Then, if F is the Fourier transform in Rd , Fξ→z operator in Lp (Rd ; H). 2 Lemma 11.10 In view of Corollary 11.7 and uν ∈ V2,β (R3+ ) such that

P (∂x , ∂x3 )uν = fν , uν,x3 |x3 = 0,

uν |x3 =a = 0,

(11.13)

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

149

we have 

 R2

R+

 2 x2β 3 |σμ (x3 )uν (x , x3 )| dx3

−pε|μ−ν|+2μp





≤ c2

R2

p/2

dx

 2 x2β 3 |ζν (x3 )f (x , x3 )| dx3

R+

(11.14)

p/2



dx .

Proof We have −1 −1 Fx →ξ ζν f, uν = Fξ→x  A(ξ)

where F denotes the Fourier transform in R2 . On the other hand, applying the Marcinkiewicz-Mikhlin result formulated in Proposition 11.9 we find that F −1 M (ξ)F , where M = σμ A(ξ)−1 ζν is 0 (R+ )). Thus using estimate (11.10) we a continuous operator in Lp (R2 ; V2,β derive the result. This concludes the proof.   Lemma 11.11 Let the assumptions of Corollary 2 uν ∈ V2,β (R3+ ). Then for p ≥ 2 and some ε1 > 0 we have 

p(β−1)−2

R3+

x3

≤ c2−|μ−μ|ε1 p

11.7

hold

and

|ζμ uν (x , x3 )|p dx dx3 

(11.15) p(β+1)−2

R3+

x3

|ζν f (x , x3 )|p dx dx3 .

Proof By the H¨older inequality we estimate the integral on the r.h.s. of (11.14) as follows 

 R2

 

 ≤

1 R2

p p−2

 p−2  p dx3

R+

supp ζν

(ν−1) p−2 2

≤2

 ≤c

 p xpβ 3 |ζν (x3 )f (x , x3 )| dx3



 R2

 R2

R+

 2 x2β 3 |ζν (x3 )f (x , x3 )| dx3

R+

R+

2/p p/2

dx dx

 p  xpβ 3 |ζν (x3 )f (x , x3 )| dx3 dx

p(β+1/2)−1

x3

p/2

|ζν (x3 )f (x , x3 )|2 dx3 dx , p

where we used that supp ζν ⊂ (2ν−1 , 2ν+1 ), x3 ∈ (2ν−1 , 2ν+1 ), 2(ν−1)( 2 −1) ∼ p −1 βp+ p −1 p(β+1/2)−1 so x3 2 = x3 . x32

150

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

Next, we deal with the l.h.s. of (11.14). Let {Qj } cover R2 , where Qj is a box with the edge equal to 2μ−1 and by 2Qj we denote a box with parallel sides to Qj and with edges equal to 2μ . The boxes 2Qj must have the same centers as Qj . Hence Qj ⊂ 2Qj . By the H¨ older inequality we have 



I1 ≡ 2Qj





1/2  

≤

12 dx3 2Qj

R+

|σμ uν |dx dx3

R+

supp σμ





μ

≤

1/2 |σμ uν |2 dx3 1/2

|σμ uν |2 dx3

2 2 +1 R+

2Qj

dx

dx ≡ I2 ,

where we used that supp σμ ⊂ (2μ−2 , 2μ+2 ). Next, we obtain 

 I2 ≤

1/2

2μ/2−βμ R+

2Qj

2 x2β 3 |σμ uν | dx3

dx ,

where we used that x3 ∼ 2μ on supp σμ , so 2−2βμ x2β 3 ∼ 1 on supp σμ . Using estimates for I1 and I2 , we have 



I1p = 2Qj

R+

|σμ uν |dx dx3 

p



≤ c2

1/2

(1/2−β)pμ R+

2Qj

2 x2β 3 |σμ uν | dx3

dx



(11.16)

p ≡ I3 .

Using the H¨older inequality we have   I3 ≤ c2

(1/2−β)pμ

1

p p−1

dx3

 p−1 p

·

2Qj





· 2Qj

p/2 R+

2 x2β 3 |σμ uν | dx3

dx

(11.17)

1/p p ≡ I4 .

Using that meas 2Qj ≤ c22μ we get 

 I4 ≤ c2

p/2

(1/2−β)μp+2μ(p−1) 2Qj

R+

2 x2β 3 |σμ uν | dx3

dx .

(11.18)

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

151

Take |μ − ν| > 3, then supp ζν ∩ supp σμ = φ. Then using (11.4) for ζj = ζν , σj = σμ , u = uν , the first term on the r.h.s. vanishes. Hence using scaling x → 2μ x we get 



|σμ uν |p dx dx3 ≤ c23μ(1−p)

R+

Qj





2Qj

R+

|σμ uν |dx dx3 =

p (11.19)

c23μ(1−p) I1p ,

because scaling on the l.h.s. gives factor 23μ and on the r.h.s. it is 23μp . From (11.16) to (11.18) we have 

 I1p

≤ c2

p/2

(1/2−β)μp+2μ(p−1) R+

2Qj

2 x2β 3 |σμ uν | dx3

dx .

(11.20)

Hence, (11.19) and (11.20) imply  R3+

|σμ uν |p dx dx3 

 ≤ c2(1/2−β)μp+2μ(p−1)+3μ(1−p)

R2

p/2 R+

2 x2β 3 |σμ uν | dx3

(11.21) dx ,

where 

     1 1 − β μp + 2μ(p − 1) + 3μ(1 − p) = −μ p β + −1 . 2 2

Next, inequalities (11.14) and (11.21) give  2μ[p(β−3/2)−1] R3+

|σμ uν |p dx dx3

≤ 2μ[p(β−3/2)−1] 2−μ[p(β+1/2)−1] · 2−pε|μ−ν|+2μp · p/2   2β  2 · x3 |ζν (x3 )f (x , x3 )| dx3 dx R2

≤ c2−pε|μ−ν| −pε|μ−ν|

≤ c2

(11.22)

R+



 

R2

R+

 2 x2β 3 |ζν (x3 )f (x , x3 )| dx3

p(β+1/2)−1

R3+

x3

|ζν f |p dx3 dx ,

p/2

dx

152

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

where we used the H¨older inequality 

 R2

 

 ≤

R+

 p−2  p 1dx3

R2

supp ζν

 ≤c

R2

2/p p/2 R+



p/2 2 x2β 3 |ζν f | dx3

p xpβ 3 |ζν f | dx3

dx dx

p

R+ ∩supp ζν



≤

p 2ν ( 2 −1) xpβ 3 |ζν f | dx3

p(β+1/2)−1

R3+

x3

|ζν f |p dx3 dx .

We multiply both sides of (11.22) by 2μ(p/2−1) , use that supp ζμ ⊂ supp σμ , and on the l.h.s. we exploit properties of supp ζμ but on the r.h.s. properties of supp ζν . Therefore, (11.22) yields  2μ(p/2−1) 2μ[p(β−3/2)−1] ≤ c2μ(p/2−1) 2−pε|μ−ν|

R3+



|ζμ uν |p dx dx3

p(β+1/2)−1

R3+

x3

|ζν f |p dx dx3 .

Continuing, we have  μ[p(β−1)−2]

2

R3+

 ≤ c2

(μ−ν)(p/2−1)−pε|μ−ν| R3+

|ζμ uν |p dx dx3

p(β+1)−2 x3 |ζν f |p dx dx3 ,

−(p/2−1)

where x3 |supp ζν = c2−ν(p/2−1) . Let μ > ν. Then the exponent of 2 on the r.h.s. equals 

 1 1 −p(μ − ν) ε − + ≡ −p(μ − ν)ε1 , 2 p and for μ < ν it equals   1 1 −pε(ν − μ) ε + − = −p(ν − μ)ε1 . 2 p

(11.23)

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

153

Hence (11.23) takes the form 

p(β−1)−2

x3

R3+

≤ c2−p|μ−ν|ε1



|ζμ uν |p dx dx3 (11.24)

p(β+1)−2

R3+

x3

|ζν f |p dx dx3 .

We note that in the case |μ − ν| < 3 we have to add on the r.h.s. of (11.19) the expression   2μp c2 |ζν f |p dx dx3 . (11.25) 2Qj

R+

Multiply (11.25) by the same factor as in (11.22) we obtain (11.25) in the form  p(β+1)−2 c22μp 2μ[p(β−3/2)−1] 2−ν[p(β+1)−2] x3 |ζν f |p dx dx3 , R3+

where the factor before the integral equals c2(μ−ν)[p(β+1/2)−1] 2−ν(p/2−1) ≤ c because p > 2 and |μ − ν| < 3. The above considerations imply (11.15) and this concludes the proof.

 

Estimate (11.15) has a local character. To obtain this estimate for whole R3+ we need the following result from [MP, Sect. 7, Lemma 7.7]. Let {ζj }∞ j=−∞ be a partition of unity introduced in Definition 11.2. Let E0 and E1 be two Banach spaces such that the multiplication of their elements by smooth scalar functions is defined. Assume that there exist numbers p and q, 1 ≤ p ≤ q ≤ ∞, such that for all u ∈ E0 and v ∈ E1 the following inequalities are valid uE0 ≤ c

  ∞

1/q ζj uqE0

(11.26)

j=−∞

and vE1 ≥ c

  ∞ j=−∞

where  Ei is the norm of Ei , i = 0, 1.

1/p ζj vpE1

,

(11.27)

154

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

Lemma 11.12 (See [MP, Section 7, Lemma 7.7]) Let ϑ : E1 → E0 be a linear operator defined on functions with compact supports such that for some ε > 0 and arbitrary integers μ and ν we have ζμ ϑζν vE0 ≤ ce−ε|μ−ν| ζν vE1 ,

(11.28)

where v is an arbitrary element of E1 . Then for any v ∈ E1 with a compact support the following inequality holds ϑvE0 ≤ cvE1 ,

(11.29)

where c does not depend on v. Proof In view of (11.26), ϑvE0

+  ∞ +  + + + = +ϑ ζν v + + ν=−∞

≤c

  ∞   ∞ μ=−∞

E0

+q 1/q   ∞ +  + ∞ + + ≤c ζμ ϑζν v + + + μ=−∞

E0

ν=−∞

q 1/q ζμ ϑζν vE0

.

ν=−∞

Hence, from (11.28) we derive ϑvE0 ≤ c

  ∞   ∞ μ=−∞

e

−ε|μ−ν|

q 1/q ζν vE1

.

ν=−∞

Employing the properties of the discrete convolution we obtain for q ≥ p ϑvE0 ≤ c

  ∞

1/p ζν vpE1

.

ν=−∞

 

Hence (11.27) implies (11.29) and concludes the proof. Lemma 11.13 Let u be a solution to the problem P (∂x , ∂x3 )u = f, u,x3 |x3 =0 = 0,

u|x3 =a = 0.

(11.30)

Let f ∈ Lβ+1−2/p (R3+ ), p ≥ 2. Then 

p(β−1)−2

R3+

x3

|u(x , x3 )|p dx dx3 ≤ c



p(β+1)−2

R3+

x3

|f (x , x3 )|p dx dx3 . (11.31)

11

The Neumann Problem (3.6) in Lp -Weighted Spaces

155

Proof To prove the lemma we apply Lemma 11.12 to inequalities (11.15). 0 0 Setting q = p, E0 (R3+ ) = Vp,β−1−2/p (R3+ ), E1 (R3+ ) = Vp,β+1−2/p (R3+ ), and assuming that ϑ is the inverse operator to the operator of problem (11.30), we write (11.15) in the form ζμ ϑζν vE0 (R3+ ) ≤ c exp(−ε|μ − ν|)ζν vE1 (R3+ ) .  

Then Lemma 11.12 implies (11.31). This concludes the proof. Lemma 11.14 Let u be a solution to (11.30). Let f ∈ Then uVp,κ 2 (R3 ) ≤ cf V 0 (R3 ) . p,κ + +

0 Vp,κ (R3+ ),

p ≥ 2.

(11.32)

Proof In view of the regularity result (11.5) we have 3 . uVp,κ 2 (R3 ) ≤ cf V 0 (R3 ) + cuV 0 p,κ + + p,κ−2 (R+ )

(11.33)

0 Now, we apply (11.31). Since f ∈ Vp,κ (R3+ ) we have that p(β + 1) − 2 = pκ so β = κ − 1 + 2/p. Then p(β − 1) − 2 = p(κ − 2). Therefore, (11.31) yields 0 uVp,κ−2 0 (R3 ) . (R3+ ) ≤ cf Vp,κ +

Combining this with (11.33) implies (11.32). This concludes the proof.

 

Chapter 12

Existence of Solutions (v, p) and (h, q)

Abstract In the last chapter we use the estimates proved in the previous parts of the book to show the existence of solutions (v, p) to NavierStokes equations in cylindrical domains. In Sect. 12.1 we use the Galerkin approximation to establish the existence of weak solutions such that v is weakly continuous with respect to t in L2 (Ω) norm and v converges to v0 = v|t=0 as t → 0 strongly in L2 (Ω) norm, as well as global weak solutions such that v ∈ V (Ω × (kT, (k + 1)T )), ∀k ∈ N0 = N ∪ {0}, vV (Ω×(kT,(k+1)T )) = ess

sup

where

|v(t)|L2 (Ω) + |∇v|L2 (Ω×(kT,(k+1)T )) .

kT 3 or p = 3, s > , p 3 3

where

1/6

di,t ∈ L2 (0, T ; W6/5 (S2 )), i = 1, 2. Then there exists a weak solution v to problem (1.1) such that v is weakly continuous with respect to t in L2 (Ω) norm and v converges to v0 as t → 0 strongly in L2 (Ω) norm. Moreover, v ∈ V (Ω T ), v · τ¯α ∈ L2 (0, T ; L2 (S1 )), for α = 1, 2 and v satisfies, for all t ≤ T v2V (Ω t ) + γ

2   α=1



t 0

v · τ¯α 2L2 (S1 ) ≤ 2f 2L2 (0,t;L6/5 (Ω))



+ϕ sup dW s−1/p (S2 ) τ ≤t

3



d2L (0,t;W 1/2 (S )) 2 2 2

+

dt 2L (0,t;W 1/6 (S )) 2 2 6/5

(12.1)

+v(0)2L2 (Ω) , where ϕ is a nonlinear positive increasing function. With the a priori estimate, we can show also the existence of global weak solutions. Theorem 12.2 Assume the compatibility condition (1.4). Let f ∈ L2 (kT, (k + 1)T ; L6/5 (Ω)), 1/2

di ∈ L∞ (R+ ; Wps−1/p (S2 )) ∩ L2 (kT, (k + 1)T ; W2 (S2 )), where

4 3 1 + ≤ s, p > 3 or p = 3, s > , p 3 3 1/6

di,t ∈ L2 (kT, (k + 1)T ; W6/5 (S2 )), i = 1, 2. Let us assume that v(0)L2 (Ω) ≤ A

12.1 Existence of Weak Solutions

159

for some constant A and 

f 2L6/5 (Ω)

2 



(k+1)T kT (k+1)T

+ ϕ sup dWps−1/p (S2 ) t



·

d2W 1/2 (S ) 2 2

kT



+



dt 2W 1/6 (S ) 2 6/5

≤ (1 − e−νT )A2

for all k ∈ N0 , where ϕ is a nonlinear positive increasing function. Then there exists a global weak solution v to (1.1) such that v ∈ V (Ω × (kT, (k + 1)T )) ∀k ∈ N0 = N ∪ {0}, and  v2V (Ω×(kT,t)) ≤ 2   +ϕ sup dWps−1/p (S2 ) τ

t



kT

t kT

f 2L6/5 (Ω) dτ + A2 (12.2) 

d2W 1/2 (S ) + dt 2W 1/6 (S 2

2

6/5

2)

dτ

for t ∈ (kT, (k + 1)T ]. We use the estimate from Lemma 3.2 w2V (Ω×(kT,t)) ≤ cA2 (T ) + exp(−νkT )|w(0)|22,Ω , where 

(k+1)T

A2 (T ) = sup k∈N0

kT

+dt (t)26/5,S2 )dt

(|f (t)|26/5,Ω + d(t)21,3,S2 ϕ(sup d(t)1,3,S2 ) < ∞, t

and (w, p) is a solution to the problem (3.8): wt + w · ∇w + w · ∇δ + δ · ∇w − div T(w, p) = f − δt − δ · ∇δ + νdiv D(δ) ≡ F (δ, t)

in Ω T ,

div w = 0

in Ω T ,

w·n ¯=0

on S T ,

νn ¯ · D(w) · τ¯α + γw · τ¯α = −ν n ¯ · D(δ) · τ¯α − γδ · τ¯α ≡ B1α (δ), α = 1, 2,

on S1T ,

n ¯ · D(w) · τ¯α = −¯ n · D(δ) · τ¯α ≡ B2α (δ), α = 1, 2,

on S2T ,

w|t=0 = v(0) − δ(0) ≡ w(0)

in Ω.

(12.3)

160

12

Existence of Solutions (v, p) and (h, q)

To prove the existence of weak solutions to the problem (3.8) we use the Galerkin method. We follow the ideas of Ladyzhenskaya [L1, Chapter 6, Sect.7]. Namely, we introduce a sequence of approximating functions wN given as wN (x, t) =

N 

CkN (t)ak (x),

k=1 2 0 where {ak }∞ k=1 is a system of orthonormal functions in L (Ω) ∩ J2 (Ω). Here,

J20 (Ω) = {f ∈ H 1 (Ω) : divf = 0} 1 and {ak }∞ k=1 is a fundamental system in H (Ω) satisfying

sup sup |ak (x)| < ∞,

sup sup |ak (x)| < ∞.

k∈N x∈Ω

k∈N x∈∂Ω

The coefficients CkN (0) are defined by CkN |t=0 = (w0 , ak ),

k = 1, . . . , N,

and the functions wN satisfy the following system with test functions ak :  

1 d N k w a + wN · ∇wN ak + δ · ∇wN · ak + wN · ∇δ · ak 2 dt Ω '  % +νD(wN )D(ak ) dx + γ wN · τ¯j ak τ¯j dS1 ⎛

=⎝

S1 2   j,σ=1

⎞



F · ak dx⎠

Bσj ak · τ¯j dSσ + Sσ

Ω

for k = 1, . . . , N. Thus, wN are weak solutions to (3.8). With  (f, g) = f gdx Ω

and  (f, g)S =

f gdS S

12.1 Existence of Weak Solutions

161

this can be rewritten as: (wtN , ak ) + (wN · ∇wN , ak ) + (δ · ∇wN , ak ) + (wN · ∇δ, ak ) ' +ν(D(wN ), D(ak )) + γ(wN · τ¯j , ak · τ¯j )S1 ⎡ ⎣

2 

=

⎤ (Bσj , ak · τ¯j )Sσ + (F, ak )⎦ ,

k = 1, . . . , N.

σ,j=1

Thus, 

d N k w ,a dt

 + (wN · ∇wN , ak ) + (δ · ∇wN , ak ) + (wN · ∇δ, ak ) +ν(D(wN ), D(ak )) + γ(wN · τ¯j , ak · τ¯j )S1 (12.4) =

2 

(Bσj , ak · τ¯j )Sσ + (F, ak ),

k = 1, . . . , N.

j,σ=1

The above equations are in fact a system of ordinary differential equations for the functions CkN (t). The properties of the sequence ak imply wN (x, t)2L2 (Ω) =

N 

2 CkN (t).

k=1

On the other hand, we can obtain a priori bounds for the approximate solutions wN of the same form as (12.3):  wN 2V (Ω T ) = sup wN L2 (Ω) + 0≤t≤T

 ˜ W s (Ω) ) ≤ ϕ( sup d p 0≤t≤T



T 0

"

T

∇wN L2 (Ω) dt 0

˜2 1 ˜ 2 d W (Ω) + dt W 1 2

6/5 (Ω)

# dt

(12.5)

T 0

f 2L6/5 (Ω) dt + wN (0)2L2 (Ω) ≤ C,

where p3 + 13 ≤ s, p > 3 or p = 3, s > 43 . Therefore, sup0≤t≤T |CkN (t)| is bounded on [0, T ] and wN are well defined for all times t. Let us now define ψN,k ≡ (wN (x, t), ak (x)). This sequence is uniformly bounded by (12.5). We can also show that it is equicontinuous. Namely, we integrate (12.4) with respect to t from t to t + Δt to obtain

162

12



t+Δt $

|wN · ∇wN |L2 (Ω) +|δ · ∇wN |L2 (Ω)

|ψN,k (t+Δt)−ψN,k (t)| ≤ sup |ak (x)| x∈Ω

Existence of Solutions (v, p) and (h, q)

t

% + |wN · ∇δ|L2 (Ω) + |F |L2 (Ω) dτ + ν|∇ak |L2 (Ω) t+Δt

+ γ sup |ak (x)| x∈S

t+Δt

|∇wN |L2 (Ω) dτ t

⎛





⎝|wN · τ¯j |L2 (S1 ) +

t

2 

⎞

|Bσj |L2 (Sσ ) ⎠ dτ

j,σ=1



√

≤ sup |a (x)| Δt sup |wN |L2 (Ω) (|∇wN |L2 (Ω T ) + |∇δ|L2 (Ω T ) ) k

x∈Ω

x∈Ω



+ sup |δ|L2 (Ω) |∇wN |L2 (Ω T ) x∈Ω



t+Δt

+ sup |a (x)| k

x∈Ω

√ |F |L2 (Ω) dτ + ν|∇ak |L2 (Ω) Δt|∇wN |L2 (Ω T )

t

⎛

√

+ γ sup |ak (x)| ⎝ Δt|∇wN |L2 (Ω T ) + x∈S



t+Δt t

⎛

√

≤ C(k) ⎝ Δt +



t+Δt

(|F |L2 (Ω) + t

2 

2 

⎞ |Bj |L2 (S) )dτ ⎠

j=1

⎞ |Bj |L2 (S) )dτ ⎠ .

j=1

We can see that for given k and N ≥ k the r.h.s. tends to zero as Δt → 0 uniformly in N. Thus, it is possible to choose a subsequence Nm such that ψNm ,k converges as m → ∞ uniformly to some continuous function ψk for any given k. Since the limit function w is defined as w(x, t) =

∞ 

ψk (t)ak (x),

k=1

we conclude that (wNm − w, ψ(x)) tends to zero as m → ∞ uniformly with respect to t ∈ [0, T ] for any ψ ∈ J20 (Ω) and w(x, t) is continuous in t in weak topology. Moreover, estimates (12.5) remain true for the limit function w. We will show that {wNm } converges strongly in L2 (Ω T ). To this end, we need to apply the following version of the Friedrichs lemma: for any ε > 0, there exists Nε such that for any u ∈ W21 (Ω) (see Lemma 6.1 from Ladyzhenskaya et al. [LSU, Ch. 5, Sect. 6]) the following inequality holds: ||u||2L2 (Ω) ≤

Nε  k=1

|(u, ak )|2 + ε||∇u||2L2 (Ω) .

12.1 Existence of Weak Solutions

163

This in terms of u = wNm − wNl reads ||wNm −wNl ||2L2 (Ω T ) ≤

Nε   k=1

T 0

|(wNm −wNl , ak )|2 dt+ε||∇wNm −∇wNl ||2L2 (Ω T ) .

By (12.5), we have ||∇wNm − ∇wNl ||2L2 (Ω T ) ≤ 2C 2 for some constant C. The first integral on the r.h.s. for a given number Nε can be arbitrarily small if only m and l are sufficiently large, so it tends to zero as m, l → ∞. Therefore, {wNm } converges strongly in L2 (Ω T ). We summarize the above convergence properties of the sequence {wNm } : 1. {wNm } → w strongly in L2 (Ω T ) for some w, 2. {wNm } → w weakly in L2 (Ω) uniformly with respect to t ∈ [0, T ], 3. ∇{wNm } → ∇w weakly in L2 (Ω T ). k j Nm For given Φk = } satisfies the j=1 dj (t)a (x), the sequence {w identities    d Nm k Nm Nm Nm Nm k Nm k w Φ +(w · ∇w +δ · ∇w +w · ∇δ)Φ +νD(w )D(Φ ) dx Ω dt   2   +γ wNm · τ¯j Φk · τ¯j dS0 = Bσj Φk · τ¯j dSσ + F Φk dx. S1

σ,j=1

Sσ

Ω

Then, we can pass to the limit with m → ∞ to obtain the identity for w. The conditions divwN = 0, wN · n ¯ |S T = 0 remain true for the limit function w as well. It remains to consider the limit limt→0 w(x, t). We note that wNm satisfies the relation (3.10) (if we use the test function wNm ). This yields 

t

wNm L2 (Ω) ≤ w0 L2 (Ω) +

(F L2 (Ω) + BL2 (S) )dt. 0

In the limit m → ∞ we obtain 

t

wL2 (Ω) ≤ w0 L2 (Ω) +

(F L2 (Ω) + BL2 (S) )dt, 0

which implies limt→0 wL2 (Ω) ≤ w0 L2 (Ω) .

164

12

Existence of Solutions (v, p) and (h, q)

On the other hand, since wNm tends to w as m → ∞, we have wNm − w0 L2 (Ω) → 0. Therefore, |wNm − w0 | → 0 weakly in L2 (Ω) as t → 0 and w0 L2 (Ω) ≤ limt→0 wL2 (Ω) . We conclude that the limit limt→0 wL2 (Ω) exists and is equal to w0 L2 (Ω) where the convergence is strong, in the L2 (Ω) norm. Consequently, we have proved the following result. Lemma 12.3 Let the assumptions of Theorem 12.1 be satisfied. Then there exists a weak solution w to problem (3.8) such that w is weakly continuous with respect to t in the L2 (Ω) norm and w converges to w0 as t → 0 strongly in the L2 (Ω) norm. Since v = w − δ we deduce the analogous existence result for v, formulated in Theorem 12.1.

12.2

Existence of Regular Solutions

Lemma 12.4 Let the assumptions of Theorem 4.14 be satisfied. Then the solution (v, p) to (1.1) and solution (h, q) to (4.6) satisfying (4.81) and (4.82) exist on (0, T ) for some T > 0. Proof To prove the existence of solutions to problem (1.1) we will use the Leray-Schauder theorem. To this end, we construct the mappings vt − div T(v, p) = −λ˜ v · ∇˜ v+f

in Ω T = Ω × (0, T ),

div v = 0 in Ω T , v·n ¯ = 0 on S1T ,

and

νn ¯ · D(v) · τ¯α + γv · τ¯α = 0, α = 1, 2,

on S1T ,

v·n ¯=d

on S2T ,

n ¯ · D(v) · τ¯α = 0, α = 1, 2,  v t=0 = v(0)

on S2T , in Ω,

(12.6)

12.2 Existence of Regular Solutions

165

˜+h ˜ · ∇˜ h,t − div T(h, q) = −λ(˜ v · ∇h v) + g

in Ω T ,

div h = 0

in Ω T ,

¯ = 0, n n·h ¯ · D(h) · τ¯α + γh · τ¯α = 0, α = 1, 2

on S1T ,

hi = −dxi , i = 1, 2, h3,x3 = Δ d  ht=0 = h(0)

on S2T ,

(12.7)

in Ω,

˜ are treated as given where g = f,x3 , Δ = ∂x21 + ∂x22 , λ ∈ [0, 1] and v˜, h ˜ functions. We assume that h = v˜,x3 , thus differentiating (12.6) with respect to x3 and subtracting from (12.7) we obtain that h = v,x3 . Problems (12.6) and (12.7) define the mappings v , λ) → (v, p), Φ1 : (˜ ˜ λ) → (h, q). Φ2 : (˜ v , h, We set Φ = (Φ1 , Φ2 ). In the previous chapters we have shown a priori estimate for a fixed point of Φ for λ = 1. On the other hand, for λ = 0 we have a unique existence of solutions to problems (12.6) and (12.7). Let us introduce the space M(Ω T ) = L2r (0, (T ; W 26η (Ω)), 3+η

η ≥ 2, r ≥ 2.

We shall find restrictions on r, η such that Φ : M(Ω (T ) × M(Ω T ) → M(Ω T ) × M(Ω T ) is a compact mapping. Assume that v˜ ∈ L2r (0, T ; W 16η (Ω)). Then 3+η



1/r

T

˜ v · ∇˜ v Lr (0,T ;Lη (Ω)) =

dt˜ v · ∇˜ v rLη (Ω)

0



1/r

T

dt˜ v rL

≤ 0



6η 3−η

v rL (Ω) ∇˜

0

(12.8)

(Ω)

1/r

T

≤c

6η 3+η

dt˜ v 2r W1

6η 3+η

(Ω)

≤ c˜ v 2L2r (0,T ;W 1

6η 3+η

(Ω)) .

166

12

Existence of Solutions (v, p) and (h, q)

In the same way we obtain ˜ · ∇˜ ˜ L (0,T ;L (Ω)) + h v Lr (0,T ;Lη (Ω)) ˜ v · ∇h r η ≤ c˜ v L2r (0,T ;W 16η

˜

(Ω)) hL2r (0,T ;W 16η (Ω)) .

3+η

(12.9)

3+η

In view of (12.8) and (12.9) can be shown that solutions to problems (12.6) 2,1 and (12.7) belong to Wη,r (Ω T ) (see Solonnikov [S1]). We are going to use the following imbeddings 2,1 (Ω T ) W22,1 (Ω T ) ⊃ Wη,r

(12.10)

W22,1 (Ω T ) ⊂ L2r (0, T ; W 16η (Ω)) ≡ M(Ω T ),

(12.11)

and 3+η

where (12.10) holds for η ≥ 2, r ≥ 2 and (12.11) is compact for r, η satisfying the inequality 3 2 5 − 6η − < 1, 2 2r 3+η which takes the form 1