Seregin Trends in Partial Differential Equations of Mathematical Physics

1. Auflage 2006
ISBN: 978-3-7643-7317-7
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 282 Seiten

Vsevolod Alekseevich Solonnikov is known as one of the outstanding mathematicians from the St. Petersburg Mathematical School. His remarkable results on exact estimates of solutions to boundary and initial-boundary value problems for linear elliptic, parabolic, Stokes and Navier-Stokes systems, his methods and contributions to the inverstigation of free boundary problems, in particular in fluid mechanics, are well known to specialists all over the world.

The International Conference on "Trends in Partial Differential Equations of Mathematical Physics" was held on the occasion of his 70th birthday in Óbidos (Portugal) from June 7 to 10, 2003. The conference consisted of thirty-eight invited and contributed lectures and gathered, in the charming and unique medieval town of Óbidos, about sixty participants from fifteen countries.

This book contains twenty original contributions on many topics related to V.A. Solonnikov's work, selected from the invited talks of the conference.

Written for: Postgraduates and researchers in analysis, pde and mathematical physics, physicists

Autoren/Hrsg.

Seregin, Gregory

Inhaltsverzeichnis

1;Table of Contents;6
2;Preface;8
2.1;On Vsevolod Alekseevich Solonnikov and his 70th birthday;9
2.2;References;12
3;List of Participants;14
4;Stopping a Viscous Fluid by a Feedback Dissipative Field: Thermal E.ects without Phase Changing;16
4.1;1. Introduction;16
4.2;2. Statement of the problem;17
4.3;3. Existence theorem;18
4.4;4. Uniqueness of weak solution;21
4.5;5. Localization e.ect;25
4.6;6. Case of a temperature depending viscosity;27
4.7;References;28
5;Ultracontractive Bounds for Nonlinear Evolution Equations Governed by the Subcritical p-Laplacian;30
5.1;1. Introduction;30
5.2;2. Entropy and logarithmic Sobolev inequalities;33
5.3;3. Preliminary results;35
5.4;References;40
6;Weighted L2-spaces and Strong Solutions of the Navier-Stokes Equations in R3;42
6.1;1. Introduction;42
6.2;2. Proof of Theorem;44
6.3;References;49
7;A Limit Model for Unidirectional Non-Newtonian Flows with Nonlocal Viscosity;52
7.1;1. Introduction;52
7.2;2. The limit problems and their formulations;54
7.3;3. Existence of weak solutions and their convergence;55
7.4;References;58
8;On the Problem of Thermocapillary Convection for Two Incompressible Fluids Separated by a Closed Interface;60
8.1;1. Statement of the problem and formulation of the main result;60
8.2;2. Linearized problems;65
8.3;3. Solvability of problem;71
8.4;(;71
8.5;References;79
9;Some Mathematical Problems in Visual Transduction;80
9.1;1. The phototransduction cascade;80
9.2;2. The physical model;83
9.3;3. The limiting equations;84
9.4;4. Main ideas in computing the homogenized limit;86
9.5;5. Weak formulation of the homogenized problem;90
9.6;6. Cytosol well-stirred in the transversal variables;91
9.7;7. Globally well-stirred cytosol;92
9.8;8. Further results and open issues;92
9.9;References;94
10;Global Regularity in Sobolev Spaces for Elliptic Problems with p-structure on Bounded Domains;96
10.1;1. Introduction;96
10.2;2. The main results;97
10.3;3. Proof of the theorem;98
10.4;References;104
11;Temperature Driven Mass Transport in Concentrated Saturated Solutions;106
11.1;1. Introduction;106
11.2;2. Description of physical system and;107
11.3;the governing di.erential equations;107
11.4;3. Modelling a speci.c mass transport process with deposition;110
11.5;4. Analysis of Stage 1;114
11.6;5. Analysis of Stage 2: a priori results;120
11.7;6. Analysis of Stage 2: weak formulation and existence;121
11.8;References;123
12;Solvability of a Free Boundary Problem for the Navier-Stokes Equations Describing the Motion of Viscous Incompressible Nonhomogeneous Fluid;124
12.1;1. Statement of the problem;124
12.2;2. Function spaces;126
12.3;3. Lagrange coordinates;128
12.4;4. Auxiliary linear problem;130
12.5;5. Model problems;132
12.6;6. Proof of Theorem;136
12.7;7. Nonlinear problem;138
12.8;References;138
13;Duality Principles for Fully Nonlinear Elliptic Equations;140
13.1;1. Introduction;140
13.2;2. Duality;142
13.3;3. A priori estimates;147
13.4;References;150
14;On the Bénard Problem;152
14.1;1. Introduction;152
14.2;2. Boussinesq approximation;154
14.3;3. Compressible scheme: layer heated from above;155
14.4;4. Compressible scheme: layer heated from below;159
14.5;References;162
15;Exact Boundary Controllability for Quasilinear Wave Equations;164
15.1;1. Introduction and main results;164
15.2;2. Reduction of equation and boundary conditions;167
15.3;3. Semi-global;169
15.4;solution for quasilinear hyperbolic;169
15.5;systems with zero eigenvalues;169
15.6;4. Proof of Theorems 1 and 2;170
15.7;5. Remarks;174
15.8;References;174
16;Regularity of Euler Equations for a Class of Three-Dimensional Initial Data;176
16.1;1. Introduction and main results;176
16.2;2. Poincar´ e-Sobolev equations in cylindrical domains;183
16.3;3. The structure and regularity of fast singular oscillating;188
16.4;limit equations;188
16.5;4. Long time regularity for .nite large;192
16.6;References;198
17;A Model of a Two-dimensional Pump;202
17.1;References;210
18;Regularity of a Weak Solution to the Navier-Stokes Equation in Dependence on Eigenvalues and Eigenvectors of the Rate of Deformation Tensor;212
18.1;1. Introduction;212
18.2;2. Regularity in dependence on eigenvalues of the rate of;215
18.3;deformation tensor;215
18.4;3. Regularity in dependence on eigenvectors of the rate of;219
18.5;deformation tensor;219
18.6;References;227
19;Free Work and Control of Equilibrium Con.gurations;228
19.1;1. Introduction;228
19.2;2. Stability in the mean;231
19.3;3. Asymptotic decay for hyperelastic, viscous materials;234
19.4;4. Nonlinear instability for hyperelastic bodies;235
19.5;References;237
20;Stochastic Geometry Approach to the Kinematic Dynamo Equation of Magnetohydrodynamics;240
20.1;1. Introduction;240
20.2;2. Riemann-Cartan-Weyl geometry of di.usions;241
20.3;3. Riemann-Cartan-Weyl di.usions on the tangent manifold;243
20.4;4. Realization of the RCW di.usions by ODE’s;244
20.5;5. RCW gradient di.usions of di.erential forms;246
20.6;6. KDE and RCW gradient di.usions;247
20.7;7. KDE and random symplectic di.usions;250
20.8;8. The Euclidean case;253
20.9;References;255
21;Quasi-Lipschitz Conditions in Euler Flows;258
21.1;1. Introduction;258
21.2;2. A quasi-Lipschitz condition for .rst-order derivatives of;260
21.3;Newtonian potentials in;260
21.4;3. The hydrodynamical equations of Euler and Helmholtz;261
21.5;4. Helmholtz and Cauchy’s vorticity equation with a discretization;262
21.6;5. The .xpoint equation;264
21.7;6. Application of the contracting mapping principle;266
21.8;References;270
22;Interfaces in Solutions of Diffusion-absorption Equations in Arbitrary Space Dimension;272
22.1;1. Introduction;273
22.2;2. Lagrangian coordinates;276
22.3;3. The weighted function spaces;281
22.4;4. The gradient .ow;284
22.5;5. Solution of the free-boundary problem;287
22.6;(;287
22.7;References;288
23;Estimates for Solutions of Fully Nonlinear Discrete Schemes;290
23.1;1. Introduction;290
23.2;2. Linear equations;292
23.3;3. Schauder estimates for nonlinear schemes;294
23.4;References;296

Leseproben

Stopping a Viscous Fluid by a Feedback Dissipative Field: Thermal E.ects without Phase Changing (p. 1)

S.N. Antontsev, J.I. D´ýaz and H.B. de Oliveira

Dedicated to Professor V.A. Solonnikov on the occasion of his 70th birthday. Abstract. We show how the action on two simultaneous e.ects (a suitable coupling about velocity and temperature and a low range of temperature but upper that the phase changing one) may be responsible of stopping a viscous .uid without any changing phase. Our model involves a system, on an unbounded pipe, given by the planar stationary Navier-Stokes equation perturbed with a sublinear term f (x, ?, u) coupled with a stationary (and possibly nonlinear) advection di.usion equation for the temperature. After proving some results on the existence and uniqueness of weak solutions we apply an energy method to show that the velocity u vanishes for x large enough.

1. Introduction

It is well known (see, for instance, [6, 8, 14]) that in phase changing .ows (as the Stefan problem) usually the solid region is assumed to remain static and so we can understand the final situation in the following way: the thermal e.ect are able to stop a viscous fluid.

The main contribution of this paper is to show how the action on two simultaneous effects (a suitable coupling about velocity and temperature and a low range of temperature but upper the phase changing one) may be responsible of stopping a viscous fld without any changing phase. This philosophy could be useful in the monitoring of many .ows problems, specially in metallurgy.