Trends in Partial Differential Equations of Mathematical Physics (eBook)

Table of Contents 6
Preface 8
On Vsevolod Alekseevich Solonnikov and his 70th birthday 9
References 12
List of Participants 14
Stopping a Viscous Fluid by a Feedback Dissipative Field: Thermal E.ects without Phase Changing 16
1. Introduction 16
2. Statement of the problem 17
3. Existence theorem 18
4. Uniqueness of weak solution 21
5. Localization e.ect 25
6. Case of a temperature depending viscosity 27
References 28
Ultracontractive Bounds for Nonlinear Evolution Equations Governed by the Subcritical p-Laplacian 30
1. Introduction 30
2. Entropy and logarithmic Sobolev inequalities 33
3. Preliminary results 35
References 40
Weighted L2-spaces and Strong Solutions of the Navier-Stokes Equations in R3 42
1. Introduction 42
2. Proof of Theorem 44
References 49
A Limit Model for Unidirectional Non-Newtonian Flows with Nonlocal Viscosity 52
1. Introduction 52
2. The limit problems and their formulations 54
3. Existence of weak solutions and their convergence 55
References 58
On the Problem of Thermocapillary Convection for Two Incompressible Fluids Separated by a Closed Interface 60
1. Statement of the problem and formulation of the main result 60
2. Linearized problems 65
3. Solvability of problem 71
( 71
References 79
Some Mathematical Problems in Visual Transduction 80
1. The phototransduction cascade 80
2. The physical model 83
3. The limiting equations 84
4. Main ideas in computing the homogenized limit 86
5. Weak formulation of the homogenized problem 90
6. Cytosol well-stirred in the transversal variables 91
7. Globally well-stirred cytosol 92
8. Further results and open issues 92
References 94
Global Regularity in Sobolev Spaces for Elliptic Problems with p-structure on Bounded Domains 96
1. Introduction 96
2. The main results 97
3. Proof of the theorem 98
References 104
Temperature Driven Mass Transport in Concentrated Saturated Solutions 106
1. Introduction 106
2. Description of physical system and 107
the governing di.erential equations 107
3. Modelling a speci.c mass transport process with deposition 110
4. Analysis of Stage 1 114
5. Analysis of Stage 2: a priori results 120
6. Analysis of Stage 2: weak formulation and existence 121
References 123
Solvability of a Free Boundary Problem for the Navier-Stokes Equations Describing the Motion of Viscous Incompressible Nonhomogeneous Fluid 124
1. Statement of the problem 124
2. Function spaces 126
3. Lagrange coordinates 128
4. Auxiliary linear problem 130
5. Model problems 132
6. Proof of Theorem 136
7. Nonlinear problem 138
References 138
Duality Principles for Fully Nonlinear Elliptic Equations 140
1. Introduction 140
2. Duality 142
3. A priori estimates 147
References 150
On the Bénard Problem 152
1. Introduction 152
2. Boussinesq approximation 154
3. Compressible scheme: layer heated from above 155
4. Compressible scheme: layer heated from below 159
References 162
Exact Boundary Controllability for Quasilinear Wave Equations 164
1. Introduction and main results 164
2. Reduction of equation and boundary conditions 167
3. Semi-global 169
solution for quasilinear hyperbolic 169
systems with zero eigenvalues 169
4. Proof of Theorems 1 and 2 170
5. Remarks 174
References 174
Regularity of Euler Equations for a Class of Three-Dimensional Initial Data 176
1. Introduction and main results 176
2. Poincar´ e-Sobolev equations in cylindrical domains 183
3. The structure and regularity of fast singular oscillating 188
limit equations 188
4. Long time regularity for .nite large 192
References 198
A Model of a Two-dimensional Pump 202
References 210
Regularity of a Weak Solution to the Navier-Stokes Equation in Dependence on Eigenvalues and Eigenvectors of the Rate of Deformation Tensor 212
1. Introduction 212
2. Regularity in dependence on eigenvalues of the rate of 215
deformation tensor 215
3. Regularity in dependence on eigenvectors of the rate of 219
deformation tensor 219
References 227
Free Work and Control of Equilibrium Con.gurations 228
1. Introduction 228
2. Stability in the mean 231
3. Asymptotic decay for hyperelastic, viscous materials 234
4. Nonlinear instability for hyperelastic bodies 235
References 237
Stochastic Geometry Approach to the Kinematic Dynamo Equation of Magnetohydrodynamics 240
1. Introduction 240
2. Riemann-Cartan-Weyl geometry of di.usions 241
3. Riemann-Cartan-Weyl di.usions on the tangent manifold 243
4. Realization of the RCW di.usions by ODE’s 244
5. RCW gradient di.usions of di.erential forms 246
6. KDE and RCW gradient di.usions 247
7. KDE and random symplectic di.usions 250
8. The Euclidean case 253
References 255
Quasi-Lipschitz Conditions in Euler Flows 258
1. Introduction 258
2. A quasi-Lipschitz condition for .rst-order derivatives of 260
Newtonian potentials in 260
3. The hydrodynamical equations of Euler and Helmholtz 261
4. Helmholtz and Cauchy’s vorticity equation with a discretization 262
5. The .xpoint equation 264
6. Application of the contracting mapping principle 266
References 270
Interfaces in Solutions of Diffusion-absorption Equations in Arbitrary Space Dimension 272
1. Introduction 273
2. Lagrangian coordinates 276
3. The weighted function spaces 281
4. The gradient .ow 284
5. Solution of the free-boundary problem 287
( 287
References 288
Estimates for Solutions of Fully Nonlinear Discrete Schemes 290
1. Introduction 290
2. Linear equations 292
3. Schauder estimates for nonlinear schemes 294
References 296

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.