New Directions in Mathematical Fluid Mechanics (eBook)

Contents 6
Preface 8
Scientific Portrait of Alexander Vasilievich Kazhikhov 10
Boundary Control Problems for Stationary Equations of Heat Convection 13
1. Introduction. Statement of extremum problem 13
2. General property of the optimality system solutions 18
3. Uniqueness and stability of solutions of extremum problems 21
4. Numerical analysis 31
References 33
Homogenization of the Poisson– Boltzmann Equation 34
1. Introduction 34
2. Basic equations 36
3. Scaling and identification of a small parameter 38
4. A priori estimates 40
5. Existence 46
6. Two-scale compactness 47
7. A corrector and error estimates 48
8. A two-scale model 50
References 50
Superconducting Vortices: Chapman Full Model 52
1. Introduction. General model 52
2. Approximation problem. Leray-Schauder fixed point argument 55
3. Limit transition on e 59
4. Limit transition on . . 0 61
References 65
Augmented Lagrangian Method and Compressible Visco- plastic Flows: Applications to Shallow Dense Avalanches 67
1. Introduction 68
2. Statement of the 3D-problem 69
3. The plane slope case 72
4. One-dimensional system and stationary solutions 76
5. A well-balanced finite volume/augmented Lagrangian algorithm 78
6. Numerical tests 87
References 93
Oscillatory Limits with Changing Eigenvalues: A Formal Study 100
1. Introduction 100
2. Wave equation 104
3. Limit equation 106
4. Almost everywhere results 107
5. Study of Sj,k 108
6. Structure of a resonant set 110
7. Formal justification of the limiting process 110
References 112
Finite-dimensional Control for the Navier– Stokes Equations 114
1. Introduction 114
2. Cauchy problem for the subdifferential inclusion 116
3. Solvability of the control problem 118
4. About existence of the strong solution of NSE for all time 119
References 119
On the Sharp Vanishing Viscosity Limit of Viscous Incompressible Fluid Flows 121
1. Introduction 121
2. Preliminaries 124
3. Proof of Theorem 125
References 129
Small P´eclet Number Approximation as a Singular Limit of the Full Navier- Stokes- Fourier System with Radiation 131
1. Introduction 131
2. Preliminaries 137
3. Main results 138
4. Uniform estimates 140
5. Convergence 147
6. Analysis of the acoustic waves 151
7. Asymptotic limit of the temperature 157
References 159
New Perspectives in Fluid Dynamics: Mathematical Analysis of a Model Proposed by Howard Brenner 161
1. Introduction 161
2. A priori estimates 167
3. Refined velocity and temperature estimates 173
4. Weak sequential stability 179
5. Global existence 181
References 186
Existence of a Regular Periodic Solution to the Rothe Approximation of the Navier– Stokes Equation in Arbitrary Dimension 188
1. Introduction 188
2. Notation and formulation of the theorem 189
3. Regularized Rothe approximation 190
4. The head pressure equation 191
5. Proof of the main theorem 194
References 198
Optimal Neumann Control for the Two- dimensional Steady- state Navier- Stokes equations 200
1. Introduction 200
2. Setting of the optimal control problem 201
3. Boundary value problems 203
4. Existence theorem for the optimal control problem 209
5. Abstract Lagrange principle 214
6. Application of the abstract Lagrange principle 218
7. The optimality system 223
8. Numerical calculations 226
References 227
On Some Boundary Value Problem for the Stokes Equations with a Parameter in an Infinite Sector 229
1. Introduction 229
2. Auxiliary propositions 231
3. Proof of Theorem 1.1 233
Appendix: Proof of Lemma 2.1. 240
References 242
Unilateral Contact Problems Between an Elastic Plate and a Beam 243
1. Case of inclined beam 243
2. Case of horizontal beam 249
References 252
On Lighthill’s Acoustic Analogy for Low Mach Number Flows 253
1. Introduction 254
2. Original and target problems, main results 257
3. Estimates for real-time and fast-time variables 267
4. Limit passage in fast-time variables and homogenous wave equation 269
5. Limit passage in the real-time variables and the Navier- Stokes equations 270
6. Proof of Lemma 5.1 and Theorem 2.1 272
7. Proof of Lemma 5.2 and Theorem 2.2 280
References 283
On the Uniqueness of Solutions to Boundary Value Problems for Non- stationary Euler Equations 286
1. Introduction 286
2. Classes of uniqueness 291
3. Additional remarks 298
References 299
On Nonlinear Stability of MHD Equilibrium Figures 305
1. Introduction 305
2. Initial boundary value problem 308
3. Energy identity 312
4. Perturbations 319
5. Nonlinear stability 326
6. Conclusions 333
References 333
Viscous Flows in Domains with a Multiply Connected Boundary 336
1. Introduction 336
2. Survey of previous results 337
3. Axially symmetric flows 340
4. Flow in a curvilinear ring 345
5. Discussion 347
References 350
Problems with Insufficient Information about Initial- boundary Data 352
Introduction 352
State equation. Linear analysis 356
Chapman–Enskog projection and Schrödinger approximation 364
Auxiliary propositions 370
References 380
On the Stability of Non-symmetric Equilibrium Figures of Rotating Self- gravitating Liquid not Subjected to Capillary Forces 382
1. Introduction 382
2. On the construction of .(t) 388
3. Scheme of the proof of Theorem 1.1 391
4. Proof of Proposition 2.2 and of the estimate (3.24) 399
5. Proof of Theorem 3.4 407
References 417
Dynamics of a Non-fixed Elastic Body 418
1. Introduction 418
2. Decomposition of the motion 420
3. Equations of motion 424
4. Model with small deformations 429
5. The energy estimate 431
6. The final formulation of the problem 433
References 434