Some Problems on Nonlinear Hyperbolic Equations and Applications (Series in Contemporary Applied Mathematics)

Contents......Page 8
Preface......Page 6
Part I Mathematical and Numerical Analyses of Strongly Nonlinear Plasma Models......Page 12
Kim Dang Phung Waves......Page 0
1 Introduction......Page 14
2.1 The constant coefficients case: derivation of the TBC......Page 17
2.2 Extensions and interpretations in the context of pseudodifferential operator calculus: introduction to the derivation of ABCs......Page 20
3.1 Two possible strategies......Page 23
3.2 Practical computation of the asymptotic expansion of the DtN operator......Page 24
3.3 Choosing the ABC in the context of strategy 1......Page 27
3.4 Choosing the ABC in the context of strategy 2......Page 29
4 Semi-discretization schemes and their properties......Page 30
4.1 Discrete convolutions-based discretizations for ABC......Page 31
4.2 Auxiliary functions-based discretizations for ABC......Page 33
5 Extensions to nonlinear problems......Page 35
6.1 Linear Schrodinger equation with variable potential......Page 38
6.2 Nonlinear Schroinger equation......Page 39
7 Conclusion......Page 43
References......Page 44
Abstract......Page 46
2.1 Generalities......Page 47
2.2 From GEO to LEO......Page 51
3 A simple ID model......Page 54
4 Analysis of the one-dimensional problem......Page 58
5 Numerical simulation of the one-dimensional problem......Page 60
6 Conclusions......Page 63
References......Page 64
1 Introduction......Page 67
2 Ergodic theory and average over a flow......Page 70
3 Well-posedness of the limit model......Page 78
4 Convergence towards the limit model......Page 87
5 The limit model in terms of prime integrals......Page 90
5.1 Examples......Page 93
References......Page 95
Abstract......Page 97
1.2 Some known results on Euler-Poisson system......Page 98
1.3 Zero-electron-mass limit for given ion density......Page 99
1.4 Formal asymptotic analysis......Page 101
2.1 Main results......Page 102
2.2 Ideas......Page 103
3.1 Main results......Page 105
3.2 Ideas......Page 106
References......Page 108
1 Introduction......Page 111
2 Particles in turbulent flows......Page 113
2.1 Modeling of turbulent flow......Page 114
2.2 Dimension analysis and asymptotic regimes......Page 117
2.2.2 Fine-particles regime......Page 119
2.3.1 Analysis of the regime (2.25)......Page 121
2.3.2 Analysis of the regime (2.26)......Page 124
2.4 Derivation of the effective equations in the highinertia particles regimes......Page 126
3 Numerical schemes for coupled fluid/particles models......Page 127
3.1 Asymptotic preserving numerical methods......Page 130
References......Page 139
Abstract......Page 142
1 Introduction and main results......Page 143
2 Preliminaries......Page 150
3 Local-in-time existence......Page 152
3.1 Construction of new problems......Page 154
3.2 Iteration scheme and local existence......Page 156
4.1 Reformulation of original problems......Page 165
4.2 The a-priori estimates......Page 166
References......Page 168
Abstract......Page 172
o Introduction......Page 173
1 Description of the Bloch decompositionbased algorithm......Page 175
1.1 Recapitulation of Bloch's decomposition method......Page 176
1.2 The Bloch decomposition(BD)-based split-step algorithm......Page 178
1.3 The classical time splitting pseudo spectral (TS) method......Page 181
1.4 Application to lattice BEe in 3D......Page 182
2.1 Numerical tests for 1D problems ( = 0)......Page 183
2.2 Numerical tests for ID NLS......Page 186
2.3 Numerical examples for lattice BEe in 3D......Page 187
3 Random coefficients: stability tests and Anderson localization......Page 190
3.1 Stability of our BD algorithm......Page 192
3.2 Numerical evidence for the Anderson's localization......Page 193
4 Conclusions......Page 195
References......Page 197
1 Introduction......Page 200
2 Asymptotic limits......Page 203
2.1.1 Construction of the asymptotic expansion......Page 204
2.1.2 Justification of the asymptotic expansion......Page 206
2.2.1 Construction of the asymptotic expansion......Page 208
2.2.2 Justification of the asymptotic expansion......Page 212
3 Numerical simulations......Page 213
3.1 Mesh and notations......Page 214
3.2 Classical finite volume scheme (VF4-scheme)......Page 215
3.3 Numerical results......Page 217
References......Page 220
1 Introduction......Page 224
2 Preliminaries......Page 225
3 Linear problems......Page 228
4 Nonlinear problems......Page 231
References......Page 235
1 Introduction......Page 236
2 Main result......Page 240
3.1 Preliminary estimates on electric field......Page 243
3.2 The proof of Theorem 2.2......Page 244
3.3 The proof of Theorem 2.5......Page 259
References......Page 266
1 Introduction......Page 269
2 Preliminaries......Page 274
3 Construction of approximation solutions......Page 276
4 Proof of the main result......Page 279
References......Page 288
Part II Exact Controllability and Observability for Quasilinear Hyperbolic Systems and Applications......Page 292
Abstract......Page 294
1 Introduction......Page 295
2 A constructive proof of Theorem 1.3......Page 298
2.1 High frequency components......Page 299
2.2 Haraux's constructive argument......Page 300
3.1 Time discrete approximations of (1.1)-(1.2)......Page 303
4.1 The continuous case......Page 308
4.2 Space semi-discrete approximation schemes......Page 311
4.3 Fully discrete approximation schemes......Page 317
5 Further comments......Page 318
References......Page 319
1 Introduction and main results......Page 321
2 Some preliminaries......Page 324
3 Interpolation inequality for an elliptic equation with an inhomogeneous boundary condition......Page 326
4 Proof of Theorem 1.2......Page 338
5 Proof of Theorem 1.1......Page 340
References......Page 341
1.1 Basic question and previous results......Page 343
1.2 The result......Page 347
2 Characteristics of DiPerna's system......Page 348
3 Proof of Proposition 1.7......Page 350
4 Proof of Proposition 1.8......Page 352
References......Page 353
1 Introduction......Page 355
2.1 Observability......Page 356
2.2 Controllability......Page 359
2.3 Linear stabilization by natural feedbacks......Page 360
2.4 Nonlinear stabilization by natural feedbacks......Page 365
3 A Petrovsky system......Page 370
4 Observability by using Fourier series......Page 375
5 A general method of stabilization......Page 377
References......Page 382
1 Introduction and known results......Page 385
1.1.2 One-sided exact boundary controllability [6]......Page 387
1.2.2 One-sided exact boundary observability [7]......Page 388
2 Remarks and open problems......Page 389
References......Page 394
1 The wave equation and observation......Page 397
2.1 The polynomial decay rate......Page 399
2.2 The approximate controllability......Page 402
2.2.1 Proof......Page 403
2.2.2 Numerical experiments......Page 405
3 Conclusion......Page 407
4.1 Monotonicity formula......Page 408
4.1.1 Proof of Lemma B......Page 409
4.1.2 Proof of Lemma A......Page 410
4.1.3 Proof of Lemma C......Page 412
4.2 Quantitative unique continuation property for the Laplacian......Page 415
4.3 Quantitative unique continuation property for the elliptic operator......Page 416
4.4 Application to the wave equation......Page 417
References......Page 422
1 Introduction......Page 424
2.1 Abstract setting......Page 426
2.3 Linearization......Page 427
3 Global controllability......Page 432
4.1 Compressible fluids......Page 433
References......Page 434
1 Introduction......Page 437
2 Main results......Page 439
3 Proof of Theorem 2.2......Page 442
References......Page 446
1 Introduction......Page 448
2 Starting point: the case of ODEs......Page 450
3 Known perturbation result of exact controllability......Page 451
5 Local exact controllability for multidimensional quasilinear hyperbolic equations......Page 453
6 Local null controllability for quasilinear parabolic equations......Page 457
7 Open problems......Page 458
References......Page 461