HCDTE Lecture Notes Part I Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations

Cover......Page 1
HCDTE Lecture Notes. Part I. Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations......Page 2
ISBN-10: 1601330146  ISBN-13: 9781601330147......Page 3
Foreword......Page 4
Contents......Page 6
1.1 Introduction......Page 8
1.2 Description of the PDE models......Page 
10
1.2.1 Waves and plates with nonlinear interior damping and critical-supercritical sources (A) Wave equation with nonlinear damping-source interaction......Page 11
1.2.2 Nonlinear waves and plates with geometrically constrained damping and critical-supercritical sources......Page 13
1.3 Well-posedness and generation of continuous  flows......Page 15
1.3.1 Wave equation with a nonlinear interior damping-model in (1.1)......Page 16
1.3.2 Von Karman equation with interior damping model (1.4)......Page 23
1.3.3 Kirchhoff -Boussinesq model with interior damping model in (1.12)......Page 27
1.3.4 Sketch of the proof......Page 29
1.3.6 Von Karman equation with boundary damping model (1.18)......Page 34
1.3.8 Kirhho -Boussinesq equation with boundary damping-model (1.21)......Page 35
1.4.1 Basic notions......Page 37
1.4.2 Criteria for asymptotic smoothness......Page 39
1.4.3 Global attractors......Page 42
1.4.4 Quasi-stable systems......Page 46
1.5 Long time behavior for canonical models described in Section 1.2......Page 55
1.5.1 Wave dynamics......Page 56
1.5.2 Von Karman plate dynamics Internal damping......Page 68
1.5.3 Kirchhoff -Boussinesq plate Interior damping......Page 85
1.6.1 Kirchhoff wave models......Page 89
1.6.2 Plate models with structural damping......Page 92
1.6.3 Mindlin-Timoshenko plates and beams......Page 93
1.6.5 Structural acoustic interactions......Page 94
1.6.6 Fluid-structure interactions......Page 95
1.6.8 Schrödinger-Boussinesq equations......Page 97
2 Stability of  finite difference schemes for hyperbolic initial boundary value problems......Page 104
2.1.1 What is and what is not inside these notes?......Page 105
2.1.3 General presentation of the stability problem......Page 106
2.2.1 Finite difference operators and stability for the discrete Cauchy problem......Page 109
2.2.2 Possible behaviors for the eigenvalues of the amplification matrix......Page 117
2.2.3 The Lax-Friedrichs and leap-frog schemes......Page 124
2.2.4 A few facts to remember in view of what follows, and a (not very interesting) conjecture......Page 128
2.3.1 Finite difference discretizations and strong stability......Page 129
2.3.2 The nomal modes analysis and the Godunov-Ryabenkii condition......Page 133
2.3.3 An equivalent form of the resolvent equation......Page 147
2.3.4 Characterization of strong stability: the main result......Page 155
2.4.1 The discrete block structure condition......Page 161
2.4.2 The construction of symmetrizers......Page 174
2.4.3 Extending the stable subspace......Page 182
2.4.4 Proof of Theorem 6......Page 185
2.4.5 Some examples: the Lax-Friedrichs and leap-frog schemes......Page 189
2.4.6 Goldberg-Tadmor's Lemma for Dirichlet boundary conditions......Page 199
2.5.1 A simple but unsu cient argument......Page 204
2.5.2 Wu's argument......Page 209
2.5.3 A more general framework for semigroup stability......Page 211
2.5.4 The Lax-Friedrichs scheme......Page 219
2.6 A partial conclusion......Page 220
3 Boundary control and boundary inverse theory for non-homogeneous second-order hyperbolic equations: A common Carleman estimate......Page 234
3.1.1 Carleman estimates for H 1 solutions of second-order hyperbolic equations with explicit boundary terms: Dirichlet and Neumann cases......Page 236
3.1.3 Sharp regularity theory for second-order hyperbolic equations of Dirichlet type......Page 241
3.1.4 Sharp regularity theory for second-order hyperbolic equations of Neumann type......Page 242
3.1.5 Notes on Carleman estimates......Page 243
3.2.2 Dirichlet case......Page 245
3.2.3 Neumann case......Page 246
3.3 Inverse theory results......Page 247
3.4.1 Problem formulation I: The original hyperbolic problem subject to an unknown damping coefficient q(x)......Page 248
3.4.2 Main results......Page 250
3.4.3 Proofs......Page 253
3.4.4 Notes and literature......Page 271
3.5.1 Problem formulation......Page 273
3.5.2 Main results......Page 275
3.5.3 Proofs Uniqueness of linear inverse problem for the u-dynamics (3......Page 277
3.5.4 Notes and literature......Page 297
3.6.1 The coupled hyperbolic system with two unknown damping coefficients......Page 298
3.6.2 Main results......Page 300
3.6.3 Proofs Uniqueness of linear inverse problem for the u; v -system (3......Page 302
3.7.1 Problem formulation......Page 319
3.7.2 Main results......Page 320
3.7.3 Proofs Sketch of proof of uniqueness Theorem 24......Page 323
3.7.4 Notes......Page 342
Back Cover......Page 351


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