Instability in Models Connected with Fluid Flows I (eBook)

Main Topics 7
Editors 9
Preface 10
1. Overview 10
2. Classification of Contributions and Comments 12
3. Methods and Tools 21
References 21
Authors 23
Content of Volume I 28
Content of Volume II 32
Solid Controllability in Fluid Dynamics 33
1. Introduction 33
2. 2D Navier–Stokes / Euler Equations Controlled by Degenerate Forcing. Definitions and Problem Setting 36
3. Geometric Control. Accessibility and Controllability via Lie Brackets 43
4. Computation of Brackets in Finite and Infinite Dimensions. Controlling along Principal Axes 49
5. Controllability and Accessibility of Galerkin Approximations of Navier–Stokes / Euler Equations on T2 51
6. Steady State Controlled Directions. Abstract Controllability Result for Navier– Stokes Equations 54
7. Navier–Stokes and Euler Equations on T2 55
8. Controllability of 2D Navier–Stokes Equations on Rectangular Domain 58
9. Controllability on Generic Riemannian Surface Diffeomorphic to Disc 59
10. Navier–Stokes / Euler Equations on Sphere S2 62
References 65
Analyticity of Periodic Solutions of the 2D Boussinesq System 68
References 83
Nonlinear Dynamics of a System of Particle- Like Wavepackets 84
1. Introduction 85
2. Statement of Results 94
3. Conditions and Definitions 110
4. Integrated Evolution Equation 120
5. Wavepacket Interaction System 127
6. Reduction of Wavepacket Interaction System to an Averaged Interaction System 136
7. Superposition Principle and Decoupling of the Wavepacket Interaction System 150
References 162
Attractors for Nonautonomous Navier– Stokes System and Other Partial Differential Equations 166
Introduction 166
1. Attractors of Autonomous Equations 169
2. Attractors of Nonautonomous Equations 192
3. Kolmogorov e-Entropy of Global Attractors 231
4. Nonautonomous 2D Navier–Stokes System with Singularly Oscillating External Force 250
5. Uniform Global Attractor of Ginzburg–Landau Equation with Singularly Oscillating Terms 271
References 287
Recent Results in Large Amplitude Monophase Nonlinear Geometric Optics 297
1. Introduction 297
2. Case of Incompressible Euler Equations 301
3. Large Amplitude Waves 305
References 317
Existence Theorems for the 3D– Navier– Stokes System Having as Initial Conditions Sums of Plane Waves 319
References 329
Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains 331
1. Introduction 332
2. Vorticity Waves and Resonances of Elementary Swirling Flows 337
3. The Strictly Resonant Euler Systems: the SO( 3) Case 345
4. Strictly Resonant Euler Systems: the Case of 3- Wave Resonances on Small-Scales 356
Appendix 363
References 365
Increased Stability in the Cauchy Problem for Some Elliptic Equations 369
1. Introduction 369
2. Energy Type Estimates in Low Frequency Zone 374
3. Some Carleman Estimates 386
4. Proofs of Theorems 1.1 and 1.2 388
References 392
Index 393