Bardos / Fursikov Instability in Models Connected with Fluid Flows I

1. Auflage 2007
ISBN: 978-0-387-75217-4
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 6, 364 Seiten

Reihe: International Mathematical Series

In this authoritative and comprehensive volume, Claude Bardos and Andrei Fursikov have drawn together an impressive array of international contributors to present important recent results and perspectives in this area. The main subjects that appear here relate largely to mathematical aspects of the theory but some novel schemes used in applied mathematics are also presented. Various topics from control theory, including Navier-Stokes equations, are covered.

Autoren/Hrsg.

Bardos, Claude

Fursikov, Andrei

Fursikov, Andrei V.

Inhaltsverzeichnis

1;Main Topics;7
2;Editors;9
3;Preface;10
3.1;1. Overview;10
3.2;2. Classification of Contributions and Comments;12
3.3;3. Methods and Tools;21
3.4;References;21
4;Authors;23
5;Content of Volume I;28
6;Content of Volume II;32
7;Solid Controllability in Fluid Dynamics;33
7.1;1. Introduction;33
7.2;2. 2D Navier–Stokes / Euler Equations Controlled by Degenerate Forcing. Definitions and Problem Setting;36
7.3;3. Geometric Control. Accessibility and Controllability via Lie Brackets;43
7.4;4. Computation of Brackets in Finite and Infinite Dimensions. Controlling along Principal Axes;49
7.5;5. Controllability and Accessibility of Galerkin Approximations of Navier–Stokes / Euler Equations on T2;51
7.6;6. Steady State Controlled Directions. Abstract Controllability Result for Navier– Stokes Equations;54
7.7;7. Navier–Stokes and Euler Equations on T2;55
7.8;8. Controllability of 2D Navier–Stokes Equations on Rectangular Domain;58
7.9;9. Controllability on Generic Riemannian Surface Diffeomorphic to Disc;59
7.10;10. Navier–Stokes / Euler Equations on Sphere S2;62
7.11;References;65
8;Analyticity of Periodic Solutions of the 2D Boussinesq System;68
8.1;References;83
9;Nonlinear Dynamics of a System of Particle- Like Wavepackets;84
9.1;1. Introduction;85
9.2;2. Statement of Results;94
9.3;3. Conditions and Definitions;110
9.4;4. Integrated Evolution Equation;120
9.5;5. Wavepacket Interaction System;127
9.6;6. Reduction of Wavepacket Interaction System to an Averaged Interaction System;136
9.7;7. Superposition Principle and Decoupling of the Wavepacket Interaction System;150
9.8;References;162
10;Attractors for Nonautonomous Navier– Stokes System and Other Partial Differential Equations;166
10.1;Introduction;166
10.2;1. Attractors of Autonomous Equations;169
10.3;2. Attractors of Nonautonomous Equations;192
10.4;3. Kolmogorov e-Entropy of Global Attractors;231
10.5;4. Nonautonomous 2D Navier–Stokes System with Singularly Oscillating External Force;250
10.6;5. Uniform Global Attractor of Ginzburg–Landau Equation with Singularly Oscillating Terms;271
10.7;References;287
11;Recent Results in Large Amplitude Monophase Nonlinear Geometric Optics;297
11.1;1. Introduction;297
11.2;2. Case of Incompressible Euler Equations;301
11.3;3. Large Amplitude Waves;305
11.4;References;317
12;Existence Theorems for the 3D– Navier– Stokes System Having as Initial Conditions Sums of Plane Waves;319
12.1;References;329
13;Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains;331
13.1;1. Introduction;332
13.2;2. Vorticity Waves and Resonances of Elementary Swirling Flows;337
13.3;3. The Strictly Resonant Euler Systems: the SO( 3) Case;345
13.4;4. Strictly Resonant Euler Systems: the Case of 3- Wave Resonances on Small-Scales;356
13.5;Appendix;363
13.6;References;365
14;Increased Stability in the Cauchy Problem for Some Elliptic Equations;369
14.1;1. Introduction;369
14.2;2. Energy Type Estimates in Low Frequency Zone;374
14.3;3. Some Carleman Estimates;386
14.4;4. Proofs of Theorems 1.1 and 1.2;388
14.5;References;392
15;Index;393