Fourier Analysis and Nonlinear Partial Differential Equations

by Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphael

ISBN13: 9783642168291

ISBN10: 3642168299

Format: Hardcover

Pub. Date: 2011-01-06

Publisher(s): Springer Nature

Other versions by this Author

List Price: ~~$149.99~~

Rent Textbook

Select for Price

Add to Cart

There was a problem. Please try again later.

New Textbook

We're Sorry
Sold Out

Used Textbook

We're Sorry
Sold Out

eTextbook

We're Sorry
Not Available

Buy from our Marketplace starting at $133.12

Summary

In recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decomposition have proved to be very efficient for the study of evolution equations. The present book aims at presenting self-contained, state- of- the- art models of those techniques with applications to different classes of partial differential equations: transport, heat, wave and Schrödinger equations. Â It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity. It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.

Basic Analysis	p. 1
Basic Real Anslysis	p. 1
Holder and Convolution Inequslities	p. 1
The Atomic Decomposition	p. 7
Proof of Refined Young Inequslityp8
A Bilinear Interpolation Theorem	p. 10
A Linear Interpolation Result	p. 11
The Hardy-Littlewood Maximal Function	p. 13
The Fourier Transform	p. 16
Fourier Transforms of Functions and the Schwartz Space	p. 16
Tempered Distributions and the Fourier Transform	p. 18
A Few Calculations of Fourier Transforms	p. 23
Homogeneous Sobolev Spaces	p. 25
Definition and Basic Properties	p. 25
Sobolev Embedding in Lebesgue Spaces	p. 29
The Limit Case H^d/2	p. 36
The Embedding Theorem in Hölder Spaces	p. 37
Nonhomogeneous Sobolev Spaces on R^d	p. 38
Definition and Basic Properties	p. 38
Embedding	p. 44
A Density Theorem	p. 47
Hardy Inequality	p. 48
References and Remarks	p. 49
Littlewood-Paley Theory	p. 51
Functions with Compactly Supported Fourier Transforms	p. 51
Bernstein-Type Lemmas	p. 52
The Smoothing Effect of Heat Flow	p. 53
The Action of a Diffeomorphism	p. 56
The Effects of Some Nonlinear Functions	p. 58
Dyadic Partition of Unity	p. 59
Homogeneous Besov Spaces	p. 63
Characterizations of Homogeneous Besov Spaces	p. 72
Besov Spaces, Lebesgue Spaces, and Refined Inequalities	p. 78
Homogeneous Paradifferential Calculus	p. 85
Homogeneous Bony Decomposition	p. 85
Action of Smooth Functions	p. 93
Time-Space Besov Spaces	p. 98
Nonhomogeneous Besov Spaces	p. 98
Nonhomogeneous Paradifferential Calculus	p. 102
The Bony Decomposition	p. 102
The Paralinearization Theorem	p. 104
Besov Spaces and Compact Embeddings	p. 108
Commutator Estimates	p. 110
Around the Space B_{&infty;,&infty;}¹	p. 116
References and Remarks	p. 120
Transport and Transport-Diffusion Equations	p. 123
Ordinary Differential Equations	p. 124
The Cauchy-Lipschitz Theorem Revisited	p. 124
Estimates for the Flow	p. 129
A Blow-up Criterion for Ordinary Differential Equations	p. 131
Transport Equations: The Lipschitz Case	p. 132
A Priori Estimates in General Besov Spaces	p. 132
Refined Estimates in Besov Spaces with Index 0	p. 135
Solving the Transport Equation in Besov Spaces	p. 136
Application to a Shallow Water Equation	p. 140
Losing Estimates for Transport Equations	p. 147
Linear Loss of Regularity in Besov Spaces	p. 147
The Exponential Loss	p. 151
Limited Loss of Regularity	p. 153
A Few Applications	p. 155
Transport-Diffusion Equations	p. 156
A Priori Estimates	p. 157
Exponential Decay	p. 163
References and Remarks	p. 166
Quasilinear Symmetric Systems	p. 169
Definition and Examples	p. 169
Linear Symmetric Systems	p. 172
The Well-posedness of Linear Symmetric Systems	p. 172
Finite Propagation Speed	p. 180
Further Well-posedness Results for Linear Symmetric Systems	p. 183
The Resolution of Quasilinear Symmetric Systems	p. 187
Paralinearization and Energy Estimates	p. 189
Convergence of the Scheme	p. 190
Completion of the Proof of Existence	p. 191
Uniqueness and Continuation Criterion	p. 192
Data with Critical Regularity and Blow-up Criteria	p. 193
Critical Besov Regularity	p. 193
A Refined Blow-up Crndition	p. 196
Continuity of the Flow Map	p. 198
References and Remarks	p. 201
The Incompressible Navier-Stokes System	p. 203
Basic Facts Concerning the Navier-Stokes System	p. 204
Well-posedness in Sobolev Spaces	p. 209
A General Result	p. 209
The Behavior of the H^d/2-1 Norm Near 0	p. 214
Results Related to the Structure of the System	p. 215
The Particular Case of Dimension Two	p. 215
The Case of Dimension Three	p. 217
An Elementary L^p Approach	p. 220
The Endpoint Space for Picard's Scheme	p. 227
The Use of the L¹-smoothing Effect of the Heat Flow	p. 233
The Cannone-Meyer-Planchon Theorem Revisited	p. 234
The Flow of the Solutions of the Navier-Stokes System	p. 236
References and Remarks	p. 242
Anisotropic Viscosity	p. 245
The Case of L² Data with One Vertical Derivative in L²	p. 246
A Global Existence Result in Anisotropic Besov Spaces	p. 254
Anisotropic Localization in Fourier Space	p. 254
The Functional Framework	p. 256
Statement of the Main Result	p. 258
Some Technical Lemmas	p. 261
The Proof of Existence	p. 266
The Proof of Uniqueness	p. 276
References and Remarks	p. 289
Euler System for Perfect Incompressible Fluids	p. 291
Local Well-posedness Results for Inviscid Fluids	p. 292
The Biot-Savart Law	p. 293
Estimates for the Pressure	p. 296
Another Formulation of the Euler System	p. 301
Local Existence of Smooth Solutions	p. 302
Uniqueness	p. 304
Continuation Criteria	p. 307
Global Existence Results in Dimension Two	p. 310
Smooth Solutions	p. 311
The Borderline Case	p. 311
The Yudovich Theorem	p. 312
The Inviscid Limit	p. 313
Regularity Results for the Navier-Stokes System	p. 314
The Smooth Case	p. 314
The Rough Case	p. 316
Viscous Vortex Patches	p. 318
Results Related to Striated Regularity	p. 19
A Stationary Estimate for the Velocity Field	p. 320
Uniform Estimates for Striated Regularity	p. 324
A Global Convergence Result for Striated Regularity	p. 326
Application to Smooth Vortex Patches	p. 330
References and Remarks	p. 331
Strichartz Estimates and Applications to Semilinear Dispersive Equations	p. 335
Examples of Dispersive Estimates	p. 336
The Dispersive Estimate for the Free Transport Equation	p. 336
The Dispersive Estimates for the Schrdillger Equation	p. 337
Integral of Oscillating Functions	p. 339
Dispersive Estimates for the Wave Equation	p. 344
The L² Boundedness of Some Fourier Integral Operators	p. 346
Billnear Methods	p. 349
The Duality Method and the TT* Argument	p. 350
Strichartz Estimates: The Case q > 2	p. 351
Strichartz Estimates: The Endpoint Case q = 2	p. 352
Application to the Cubic Semilinear Schrödinger Equation	p. 355
Strichartz Estimates for the Wave Equation	p. 359
The Basic Strichartz Estimate	p. 359
The Refined Strichartz Estimate	p. 362
The Qulntic Wave Equation in R³	p. 368
The Cubic Wave Equation in R³	p. 370
Solutions in H¹	p. 370
Local and Global Well-posedness for Rough Data	p. 372
The Nonlinear Interpolation Method	p. 374
Application to a Class of Semilinear Wave Equations	p. 381
References and Remarks	p. 386
Smoothing Effect in Quasilinear Wave Equations	p. 389
A Well-posedness Result Based on an Energy Method	p. 391
The Main Statement and the Strategy of its Proof	p. 401
Refined Paralinearization of the Wave Equation	p. 403
Reduction to a Microlocal Strichartz Estimate	p. 406
Microlocal Strichartz Estimates	p. 413
A Rather General Statement	p. 413
Geometrical Optics	p. 414
The Solution of the Eikonal Equation	p. 415
The Transport Equation	p. 419
The Approximation Theorem	p. 421
The Proof of Theorem 9.16	p. 423
References and Remarks	p. 427
The Compressible Navier-Stokes System	p. 429
About the Model	p. 429
General Overview	p. 430
The Barotropic Navier-Stokes Equations	p. 432
Local Theory for Data with Critical Regularity	p. 433
Scaling Invariance and Statement of the Main Result	p. 433
A Priori Estimates	p. 435
Existence of a Local Solution	p. 440
Uniqueness	p. 445
A Continuation Criterion	p. 450
Local Theory for Data Bounded Away from the Vacuum	p. 451
A Priori Estimates for the Linearized Momentum Equation	p. 451
Existence of a Local Solution	p. 457
Uniqueness	p. 460
A Continuation Criterion	p. 462
Global Existence for Small Data	p. 462
Statement of the Results	p. 463
A Spectral Analysis of the Linearized Equation	p. 464
A Prioli Estimates for the Linearized Equation	p. 466
Proof of Global Existence	p. 473
The Incompressible Limit	p. 475
Main Results	p. 475
The Case of Small Data with Critical Regularity	p. 477
The Case of Large Data with More Regularity	p. 483
References and Remarks	p. 492
References	p. 497
List of Notations	p. 523
Index	p. 527
Table of Contents provided by Ingram. All Rights Reserved.

Fourier Analysis and Nonlinear Partial Differential Equations

Rent Textbook

New Textbook

Used Textbook

eTextbook

How Marketplace Works:

Summary

Table of Contents

Digital License