Fourier Analysis and Nonlinear Partial Differential Equations (eBook)

Preface 6
Contents 10
Basic Analysis 15
Basic Real Analysis 15
Hölder and Convolution Inequalities 15
The Atomic Decomposition 21
Proof of Refined Young Inequality 22
A Bilinear Interpolation Theorem 24
A Linear Interpolation Result 25
The Hardy-Littlewood Maximal Function 27
The Fourier Transform 30
Fourier Transforms of Functions and the Schwartz Space 30
Tempered Distributions and the Fourier Transform 32
A Few Calculations of Fourier Transforms 37
Homogeneous Sobolev Spaces 39
Definition and Basic Properties 39
Sobolev Embedding in Lebesgue Spaces 43
The Limit Case Hd2 50
The Embedding Theorem in Hölder Spaces 51
Nonhomogeneous Sobolev Spaces on Rd 52
Definition and Basic Properties 52
Embedding 58
A Density Theorem 61
Hardy Inequality 62
References and Remarks 63
Littlewood-Paley Theory 65
Functions with Compactly Supported Fourier Transforms 65
Bernstein-Type Lemmas 66
The Smoothing Effect of Heat Flow 67
The Action of a Diffeomorphism 70
The Effects of Some Nonlinear Functions 72
Dyadic Partition of Unity 73
Homogeneous Besov Spaces 77
Characterizations of Homogeneous Besov Spaces 86
Besov Spaces, Lebesgue Spaces, and Refined Inequalities 92
Homogeneous Paradifferential Calculus 99
Homogeneous Bony Decomposition 99
Action of Smooth Functions 107
Time-Space Besov Spaces 112
Nonhomogeneous Besov Spaces 112
Nonhomogeneous Paradifferential Calculus 116
The Bony Decomposition 116
The Paralinearization Theorem 118
Besov Spaces and Compact Embeddings 122
Commutator Estimates 124
Around the Space B1infty,infty 130
References and Remarks 134
Transport and Transport-Diffusion Equations 136
Ordinary Differential Equations 137
The Cauchy-Lipschitz Theorem Revisited 137
Estimates for the Flow 142
A Blow-up Criterion for Ordinary Differential Equations 144
Transport Equations: The Lipschitz Case 145
A Priori Estimates in General Besov Spaces 145
Refined Estimates in Besov Spaces with Index 0 148
Solving the Transport Equation in Besov Spaces 149
Application to a Shallow Water Equation 153
First Step: Constructing Approximate Solutions 156
Second Step: Uniform Bounds 156
Third Step: Convergence 156
Final Step: Conclusion 157
Losing Estimates for Transport Equations 160
Linear Loss of Regularity in Besov Spaces 160
The Exponential Loss 164
Limited Loss of Regularity 166
A Few Applications 168
Transport-Diffusion Equations 169
A Priori Estimates 170
Exponential Decay 176
References and Remarks 179
Quasilinear Symmetric Systems 181
Definition and Examples 181
Linear Symmetric Systems 184
The Well-posedness of Linear Symmetric Systems 184
Finite Propagation Speed 192
Further Well-posedness Results for Linear Symmetric Systems 195
The Resolution of Quasilinear Symmetric Systems 199
Paralinearization and Energy Estimates 201
Convergence of the Scheme 202
Completion of the Proof of Existence 203
Uniqueness and Continuation Criterion 204
Data with Critical Regularity and Blow-up Criteria 205
Critical Besov Regularity 205
A Refined Blow-up Condition 208
Continuity of the Flow Map 210
References and Remarks 213
The Incompressible Navier-Stokes System 215
Basic Facts Concerning the Navier-Stokes System 216
Well-posedness in Sobolev Spaces 221
A General Result 221
The Behavior of the Hd2-1 Norm Near 0 226
Results Related to the Structure of the System 227
The Particular Case of Dimension Two 227
The Case of Dimension Three 229
An Elementary Lp Approach 232
The Endpoint Space for Picard's Scheme 239
The Use of the L1-smoothing Effect of the Heat Flow 245
The Cannone-Meyer-Planchon Theorem Revisited 246
The Flow of the Solutions of the Navier-Stokes System 248
References and Remarks 254
Anisotropic Viscosity 256
The Case of L2 Data with One Vertical Derivative in L2 257
A Global Existence Result in Anisotropic Besov Spaces 265
Anisotropic Localization in Fourier Space 265
The Functional Framework 267
Statement of the Main Result 269
Some Technical Lemmas 272
The Proof of Existence 277
The Proof of Uniqueness 287
References and Remarks 300
Euler System for Perfect Incompressible Fluids 301
Local Well-posedness Results for Inviscid Fluids 302
The Biot-Savart Law 303
Estimates for the Pressure 306
Another Formulation of the Euler System 311
Local Existence of Smooth Solutions 312
First Step: Construction of Approximate Solutions 312
Second Step: A Priori Estimates 312
Third Step: Convergence of the Sequence 313
Fourth Step: Passing to the Limit 314
Uniqueness 314
Continuation Criteria 317
Global Existence Results in Dimension Two 320
Smooth Solutions 321
The Borderline Case 321
The Yudovich Theorem 322
The Inviscid Limit 323
Regularity Results for the Navier-Stokes System 324
The Smooth Case 324
The Rough Case 326
Viscous Vortex Patches 328
Results Related to Striated Regularity 329
A Stationary Estimate for the Velocity Field 330
Uniform Estimates for Striated Regularity 334
A Global Convergence Result for Striated Regularity 336
First Step: Construction of Smooth Solutions 337
Second Step: Uniform Estimates for Striated Regularity 337
Third Step: Convergence of Smooth Solutions 339
Final Step: The Inviscid Limit 340
Application to Smooth Vortex Patches 340
References and Remarks 341
Strichartz Estimates and Applications to Semilinear Dispersive Equations 344
Examples of Dispersive Estimates 345
The Dispersive Estimate for the Free Transport Equation 345
The Dispersive Estimates for the Schrödinger Equation 346
Integral of Oscillating Functions 348
Dispersive Estimates for the Wave Equation 353
The L2 Boundedness of Some Fourier Integral Operators 355
Bilinear Methods 358
The Duality Method and the TT Argument 359
Strichartz Estimates: The Case q> 2
Strichartz Estimates: The Endpoint Case q=2 361
Application to the Cubic Semilinear Schrödinger Equation 364
Strichartz Estimates for the Wave Equation 368
The Basic Strichartz Estimate 368
The Refined Strichartz Estimate 371
The Quintic Wave Equation in R3 377
The Cubic Wave Equation in R3 379
Solutions in H1 379
Local and Global Well-posedness for Rough Data 381
The Nonlinear Interpolation Method 383
Application to a Class of Semilinear Wave Equations 390
References and Remarks 395
Smoothing Effect in Quasilinear Wave Equations 397
A Well-posedness Result Based on an Energy Method 399
First Step: Uniform Bounds in Large Norm 404
Second Step: Convergence of the Approximate Sequence 405
Third Step: Time Continuity of the Solution 406
Fourth Step: Uniqueness 407
Fifth Step: The Blow-up Criterion 407
Final Step: Additional Regularity 408
The Main Statement and the Strategy of its Proof 409
Refined Paralinearization of the Wave Equation 411
Reduction to a Microlocal Strichartz Estimate 414
Microlocal Strichartz Estimates 421
A Rather General Statement 421
Geometrical Optics 422
The Solution of the Eikonal Equation 423
The Transport Equation 427
The Approximation Theorem 429
The Proof of Theorem 9.16 431
References and Remarks 435
The Compressible Navier-Stokes System 437
About the Model 437
General Overview 438
The Barotropic Navier-Stokes Equations 440
Local Theory for Data with Critical Regularity 441
Scaling Invariance and Statement of the Main Result 441
A Priori Estimates 443
Existence of a Local Solution 448
First Step: Friedrichs Approximation 448
Second Step: Uniform Estimates 449
Third Step: Time Derivatives 450
Fourth Step: Compactness and Convergence 450
Final Step: Completion of the Proof 451
Uniqueness 453
A Continuation Criterion 458
Local Theory for Data Bounded Away from the Vacuum 459
A Priori Estimates for the Linearized Momentum Equation 459
Existence of a Local Solution 465
First Step: Friedrichs Approximation 465
Second Step: Uniform Estimates 466
Third Step: Time Derivatives 466
Fourth Step: Compactness and Convergence 467
Uniqueness 468
A Continuation Criterion 470
Global Existence for Small Data 470
Statement of the Results 471
A Spectral Analysis of the Linearized Equation 472
A Priori Estimates for the Linearized Equation 474
First Step: The Incompressible Part 475
Second Step: The Compressible Part 476
Third Step: Global A Priori Estimates 479
Fourth Step: The Parabolic Behavior of u 479
Proof of Global Existence 481
The Incompressible Limit 483
Main Results 483
The Case of Small Data with Critical Regularity 485
Step 1. Proof of Global Existence with Uniform Estimates 487
Step 2. Convergence to Zero of the Compressible Modes 488
Step 3. Convergence of the Incompressible Part 490
The Case of Large Data with More Regularity 491
First Step: Dispersive Estimates for (bepsilon,Puepsilon) 493
Second Step: Estimates for wepsilon 494
Third Step: Estimates for (bepsilon,uepsilon) in Eepsilonnu,Td2+beta 496
Fourth Step: Bootstrap 498
Last Step: Continuation Argument 500
References and Remarks 500
References 505
List of Notations 524
Index 527