Fourier analysis and nonlinear partial differential equations / Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin.

Format:

Book

Author/Creator:

Bahouri, Hajer.

Contributor:

Chemin, Jean-Yves.

Danchin, Raphaël.

Series:

Grundlehren der mathematischen Wissenschaften ; 343.

Grundlehren der mathematischen Wissenschaften

Language:

English

Subjects (All):

Fourier analysis.

Differential equations, Partial.

Physical Description:

xv, 523 pages ; 24 cm.

Place of Publication:

Heidelberg ; New York : Springer, [2011]

Summary:

In recent years, the Fourier analysis methods have experienced 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. 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. Book jacket.

Contents:

1 Basic Analysis 1

1.1 Basic Real Anslysis 1

1.1.1 Holder and Convolution Inequslities 1

1.1.2 The Atomic Decomposition 7

1.1.3 Proof of Refined Young Inequslityp8

1.1.4 A Bilinear Interpolation Theorem 10

1.1.5 A Linear Interpolation Result 11

1.1.6 The Hardy-Littlewood Maximal Function 13

1.2 The Fourier Transform 16

1.2.1 Fourier Transforms of Functions and the Schwartz Space 16

1.2.2 Tempered Distributions and the Fourier Transform 18

1.2.3 A Few Calculations of Fourier Transforms 23

1.3 Homogeneous Sobolev Spaces 25

1.3.1 Definition and Basic Properties 25

1.3.2 Sobolev Embedding in Lebesgue Spaces 29

1.3.3 The Limit Case H^d/² 36

1.3.4 The Embedding Theorem in Hölder Spaces 37

1.4 Nonhomogeneous Sobolev Spaces on R^d 38

1.4.1 Definition and Basic Properties 38

1.4.2 Embedding 44

1.4.3 A Density Theorem 47

1.4.4 Hardy Inequality 48

1.5 References and Remarks 49

2 Littlewood-Paley Theory 51

2.1 Functions with Compactly Supported Fourier Transforms 51

2.1.1 Bernstein-Type Lemmas 52

2.1.2 The Smoothing Effect of Heat Flow 53

2.1.3 The Action of a Diffeomorphism 56

2.1.4 The Effects of Some Nonlinear Functions 58

2.2 Dyadic Partition of Unity 59

2.3 Homogeneous Besov Spaces 63

2.4 Characterizations of Homogeneous Besov Spaces 72

2.5 Besov Spaces, Lebesgue Spaces, and Refined Inequalities 78

2.6 Homogeneous Paradifferential Calculus 85

2.6.1 Homogeneous Bony Decomposition 85

2.6.2 Action of Smooth Functions 93

2.6.3 Time-Space Besov Spaces 98

2.7 Nonhomogeneous Besov Spaces 98

2.8 Nonhomogeneous Paradifferential Calculus 102

2.8.1 The Bony Decomposition 102

2.8.2 The Paralinearization Theorem 104

2.9 Besov Spaces and Compact Embeddings 108

2.10 Commutator Estimates 110

2.11 Around the Space B∞,∞¹ 116

2.12 References and Remarks 120

3 Transport and Transport-Diffusion Equations 123

3.1 Ordinary Differential Equations 124

3.1.1 The Cauchy-Lipschitz Theorem Revisited 124

3.1.2 Estimates for the Flow 129

3.1.3 A Blow-up Criterion for Ordinary Differential Equations 131

3.2 Transport Equations: The Lipschitz Case 132

3.2.1 A Priori Estimates in General Besov Spaces 132

3.2.2 Refined Estimates in Besov Spaces with Index 0 135

3.2.3 Solving the Transport Equation in Besov Spaces 136

3.2.4 Application to a Shallow Water Equation 140

3.3 Losing Estimates for Transport Equations 147

3.3.1 Linear Loss of Regularity in Besov Spaces 147

3.3.2 The Exponential Loss 151

3.3.3 Limited Loss of Regularity 153

3.3.4 A Few Applications 155

3.4 Transport-Diffusion Equations 156

3.4.1 A Priori Estimates 157

3.4.2 Exponential Decay 163

3.5 References and Remarks 166

4 Quasilinear Symmetric Systems 169

4.1 Definition and Examples 169

4.2 Linear Symmetric Systems 172

4.2.1 The Well-posedness of Linear Symmetric Systems 172

4.2.2 Finite Propagation Speed 180

4.2.3 Further Well-posedness Results for Linear Symmetric Systems 183

4.3 The Resolution of Quasilinear Symmetric Systems 187

4.3.1 Paralinearization and Energy Estimates 189

4.3.2 Convergence of the Scheme 190

4.3.3 Completion of the Proof of Existence 191

4.3.4 Uniqueness and Continuation Criterion 192

4.4 Data with Critical Regularity and Blow-up Criteria 193

4.4.1 Critical Besov Regularity 193

4.4.2 A Refined Blow-up Crndition 196

4.5 Continuity of the Flow Map 198

4.6 References and Remarks 201

5 The Incompressible Navier-Stokes System 203

5.1 Basic Facts Concerning the Navier-Stokes System 204

5.2 Well-posedness in Sobolev Spaces 209

5.2.1 A General Result 209

5.2.2 The Behavior of the H^d/²⁻¹ Norm Near 0 214

5.3 Results Related to the Structure of the System 215

5.3.1 The Particular Case of Dimension Two 215

5.3.2 The Case of Dimension Three 217

5.4 An Elementary L^p Approach 220

5.5 The Endpoint Space for Picard's Scheme 227

5.6 The Use of the L¹-smoothing Effect of the Heat Flow 233

5.6.1 The Cannone-Meyer-Planchon Theorem Revisited 234

5.6.2 The Flow of the Solutions of the Navier-Stokes System 236

5.7 References and Remarks 242

6 Anisotropic Viscosity 245

6.1 The Case of L² Data with One Vertical Derivative in L² 246

6.2 A Global Existence Result in Anisotropic Besov Spaces 254

6.2.1 Anisotropic Localization in Fourier Space 254

6.2.2 The Functional Framework 256

6.2.3 Statement of the Main Result 258

6.2.4 Some Technical Lemmas 261

6.3 The Proof of Existence 266

6.4 The Proof of Uniqueness 276

6.5 References and Remarks 289

7 Euler System for Perfect Incompressible Fluids 291

7.1 Local Well-posedness Results for Inviscid Fluids 292

7.1.1 The Biot-Savart Law 293

7.1.2 Estimates for the Pressure 296

7.1.3 Another Formulation of the Euler System 301

7.1.4 Local Existence of Smooth Solutions 302

7.1.5 Uniqueness 304

7.1.6 Continuation Criteria 307

7.2 Global Existence Results in Dimension Two 310

7.2.1 Smooth Solutions 311

7.2.2 The Borderline Case 311

7.2.3 The Yudovich Theorem 312

7.3 The Inviscid Limit 313

7.3.1 Regularity Results for the Navier-Stokes System 314

7.3.2 The Smooth Case 314

7.3.3 The Rough Case 316

7.4 Viscous Vortex Patches 318

7.4.1 Results Related to Striated Regularity 19

7.4.2 A Stationary Estimate for the Velocity Field 320

7.4.3 Uniform Estimates for Striated Regularity 324

7.4.4 A Global Convergence Result for Striated Regularity 326

7.4.5 Application to Smooth Vortex Patches 330

7.5 References and Remarks 331

8 Strichartz Estimates and Applications to Semilinear Dispersive Equations 335

8.1 Examples of Dispersive Estimates 336

8.1.1 The Dispersive Estimate for the Free Transport Equation 336

8.1.2 The Dispersive Estimates for the Schrdillger Equation 337

8.1.3 Integral of Oscillating Functions 339

8.1.4 Dispersive Estimates for the Wave Equation 344

8.1.5 The L² Boundedness of Some Fourier Integral Operators 346

8.2 Billnear Methods 349

8.2.1 The Duality Method and the TT* Argument 350

8.2.2 Strichartz Estimates: The Case q > 2 351

8.2.3 Strichartz Estimates: The Endpoint Case q = 2 352

8.2.4 Application to the Cubic Semilinear Schrödinger Equation 355

8.3 Strichartz Estimates for the Wave Equation 359

8.3.1 The Basic Strichartz Estimate 359

8.3.2 The Refined Strichartz Estimate 362

8.4 The Qulntic Wave Equation in R³ 368

8.5 The Cubic Wave Equation in R³ 370

8.5.1 Solutions in H¹ 370

8.5.2 Local and Global Well-posedness for Rough Data 372

8.5.3 The Nonlinear Interpolation Method 374

8.6 Application to a Class of Semilinear Wave Equations 381

8.7 References and Remarks 386

9 Smoothing Effect in Quasilinear Wave Equations 389

9.1 A Well-posedness Result Based on an Energy Method 391

9.2 The Main Statement and the Strategy of its Proof 401

9.3 Refined Paralinearization of the Wave Equation 403

9.4 Reduction to a Microlocal Strichartz Estimate 406

9.5 Microlocal Strichartz Estimates 413

9.5.1 A Rather General Statement 413

9.5.2 Geometrical Optics 414

9.5.3 The Solution of the Eikonal Equation 415

9.5.4 The Transport Equation 419

9.5.5 The Approximation Theorem 421

9.5.6 The Proof of Theorem 9.16 423

9.6 References and Remarks 427

10 The Compressible Navier-Stokes System 429

10.1 About the Model 429

10.1.1 General Overview 430

10.1.2 The Barotropic Navier-Stokes Equations 432

10.2 Local Theory for Data with Critical Regularity 433

10.2.1 Scaling Invariance and Statement of the Main Result 433

10.2.2 A Priori Estimates 435

10.2.3 Existence of a Local Solution 440

10.2.4 Uniqueness 445

10.2.5 A Continuation Criterion 450

10.3 Local Theory for Data Bounded Away from the Vacuum 451

10.3.1 A

Priori Estimates for the Linearized Momentum Equation 451

10.3.2 Existence of a Local Solution 457

10.3.3 Uniqueness 460

10.3.4 A Continuation Criterion 462

10.4 Global Existence for Small Data 462

10.4.1 Statement of the Results 463

10.4.2 A Spectral Analysis of the Linearized Equation 464

10.4.3 A Prioli Estimates for the Linearized Equation 466

10.4.4 Proof of Global Existence 473

10.5 The Incompressible Limit 475

10.5.1 Main Results 475

10.5.2 The Case of Small Data with Critical Regularity 477

10.5.3 The Case of Large Data with More Regularity 483

10.6 References and Remarks 492.

Notes:

Includes bibliographical references (pages 497-515) and index.

ISBN:

9783642168291

3642168299

OCLC:

690089160

Online:

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