INTRO THEORY OF WAVE MAPS AND RELATED GEOMETRIC PROBLEMS, AN

by Dan-andrei Geba, Manoussos G Grillakis

Overview

The wave maps system is one of the most beautiful and challenging nonlinear hyperbolic systems, which has captured the attention of mathematicians for more than thirty years now. In the study of its various issues, such as the well-posedness theory, the formation of singularities, and the stability of the solitons, in order to obtain optimal results, one has to use intricate tools coming not only from analysis, but also from geometry and topology. Moreover, the wave maps system is nothing other than the Euler-Lagrange system for the nonlinear sigma model, which is one of the fundamental problems in classical field theory. One of the goals of our book is to give an up-to-date and almost self-contained overview of the main regularity results proved for wave maps. Another one is to introduce, to a wide mathematical audience, physically motivated generalizations of the wave maps system (e.g., the Skyrme model), which are extremely interesting and difficult in their own right.

Product Details

ISBN-13:	9789814713924
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	08/18/2016
Sold by:	Barnes & Noble
Format:	eBook
Pages:	496
File size:	37 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface vii

Acknowledgments ix

List of Figures xvii

1 Introduction 1

1.1 Physical description and motivation 1

1.1.1 Nonlinear sigma models 1

1.1.2 Skyrme and Skyrme-like models 4

1.2 Mathematical formulation and basic properties 9

1.2.1 The wave map problem: initial formulation and the Eider-Lagrange equations 10

1.2.2 The wave map problem: symmetries of the base and the energy-momentum tensor 12

1.2.3 The wave map problem: energy conservation and the finite speed of propagation property 18

1.2.4 The wave map problem: scaling invariance and energy criticality 22

1.2.5 The wave map problem: general discussion of symmetries on the target 23

1.2.6 The wave map problem: discussion of symmetries on the target when N = S² or N = H² 24

1.2.7 The wave map problem: initial considerations on the associated Hodge system 26

1.2.8 The wave map problem: the Hodge system in the general case of a parallelizable target manifold 32

1.2.9 The wave map problem: static solutions for the 2 + 1-dimensional case 36

1.2.10 The wave map problem: the completely integrable 1 + 1-dimensional case 39

1.2.11 The Skyrme model: its energy-momentum tensor and associated Hodge system 39

1.2.12 The Einstein vacuum equations: a symmetry reduction to a system containing wave map type problems 43

2 Analytic tools 57

2.1 Preliminaries 57

2.1.1 Littlewood-Paley theory 58

2.1.2 Function spaces 59

2.2 Strichartz estimates 64

2.2.1 Homogeneous bounds 72

2.2.2 Inhomogeneous bounds 75

2.3 Hyperbolic Sobolev spaces 80

2.4 Tataru's F-spaces 106

3 Local and small data global well-posedness theory for wave maps 123

3.1 The Cauchy problem, the concept of well-posedness, and scaling heuristics 123

3.2 Abstract existence theory 126

3.3 Local well-posedness results 128

3.3.1 Energy arguments 129

3.3.2 Strichartz-type arguments 130

3.3.3 Hyperbolic Sobolev spaces approach 132

3.4 Small data global well-posedness results 134

3.4.1 Semilinear approach 134

3.4.2 Fully nonlinear approach 137

3.4.3 Tao's result 142

3.1.1 Shatah-Struwe's result 170

4 Large data regularity and scattering for 2+1-dimensional wave maps 191

4.1 Introduction 191

4.2 Informal discussion of Sterbenz-Tataru's results 193

4.3 Basic setup of the problem 201

4.4 Outline of the argument proving Theorems 4.4 and 4.7 205

4.4.1 Extension and scaling (first step) 205

4.4.2 Energy-type estimates (second step) 206

4.4.3 Elimination of the null concentration scenario (third step) 208

4.4.4 Uniform time propagation of nontrivial energy (fourth step) 209

4.4.5 Final rescaling (fifth step) 210

4.4.6 Isolating concentration scales (sixth step) 210

4.4.7 The compactness argument (final step) 211

4.5 The extension procedure 212

4.5.1 The extension procedure for the blow-up result 213

4.5.2 The extension procedure for the scattering result 214

4.6 Generalized energy estimates and applications 216

4.6.1 Energy-type integral identity on hyperboloids 217

4.6.2 The hyperbolic projection 220

4.6.3 Partial non-concentration of energy and time decay estimates I 223

4.6.4 Partial non-concentration of energy and time decay estimates II 228

4.6.5 Propagation of energy concentration 235

4.6.6 Energy concentration scenarios 240

4.7 Energy dispersion and non-concentration of energy 242

4.7.1 Preliminaries and formulation of the main goal 243

4.7.2 A reduction step and initial gauge construction 245

4.7.3 The low frequency case 248

4.7.4 The high frequency case 249

4.8 The concentration of energy scenario 254

4.8.1 Initial considerations and final rescaling 254

4.8.2 Refined concentration scenarios 257

4.9 The final compactness argument 266

4.9.1 Analysis of scenario (ii) in Lemma 4.7 269

4.9.2 Analysis of scenario (i) in Lemma 4.7 271

5 General well-posedness issues for Skyrme and Skyrme-like models 273

5.1 Relevant results 273

5.2 Wong's result 274

5.2.1 Preliminaries 275

5.2.2 Dominant energy condition 276

5.2.3 Regular hyperbolicity 282

5.3 Lei-Lin-Zhou's result 291

5.3.1 Preliminaries 296

5.3.2 Proof of (5.45) 301

5.3.3 Proof of (5.47) 309

6 Equivariant results 315

6.1 Prologue 315

6.2 Non-concentration of energy for classical equivariant wave maps 317

6.2.1 Differential identities 318

6.2.2 Standard energy estimates and applications 322

6.2.3 Conditional regularity 327

6.3 Small data global well-posedness for Skyrme and Skyrme-like models 337

6.3.1 Preliminaries 337

6.3.2 Main analysis for the Adkins-Nappi problem 347

6.3.3 The analysis for the pure power nonlincaritics in the Faddeev problem 348

6.3.4 The analysis for the null-form nonlinearity in the Skyrme and Faddeev problems 354

6.4 Large data global regularity for the 2 + 1-dimensional Faddeev model 360

6.4.1 Statement of the main result and initial reductions 360

6.4.2 Motivation for the construction of the auxiliary function 363

6.4.3 Analysis of Φ based on energy-type arguments 367

6.4.4 Analysis of Φ based on Strichartz estimates 369

6.4.5 Confirmation of the continuation criterion 373

7 The question of collapse for wave maps 375

7.1 Introduction 377

7.2 The associated Hodge system in complex coordinates 386

7.3 The equivariant reduction 388

7.4 Differential identities and applications 391

7.4.1 The second-order energy 392

7.4.2 Conformal identity and other conservation laws 394

7.5 Initial setup, key energy facts, and stability issues 395

7.5.1 Notational conventions and initial considerations 395

7.5.2 Energy formulas 398

7.5.3 A differential equation for the speed 401

7.5.4 Preliminary assumptions on the initial data and the stability argument 404

7.6 Operator estimates and comparison estimates 409

7.7 Estimating the second-order energy and the acceleration 422

7.8 The last step in the argument for collapse 430

Appendix A Tools from geometry: invariance, symmetry, and curvature 435

A.1 Background in differential geometry 436

A.2 Calculus on manifolds 438

A.3 The wave operator and the Lagrangian of a scalar field 441

A.4 The concept of Lie derivative 441

A.5 The energy-momentum tensor 444

A.6 Maxwell's equations 446

A.7 The Riemann curvature tensor 448

A.8 The energy-momentum tensor for Maxwell's equations 451

A.9 The Ricci tensor and general relativity 453

Bibliography 459

Index 471