**Mathematical Surveys and Monographs**

Volume: 214;
2016;
515 pp;
Hardcover

MSC: Primary 35;

**Print ISBN: 978-1-4704-2857-0
Product Code: SURV/214**

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**Electronic ISBN: 978-1-4704-3564-6
Product Code: SURV/214.E**

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AMS Member Price: $88.00

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#### Supplemental Materials

# Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations

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*Jared Speck*

In 1848 James Challis showed that smooth
solutions to the compressible Euler equations can become multivalued,
thus signifying the onset of a shock singularity. Today it is known
that, for many hyperbolic systems, such singularities often
develop. However, most shock-formation results have been proved only
in one spatial dimension. Serge Alinhac's groundbreaking work on wave
equations in the late 1990s was the first to treat more than one
spatial dimension. In 2007, for the compressible Euler equations in
vorticity-free regions, Demetrios Christodoulou remarkably sharpened
Alinhac's results and gave a complete description of shock
formation.

In this monograph, Christodoulou's framework is extended to two
classes of wave equations in three spatial dimensions. It is shown
that if the nonlinear terms fail to satisfy the null condition, then
for small data, shocks are the only possible singularities that can
develop. Moreover, the author exhibits an open set of small data whose
solutions form a shock, and he provides a sharp description of the
blow-up. These results yield a sharp converse of the fundamental
result of Christodoulou and Klainerman, who showed that small-data
solutions are global when the null condition is satisfied.

Readers who master the material will have acquired tools on the
cutting edge of PDEs, fluid mechanics, hyperbolic conservation laws,
wave equations, and geometric analysis.

#### Readership

Graduate students and researchers interested in nonlinear PDE and shock formation.

#### Reviews & Endorsements

This is a well written monograph and is a welcome addition to the literature on three-dimensional quasilinear wave equations.

-- Vishnu Dutt Sharma, Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface xv16
- Acknowledgments xxiii24
- Chapter 1. Introduction 126
- Chapter 2. Overview of the Two Main Theorems 3156
- 2.1. First description of the two theorems 3358
- 2.2. The basic structure of the equations 3560
- 2.3. The structure of the equation relative to rectangular coordinates 3863
- 2.4. The (classic) null condition 3863
- 2.5. Basic geometric constructions 4166
- 2.6. The rescaled frame and dispersive sup-norm estimates 4469
- 2.7. The basic structure of the coupled system and sup-norm estimates for the eikonal function quantities 4671
- 2.8. Lower bounds for the rescaled radial derivative of the solution in the case of shock formation 4772
- 2.9. The main ideas behind the vanishing of the inverse foliation density 4873
- 2.10. The role of Theorem 22.1 in justifying the heuristics 4974
- 2.11. Comparison with related work 6186
- 2.12. Outline of the monograph 76101
- 2.13. Suggestions on how to read the monograph 78103

- Chapter 3. Initial Data, Basic Geometric Constructions, and the Future Null Condition Failure Factor 81106
- 3.1. Initial data 81106
- 3.2. The eikonal function and the geometric radial variable 82107
- 3.3. First fundamental forms and Levi-Civita connections 83108
- 3.4. Frame vectorfields and the inverse foliation density 84109
- 3.5. Geometric coordinates 87112
- 3.6. Frames 89114
- 3.7. The future null condition failure factor 90115
- 3.8. Contraction and component notation 90115
- 3.9. Projection operators and tensors along submanifolds 91116
- 3.10. Expressions for the metrics and volume form factors 93118
- 3.11. The trace and trace-free parts of tensors 95120
- 3.12. Angular differential 96121
- 3.13. Musical notation 97122
- 3.14. Pointwise norms 97122
- 3.15. Lie derivatives and projected Lie derivatives 97122
- 3.16. Second fundamental forms 99124
- 3.17. Frame components, relative to the nonrescaled frame, of the derivatives of the metric with respect to the solution 100125
- 3.18. The change of variables map 101126
- 3.19. Area forms, volume forms, and norms 102127
- 3.20. Schematic notation 104129

- Chapter 4. Transport Equations for the Eikonal Function Quantities 107132
- 4.1. Re-centered variables and the eikonal function quantities 107132
- 4.2. Covariant derivatives and Christoffel symbols relative to the rectangular coordinates 108133
- 4.3. Transport equation for the inverse foliation density 108133
- 4.4. Transport equations for the rectangular components of the frame vectorfields 109134
- 4.5. An expression for the re-centered null second fundamental form in terms of other quantities 111136
- 4.6. Identities involving deformation tensors and Lie derivatives 112137

- Chapter 5. Connection Coefficients of the Rescaled Frames and Geometric Decompositions of the Wave Operator 117142
- Chapter 6. Construction of the Rotation Vectorfields and Their Basic Properties 123148
- Chapter 7. Definition of the Commutation Vectorfields and Deformation Tensor Calculations 127152
- Chapter 8. Geometric Operator Commutator Formulas and Schematic Notation for Repeated Differentiation 135160
- Chapter 9. The Structure of the Wave Equation Inhomogeneous Terms After One Commutation 143168
- Chapter 10. Energy and Cone Flux Definitions and the Fundamental Divergence Identities 151176
- Chapter 11. Avoiding Derivative Loss and Other Difficulties via Modified Quantities 167192
- 11.1. Preliminary structural identities 168193
- 11.2. Full modification of the trace of the re-centered null second fundamental form 173198
- 11.3. Partial modification of the trace of the re-centered null second fundamental form 177202
- 11.4. Partial modification of the angular gradient of the inverse foliation density 178203

- Chapter 12. Small Data, Sup-Norm Bootstrap Assumptions, and First Pointwise Estimates 181206
- 12.1. Restricting the analysis to solutions of the evolution equations 181206
- 12.2. Small data 182207
- 12.3. Fundamental positivity bootstrap assumption for the inverse foliation density 182207
- 12.4. Sup-norm bootstrap assumptions 182207
- 12.5. Basic estimates for the geometric radial variable 184209
- 12.6. Basic estimates for the rectangular spatial coordinate functions 184209
- 12.7. Estimates for the rectangular components of the metrics and the spherical projection tensorfield 185210
- 12.8. The behavior of quantities along the initial data hypersurface 187212
- 12.9. Estimates for the derivatives of rectangular components of various vectorfields and the radial component of the Euclidean rotations 190215
- 12.10. Estimates for the rectangular components of the metric dual of the unit-length radial vectorfield 192217
- 12.11. Precise pointwise estimates for the rotation vectorfields 193218
- 12.12. Precise pointwise differential operator comparison estimates 197222
- 12.13. Useful estimates for avoiding detailed commutators 199224
- 12.14. Estimates for the derivatives of the angular differential of the rectangular spatial coordinate functions 200225
- 12.15. Pointwise estimates for the Lie derivatives of the frame components of the derivative of the rectangular components of the metric with respect to the solution 200225
- 12.16. Crude pointwise estimates for the Lie derivatives of the angular components of the deformation tensors 201226
- 12.17. Two additional crude differential operator comparison estimates 203228
- 12.18. Pointwise estimates for the derivatives of the re-centered null second fundamental form in terms of other quantities 204229
- 12.19. Pointwise estimates for the Lie derivatives of the rotation vectorfields 206231
- 12.20. Pointwise estimates for the angular one-forms and vectorfields corresponding to the commutation vectorfield deformation tensors 207232
- 12.21. Preliminary Lie derivative commutator estimates 209234
- 12.22. Commutator estimates for vectorfields acting on functions and spherical covariant tensorfields 210235
- 12.23. Commutator estimates for vectorfields acting on the covariant angular derivative of a spherical tensorfield 215240
- 12.24. Commutator estimates for vectorfields acting on the angular Hessian of a function 215240
- 12.25. Commutator estimates involving the trace and trace-free parts 218243
- 12.26. Pointwise estimates, in terms of other quantities, for the Lie derivatives of the re-centered null second fundamental form involving an outgoing null differentiation 223248
- 12.27. Improvement of the auxiliary bootstrap assumptions 225250
- 12.28. Sharp pointwise estimates for a frame component of the derivative of the metric with respect to the solution 230255
- 12.29. Pointwise estimates for the angular Laplacian of the derivatives of the rectangular components of the re-centered version of the outgoing null vectorfield 231256
- 12.30. Estimates related to integrals over the spheres 233258
- 12.31. Faster than expected decay for certain wave-variable-related quantities 236261
- 12.32. Pointwise estimates for the vectorfield Xi 239264
- 12.33. Estimates for the components of the commutation vectorfields relative to the geometric coordinates 241266
- 12.34. Estimates for the rectangular spatial derivatives of the eikonal function 243268

- Chapter 13. Sharp Estimates for the Inverse Foliation Density 245270
- Chapter 14. Square Integral Coerciveness and the Fundamental Square-Integral-Controlling Quantities 277302
- Chapter 15. Top-Order Pointwise Commutator Estimates Involving the Eikonal Function 283308
- Chapter 16. Pointwise Estimates for the Easy Error Integrands and Identification of the Difficult Error Integrands Corresponding to the Commuted Wave Equation 295320
- 16.1. Preliminary analysis and the definition of harmless terms 295320
- 16.2. The important terms in the top-order derivatives of the deformation tensors of the commutation vectorfields 299324
- 16.3. Crude pointwise estimates for the below-top-order derivatives of the deformation tensors of the commutation vectorfields 309334
- 16.4. Pointwise estimates for the top-order derivatives of the outgoing null derivative of the commutation vectorfield deformation tensors 312337
- 16.5. Proof of Proposition 16.4 313338
- 16.6. Proof of Corollary 16.5 319344
- 16.7. Pointwise estimates for the error integrands involving the deformation tensors of the multiplier vectorfields 320345
- 16.8. Pointwise estimates needed to close the elliptic estimates 324349

- Chapter 17. Pointwise Estimates for the Difficult Error Integrands Corresponding to the Commuted Wave Equation 327352
- 17.1. Preliminary pointwise estimates for the derivatives of the inhomogeneous terms in the transport equations for the fully modified quantities 327352
- 17.2. Preliminary pointwise estimates for the derivatives of the inhomogeneous terms in the transport equations for the partially modified quantities 332357
- 17.3. Solving the transport equation satisfied by the fully modified version of the spatial derivatives of the trace of the re-centered null second fundamental form 336361
- 17.4. Pointwise estimates for the difficult error integrands requiring full modification 340365
- 17.5. Pointwise estimates for the difficult error integrands requiring partial modification 349374

- Chapter 18. Elliptic Estimates and Sobolev Embedding on the Spheres 353378
- Chapter 19. Square Integral Estimates for the Eikonal Function Quantities that Do Not Rely on Modified Quantities 361386
- Chapter 20. A Priori Estimates for the Fundamental Square-Integral-Controlling Quantities 365390
- 20.1. Bootstrap assumptions for the fundamental square-integral-controlling quantities 365390
- 20.2. Statement of the two main propositions and the fundamental Gronwall lemma 366391
- 20.3. Estimates for all but the most difficult error integrals 372397
- 20.4. Difficult top-order error integral estimates 388413
- 20.5. Proof of Lemma 20.20 407432
- 20.6. Proof of Lemma 20.25 410435
- 20.7. Proof of Lemma 20.26 421446
- 20.8. Proof of Proposition 20.8 426451
- 20.9. Proof of Proposition 20.9 429454
- 20.10. Proof of Lemma 20.10 431456

- Chapter 21. Local Well-Posedness and Continuation Criteria 447472
- Chapter 22. The Sharp Classical Lifespan Theorem 453478
- Chapter 23. Proof of Shock Formation for Nearly Spherically Symmetric Data 467492
- Appendix A. Extension of the Results to a Class of Non-Covariant Wave Equations 479504
- Appendix B. Summary of Notation and Conventions 489514
- B.1. Coordinates 489514
- B.2. Indices 490515
- B.3. Constants 490515
- B.4. Spacetime subsets 490515
- B.5. Metrics, musical notation, and inner products 491516
- B.6. Eikonal function quantities 492517
- B.7. Additional tensorfields related to the connection coefficients 492517
- B.8. Frame vectorfields and the timelike unit normal to the constant-time hypersurfaces 493518
- B.9. Contraction and component notation 494519
- B.10. Projection operators and frame components 494519
- B.11. Tensor products, traces, and contractions 495520
- B.12. The size of the data and the bootstrap parameter 495520
- B.13. Commutation vectorfields 495520
- B.14. Differential operators and commutator notation 496521
- B.15. Floor and ceiling functions and repeated differentiation 497522
- B.16. Area and volume forms 498523
- B.17. Norms 498523
- B.18. Energy-momentum tensorfield, multiplier vectorfields, and compatible currents 499524
- B.19. Square-integral-controlling quantities 499524
- B.20. Modified quantities 500525
- B.21. Curvature tensors 500525
- B.22. Omission of the independent variables in some expressions 501526
- B.23. Data and functions relevant for the proof of shock formation 501526

- Bibliography 503528
- Index 507532
- Back Cover Back Cover1544