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

### About this Title

**Jared Speck**, *Massachusetts Institute of Technology, Cambridge, MA*

Publication: Mathematical Surveys and Monographs

Publication Year:
2016; Volume 214

ISBNs: 978-1-4704-2857-0 (print); 978-1-4704-3564-6 (online)

DOI: https://doi.org/10.1090/surv/214

MathSciNet review: MR3561670

MSC: Primary 35-02; Secondary 35B30, 35F20, 35L67, 35L70, 35Q31

### Table of Contents

**Front/Back Matter**

**Chapters**

- Chapter 1. Introduction
- Chapter 2. Overview of the two main theorems
- Chapter 3. Initial data, basic geometric constructions, and the future null condition failure factor
- Chapter 4. Transport equations for the Eikonal function quantities
- Chapter 5. Connection coefficients of the rescaled frames and geometric decompositions of the wave operator
- Chapter 6. Construction of the rotation vectorfields and their basic properties
- Chapter 7. Definition of the commutation vectorfields and deformation tensor calculations
- Chapter 8. Geometric operator commutator formulas and schematic notation for repeated differentiation
- Chapter 9. The structure of the wave equation inhomogeneous terms after one commutation
- Chapter 10. Energy and cone flux definitions and the fundamental divergence identities
- Chapter 11. Avoiding derivative loss and other difficulties via modified quantities
- Chapter 12. Small data, sup-norm bootstrap assumptions, and first pointwise estimates
- Chapter 13. Sharp estimates for the inverse foliation density
- Chapter 14. Square integral coerciveness and the fundamental square-integral-controlling quantities
- Chapter 15. Top-order pointwise commutator estimates involving the Eikonal function
- Chapter 16. Pointwise estimates for the easy error integrands and identification of the difficult error integrands corresponding to the commuted wave equation
- Chapter 17. Pointwise estimates for the difficult error integrands corresponding to the commuted wave equation
- Chapter 18. Elliptic estimates and Sobolev embedding on the spheres
- Chapter 19. Square integral estimates for the Eikonal function quantities that do not rely on modified quantities
- Chapter 20. A priori estimates for the fundamental square-integral-controlling quantities
- Chapter 21. Local well-posedness and continuation criteria
- Chapter 22. The sharp classical lifespan theorem
- Chapter 23. Proof of shock formation for nearly spherically symmetric data
- Appendix A. Extension of the results to a class of non-covariant wave equations
- Appendix B. Summary of notation and conventions