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Hardcover ISBN:  9781470428570 
Product Code:  SURV/214 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470435646 
Product Code:  SURV/214.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470428570 
eBook ISBN:  9781470435646 
Product Code:  SURV/214.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 214; 2016; 515 ppMSC: Primary 35
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 shockformation 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 vorticityfree 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 blowup. These results yield a sharp converse of the fundamental result of Christodoulou and Klainerman, who showed that smalldata 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.
ReadershipGraduate students and researchers interested in nonlinear PDE and shock formation.

Table of Contents

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, supnorm bootstrap assumptions, and first pointwise estimates

Chapter 13. Sharp estimates for the inverse foliation density

Chapter 14. Square integral coerciveness and the fundamental squareintegralcontrolling quantities

Chapter 15. Toporder 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 squareintegralcontrolling quantities

Chapter 21. Local wellposedness 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 noncovariant wave equations

Appendix B. Summary of notation and conventions


Additional Material

Reviews

This is a well written monograph and is a welcome addition to the literature on threedimensional quasilinear wave equations.
Vishnu Dutt Sharma, Zentralblatt MATH


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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 shockformation 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 vorticityfree 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 blowup. These results yield a sharp converse of the fundamental result of Christodoulou and Klainerman, who showed that smalldata 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.
Graduate students and researchers interested in nonlinear PDE and shock formation.

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, supnorm bootstrap assumptions, and first pointwise estimates

Chapter 13. Sharp estimates for the inverse foliation density

Chapter 14. Square integral coerciveness and the fundamental squareintegralcontrolling quantities

Chapter 15. Toporder 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 squareintegralcontrolling quantities

Chapter 21. Local wellposedness 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 noncovariant wave equations

Appendix B. Summary of notation and conventions

This is a well written monograph and is a welcome addition to the literature on threedimensional quasilinear wave equations.
Vishnu Dutt Sharma, Zentralblatt MATH