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| Product Code: | SURV/214 |
| List Price: | $129.00 |
| MAA Member Price: | $116.10 |
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| eBook ISBN: | 978-1-4704-3564-6 |
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| Hardcover ISBN: | 978-1-4704-2857-0 |
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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 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.
ReadershipGraduate students and researchers interested in nonlinear PDE and shock formation.
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Table of Contents
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Chapters
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Chapter 1. Introduction
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Chapter 2. Overview of the two main theorems
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Chapter 3. Initial data, basic geometric constructions, and the future null condition failure factor
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Chapter 4. Transport equations for the Eikonal function quantities
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Chapter 5. Connection coefficients of the rescaled frames and geometric decompositions of the wave operator
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Chapter 6. Construction of the rotation vectorfields and their basic properties
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Chapter 7. Definition of the commutation vectorfields and deformation tensor calculations
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Chapter 8. Geometric operator commutator formulas and schematic notation for repeated differentiation
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Chapter 9. The structure of the wave equation inhomogeneous terms after one commutation
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Chapter 10. Energy and cone flux definitions and the fundamental divergence identities
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Chapter 11. Avoiding derivative loss and other difficulties via modified quantities
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Chapter 12. Small data, sup-norm bootstrap assumptions, and first pointwise estimates
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Chapter 13. Sharp estimates for the inverse foliation density
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Chapter 14. Square integral coerciveness and the fundamental square-integral-controlling quantities
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Chapter 15. Top-order pointwise commutator estimates involving the Eikonal function
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Chapter 16. Pointwise estimates for the easy error integrands and identification of the difficult error integrands corresponding to the commuted wave equation
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Chapter 17. Pointwise estimates for the difficult error integrands corresponding to the commuted wave equation
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Chapter 18. Elliptic estimates and Sobolev embedding on the spheres
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Chapter 19. Square integral estimates for the Eikonal function quantities that do not rely on modified quantities
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Chapter 20. A priori estimates for the fundamental square-integral-controlling quantities
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Chapter 21. Local well-posedness and continuation criteria
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Chapter 22. The sharp classical lifespan theorem
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Chapter 23. Proof of shock formation for nearly spherically symmetric data
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Appendix A. Extension of the results to a class of non-covariant wave equations
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Appendix B. Summary of notation and conventions
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Additional Material
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Reviews
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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
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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 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.
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, 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
-
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
