Muutke küpsiste eelistusi

Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations [Kõva köide]

4.00/5 (2 hinnangut Goodreads-ist)
  • Formaat: Hardback, 518 pages, kõrgus x laius: 229x152 mm, kaal: 1085 g
  • Sari: Mathematical Surveys and Monographs
  • Ilmumisaeg: 30-Mar-2017
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470428571
  • ISBN-13: 9781470428570
Teised raamatud teemal:
Shock Formation in Small-Data Solutions to 3D Quasilinear Wave EquationsSuurem pilt
  • Formaat: Hardback, 518 pages, kõrgus x laius: 229x152 mm, kaal: 1085 g
  • Sari: Mathematical Surveys and Monographs
  • Ilmumisaeg: 30-Mar-2017
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470428571
  • ISBN-13: 9781470428570
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781470428570.html
Märksõnad:
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 Alinhacs 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 Alinhacs results and gave a complete description of shock formation.

In this monograph, Christodoulous 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-datasolutions 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.
Preface xv
Acknowledgments xxiii
Chapter 1 Introduction 1(30)
1.1 Shock formation in one spatial dimension
1(10)
1.2 New aspects in more than one spatial dimension
11(20)
Chapter 2 Overview of the Two Main Theorems 31(50)
2.1 First description of the two theorems
33(2)
2.2 The basic structure of the equations
35(3)
2.3 The structure of the equation relative to rectangular coordinates
38(1)
2.4 The (classic) null condition
38(3)
2.5 Basic geometric constructions
41(3)
2.6 The resealed frame and dispersive sup-norm estimates
44(2)
2.7 The basic structure of the coupled system and sup-norm estimates for the eikonal function quantities
46(1)
2.8 Lower bounds for the resealed radial derivative of the solution in the case of shock formation
47(1)
2.9 The main ideas behind the vanishing of the inverse foliation density
48(1)
2.10 The role of Theorem 22.1 in justifying the heuristics
49(12)
2.11 Comparison with related work
61(15)
2.12 Outline of the monograph
76(2)
2.13 Suggestions on how to read the monograph
78(3)
Chapter 3 Initial Data, Basic Geometric Constructions, and the Future Null Condition Failure Factor 81(26)
3.1 Initial data
81(1)
3.2 The eikonal function and the geometric radial variable
82(1)
3.3 First fundamental forms and Levi-Civita connections
83(1)
3.4 Frame vectorfields and the inverse foliation density
84(3)
3.5 Geometric coordinates
87(2)
3.6 Frames
89(1)
3.7 The future null condition failure factor
90(1)
3.8 Contraction and component notation
90(1)
3.9 Projection operators and tensors along submanifolds
91(2)
3.10 Expressions for the metrics and volume form factors
93(2)
3.11 The trace and trace-free parts of tensors
95(1)
3.12 Angular differential
96(1)
3.13 Musical notation
97(1)
3.14 Pointwise norms
97(1)
3.15 Lie derivatives and projected Lie derivatives
97(2)
3.16 Second fundamental forms
99(1)
3.17 Frame components, relative to the nonrescaled frame, of the derivatives of the metric with respect to the solution
100(1)
3.18 The change of variables map
101(1)
3.19 Area forms, volume forms, and norms
102(2)
3.20 Schematic notation
104(3)
Chapter 4 Transport Equations for the Eikonal Function Quantities 107(10)
4.1 Re-centered variables and the eikonal function quantities
107(1)
4.2 Covariant derivatives and Christoffel symbols relative to the rectangular coordinates
108(1)
4.3 Transport equation for the inverse foliation density
108(1)
4.4 Transport equations for the rectangular components of the frame vectorfields
109(2)
4.5 An expression for the re-centered null second fundamental form in terms of other quantities
111(1)
4.6 Identities involving deformation tensors and Lie derivatives
112(5)
Chapter 5 Connection Coefficients of the Rescaled Frames and Geometric Decompositions of the Wave Operator 117(6)
5.1 Connection coefficients of the rescaled frame
117(2)
5.2 Connection coefficients of the rescaled null frame
119(1)
5.3 Frame decomposition of the inverse-foliation-density-weighted wave operator
119(4)
Chapter 6 Construction of the Rotation Vectorfields and Their Basic Properties 123(4)
6.1 Construction of the rotation vectorfields
123(1)
6.2 Basic properties of the rotation vectorfields
123(4)
Chapter 7 Definition of the Commutation Vectorfields and Deformation Tensor Calculations 127(8)
7.1 The commutation vectorfields
127(1)
7.2 Deformation tensor calculations
128(7)
Chapter 8 Geometric Operator Commutator Formulas and Schematic Notation for Repeated Differentiation 135(8)
8.1 Definitions of various differential operators
135(1)
8.2 Operator commutator identities
136(4)
8.3 Notation for repeated differentiation
140(3)
Chapter 9 The Structure of the Wave Equation Inhomogeneous Terms After One Commutation 143(8)
9.1 Preliminary calculations
143(3)
9.2 Frame decomposition of the commutation current
146(5)
Chapter 10 Energy and Cone Flux Definitions and the Fundamental Divergence Identities 151(16)
10.1 Preliminary calculations
151(5)
10.2 The energy-cone flux integral identities
156(4)
10.3 Integration by parts identities for the top-order square integral estimates
160(4)
10.4 Error integrands arising from the deformation tensors of the multiplier vectorfields
164(3)
Chapter 11 Avoiding Derivative Loss and Other Difficulties via Modified Quantities 167(14)
11.1 Preliminary structural identities
168(5)
11.2 Full modification of the trace of the re-centered null second fundamental form
173(4)
11.3 Partial modification of the trace of the re-centered null second fundamental form
177(1)
11.4 Partial modification of the angular gradient of the inverse foliation density
178(3)
Chapter 12 Small Data, Sup-Norm Bootstrap Assumptions, and First Pointwise Estimates 181(64)
12.1 Restricting the analysis to solutions of the evolution equations
181(1)
12.2 Small data
182(1)
12.3 Fundamental positivity bootstrap assumption for the inverse foliation density
182(1)
12.4 Sup-norm bootstrap assumptions
182(2)
12.5 Basic estimates for the geometric radial variable
184(1)
12.6 Basic estimates for the rectangular spatial coordinate functions
184(1)
12.7 Estimates for the rectangular components of the metrics and the spherical projection tensorfield
185(2)
12.8 The behavior of quantities along the initial data hypersurface
187(3)
12.9 Estimates for the derivatives of rectangular components of various vectorfields and the radial component of the Euclidean rotations
190(2)
12.10 Estimates for the rectangular components of the metric dual of the unit-length radial vectorfield
192(1)
12.11 Precise pointwise estimates for the rotation vectorfields
193(4)
12.12 Precise pointwise differential operator comparison estimates
197(2)
12.13 Useful estimates for avoiding detailed commutators
199(1)
12.14 Estimates for the derivatives of the angular differential of the rectangular spatial coordinate functions
200(1)
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
200(1)
12.16 Crude pointwise estimates for the Lie derivatives of the angular components of the deformation tensors
201(2)
12.17 Two additional crude differential operator comparison estimates
203(1)
12.18 Pointwise estimates for the derivatives of the re-centered null second fundamental form in terms of other quantities
204(2)
12.19 Pointwise estimates for the Lie derivatives of the rotation vectorfields
206(1)
12.20 Pointwise estimates for the angular one-forms and vectorfields corresponding to the commutation vectorfield deformation tensors
207(2)
12.21 Preliminary Lie derivative commutator estimates
209(1)
12.22 Commutator estimates for vectorfields acting on functions and spherical covariant tensorfields
210(5)
12.23 Commutator estimates for vectorfields acting on the covariant angular derivative of a spherical tensorfield
215(1)
12.24 Commutator estimates for vectorfields acting on the angular Hessian of a function
215(3)
12.25 Commutator estimates involving the trace and trace-free parts
218(5)
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
223(2)
12.27 Improvement of the auxiliary bootstrap assumptions
225(5)
12.28 Sharp pointwise estimates for a frame component of the derivative of the metric with respect to the solution
230(1)
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
231(2)
12.30 Estimates related to integrals over the spheres
233(3)
12.31 Faster than expected decay for certain wave-variable-related quantities
236(3)
12.32 Pointwise estimates for the vectorfield Xi
239(2)
12.33 Estimates for the components of the commutation vectorfields relative to the geometric coordinates
241(2)
12.34 Estimates for the rectangular spatial derivatives of the eikonal function
243(2)
Chapter 13 Sharp Estimates for the Inverse Foliation Density 245(32)
13.1 Basic ingredients in the analysis
245(4)
13.2 Sharp pointwise estimates for the inverse foliation density
249(17)
13.3 Fundamental estimates for time integrals involving the foliation density
266(11)
Chapter 14 Square Integral Coerciveness and the Fundamental Square- Integral-Controlling Quantities 277(6)
14.1 Coerciveness of the energies and cone fluxes
277(2)
14.2 Definitions of the fundamental square-integral-controlling quantities
279(1)
14.3 Coerciveness of the fundamental square-integral-controlling quantities
280(3)
Chapter 15 Top-Order Pointwise Commutator Estimates Involving the Eikonal Function 283(12)
15.1 Top-order pointwise commutator estimates connecting the angular Hessian of the inverse foliation density to the radial Lie derivative of the re-centered null second fundamental form
283(7)
15.2 Top-order pointwise commutator estimates corresponding to the spherical Codazzi equations
290(5)
Chapter 16 Pointwise Estimates for the Easy Error Integrands and Identification of the Difficult Error Integrands Corresponding to the Commuted Wave Equation 295(32)
16.1 Preliminary analysis and the definition of harmless terms
295(4)
16.2 The important terms in the top-order derivatives of the deformation tensors of the commutation vectorfields
299(10)
16.3 Crude pointwise estimates for the below-top-order derivatives of the deformation tensors of the commutation vectorfields
309(3)
16.4 Pointwise estimates for the top-order derivatives of the outgoing null derivative of the commutation vectorfield deformation tensors
312(1)
16.5 Proof of Proposition 16.4
313(6)
16.6 Proof of Corollary 16.5
319(1)
16.7 Pointwise estimates for the error integrands involving the deformation tensors of the multiplier vectorfields
320(4)
16.8 Pointwise estimates needed to close the elliptic estimates
324(3)
Chapter 17 Pointwise Estimates for the Difficult Error Integrands Corresponding to the Commuted Wave Equation 327(26)
17.1 Preliminary pointwise estimates for the derivatives of the inhomogeneous terms in the transport equations for the fully modified quantities
327(5)
17.2 Preliminary pointwise estimates for the derivatives of the inhomogeneous terms in the transport equations for the partially modified quantities
332(4)
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
336(4)
17.4 Pointwise estimates for the difficult error integrands requiring full modification
340(9)
17.5 Pointwise estimates for the difficult error integrands requiring partial modification
349(4)
Chapter 18 Elliptic Estimates and Sobolev Embedding on the Spheres 353(8)
18.1 Elliptic estimates
353(5)
18.2 Sobolev embedding
358(3)
Chapter 19 Square Integral Estimates for the Eikonal Function Quantities that Do Not Rely on Modified Quantities 361(4)
19.1 Square integral estimates for the eikonal function quantities that do not rely on modified quantities
361(4)
Chapter 20 A Priori Estimates for the Fundamental Square-Integral- Controlling Quantities 365(82)
20.1 Bootstrap assumptions for the fundamental square-integral- controlling quantities
365(1)
20.2 Statement of the two main propositions and the fundamental Gronwall lemma
366(6)
20.3 Estimates for all but the most difficult error integrals
372(16)
20.4 Difficult top-order error integral estimates
388(19)
20.5 Proof of Lemma 20.20
407(3)
20.6 Proof of Lemma 20.25
410(11)
20.7 Proof of Lemma 20.26
421(5)
20.8 Proof of Proposition 20.8
426(3)
20.9 Proof of Proposition 20.9
429(2)
20.10 Proof of Lemma 20.10
431(16)
Chapter 21 Local Well-Posedness and Continuation Criteria 447(6)
21.1 Local well-posedness and continuation criteria
447(6)
Chapter 22 The Sharp Classical Lifespan Theorem 453(14)
22.1 The sharp classical lifespan theorem
453(10)
22.2 More precise control over angular derivatives
463(4)
Chapter 23 Proof of Shock Formation for Nearly Spherically Symmetric Data 467(12)
23.1 Preliminary pointwise estimates based on approximate transport equations
468(2)
23.2 Existence of small, stable, shock-generating data
470(7)
23.3 Proof of shock formation for small, nearly spherically symmetric data
477(2)
Appendix A: Extension of the Results to a Class of Non-Covariant Wave Equations 479(10)
A.1 From the scalar quasilinear wave equation to the equivalent system of covariant wave equations
479(5)
A.2 The main new estimate needed at the top order
484(5)
Appendix B: Summary of Notation and Conventions 489(14)
B.1 Coordinates
489(1)
B.2 Indices
490(1)
B.3 Constants
490(1)
B.4 Spacetime subsets
490(1)
B.5 Metrics, musical notation, and inner products
491(1)
B.6 Eikonal function quantities
492(1)
B.7 Additional tensorfields related to the connection coefficients
492(1)
B.8 Frame vectorfields and the timelike unit normal to the constant-time hypersurfaces
493(1)
B.9 Contraction and component notation
494(1)
B.10 Projection operators and frame components
494(1)
B.11 Tensor products, traces, and contractions
495(1)
B.12 The size of the data and the bootstrap parameter
495(1)
B.13 Commutation vectorfields
495(1)
B.14 Differential operators and commutator notation
496(1)
B.15 Floor and ceiling functions and repeated differentiation
497(1)
B.16 Area and volume forms
498(1)
B.17 Norms
498(1)
B.18 Energy-momentum tensorfield, multiplier vectorfields, and compatible currents
499(1)
B.19 Square-integral-controlling quantities
499(1)
B.20 Modified quantities
500(1)
B.21 Curvature tensors
500(1)
B.22 Omission of the independent variables in some expressions
501(1)
B.23 Data and functions relevant for the proof of shock formation
501(2)
Bibliography 503(4)
Index 507
Jared Speck, Massachusetts Institute of Technology, Cambridge, MA, USA.