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E-grāmata: Mathematics of Shock Reflection-Diffraction and von Neumann's Conjectures

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  • Formāts: 832 pages
  • Sērija : Annals of Mathematics Studies
  • Izdošanas datums: 27-Feb-2018
  • Izdevniecība: Princeton University Press
  • Valoda: eng
  • ISBN-13: 9781400885435
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Mathematics of Shock Reflection-Diffraction and von Neumann's ConjecturesPalielināt
  • Formāts: 832 pages
  • Sērija : Annals of Mathematics Studies
  • Izdošanas datums: 27-Feb-2018
  • Izdevniecība: Princeton University Press
  • Valoda: eng
  • ISBN-13: 9781400885435
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This book offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of original mathematical proofs of von Neumann's conjectures for potential flow, and a collection of related results and new techniques in the analysis of partial differential equations (PDEs), as well as a set of fundamental open problems for further development.

Shock waves are fundamental in nature. They are governed by the Euler equations or their variants, generally in the form of nonlinear conservation laws—PDEs of divergence form. When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, such as the structure and transition criteria of different configuration patterns, it is essential to establish the global existence, regularity, and structural stability of shock reflection-diffraction solutions. This involves dealing with several core difficulties in the analysis of nonlinear PDEs—mixed type, free boundaries, and corner singularities—that also arise in fundamental problems in diverse areas such as continuum mechanics, differential geometry, mathematical physics, and materials science. Presenting recently developed approaches and techniques, which will be useful for solving problems with similar difficulties, this book opens up new research opportunities.

Preface xi
I Shock Reflection-Diffraction, Nonlinear Conservation Laws of Mixed Type, and von Neumann's Conjectures 1(66)
1 Shock Reflection-Diffraction, Nonlinear Partial Differential Equations of Mixed Type, and Free Boundary Problems
3(13)
2 Mathematical Formulations and Main Theorems
16(21)
2.1 The potential flow equation
16(3)
2.2 Mathematical problems for shock reflection-diffraction
19(4)
2.3 Weak solutions of Problem 2.2.1 and Problem 2.2.3
23(1)
2.4 Structure of solutions: Regular reflection-diffraction configurations
24(3)
2.5 Existence of state (2) and continuous dependence on the parameters
27(1)
2.6 Von Neumann's conjectures, Problem 2.6.1 (free boundary problem), and main theorems
28(9)
3 Main Steps and Related Analysis in the Proofs of the Main Theorems
37(30)
3.1 Normal reflection
37(1)
3.2 Main steps and related analysis in the proof of the sonic conjecture
37(18)
3.3 Main steps and related analysis in the proof of the detachment conjecture
55(10)
3.4 Appendix: The method of continuity and fixed point theorems
65(2)
II Elliptic Theory and Related Analysis for Shock Reflection-Diffraction 67(546)
4 Relevant Results for Nonlinear Elliptic Equations of Second Order
69(147)
4.1 Notations: Holder norms and ellipticity
69(3)
4.2 Quasilinear uniformly elliptic equations
72(33)
4.3 Estimates for Lipschitz solutions of elliptic boundary value problems
105(37)
4.4 Comparison principle-for a mixed boundary value problem in a domain with corners
142(3)
4.5 Mixed boundary value problems in a domain with corners for uniformly elliptic equations
145(47)
4.6 Holder spaces with parabolic scaling
192(5)
4.7 Degenerate elliptic equations
197(10)
4.8 Uniformly elliptic equations in a curved triangle-shaped domain with one-point Dirichlet condition
207(9)
5 Basic Properties of the Self-Similar Potential Flow Equation
216(13)
5.1 Some basic facts and formulas for the potential flow equation
216(6)
5.2 Interior ellipticity principle for self-similar potential flow
222(5)
5.3 Ellipticity principle for self-similar potential flow with slip condition on the flat boundary
227(2)
III Proofs of the Main Theorems for the Sonic Conjecture and Related Analysis
229(384)
6 Uniform States and Normal Reflection
231(11)
6.1 Uniform states for self-similar potential flow
231(7)
6.2 Normal reflection and its uniqueness
238(1)
6.3 The self-similar potential flow equation in the coordinates flattening the sonic circle of a uniform state
239(3)
7 Local Theory and von Neumann's Conjectures
242(39)
7.1 Local regular reflection and state (2)
242(3)
7.2 Local theory of shock reflection for large-angle wedges
245(3)
7.3 The shock polar for steady potential flow and its properties
248(15)
7.4 Local theory for shock reflection: Existence of the weak and strong state (2) up to the detachment angle
263(10)
7.5 Basic properties of the weak state (2) and the definition of supersonic and subsonic wedge angles
273(6)
7.6 Von Neumann's sonic and detachment conjectures
279(2)
8 Admissible Solutions and Features of Problem 2.6.1
281(38)
8.1 Definition of admissible solutions
281(5)
8.2 Strict directional monotonicity for admissible solutions
286(19)
8.3 Appendix: Properties of solutions of Problem 2.6.1 for large-angle wedges
305(14)
9 Uniform Estimates for Admissible Solutions
319(63)
9.1 Bounds of the elliptic domain Omega and admissible solution phi in Omega
319(3)
9.2 Regularity of admissible solutions away from Gammashock unionGammasonic union {P3}
322(17)
9.3 Separation of Gammashock from Gammasym
339(2)
9.4 Lower bound for the distance between Gammashock and Gammawedge
341(13)
9.5 Uniform positive lower bound for the distance between Gammashock and the sonic circle of state (1)
354(15)
9.6 Uniform estimates of the ellipticity constant in Omega/Gammasonic
369(13)
10 Regularity of Admissible Solutions away from the Sonic Arc
382(14)
10.1 Gammashock as a graph in the radial directions with respect to state (1)
382(3)
10.2 Boundary conditions on Gammashock for admissible solutions
385(2)
10.3 Local estimates near Gammashock
387(2)
10.4 The critical angle and the distance between Gammashock and Gammawedge
389(1)
10.5 Regularity of admissible solutions away from Gammasonic
390(2)
10.6 Regularity of the limit of admissible solutions away from Gammasonic
392(4)
11 Regularity of Admissible Solutions near the Sonic Arc
396(44)
11.1 The equation near the sonic arc and structure of elliptic degeneracy
396(2)
11.2 Structure of the neighborhood of Gammasonic sonic Omega and estimates of (psi,Dpsi)
398(15)
11.3 Properties of the Rankine-Hugoniot condition on Gammashock near Gammasonic
413(8)
11.4 C2alpha-estimates in the scaled Holder norms near Gammasonic
421(10)
11.5 The reflected-diffracted shock is C2,alpha near P1
431(3)
11.6 Compactness of the set of admissible solutions
434(6)
12 Iteration Set and Solvability of the Iteration Problem
440(84)
12.1 Statement of the existence results
440(1)
12.2 Mapping to the iteration region
440(21)
12.3 Definition of the iteration set
461(8)
12.4 The equation for the iteration
469(16)
12.5 Assigning a boundary condition on the shock for the iteration
485(19)
12.6 Normal reflection, iteration set, and admissible solutions
504(1)
12.7 Solvability of the iteration problem and estimates of solutions
505(15)
12.8 Openness of the iteration set
520(4)
13 Iteration Map, Fixed Points, and Existence of Admissible Solutions up to the Sonic Angle
524(62)
13.1 Iteration map
524(4)
13.2 Continuity and compactness of the iteration map
528(2)
13.3 Normal reflection and the iteration map for Thetaw = pi/2
530(1)
13.4 Fixed points of the iteration map for Thetaw < pi/2 are admissible solutions
531(26)
13.5 Fixed points cannot lie on the boundary of the iteration set
557(2)
13.6 Proof of the existence of solutions up to the sonic angle or the critical angle
559(1)
13.7 Proof of Theorem 2.6.2: Existence of global solutions up to the sonic angle when u1 < c1
559(3)
13.8 Proof of Theorem 2.6.4: Existence of global solutions when u1 > c1
562(2)
13.9 Appendix: Extension of the functions in weighted spaces
564(22)
14 Optimal Regularity of Solutions near the Sonic Circle
586(29)
14.1 Regularity of solutions near the degenerate boundary for nonlinear degenerate elliptic equations of second order
586(13)
14.2 Optimal regularity of solutions across Gammasonic
599(14)
IV Subsonic Regular Reflection-Diffraction and Global Existence of Solutions up to the Detachment Angle 613(142)
15 Admissible Solutions and Uniform Estimates up to the Detachment Angle
615(14)
15.1 Definition of admissible solutions for the supersonic and subsonic reflections
615(2)
15.2 Basic estimates for admissible solutions up to the detachment angle
617(1)
15.3 Separation of Gammashock from Gammasym
618(1)
15.4 Lower bound for the distance between Gammashock and Gammawedge away from P0
618(3)
15.5 Uniform positive lower bound for the distance between Gammashock and the sonic circle of state (1)
621(1)
15.6 Uniform estimates of the ellipticity constant
622(3)
15.7 Regularity of admissible solutions away from Gammasonic
625(4)
16 Regularity of Admissible Solutions near the Sonic Arc and the Reflection Point
629(61)
16.1 Pointwise and gradient estimates near Gammasonic and the reflection point
629(4)
16.2 The Rankine-Hugoniot condition on Gammashock near Gammasonic and the reflection point
633(2)
16.3 A priori estimates near Gammasonic in the supersonic-away-from-sonic case
635(1)
16.4 A priori estimates near Gammasonic in the supersonic-near-sonic case
636(20)
16.5 A priori estimates near the reflection point in the subsonic-near- sonic case
656(9)
16.6 A priori estimates near the reflection point in the subsonic-away- from-sonic case
665(25)
17 Existence of Global Regular Reflection-Diffraction Solutions up to the Detachment Angle
690(67)
17.1 Statement of the existence results
690(1)
17.2 Mapping to the iteration region
690(17)
17.3 Iteration set
707(18)
17.4 Existence and estimates of solutions of the iteration problem
725(12)
17.5 Openness of the iteration set
737(4)
17.6 Iteration map and its properties
741(4)
17.7 Compactness of the iteration map
745(2)
17.8 Normal reflection and the iteration map for Thetaw = pi/2
747(1)
17.9 Fixed points of the iteration map for Thetaw < pi/2 admissible solutions
747(5)
17.10 Fixed points cannot lie on the boundary of the iteration set
752(1)
17.11 Proof of the existence of solutions up to the critical angle
753(1)
17.12 Proof of Theorem 2.6.6: Existence of global solutions up to the detachment angle when u1 < or = to c1
753(1)
17.13 Proof of Theorem 2.6.8: Existence of global solutions when u1 > c1
753(2)
V Connections and Open Problems 755(39)
18 The Full Euler Equations and the Potential Flow Equation
757(28)
18.1 The full Euler equations
757(4)
18.2 Mathematical formulation I: Initial-boundary value problem
761(1)
18.3 Mathematical formulation II: Boundary value problem
762(6)
18.4 Normal reflection
768(1)
18.5 Local theory for regular reflection near the reflection point
769(8)
18.6 Von Neumann's conjectures
777(4)
18.7 Connections with the potential flow equation
781(4)
19 Shock Reflection-Diffraction and New Mathematical Challenges
785(9)
19.1 Mathematical theory for multidimensional conservation laws
785(3)
19.2 Nonlinear partial differential equations of mixed elliptic-hyperbolic type
788(2)
19.3 Free boundary problems and techniques
790(1)
19.4 Numerical methods for multidimensional conservation laws
791(3)
Bibliography 794(21)
Index 815
Gui-Qiang G. Chen is the Statutory Professor in the Analysis of Partial Differential Equations at the Mathematical Institute of the University of Oxford, where he is also professorial fellow at Keble College. Mikhail Feldman is professor of mathematics at the University of Wisconsin-Madison.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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