Description
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.
Chapter
4.4 Comparison principle for a mixed boundary value problem in a domain with corners
4.5 Mixed boundary value problems in a domain with corners for uniformly elliptic equations
4.6 Hölder spaces with parabolic scaling
4.7 Degenerate elliptic equations
4.8 Uniformly elliptic equations in a curved triangle-shaped domain with one-point Dirichlet condition
5 Basic Properties of the Self-Similar Potential Flow Equation
5.1 Some basic facts and formulas for the potential flow equation
5.2 Interior ellipticity principle for self-similar potential flow
5.3 Ellipticity principle for self-similar potential flow with slip condition on the flat boundary
III Proofs of the Main Theorems for the Sonic Conjecture and Related Analysis
6 Uniform States and Normal Reflection
6.1 Uniform states for self-similar potential flow
6.2 Normal reflection and its uniqueness
6.3 The self-similar potential flow equation in the coordinates flattening the sonic circle of a uniform state
7 Local Theory and von Neumann’s Conjectures
7.1 Local regular reflection and state (2)
7.2 Local theory of shock reflection for large-angle wedges
7.3 The shock polar for steady potential flow and its properties
7.4 Local theory for shock reflection: Existence of the weak and strong state (2) up to the detachment angle
7.5 Basic properties of the weak state (2) and the definition of supersonic and subsonic wedge angles
7.6 Von Neumann’s sonic and detachment conjectures
8 Admissible Solutions and Features of Problem 2.6.1
8.1 Definition of admissible solutions
8.2 Strict directional monotonicity for admissible solutions
8.3 Appendix: Properties of solutions of Problem 2.6.1 for large-angle wedges
9 Uniform Estimates for Admissible Solutions
9.1 Bounds of the elliptic domain Ω and admissible solution φ in Ω
9.2 Regularity of admissible solutions away from Г shock ∪Гsonic ∪{P3}
9.3 Separation of Гshock from Гsym
9.4 Lower bound for the distance between Гshock and Гwedge
9.5 Uniform positive lower bound for the distance between Гshock and the sonic circle of state (1)
9.6 Uniform estimates of the ellipticity constant in Ω\Γsonic
10 Regularity of Admissible Solutions away from the Sonic Arc
10.1 Γshock as a graph in the radial directions with respect to state (1)
10.2 Boundary conditions on Γshock for admissible solutions
10.3 Local estimates near Γshock
10.4 The critical angle and the distance between Γshock and Γwedge
10.5 Regularity of admissible solutions away from Γsonic
10.6 Regularity of the limit of admissible solutions away from Γsonic
11 Regularity of Admissible Solutions near the Sonic Arc
11.1 The equation near the sonic arc and structure of elliptic degeneracy
11.2 Structure of the neighborhood of Γsonic in Ω and estimates of (Ψ, DΨ)
11.3 Properties of the Rankine-Hugoniot condition on Γshock near Γsonic
11.4 C2,α –estimates in the scaled Hölder norms near Γsonic
11.5 The reflected-diffracted shock is C2,α near P1
11.6 Compactness of the set of admissible solutions
12 Iteration Set and Solvability of the Iteration Problem
12.1 Statement of the existence results
12.2 Mapping to the iteration region
12.3 Definition of the iteration set
12.4 The equation for the iteration
12.5 Assigning a boundary condition on the shock for the iteration
12.6 Normal reflection, iteration set, and admissible solutions
12.7 Solvability of the iteration problem and estimates of solutions
12.8 Openness of the iteration set
13 Iteration Map, Fixed Points, and Existence of Admissible Solutions up to the Sonic Angle
13.2 Continuity and compactness of the iteration map
13.3 Normal reflection and the iteration map for θw = π/2
13.4 Fixed points of the iteration map for θw < π/2 are admissible solutions
13.5 Fixed points cannot lie on the boundary of the iteration set
13.6 Proof of the existence of solutions up to the sonic angle or the critical angle
13.7 Proof of Theorem 2.6.2: Existence of global solutions up to the sonic angle when u1 ≤ c1
13.8 Proof of Theorem 2.6.4: Existence of global solutions when u1 > c1
13.9 Appendix: Extension of the functions in weighted spaces
14 Optimal Regularity of Solutions near the Sonic Circle
14.1 Regularity of solutions near the degenerate boundary for nonlinear degenerate elliptic equations of second order
14.2 Optimal regularity of solutions across Γsonic
IV Subsonic Regular Reflection-Diffraction and Global Existence of Solutions up to the Detachment Angle
15 Admissible Solutions and Uniform Estimates up to the Detachment Angle
15.1 Definition of admissible solutions for the supersonic and subsonic reflections
15.2 Basic estimates for admissible solutions up to the detachment angle
15.3 Separation of Γshock from Γsym
15.4 Lower bound for the distance between Γshock and Γwedge away from P0
15.5 Uniform positive lower bound for the distance between Γshock and the sonic circle of state (1)
15.6 Uniform estimates of the ellipticity constant
15.7 Regularity of admissible solutions away from Γsonic
16 Regularity of Admissible Solutions near the Sonic Arc and the Reflection Point
16.1 Pointwise and gradient estimates near Γsonic and the reflection point
16.2 The Rankine-Hugoniot condition on Γshock near Γsonic and the reflection point
16.3 A priori estimates near Γsonic in the supersonic-away-from-sonic case
16.4 A priori estimates near Γsonic in the supersonic-near-sonic case
16.5 A priori estimates near the reflection point in the subsonic-near-sonic case
16.6 A priori estimates near the reflection point in the subsonic-away-from-sonic case
17 Existence of Global Regular Reflection-Diffraction Solutions up to the Detachment Angle
17.1 Statement of the existence results
17.2 Mapping to the iteration region
17.4 Existence and estimates of solutions of the iteration problem
17.5 Openness of the iteration set
17.6 Iteration map and its properties
17.7 Compactness of the iteration map
17.8 Normal reflection and the iteration map for θw = π/2
17.10 Fixed points cannot lie on the boundary of the iteration set
17.11 Proof of the existence of solutions up to the critical angle
17.12 Proof of Theorem 2.6.6: Existence of global solutions up to the detachment angle when u1 ≤ c1
17.13 Proof of Theorem 2.6.8: Existence of global solutions when u1 > c1
V Connections and Open Problems
18 The Full Euler Equations and the Potential Flow Equation
18.1 The full Euler equations
18.2 Mathematical formulation I: Initial-boundary value problem
18.3 Mathematical formulation II: Boundary value problem
18.5 Local theory for regular reflection near the reflection point
18.6 Von Neumann’s conjectures
18.7 Connections with the potential flow equation
19 Shock Reflection-Diffraction and New Mathematical Challenges
19.1 Mathematical theory for multidimensional conservation laws
19.2 Nonlinear partial differential equations of mixed elliptic-hyperbolic type
19.3 Free boundary problems and techniques
19.4 Numerical methods for multidimensional conservation laws
The Mathematics of Shock Reflection-Diffraction and von Neumann's Conjectures
( Annals of Mathematics Studies )
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