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.

In English.

Online resource; title from PDF title page (publisher's Web site, viewed Feb. 24, 2017).

Includes bibliographical references and index.

I. Shock reflection-diffraction, nonlinear conservation laws of mixed type, and von Neumann's conjectures -- Shock reflection-diffraction, nonlinear partial differential equations of mixed type, and free boundary problems -- Mathematical formulations and main theorems -- Main steps and related analysis in the proofs of the main theorems -- II. Elliptic theory and related analysis for shock reflection-diffraction -- Relevant results for nonlinear elliptic equations of second order -- Basic properties of the self-similar potential flow equation -- III. Proofs of the main theorems for the sonic conjecture and related analysis -- Uniform states and normal reflection -- Local theory and von Neumann's conjectures -- Admissible solutions and features of problem 2.6.1 -- Uniform estimates for admissible solutions -- Regularity of admissible solutions away from the sonic arc -- Regularity of admissible solutions near the sonic arc -- Iteration set and solvability of the iteration problem -- Iteration map, fixed points, and existence of admissible solutions up to the sonic angle -- Optimal regularity of solutions near the sonic circle -- IV. Subsonic regular reflection-diffraction and global existence of solutions up to the detachment angle -- Regularity of admissible solutions near the sonic arc and the reflection point -- Existence of global regular reflection-diffraction solutions up to the detachment angle -- V. Connections and open problems -- The full Euler equation and the potential flow equation -- Shock reflection-diffraction and new mathematical challenges.

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