Shock Diffraction by Convex Cornered Wedges for the Nonlinear Wave System

TL;DR

This paper provides a rigorous mathematical analysis of shock diffraction by convex cornered wedges in nonlinear wave systems, establishing existence, regularity, and optimal regularity results for the associated boundary value problems.

Contribution

It develops the first global theory of existence and regularity for shock diffraction by convex wedges in nonlinear wave systems, including optimal regularity across sonic boundaries.

Findings

Proved existence of solutions for the shock diffraction problem.

Established $C^{0,1}$ regularity across the sonic boundary.

Developed new mathematical techniques applicable to similar PDE problems.

Abstract

We are concerned with rigorous mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow governed by the nonlinear wave system. This shock diffraction problem can be formulated as a boundary value problem for second-order nonlinear partial differential equations of mixed elliptic-hyperbolic type in an unbounded domain. It can be further reformulated as a free boundary problem for nonlinear degenerate elliptic equations of second order. We establish a first global theory of existence and regularity for this shock diffraction problem. In particular, we establish that the optimal regularity for the solution is $C^{0,1}$ across the degenerate sonic boundary. To achieve this, we develop several mathematical ideas and techniques, which are also useful for other related problems involving similar analytical difficulties.