WEAK REGULAR REFLECTION FOR THE NONLINEAR WAVE SYSTEM

Abstract

We prove local existence of a solution to a Riemann problem for the two-dimensional nonlinear wave system using the approach by Čanić, Keyfitz, Kim and Lieberman. We consider initial data resulting in weak regular shock reflection. By writing the problem in self-similar coordinates, we obtain a mixed type system and a free boundary problem for the subsonic flow and the position of the reflected shock. We reformulate this problem using a second order equation for density with mixed boundary conditions and an ordinary differential equation describing the location of the reflected shock. The main difficulty in the study of this second order problem for density is that the operator is degenerate elliptic along the sonic circle. We regularize and modify the operator, and we show existence of solutions to the regularized problems. We prove that the sequence of solutions to the regularized problems is precompact, and, using uniform local ellipticity, covering and diagonalization techniques, we obtain a local solution to the original problem.