A Free Boundary Problem for a Quasi-linear Degenerate Elliptic Equation: Regular Reflection of Weak Shocks

We prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance (UTSD) model for shock reflection by a wedge. In weak regular reflection, the state immediately behind the reflected shock is supersonic and constant. The flow becomes subsonic further downstream; the equation in self-similar coordinates is degenerate at the sonic line. The reflected shock becomes transonic and begins to curve there; its position is the solution to a free boundary problem for the degenerate equation. Using the Rankine-Hugoniot conditions along the reflected shock, we derive an evolution equation for the transonic shock, and an oblique derivative boundary condition at the unknown shock position. By regularizing the degenerate problem, we construct uniform bounds; we apply local compactness arguments to extract a limit that solves the problem. The solution is smooth in the interior and continuous up to the degenerate boundary. This work completes a stage in our program to construct self-similar solutions of two-dimensional Riemann problems. In a series of papers, we developed techniques for solving the degenerate elliptic equations that arise in self-similar reductions of hyperbolic conservation laws. In other papers, especially in joint work with Gary Lieberman, we developed techniques for solving free boundary problems of the type that arise from Rankine-Hugoniot relations. For the first time, in this paper, we combine these approaches and show that they are compatible. Although our construction is limited to a finite part of the unbounded subsonic region, it suggests that this approach has the potential to solve a variety of problems in weak shock reflection, including Mach and von Neumann reflection in the UTSD equation, and the analogous problems for the unsteady full potential equation. c © 2002 John Wiley & Sons, Inc. Communications on Pure and Applied Mathematics, Vol. LV, 0001–0022 (2002) c © 2002 John Wiley & Sons, Inc. 2 S. ČANIĆ, B. L. KEYFITZ, AND E. H. KIM