Free Boundary Problems for Nonlinear Wave Systems: Mach Stems for Interacting Shocks

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We study a family of two-dimensional Riemann problems for compressible flow modeled by the nonlinear wave system. The initial constant states are separated by two jump discontinuities, $x = \pm \kappa_a y$, which develop into two interacting shock waves. We consider shock angles in a range where regular reflection is not possible. The solution is symmetric about the y-axis and on each side of the y-axis consists of an incident shock, a reflected compression wave, and a Mach stem. This has a clear analogy with the problem of shock reflection by a ramp. It is well known that no triple point structure exists in which incident, reflected, and Mach stem shocks meet at a point. In this paper, we model the reflected wave by a continuous function with a singularity in the derivative. This fails to be a weak solution across the sonic line. We show that a solution to the free boundary problem for the Mach stem exists, and we conjecture that the global solution can be completed by the construction of a reflected shock, by a similar free boundary technique. The point of our paper is the capability to deal analytically with a Mach stem by solving a free boundary problem. The difficulties associated with the analysis of solutions containing Mach stems include (1) loss of obliqueness in the derivative boundary condition corresponding to the jump conditions across the Mach stem, and (2) loss of ellipticity at the formation point of the Mach stem. We use barrier functions to show that for sufficiently large values of $\kappa_a$ the subsonic solution is continuous up to the sonic line at the Mach stem.