Existence of Undercompressive Shock Wave Solutions to the Euler Equations

Abstract

The sharp-interface dynamics of compressible inviscid liquid–vapor flows with constant temperature can be described by the isothermal Euler equations using a non-monotone pressure function. The motion of the discontinuous phase boundaries is constrained besides mass conservation by the dynamical Young–Laplace law and the prescribed entropy dissipation rate. We consider the initial value problem for a two-phase configuration in multiple space dimensions, such that the smooth bulk state data are separated by a subsonic phase boundary which can be understood as a non-Laxian, undercompressive shock wave. It is proven that the associated free boundary problem admits a piecewise classical solution for short times. This strongly nonlinear problem will be formulated as an abstract combination of a hyperbolic initial boundary value problem for the hydromechanical unknowns and a parabolic evolution equation for the front position. By an iteration scheme (local-in-time), the well-posedness of the nonlinear problem is established.