Existence of undercompressive weak solutions to the Euler equations

Abstract

AbstractWe prove the local existence of an undercompressive hydrodynamical shock to the isothermal Euler equations with a non‐monotone pressure function. This nonlinear problem will be formulated as an abstract hyperbolic initial boundary value problem. The existence of a weak solution to a linearized version of the problem is shown with the use of Riesz theorem. Using the results of the linear system yields by an iteration scheme (local in time) well‐posedness of the nonlinear problem. The system of equations is obtained by modeling the motion of sharp liquid‐vapor interfaces including configurational forces as well as surface tension. The considered non‐viscous Van der Waals fluid is compressible and allows phase transitions. The propagating phase boundary is controlled by a modified version of the Rankine‐Hugoniot jump condition obtained by the Young‐Laplace law. Entropy dissipation at the interface is precisely described by a kinetic relation. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)