Existence of undercompressive weak solutions to the Euler equations

We 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)