Existence and spectral stability analysis of viscous-dispersive shock profiles for isentropic compressible fluids of Korteweg type

TL;DR

This paper proves the existence and spectral stability of viscous-dispersive shock profiles in a general isentropic compressible fluid model with nonlinear viscosity and capillarity, including stability conditions for small shocks.

Contribution

It establishes the existence, uniqueness, and spectral stability of viscous-dispersive shock profiles in a highly general setting with nonlinear coefficients.

Findings

Traveling wave solutions connect constant states satisfying Rankine-Hugoniot and Lax conditions.

Essential spectrum of the linearized operator is stable regardless of shock strength.

Point spectrum stability is proven for small shock amplitudes under certain conditions.

Abstract

The system describing the dynamics of a compressible isentropic fluid exhibiting viscosity and internal capillarity in one space dimension and in Lagrangian coordinates, is considered. It is assumed that the viscosity and the capillarity coefficients are nonlinear smooth, positive functions of the specific volume, making the system the most general case possible. It is shown, under very general circumstances, that the system admits traveling wave solutions connecting two constant states and traveling with a certain speed that satisfy the classical Rankine-Hugoniot and Lax entropy conditions, and hence called viscous-dispersive shock profiles. These traveling wave solutions are unique up to translations and have arbitrary amplitude. The spectral stability of such viscous-dispersive profiles is also considered. It is shown that the essential spectrum of the linearized operator around the profile (posed on an appropriate energy space) is stable, independently of the shock strength. With the aid of energy estimates, it is also proved that the point spectrum is also stable, provided that the shock amplitude is sufficiently small and a structural condition on the inviscid shock is fulfilled.