Singular limits in phase dynamics with physical viscosity and capillarity

Abstract

Following pioneering work by Fan and Slemrod who studied the effect of artificial viscosity terms, we consider the system of conservation laws arising in liquid-vapor phase dynamics with physical viscosity and capillarity effects taken into account. Following Dafermos we consider self-similar solutions to the Riemann problem and establish uniform total variation bounds, allowing us to deduce new existence results. Our analysis cover both the hyperbolic and the hyperbolic-elliptic regimes and apply to arbitrarily large Riemann data. The proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system. Strong L¹ convergence to a weak solution of bounded variation is established in the hyperbolic regime, while in the hyperbolicelliptic regime a stationary singularity near the axis separating the two wave fans, or more generally an almost-stationary oscillating wave pattern (of thickness depending upon the capillarityviscosity ratio) are observed which prevent the solution to have globally bounded variation.