Stability of Subsonic Planar Phase Boundaries in a van der Waals Fluid

Abstract:

We are concerned with the structural stability of dynamic phase changes occurring across sharp interfaces in a multidimensional van der Waals fluid. Such phase transitions can be viewed as propagating discontinuities. However, they are usually subsonic, and thus undercompressive. The lacking information lies in an additional jump condition, which may be derived from the viscosity-capillarity criterion. This condition is rather simple in the case of reversible phase transitions, since it reduces to a generalized equal area rule. In a previous work, I proved that reversible planar phase boundaries are weakly linearly stable, in the sense introduced by Majda for shock fronts. This means that they satisfy a generalized Lopatinsky condition but not a uniform one. The aim of this paper is to point out the influence of viscosity on the stability analysis, in order to deal with the more realistic case of dissipative phase transitions. The main difficulty lies in the additional jump condition, which is no longer explicit and depends on the (unknown) internal structure of the interface. We overcome it by using bifurcation arguments on the nondimensional parameter measuring the competition between viscosity and capillarity. We show by perturbation that the positivity of this parameter stabilizes the phase transitions. As a conclusion, we find that dissipative planar phase boundaries are uniformly linearly stable, in the sense of the uniform Lopatinsky condition.