Effects of Surface Tension on the Stability of Dynamical Liquid-Vapor Interfaces

Abstract

The persistence of sharp and flat interfaces under multidimensional perturbations is investigated from a viewpoint analogous to Majda’s treatment of classical shocks. The main novelty here is that jump conditions for the free interfaces take into account surface tension. This means that, unlike classical jump conditions, they are non-homogeneous, containing first-order and second-order differentials of the front. A normal modes analysis shows that neutral modes may propagate along the front. In the standard setting, this would imply a weak stability result, involving energy estimates with loss of derivatives. In our case the lack of homogeneity of the underlying boundary value problem implies that neutral modes can only be of large enough wave length. Suitable frequency cut-offs then yield energy estimates without loss of derivatives – for the constant-coefficients linearized problem, as in the case of uniformly stable classical shocks.