Stability of diffuse fluidic phase boundaries

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In continuum models for the dynamics of two-phase compressible fluids, microforces between the phases are expressed as stress tensors that depend on the gradient of an order parameter, e. g., the mixture ratio of the phases. The purpose of this project consists in capturing the stability of reference configurations that realise both phases and contain corresponding interfaces, such as a droplet of one phase surrounded by a region filled with the other phase. Besides classical Korteweg models, the project will primarily be devoted to systems with phase field, as well as finally, with different techniques, to sharp-interface models. Regarding Korteweg and phase-field models, the stability question poses itself as one on the long-time asymptotics of solutions near the reference solutions; regarding sharp-interface models it concerns the finite-time persistence of discontinuity fronts.