Stability of diffuse fluidic phase boundaries

在两相可压缩流体动力学的连续介质模型中，相间微力被表示为依赖于序参数的梯度的应力张量，例如，相的混合比。该项目的目的在于捕获实现两个相并包含相应界面的参考配置的稳定性，例如一个相的液滴被另一个相填充的区域包围。除了经典的Korteweg模型外，该项目将主要致力于具有相场的系统，以及最后，使用不同的技术，研究尖锐界面模型。对于Korteweg模型和相场模型，稳定性问题表现为关于参考解附近解的长时间渐近性的问题；对于尖锐界面模型，稳定性问题涉及不连续前沿的有限时间持久性。

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.