ABSTRACT

We study, in this article, a stochastic version of a coupled Allen–Cahn–Navier–Stokes model in a two- or three-dimensional bounded domain. The model consists of the Navier–Stokes equations for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter. These equations are motivated by the dynamic of binary fluids under the influence of stochastic external forces. We prove the existence of a probabilistic weak solutions. The proof relies on a Galerkin approximation as well as some compactness results. In the two-dimensional case, we prove the uniqueness of the weak solutions.

中文翻译：

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关于随机两相流模型的弱解

摘要

在本文中，我们研究了二维或三维有界域中耦合 Allen-Cahn-Navier-Stokes 模型的随机版本。该模型由速度的 Navier-Stokes 方程和顺序（相位）参数的 Allen-Cahn 模型组成。这些方程是由在随机外力影响下的二元流体动力学推动的。我们证明了概率弱解的存在。证明依赖于 Galerkin 近似以及一些紧致性结果。在二维情况下，我们证明了弱解的唯一性。