A stochastic Allen-Cahn-Navier-Stokes system with singular potential

TL;DR

This paper studies a stochastic coupled Allen-Cahn-Navier-Stokes system with singular potential, proving existence, uniqueness, and pressure results for solutions in two and three dimensions under random initial conditions.

Contribution

It introduces a stochastic model with singular potential, establishing existence and uniqueness of solutions, including pressure, in a novel stochastic PDE framework.

Findings

Existence of weak martingale solutions in 2D and 3D.

Pathwise uniqueness and strong solutions in 2D.

Existence and uniqueness of pressure via stochastic De Rham theorem.

Abstract

We investigate a stochastic version of the Allen-Cahn-Navier-Stokes system in a smooth two- or three-dimensional domain with random initial data. The system consists of a Navier-Stokes equation coupled with a convective Allen-Cahn equation, with two independent sources of randomness given by general multiplicative-type Wiener noises. In particular, the Allen-Cahn equation is characterized by a singular potential of logarithmic type as prescribed by the classical thermodynamical derivation of the model. The problem is endowed with a no-slip boundary condition for the (volume averaged) velocity field, as well as a homogeneous Neumann condition for the order parameter. We first prove the existence of analytically weak martingale solutions in two and three spatial dimensions. Then, in two dimensions, we also estabilish pathwise uniqueness and the existence of a unique probabilistically-strong solution. Eventually, by exploiting a suitable generalisation of the classical De Rham theorem to stochastic processes, existence and uniqueness of a pressure is also shown.