Existence, uniqueness and asymptotic stability of invariant measures for the stochastic Allen-Cahn-Navier-Stokes system with singular potential

TL;DR

This paper investigates the long-term behavior of a stochastic fluid mixture model, establishing existence and uniqueness of invariant measures and demonstrating stability under certain noise and dissipation conditions.

Contribution

It proves the existence of ergodic invariant measures for the stochastic Allen-Cahn-Navier-Stokes system and shows conditions for their uniqueness and asymptotic stability.

Findings

Existence of ergodic invariant measures for the system.

Conditions for uniqueness and stability of invariant measures.

Characterization of the support of invariant measures.

Abstract

We study the long-time behaviour of a stochastic Allen-Cahn-Navier-Stokes system modelling the dynamics of binary mixtures of immiscible fluids. The model features two stochastic forcings, one on the velocity in the Navier-Stokes equation and one on the phase variable in the Allen-Cahn equation, and includes the thermodynamically-relevant Flory-Huggins logarithhmic potential. We first show existence of ergodic invariant measures and characterise their support by exploiting ad-hoc regularity estimates and suitable Feller-type and Markov properties. Secondly, we prove that if the noise acting in the Navier-Stokes equation is non-degenerate along a sufficiently large number of low modes, and the Allen-Cahn equation is highly dissipative, then the stochastic flow admits a unique invariant measure and is asymptotically stable with respect to a suitable Wasserstein metric.