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

Andrea Di Primio, Luca Scarpa, Margherita Zanella

TL;DR

This work analyzes the long-time behavior of a 2D stochastic diffuse-interface model for binary fluids, coupling the Allen–Cahn and Navier–Stokes equations with a singular Flory–Huggins potential $F_{\text{log}}$ and stochastic forcings on velocity and phase. The authors establish existence of ergodic invariant measures and characterize their support, and under a non-degenerate low-mode Navier–Stokes forcing together with strong Allen–Cahn dissipation (large $\beta$), prove uniqueness and asymptotic stability in the Wasserstein sense via a Foias–Prodi-type framework. The approach hinges on constructing a Markov transition semigroup, proving Feller/sequential Feller properties, and employing Krylov–Bogoliubov arguments for invariant measures, complemented by a refined energy-method-based continuous dependence and generalized-coupling arguments (GHMR17/KS). The results provide quantitative mixing and ergodic properties for stochastic diffuse-interface flows with singular potentials, offering rigorous insights into the stability and uniqueness of stationary states in complex fluid mixtures.

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.

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

TL;DR

This work analyzes the long-time behavior of a 2D stochastic diffuse-interface model for binary fluids, coupling the Allen–Cahn and Navier–Stokes equations with a singular Flory–Huggins potential and stochastic forcings on velocity and phase. The authors establish existence of ergodic invariant measures and characterize their support, and under a non-degenerate low-mode Navier–Stokes forcing together with strong Allen–Cahn dissipation (large ), prove uniqueness and asymptotic stability in the Wasserstein sense via a Foias–Prodi-type framework. The approach hinges on constructing a Markov transition semigroup, proving Feller/sequential Feller properties, and employing Krylov–Bogoliubov arguments for invariant measures, complemented by a refined energy-method-based continuous dependence and generalized-coupling arguments (GHMR17/KS). The results provide quantitative mixing and ergodic properties for stochastic diffuse-interface flows with singular potentials, offering rigorous insights into the stability and uniqueness of stationary states in complex fluid mixtures.

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.
Paper Structure (36 sections, 31 theorems, 339 equations)

This paper contains 36 sections, 31 theorems, 339 equations.

Table of Contents

  1. Introduction
  2. Mathematical setting
  3. Notation
  4. Functional spaces and operators
  5. Embeddings and interpolation inequalities
  6. Weak topologies and weak continuity
  7. Spaces of trajectories
  8. Assumptions
  9. Results on the Allen--Cahn--Navier--Stokes system
  10. Existence and uniqueness of solutions
  11. Refined well-posedness results
  12. Existence of invariant measures
  13. The transition semigroup
  14. Feller properties
  15. The Markov property
  16. ...and 21 more sections

Key Result

Theorem 3.5

Let Assumptions hyp:structural-hyp:diffusionAC hold with $Z = {\boldsymbol{V}_\sigma^*}$ and let $T > 0$ be fixed. Then, for every initial datum $({\boldsymbol u}_0,\varphi_0)$ satisfying eq:u0-eq:phi0 with $p > 2$, there exists a unique probabilistically-strong solution $({\boldsymbol u}, \varphi)$

Theorems & Definitions (75)

  • Remark 2.1
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Theorem 3.5
  • Remark 3.6
  • Remark 3.7
  • Lemma 3.8
  • proof
  • ...and 65 more