Existence, uniqueness and asymptotic stability of invariant measures for the stochastic Allen-Cahn-Navier-Stokes system with singular potential
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| Format: | Preprint |
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2025
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| _version_ | 1866912183300915200 |
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| author | Di Primio, Andrea Scarpa, Luca Zanella, Margherita |
| author_facet | Di Primio, Andrea Scarpa, Luca Zanella, Margherita |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_06174 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence, uniqueness and asymptotic stability of invariant measures for the stochastic Allen-Cahn-Navier-Stokes system with singular potential Di Primio, Andrea Scarpa, Luca Zanella, Margherita Probability Analysis of PDEs 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. |
| title | Existence, uniqueness and asymptotic stability of invariant measures for the stochastic Allen-Cahn-Navier-Stokes system with singular potential |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2501.06174 |