Ergodicity for stochastic equations of Navier–Stokes type

Brzeźniak, Zdzisław; Komorowski, Tomasz; Peszat, Szymon

doi:10.1214/21-ecp443

Cited by 3 publications

(1 citation statement)

References 28 publications

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“…It is well known that studying ergodic properties of dynamical systems is of profound importance from both applied and theoretical standpoints, [10]. Some of these properties are the existence and uniqueness of invariant probability measures.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Log-harnack Inequality for the 2d Stochastic Cahn-hilliard-navier-stokes System With Degenerate Noise

Medjo

2023

Preprint

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In this article, we consider a Cahn-Hilliard-Navier-Stokes equations in a two dimensional bounded domain and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with the Cahn-Hilliard-Navier-Stokes system driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity. Mathematics Subject Classification. 35R60,35Q35,60H15,76M35,86A05.

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Section: Introductionmentioning

confidence: 99%

Asymptotic Log-harnack Inequality for the 2d Stochastic Cahn-hilliard-navier-stokes System With Degenerate Noise

Medjo

2023

Preprint

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New a Priori estimate for stochastic 2D Navier–Stokes equations with applications to invariant measure

Ferrari

2024

Annali di Matematica

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The paper deals with the stochastic two-dimensional Navier–Stokes equations for homogeneous and incompressible fluids, set in a bounded domain with Dirichlet boundary conditions. We consider additive noise in the form $$G\,\textrm{d}W$$ G d W , where W is a cylindrical Wiener process and G a bounded linear operator with range dense in the domain of $$A^\gamma $$ A γ , A being the Stokes operator. While it is known that existence of invariant measure holds for $$\gamma >1/4$$ γ > 1 / 4 , previous results show its uniqueness only for $$\gamma > 3/8$$ γ > 3 / 8 . We fill this gap and prove uniqueness and strong mixing property in the range $$\gamma \in (1/4, 3/8]$$ γ ∈ ( 1 / 4 , 3 / 8 ] by adapting the so-called Sobolevskiĭ-Kato-Fujita approach to the stochastic N–S equations. This method provides new a priori estimates, which entail both better regularity in space for the solution and strong Feller and irreducibility properties for the associated Markov semigroup.

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Asymptotic Log-Harnack inequality for the 2D Ericksen–Leslie model system with degenerate noise and damping

Tachim Medjo

2024

Stoch. Dyn.

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In this paper, we consider a 2D liquid crystal model with damping and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with a simplified liquid crystal model driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity.

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Ergodicity for stochastic equations of Navier–Stokes type

Cited by 3 publications

References 28 publications

Asymptotic Log-harnack Inequality for the 2d Stochastic Cahn-hilliard-navier-stokes System With Degenerate Noise

Asymptotic Log-harnack Inequality for the 2d Stochastic Cahn-hilliard-navier-stokes System With Degenerate Noise

New a Priori estimate for stochastic 2D Navier–Stokes equations with applications to invariant measure

Asymptotic Log-Harnack inequality for the 2D Ericksen–Leslie model system with degenerate noise and damping

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