Analysis of Equilibrium States of Markov Solutions to the 3D Navier-Stokes Equations Driven by Additive Noise

Romito, Marco

doi:10.1007/s10955-007-9477-8

Cited by 31 publications

(45 citation statements)

References 19 publications

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“…Recently, in [4,5,25], Da-Prato, Debussche and Odasso proved the existence and ergodicity of Markov solutions for 3D SNSE without the taming term g N , which are obtained as limits of Galerkin's approximations. Similar results were obtained by Flandoli and Romito in [12,29] for all Markov solutions. Moreover, using stochastic cascades, Bakhtin [1] explicitly constructed a stationary solution of 3D Navier-Stokes system and proved a uniqueness theorem.…”

Section: Du(t) = ∆U(t) − (U(t) · ∇)U(t) + ∇P(t) − G N (|U(t)|supporting

confidence: 91%

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Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity

Röckner

Zhang

2008

Probab. Theory Relat. Fields

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Abstract. In this paper, we prove the existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly [15], we prove the uniqueness of invariant measures for the corresponding transition semigroup.

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Section: Du(t) = ∆U(t) − (U(t) · ∇)U(t) + ∇P(t) − G N (|U(t)|supporting

confidence: 91%

“…29 Note that v(t) 2 H 0 depends on ǫ through (5.18). By Lemma 6.1 in the Appendix, for any δ, h > 0, we may choose a T 0 > 0 sufficiently large and an ǫ small enough such that…”

Section: Lemma 54 (I) For Anymentioning

confidence: 99%

Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity

Röckner

Zhang

2008

Probab. Theory Relat. Fields

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“…It is known that the above problem admits global weak solutions, as well as unique local strong solutions, as in the deterministic case. Nevertheless the presence of noise allows to prove additional properties, such as continuous dependence on initial data [5,6,[12][13][14], as well as convergence to equilibrium [20,22]. See also the recent surveys [2,14] for a general introduction to the problem.…”

Section: Introductionmentioning

confidence: 99%

Hölder regularity of the densities for the Navier–Stokes equations with noise

Romito

2016

Stoch PDE: Anal Comp

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We prove that the densities of the finite dimensional projections of weak solutions of the Navier-Stokes equations driven by Gaussian noise are bounded and Hölder continuous, thus improving the results of Debussche and Romito [8]. The proof is based on analytical estimates on a conditioned Fokker-Planck equation solved by the density, that has a "non-local" term that takes into account the influence of the rest of the infinite dimensional dynamics over the finite subspace under observation.

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“…See however some progresses in [3], [10], [16]. The attempts made until now on the stochastic 3D Navier-Stokes equations have been "abstract", in nature, based on the generalization to infinite dimensions of tools and theories developed for stochastic ordinary equations.…”

Section: Introductionmentioning

confidence: 99%

The Interaction Between Noise and Transport Mechanisms in PDEs

Flandoli

2011

Milan J. Math.

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The understanding of transport mechanisms in PDEs is at the core of some of the main open problems in fluid dynamics. The introduction of random perturbations in these equations may have an influence on the properties of transport. We report here some recent progresses on the interaction between noise and transport. The Lagrangian viewpoint is used to investigate the problem. In some example we observe that noise improves the properties of the Lagrangian motion and restores uniqueness or prevents blow-up of the associated PDE; in other examples this does not happen.

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Analysis of Equilibrium States of Markov Solutions to the 3D Navier-Stokes Equations Driven by Additive Noise

Cited by 31 publications

References 19 publications

Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity

Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity

Hölder regularity of the densities for the Navier–Stokes equations with noise

The Interaction Between Noise and Transport Mechanisms in PDEs

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