Martingale Solution to Equations for Differential Type Fluids of Grade Two Driven by Random Force of Lévy Type

Hausenblas, Erika; Razafimandimby, Paul André; Sango, Mamadou

doi:10.1007/s11118-012-9316-7

Cited by 26 publications

(22 citation statements)

References 39 publications

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“…Using the estimates (22), (31), the convergence (69) and the uniqueness of u, we conclude that the whole sequence (u n ) n of the Galerkin approximation (28) …”

Section: Proof Of Theoremmentioning

confidence: 79%

“…Suppose that the hypotheses of Theorem 1 hold. Let (u n ) n∈N the unique solution of the Galerkin approximation of problem (1) (see (28) below) and u the unique solution of problem (1). We have the following convergence results: (2)-Note that if a process u(t), t ∈ [0, T ] is adapted and càdlàg, then the left-limit process u(t−) is left continuous and adapted .…”

Section: The First Main Resultsmentioning

confidence: 99%

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Strong solutions for the stochastic 3D LANS-α model driven by non-Gaussian Lévy noise

Deugoué

Sango

2015

Stoch. Dyn.

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We establish the existence, uniqueness and approximation of the strong solutions for the stochastic 3D LANS-α model driven by a non-Gaussian Lévy noise. Moreover, we also study the stability of solutions. In particular, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solution.

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“…Using the estimates (22), (31), the convergence (69) and the uniqueness of u, we conclude that the whole sequence (u n ) n of the Galerkin approximation (28) …”

Section: Proof Of Theoremmentioning

confidence: 79%

Section: The First Main Resultsmentioning

confidence: 99%

Strong solutions for the stochastic 3D LANS-α model driven by non-Gaussian Lévy noise

Deugoué

Sango

2015

Stoch. Dyn.

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“…Now, the existence of solution u will follow from a similar argument as in [27] (see also [28], [36]). The uniqueness of the solution can be proved by arguing as in [27] or [10].…”

Section: Ergodicity Of the Stochastic Shell Modelsmentioning

confidence: 99%

Ergodicity of Stochastic Shell Models Driven by Pure Jump Noise

Bessaih¹,

Hausenblas²,

Razafimandimby³

2016

SIAM J. Math. Anal.

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Abstract. In the present paper we study a stochastic evolution equation for shell (SABRA & GOY) models with pure jump Lévy noise L = ∞ k=1 l k (t)e k on a Hilbert space H. Here {l k ; k ∈ N} is a family of independent and identically distributed (i.i.d.) real-valued pure jump Lévy processes and {e k ; k ∈ N} is an orthonormal basis of H. We mainly prove that the stochastic system has a unique invariant measure. For this aim we show that if the Lévy measure of each component l k (t) of L satisfies a certain order and a nondegeneracy condition and is absolutely continuous with respect to the Lebesgue measure, then the Markov semigroup associated with the unique solution of the system has the strong Feller property. If, furthermore, each l k (t) satisfies a small deviation property, then 0 is accessible for the dynamics independently of the initial condition. Examples of noises satisfying our conditions are a family of i.i.d tempered Lévy noisesis a sequence of i.i.d subordinator Gamma (resp., real-valued Wiener) processes with Lévy density f G (z) = (ϑz) −1 e − z ϑ 1 z>0 . The proof of the strong Feller property relies on the truncation of the nonlinearity and the use of a gradient estimate for the Galerkin system of the truncated equation. The gradient estimate is a consequence of a Bismut-Elworthy-Li (BEL) type formula that we prove in the Appendix A of the paper.

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“…Recently, the stochastic models of two-dimensional second grade fluids (1.1) have been studied in [37], [38] and [39], where the authors obtained the existence and uniqueness of solutions and investigated the behavior of the solutions. The martingale solution of such system driven by Lévy noise is studied in [27].…”

Section: Introductionmentioning

confidence: 99%

Moderate deviations for stochastic models of two-dimensional second grade fluids

Zhai

Zhang²,

Zheng

2018

Stoch. Dyn.

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Martingale Solution to Equations for Differential Type Fluids of Grade Two Driven by Random Force of Lévy Type

Cited by 26 publications

References 39 publications

Strong solutions for the stochastic 3D LANS-α model driven by non-Gaussian Lévy noise

Strong solutions for the stochastic 3D LANS-α model driven by non-Gaussian Lévy noise

Ergodicity of Stochastic Shell Models Driven by Pure Jump Noise

Moderate deviations for stochastic models of two-dimensional second grade fluids

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