2D stochastic Navier–Stokes equations driven by jump noise

Brzeźniak, Zdzisław; Hausenblas, Erika; Zhu, Jiahui

doi:10.1016/j.na.2012.10.011

Cited by 79 publications

(78 citation statements)

References 24 publications

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“…(2) In [5], the Galerkin scheme of problem (1) is nonlinear. Therefore it is complicated for numerical developments.…”

Section: The Approximate Linear Problem and The Main Resultsmentioning

confidence: 99%

“…Therefore it is complicated for numerical developments. (3) Problem (3.1) is a linear evolution equations which is easier to study numerically than the Galerkin scheme in [5].…”

Section: The Approximate Linear Problem and The Main Resultsmentioning

confidence: 99%

“…Under the Conditions (A) and (B), Theorem 4.4 in [5] implies the existence and uniqueness of the solution of problem (1.1). …”

Section: Condition (B)mentioning

confidence: 97%

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Approximation of the stochastic 2D hydrodynamical type systems driven by non-Gaussian Lévy noise

Deugoué

2017

Stoch. Dyn.

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“…(2) In [5], the Galerkin scheme of problem (1) is nonlinear. Therefore it is complicated for numerical developments.…”

Section: The Approximate Linear Problem and The Main Resultsmentioning

confidence: 99%

“…Therefore it is complicated for numerical developments. (3) Problem (3.1) is a linear evolution equations which is easier to study numerically than the Galerkin scheme in [5].…”

Section: The Approximate Linear Problem and The Main Resultsmentioning

confidence: 99%

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Approximation of the stochastic 2D hydrodynamical type systems driven by non-Gaussian Lévy noise

Deugoué

2017

Stoch. Dyn.

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“…for any ∈ V. If Assumptions 2.1 and the estimate (6) in Assumption 2.5(iii) are verified, then, by using the Galerkin approximation, one can prove as in Proposition 3.3, see also [9], that (7) has a unique weak solution Z satisfying…”

Section: Now We Can Argue As Above To Show That For Anymentioning

confidence: 91%

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

Fernando

Hausenblas

Razafimandimby

2016

Commun. Math. Phys.

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Abstract:In this paper, we derive several results related to the long-time behavior of a class of stochastic semilinear evolution equations in a separable Hilbert space H:Here A is a positive self-adjoint operator and B is a bilinear map, and the driving noise L is basically a D(A −1/2 )-valued Lévy process satisfying several technical assumptions. By using a density transformation theorem type for Lévy measure, we first prove a support theorem and an irreducibility property of the Ornstein-Uhlenbeck processes associated to the nonlinear stochastic problem. Second, by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain γUsing a coupling argument, the exponential ergodicity is also proved under the stronger assumption that B is continuous on H × H. While the latter condition is only satisfied by the nonlinearities of GOY and Sabra shell models, the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2D Navier-Stokes, Magnetohydrodynamics equations, the 3D Leray-α model, the GOY and Sabra shell models.

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“…[7], [8], [15], [17], [25], [16], [38], [37], [41], [42] and [13]. The noise term of Poissonian type is considered in the papers [20], [19], [21] and [12], and more general Lévy noise in [39] and [45]. We consider these equations because of their importance in other hydrodynamic models, e.g.…”

Section: G(t U(t)) Dw(t) =mentioning

confidence: 99%

Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—Abstract framework and applications

Motyl

2014

Stochastic Processes and their Applications

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The existence of martingale solutions of the hydrodynamic-type equations in 3D possibly unbounded domains is proved. The construction of the solution is based on the Faedo-Galerkin approximation. To overcome the difficulty related to the lack of the compactness of Sobolev embeddings in the case of unbounded domain we use certain Fréchet space. We use also compactness and tightness criteria in some nonmetrizable spaces and a version of the Skorokhod Theorem in non-metric spaces. The general framework is applied to the stochastic Navier-Stokes, magneto-hydrodynamic (MHD) and the Boussinesq equations.

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2D stochastic Navier–Stokes equations driven by jump noise

Cited by 79 publications

References 24 publications

Approximation of the stochastic 2D hydrodynamical type systems driven by non-Gaussian Lévy noise

Approximation of the stochastic 2D hydrodynamical type systems driven by non-Gaussian Lévy noise

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—Abstract framework and applications

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