Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations

Röckner, Michael; Zhang, Tusheng; Zhang, Xicheng

doi:10.1007/s00245-009-9089-6

Cited by 86 publications

(52 citation statements)

References 26 publications

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“…where K, C, κ are some positive constants, r ∈ [1, ∞) and u · v stands for the inner product in R 3 . When the forcing is of random type, that is f = σ(t, u)dW (t), M. Röckner, T. Zhang and X. Zhang tackled a stochastic version of a modification of the previous model (1.1), that they called the tamed stochastic Navier-Stokes equations, in several papers such as [23], and [24]. Let us mention that in both the deterministic and the stochastic versions of (1.1), the solutions are investigated when the regularity of initial condition is at least H 1 and the viscosity acts in all three directions.…”

Section: Introductionmentioning

confidence: 99%

On stochastic modified 3D Navier–Stokes equations with anisotropic viscosity

Bessaih

Millet

2018

Journal of Mathematical Analysis and Applications

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Navier-Stokes equations in the whole space R 3 subject to an anisotropic viscosity and a random perturbation of multiplicative type is described. By adding a term of Brinkman-Forchheimer type to the model, existence and uniqueness of global weak solutions in the PDE sense are proved. These are strong solutions in the probability sense. The Brinkman-Forchheirmer term provides some extra regularity in the space L 2α+2 (R 3 ), with α > 1. As a consequence, the nonlinear term has better properties which allow to prove uniqueness. The proof of existence is performed through a control method. A Large Deviations Principle is given and proven at the end of the paper. ∂ t u − ν ∆u + u · ∇u + ∇p + a u 2α u = f, divu = 0, u| t=0 = u 0 , 2000 Mathematics Subject Classification. Primary 60H15, 60F10; Secondary 76D06, 76M35.

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Section: Introductionmentioning

confidence: 99%

On stochastic modified 3D Navier–Stokes equations with anisotropic viscosity

Bessaih

Millet

2018

Journal of Mathematical Analysis and Applications

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“…The last example is a tamed version of backward stochastic 3D Navier-Stokes equation. Stochastic tamed 3D Navier-Stokes equation has been investigated in a series of works of Röckner et al [31,32,33,34]. The classical 3D Navier-Stokes equations (i.e.…”

Section: Backward Stochastic Tamed 3d Navier-stokes Equationmentioning

confidence: 99%

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

Liu

Zhu

2020

Forum Mathematicum

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In this paper we show the existence and uniqueness of strong solutions for a large class of backward SPDE where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE 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].…”

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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Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations

Abstract: Abstract. In this paper, using weak convergence method, we prove a large deviation principle of Freidlin-Wentzell type for the stochastic tamed 3D Navier-Stokes equations driven by multiplicative noise, which was investigated in [21].

Cited by 86 publications

References 26 publications

On stochastic modified 3D Navier–Stokes equations with anisotropic viscosity

On stochastic modified 3D Navier–Stokes equations with anisotropic viscosity

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

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

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