Well-posedness and large deviations for the stochastic primitive equations in two space dimensions

Gao, Hongjun; Sun, Chengfeng

doi:10.4310/cms.2012.v10.n2.a8

Cited by 23 publications

(16 citation statements)

References 40 publications

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“…The authors use continuous martingale theory and stopping time arguments to treat the primitive equations with physical boundary conditions without the above-mentioned cancellation by establishing stronger convergence of the Galerkin approximations. Furthermore, a large deviation principle [20] and a central limit theorem [29] are known to hold.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

Saal¹,

Slavík²

2021

Preprint

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We establish the existence and uniqueness of pathwise strong solutions to the stochastic 3D primitive equations with only horizontal viscosity and diffusivity driven by transport noise on a cylindrical domain M = (−h, 0) × G, G ⊂ R 2 bounded and smooth, with the physical Dirichlet boundary conditions on the lateral part of the boundary. Compared to the deterministic case where the uniqueness of z-weak solutions holds in L 2 , more regular initial data are necessary to establish uniqueness in the anisotropic space H 1 z L 2 xy so that the existence of local pathwise solutions can be deduced from the Gyöngy-Krylov theorem. Global existence is established using the logarithmic Sobolev embedding, the stochastic Gronwall lemma and an iterated stopping time argument.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

Saal¹,

Slavík²

2021

Preprint

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“…Concerning the exponential behavior of SPEs, Medjo 42 established the stability result for SPEs when the noise is Brownian motion. Gao and Sun 43‐45 also studied the long‐time behavior and asymptotic and regular properties of SPEs. It is pointed out in Majda et al and Palmer 46,47 that studies from climate show that the complex multiscale nature of the earth's climate system results in many uncertainties that should be accounted for in the basic dynamical models of atmospheric and oceanic processes.…”

Section: Introductionmentioning

confidence: 99%

The exponential behavior of 3D stochastic primitive equations driven by fractional noise

Wang

Zhou

Guo

2021

Math Methods in App Sciences

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The long‐time behavior of flows is a very interesting and important problem in the theory of stochastic hydrodynamic equations. Inspired by the theory of dynamic system, we develop a new method to the study of the pathwise exponential behavior of 3D stochastic primitive equations driven by fractional Brownian motion. That is, we show that the weak solutions to the 3D stochastic primitive equations converge exponentially to the unique stationary solution. The result and method presented in this article can be widely applied to the study of long‐time behavior of other hydrodynamic equations with noises including Brownian motion and Lévy noise.

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“…Moreover, they established a large deviation principle for this model in [6]. For 2D stochastic primitive equations, Gao and Sun [9] obtained its global well-posedness and Freidlin-Wentzell's large deviations.…”

Section: Introductionmentioning

confidence: 99%

Splitting up method for 2D stochastic primitive equations with multiplicative noise

Peng

Zhang

2019

Communications in Mathematical Sciences

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This paper concerns the convergence of an iterative scheme for 2D stochastic primitive equations on a bounded domain. The stochastic system is split into two equations: a deterministic 2D primitive equations with random initial value and a linear stochastic parabolic equation, which are both simpler for numerical computations. An estimate of approximation error is given, which implies that the strong speed rate of the convergence in probability is almost 1 2 . MSC: Primary 60H15, 60H30; Secondary 76D06, 76M35.

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Well-posedness and large deviations for the stochastic primitive equations in two space dimensions

Abstract: The two dimensional primitive equations with multiplicative noise are studied in this paper. The existence and uniqueness of solutions in a fixed probability space and a Wentzell-Freidlin type large deviation principle for small multiplicative noise by weak convergence method are obtained.

Cited by 23 publications

References 40 publications

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

The exponential behavior of 3D stochastic primitive equations driven by fractional noise

Splitting up method for 2D stochastic primitive equations with multiplicative noise

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