Stochastic Solutions of the Two-Dimensional Primitive Equations of the Ocean and Atmosphere With an Additive Noise

Ewald, Brian D.; Petcu, Mădălina; Témam, Roger

doi:10.1142/s0219530507000948

Cited by 29 publications

(17 citation statements)

References 11 publications

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“…We do not expand further on these latter applications in this article in order to avoid excessive developments. The stochastic primitive equations of the ocean have been previously studied in [26,22,24,15,27] but none of these works address the full 3-d system in the context of a nonlinear multiplicative noise. The deterministic primitive equations are widely seen as a fundamental model for large scale oceanic and atmospheric systems.…”

Section: Introductionmentioning

confidence: 99%

Local martingale and pathwise solutions for an abstract fluids model

Debussche

Glatt-Holtz

Témam

2011

Physica D: Nonlinear Phenomena

147

186

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We establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system. The primary application of this abstract framework is to infer the local existence of strong, pathwise solutions to the 3D primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise. Instead of developing our results specifically for the 3D primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations (atmosphere, oceans, coupled atmosphere-ocean, on the sphere, on the β-plane approximation etc and the incompressible Navier-Stokes equations). In applications, all of the details are given for the β-plane approximation of the oceans equations.

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Section: Introductionmentioning

confidence: 99%

Local martingale and pathwise solutions for an abstract fluids model

Debussche

Glatt-Holtz

Témam

2011

Physica D: Nonlinear Phenomena

147

186

View full text Add to dashboard Cite

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“…This complicates the a priori estimates which in turn prevent the usage of more direct compactness arguments adopted in, [8], [23]. On the other hand, due to the nonlinear multiplicative noise structure, the system may not be transformed into a random PDE as in [18]. For this reason we are not able to treat the probabilistic dependence as a parameter in the problem.…”

Section: Introductionmentioning

confidence: 99%

“…

In this work we consider a stochastic version of the Primitive Equations (PEs) of the ocean and the atmosphere and establish the existence and uniqueness of pathwise, strong solutions. The analysis employs novel techniques in contrast to previous works [18], [23] in order to handle a general class of nonlinear noise structures and to allow for physically relevant boundary conditions. The proof relies on Cauchy estimates, stopping time arguments and anisotropic estimates.

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mentioning

confidence: 99%

Pathwise Solutions of the 2-D Stochastic Primitive Equations

Glatt-Holtz

Témam

2010

Appl Math Optim

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“…Despite the developments in the deterministic case, the theory for the stochastic PEs remains underdeveloped. B. Ewald, M. Petcu, R. Teman [13] and N. Glatt-Holtz, M. Ziane [21] considered a two-dimensional stochastic PEs. Then N. Glatt-Holtz and R. Temam [22,23] extended the case to the greater generality of physically relevant boundary conditions and nonlinear multiplicative noise.…”

Section: Introductionmentioning

confidence: 99%

Random attractor for the 3D viscous primitive equations driven by fractional noises

Zhou

2019

Journal of Differential Equations

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We develop a new and general method to prove the the existence of the random attractor (strong attractor) for the primitive equations (PEs) of large-scale ocean and atmosphere dynamics under non-periodic boundary conditions and driven by infinitedimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the uniform a priori estimates in some Sobolev space whose regularity is high enough. But this is very complicated for the 3D stochastic PEs with the non-periodic boundary conditions. Therefore, the existence of universal attractor ( weak attractor) was established in previous work. The main idea of our method is that we first derive that P-almost surely the solution operator of stochastic PEs is compact. Then we construct a compact absorbing set by virtue of the compact property of the the solution operator and the existence of a absorbing set. We should point out that our method has some advantages over the common method of using compact Sobolev embedding theorem, i.e., if the random attractor in some Sobolev space do exist in view of the common method, our method would then further implies the existence of random attractor in this space. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations and improves the existing results concerning random attractor of stochastic PEs. In a forth coming paper, we use this new method to prove the existence of strong attractor for the stochastic moist primitive equations, improving the results, the existence of weak (universal) attractor of the deterministic model. M × (−h, 0) ⊂ R 3 , and consider the following 3D stochastic PEs of Geophysical Fluid Dynamics.The unknowns for the 3D stochastic viscous PEs are the fluid velocity field (υ, w) = (υ 1 , υ 2 , w) ∈ R 3 with υ = (υ 1 , υ 2 ) and υ ⊥ = (−υ 2 , υ 1 ) being horizontal, the temperature T and the pressure p. f = f 0 (β + y) is the given Coriolis parameter, Q is a given heat source. The viscosity and the heat diffusion operators L 1 and L 2 are given byHere the positive constants ν 1 , µ 1 are the horizontal and vertical Reynolds numbers, respectively, and ν 2 , µ 2 are positive constants which stand for the horizontal and vertical heat diffusivity, respectively. To simplify the nations, we assume ν i = µ i = 1, i = 1, 2. The results in this paper are still valid when we consider the general cases. We set ∇ = (∂ x , ∂ y ) to be the horizontal gradient operator and ∆ = ∂ 2x + ∂ 2 y to be the horizontal Laplacian. Here, we takeẆ H i (t, x, y, z), i = 1, 2, the informal derivative for the fractional Wiener process W H i given below. The boundary of ℧ is partitioned into three parts:Here h is a sufficiently smooth function. Without loss generality, we assume h = 1. We consider the following boundary conditions of the stochastic 3D viscous PEs.where η(x, y) is the wind stress on the surface of the ocean, α is a positive constant, τ is t...

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Stochastic Solutions of the Two-Dimensional Primitive Equations of the Ocean and Atmosphere With an Additive Noise

Cited by 29 publications

References 11 publications

Local martingale and pathwise solutions for an abstract fluids model

Local martingale and pathwise solutions for an abstract fluids model

Pathwise Solutions of the 2-D Stochastic Primitive Equations

Random attractor for the 3D viscous primitive equations driven by fractional noises

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