Pathwise Solutions of the 2-D Stochastic Primitive Equations

Glatt-Holtz, Nathan; Témam, Roger

doi:10.1007/s00245-010-9126-5

Cited by 36 publications

(22 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [23] the necessary compactness is established by directly showing that the sequence of Galerkin solutions are Cauchy. For an application of this approach to the 2D Primitive equations see [24,25]. In any case the work here provides an alternative proof of the results in [23].…”

Section: Example: the Primitive Equations Of The Oceansupporting

confidence: 59%

Local martingale and pathwise solutions for an abstract fluids model

Debussche

Glatt-Holtz

Témam

2011

Physica D: Nonlinear Phenomena

Self Cite

147

186

View full text Add to dashboard Cite

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.

show abstract

Section: Example: the Primitive Equations Of The Oceansupporting

confidence: 59%

Local martingale and pathwise solutions for an abstract fluids model

Debussche

Glatt-Holtz

Témam

2011

Physica D: Nonlinear Phenomena

Self Cite

147

186

View full text Add to dashboard Cite

show abstract

“…Notwithstanding these extensive results in the deterministic case, the theory for the stochastic Primitive Equations remains underdeveloped. A two dimensional version of the PEs has been studied in a simplified form in [EPT07,GHZ08] and more recently in [GHT11a,GHT11b] in the greater generality of physically relevant boundary conditions and nonlinear multiplicative noise. While the full three dimensional system has been studied in [GH09] following the methods in [CT07], this work covers only the case of additive noise.…”

Section: Previous Mathematical Workmentioning

confidence: 99%

Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise

et al. 2012

View full text Add to dashboard Cite

The Primitive Equations are a basic model in the study of large scale Oceanic and Atmospheric dynamics. These systems form the analytical core of the most advanced General Circulation Models. For this reason and due to their challenging nonlinear and anisotropic structure the Primitive Equations have recently received considerable attention from the mathematical community.On the other hand, in view of the complex multi-scale nature of the earth's climate system, many uncertainties appear that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. In the climate community stochastic methods have come into extensive use in this connection. For this reason there has appeared a need to further develop the foundations of nonlinear stochastic partial differential equations in connection with the Primitive Equations and more generally.In this work we study a stochastic version of the Primitive Equations. We establish the global existence and uniqueness of strong, pathwise solutions for these equations in dimension 3 for the case of a nonlinear multiplicative noise. The proof makes use of anisotropic estimates, L p t L q x estimates on the pressure and stopping time arguments.MSC2010: 35Q86, 60H15, 35Q35 * To appear in Nonlinearity.1 Physical uncertainties on the boundary of the oceanic or coupled oceanic-atmospheric system are also present and are responsible for another source of stochasticity in these models. The boundary conditions (1.4) on Γ i (which is physically the sea surface-air interface) are a simplified version of the more realistic boundary conditions(1.1)In the form (1.1) the boundary condition expresses two fundamental aspects of oceanic-atmospheric interaction namely the driving force of the wind (the term gv) and the heating or cooling of the air by the ocean (the term g T ). These functions gv and g T are not well known and are estimated by modelers through very rough averages. Thus the uncertainties of these functions lead to stochastic PDEs with white noise on the boundary a subject which we intend to pursue in future work.

show abstract

“…Ewald et al 32 and Glatt‐Holtz and Ziane 33 considered a two‐dimensional SPEs. Then Glatt‐Holtz and Temam 34,35 extended the case to the greater generality of physically relevant boundary conditions and nonlinear multiplicative noise. Following the methods similar to Cao and Titi, 24 Guo and Huang 36 studied the global well‐posedness and long‐time behavior of the 3D system with additive noise.…”

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

View full text Add to dashboard Cite

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.

show abstract

Pathwise Solutions of the 2-D Stochastic Primitive Equations

Cited by 36 publications

References 38 publications

Local martingale and pathwise solutions for an abstract fluids model

Local martingale and pathwise solutions for an abstract fluids model

Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise

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

Contact Info

Product

Resources

About