Stochastic Primitive Equations Research Articles

Stochastic primitive equations with horizontal viscosity and diffusivity

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⊂R2 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 L2, more regular initial data are necessary to establish uniqueness in the anisotropic space Hz1Lxy2 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.

The stochastic primitive equations with transport noise and turbulent pressure

In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-regularity, the theory of critical spaces for stochastic evolution equations, and global a priori bounds. Compared to other results in this direction, we do not need any smallness assumption on the transport noise which acts directly on the velocity field and we also allow rougher noise terms. The adaptation to Stratonovich type noise and, more generally, to variable viscosity and/or conductivity are discussed as well.

Well-posedness of the 3D stochastic primitive equations with multiplicative and transport noise

We show that the stochastic 3D primitive equations with the Neumann boundary condition on the top, the lateral Dirichlet boundary condition and either the Dirichlet or the Neumann boundary condition on the bottom driven by multiplicative gradient-dependent white noise have unique maximal strong solutions both in the stochastic and PDE senses under certain assumptions on the growth of the noise. For the case of the Neumann boundary condition on the bottom, global existence is established by using the decomposition of the vertical velocity to the barotropic and baroclinic modes and an iterated stopping time argument. An explicit example of non-trivial infinite dimensional noise depending on the vertical average of the horizontal gradient of horizontal velocity is presented.

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

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.

ON THE WELL-POSEDNESS OF THE STOCHASTIC 2D PRIMITIVE EQUATIONS

<abstract> Under non-Lipschitz conditions for the external force term and noise term, the two-dimensional stochastic primitive equations are studied in this paper. Based on Galerkin method, iterative method and the moment estimations, we prove the existence and uniqueness of the solutions in a fixed probability space. </abstract>

Asymptotic behavior of 3-D stochastic primitive equations of large-scale moist atmosphere with additive noise

Asymptotic behavior of 3-D stochastic primitive equations of large-scale moist atmosphere with additive noise

Global well-posedness of stochastic 2D primitive equations with random initial conditions

In this article, we consider 2D stochastic primitive equations (PEs) driven by affine-linear multiplicative white noise and with random initial conditions. We obtain the global well-posedness of the stochastic PEs when the random initial condition satisfies sufficient Malliavin regularity. In the proof process of the global existence of solutions to 2D stochastic PEs, the Malliavin calculus plays a key role.

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

We develop a new and general method to prove 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 infinite-dimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the time-uniform a priori estimates in some Sobolev space whose regularity is higher than the solution space. But this method can not be applied to the 3D stochastic PEs with the non-periodic boundary conditions. Therefore, the existence of universal attractor (weak attractor) was established in previous works (see [15,16]). 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 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., using our method we only need to obtain time-uniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system, while the common method need to establish time-uniform a priori estimates in a more regular functional space than the solution space. Take the stochastic PEs for example, as the unique strong solution to the stochastic PEs belongs to C([0,T];(H1(℧))3), in view of our method, we only need to obtain the time-uniform a priori estimates in the solution space (H1(℧))3 to prove the existence of random attractor for this stochastic system, while the common method need to establish time-uniform a priori estimates for the solution in the functional space (H2(℧))3. However, time-uniform a priori estimates in (H2(℧))3 for the solution to stochastic PEs are too difficult to be established. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by Wiener noises, fractional noises and even jump noises. In a forth coming paper, using this new method we [46] prove the existence of random attractor for the stochastic nematic liquid crystals equations. This is the first result about the long-time behavior of stochastic nematic liquid crystals equations.

Stochastic 2D primitive equations: Central limit theorem and moderate deviation principle

In this paper, we establish a central limit theorem and a moderate deviation for two-dimensional stochastic primitive equations driven by multiplicative noise. This is the first result about the limit theorem and the moderate deviations for stochastic primitive equations. The proof of the results relies on the weak convergence method and some delicate and careful apriori estimates.

On the backward uniqueness of the stochastic primitive equations with additive noise

The previous works focus on the uniqueness for the initial-value problems of stochastic primitive equations. Uniqueness for the initial-value problems means that if the two initial conditions are the same, then the two solutions coincide with each other. However there is no work to answer what will happen to the solutions if the two initial conditions are different. This problem for the stochastic three dimensional primitive equations is addressed by the backward uniqueness established in this article. The backward uniqueness means that if two solutions intersect at time $t>0, $ then they are equal everywhere on the interval $(0, t).$ In other words, given two different initial-value conditions, the corresponding two solutions will never cross in the future. Hence this article can be viewed as a further study of the dependence of the solutions on the initial data.

On the small‐time asymptotics of 3D stochastic primitive equations

In this paper, we establish a small‐time large deviation principle for the strong solution of three‐dimensional stochastic primitive equations driven by multiplicative noise, which involves not only the study of small noise but also the challenging nonlinear drift terms.

Large deviation for the stochastic 2D primitive equations with additive Lévy noise

Large deviation for the stochastic 2D primitive equations with additive Lévy noise

Markov selection and W-strong Feller for 3D stochastic primitive equations

This paper studies some analytical properties of weak solutions of 3D stochastic primitive equations with periodic boundary conditions. The martingale problem associated to this model is shown to have a family of solutions satisfying the Markov property, which is achieved by means of an abstract selection principle. The Markov property is crucial to extend the regularity of the transition semigroup from small times to arbitrary times. Thus, under a regular additive noise, every Markov solution is shown to have a property of continuous dependence on initial conditions, which follows from employing the weak-strong uniqueness principle and the Bismut-Elworthy-Li formula.

Large deviation principles for 3D stochastic primitive equations

In this paper, we establish the Freidlin–Wentzell's large deviations for 3D stochastic primitive equations with small noise perturbation. The weak convergence approach plays an important role.

Time discrete approximation of weak solutions to stochastic equations of geophysical fluid dynamics and applications

As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans, the time discretization of these equations by an implicit Euler scheme is studied. From the deterministic point of view, the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions. From the probabilistic viewpoint, this paper deals with a wide class of nonlinear, state dependent, white noise forcings which may be interpreted in either the Ito or the Stratonovich sense. The proof of convergence of the Euler scheme, which is carried out within an abstract framework, covers the equations for the oceans, the atmosphere, the coupled oceanic-atmospheric system as well as other related geophysical equations. The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense, a result which is new by itself to the best of our knowledge.

Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions

Three dimensional primitive equations with a small multiplicative noise are studied in this paper. The existence and uniqueness of solutions with small initial value in a fixed probability space are obtained. The proof is based on Galerkin approximation, Ito's formula and weak convergence methods.

Diffusion limit of 3D primitive equations of the large-scale ocean under fast oscillating random force

The three-dimensional (3D) viscous primitive equations describing the large-scale oceanic motions under fast oscillating random perturbation are studied. Under some assumptions on the random force, the solution to the initial boundary value problem (IBVP) of the 3D random primitive equations converges in distribution to that of IBVP of the limiting equations, which are the 3D stochastic primitive equations describing the large-scale oceanic motions under a white in time noise forcing. This also implies the convergence of the stationary solution of the 3D random primitive equations.

Study on some partial differential equations in the atmospheric and oceanic dynamics

We review some development of study on some partial differential equations in the atmospheric and oceanic dynamics. Firstly, we give some mathematical results of the surface quasi-geostrophic equations. Secondly, we recall some results on the global well-posedness and long-time dynamics for the three-dimensional viscous primitive equations describing the large-scale atmospheric and oceanic motions. At last, we review some results of stochastical partial differential equations in the atmospheric and oceanic dynamics, which include the existence and uniqueness of global strong solutions to the initial boundary value problem for the stochastic primitive equations, and the existence of random attractors for the corresponding random dynamical system.

Existence and Regularity of Invariant Measures for the Three Dimensional Stochastic Primitive Equations

We establish the continuity of the Markovian semigroup associated with strong solutions of the stochastic 3D Primitive Equations, and prove the existence of an invariant measure. The proof is based on new moment bounds for strong solutions. The invariant measure is supported on strong solutions and is furthermore shown to have higher regularity properties.

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

The primitive equations (PEs) 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 PEs 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 PEs and more generally.In this work we study a stochastic version of the PEs. 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, estimates on the pressure and stopping time arguments.