Ergodicity of two-dimensional primitive equations of large scale ocean in geophysics driven by degenerate noise

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Abstract

The current paper is devoted to ergodicity of the stochastic two-dimensional primitive equations of large scale ocean in geophysics driven by degenerate noise. The central challenge is to prove the asymptotically strong Feller property of the probability transition semigroups.

Introduction

In this paper, we study the following two-dimensional primitive equations of large scale ocean in geophysics driven by degenerate noise ut+uux+(z0ux)uz1εv+1εpsx1εz0(μ1Txμ2Sx)=γ1Δu+QdWt,vt+uvx+(z0ux)vz+1εu=γ2Δv+QdWt,Tt+uTx+(z0ux)Tz=γ3ΔT+QdWt,St+uSx+(z0ux)Sz=γ4ΔS+QdWt.with the homogeneous boundary value conditions uz=0,vz=0,Tz=0,Sz=0ons,(u,v)=0,T=0,S=0onb,(u,v)=0,Tx=0,Sx=0onl,and the initial conditions U0=(u,v,T,S)t=0=(u0,v0,T0,S0),where the space domain of the above equations is D=(0,1)×(1,0), and the boundary is defined by D=slb with s=(0,1)×{0},l=0×(1,0)1×(1,0),b=(0,1)×{1}.The driving noise process W(t) is a Wiener process defined on a filtered probability space (Ω,,P). Let {ei,i=1,2,} be a set of normalized eigenfunctions corresponding to eigenvalues of 0<λ1λ2 of Δ. For some NN, we set QW(t)=k=1Nqkβk(t)ek, where {βk(t)} is a class of independent real-valued Brownian motions. The noise is degenerate in the sense that it drives the system only in the first N Fourier modes.

The analysis of the Primitive Equations dates back to 1990s. We refer the reader to [1] and references therein for the background of the mathematical theory for the deterministic Primitive Equations. Our considered equations are a stochastic version of the ones in [2], which then were discussed in [3] wherein z-weak solutions were obtained in the case of additive noise. Later [4], [5] addressed the case of physical boundary conditions and nonlinear multiplicative noise by establishing the global existence and uniqueness of pathwise solutions. The analysis of three dimensional case is also worth mentioning in that the global existence and uniqueness of strong pathwise solutions was obtained in [6], as well as the existence and regularity of invariant measure was obtained recently in [7].

Despite these extensive results, the theory for the stochastic Primitive Equations still needs to be developed, especially in the direction of establishing a general framework to describe the long time behavior of oceanic and atmospheric processes. In this sense, statistically invariant measures provide the initial foundations, and the issue of unique ergodicity is critical. This leads to our main theorem.

Theorem 1.1

Let {Pt} be the transition semigroup associated to the primitive equations of large scale ocean in geophysics (1.1), then for any sufficiently large N, there exists a unique invariant probability measure associated with{Pt}t0.

Making use of the classical Krylov–Bogoliubov averaging procedure, or proving in a similar yet simpler manner like [7], one can establish the existence of invariant measures for Eqs. (1.1). However, the uniqueness of invariant measures is not easy to prove, especially for infinite-dimensional systems under the degenerate noise. Thanks to [8], one can adopt the strategy of combining the asymptotic strong Feller property of the associated Markov semigroup Pt, and irreducibility of the dynamics. Since solutions of deterministic equations converge to zero, due to dissipation, the proof of irreducibility is more or less straightforward. Thus we aim at establishing sufficient smoothing properties of Pt in the manuscript.

Section snippets

Preliminaries and key lemma

Without loss of generality, we set all the γk’s to be 1. We first give some function space H1{u;uL2(D),10udz=0},V1{u;uH1(D),10udz=0,u|bl=0}, V2{u;uH1(D),u|bl=0},V3{u;uH1(D),u|b=0}. VV1×V2×V3×V3 and HH1×L2(D)×L2(D)×L2(D) are equipped with their usual Sobolev norm, respectively. The existence and uniqueness of pathwise, z-weak solutions in C([0,T];H)L2([0,T];V) are established in [3]. Next, we give some preliminaries associated with anisotropic spaces.

Definition 2.1

Anisotropic Spaces [9]

Given p,q[1,], we say

Asymptotically strong Feller

In the sequel, denote Ll2 the finite-dimensional “low-frequency” subspace of L2(D), i.e. Ll2span{e1,,eN}, and Lh2 the corresponding “high-frequency” subspace of L2(D) such that the direct sum decomposition holds: L2(D)=Ll2Lh2. This decomposition naturally associates the projecting operator πl:L2(D)Ll2 and πh=Iπl. For any uL2(D), let ulπlu and uhπhu.

The asymptotically strong Feller is straightforward once the following type of gradient inequality is established.

Proposition 3.1

Let {Pt}t0 be the

CRediT authorship contribution statement

Daiwen Huang: Conceptualization, Methodology. Tianlong Shen: Writing - original draft. Yan Zheng: Writing - review & editing.

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This work was supported by the National Natural Science Foundation of China (No. 11771449, 11801563), NSF of Hunan, China (No.2018JJ2468), and Fundamental Program of NUDT, China (No.ZK17-03-19).

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