Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

TL;DR

This paper proves the existence and uniqueness of solutions to stochastic 3D primitive equations with horizontal viscosity and diffusivity, advancing understanding of their mathematical properties under stochastic influences.

Contribution

It establishes the first rigorous results on pathwise strong solutions for stochastic primitive equations with only horizontal viscosity and diffusivity.

Findings

Existence of local pathwise solutions using Gy"{o}ngy-Krylov theorem.

Global solutions established via stochastic Gronwall lemma.

Uniqueness requires more regular initial data in anisotropic spaces.

Abstract

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 $M=(-h,0) \times G$, $G\subset \mathbb{R}^2$ 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 Gy\"{o}ngy-Krylov theorem. Global existence is established using the logarithmic Sobolev embedding, the stochastic Gronwall lemma and an iterated stopping time argument.