The primitive equations with stochastic wind driven boundary conditions

TL;DR

This paper investigates the primitive equations used in geophysical fluid dynamics under stochastic boundary conditions driven by wind, establishing existence and uniqueness of solutions in critical function spaces.

Contribution

It introduces a novel approach to handle stochastic boundary conditions in primitive equations, proving local well-posedness in anisotropic critical spaces.

Findings

Unique local pathwise solutions exist for the stochastic primitive equations.

Solutions are constructed in anisotropic $L^q_t$-$H^{-1,p}_zL^p_{xy}$-spaces.

The approach combines stochastic and deterministic methods for boundary value problems.

Abstract

The primitive equations for geophysical flows are studied under the influence of {\em stochastic wind driven boundary conditions} modeled by a cylindrical Wiener process. We adapt an approach by Da Prato and Zabczyk for stochastic boundary value problems to define a notion of solutions. Then a rigorous treatment of these stochastic boundary conditions, which combines stochastic and deterministic methods, yields that these equations admit a unique, local pathwise solution within the anisotropic $L^q_t$-$H^{-1,p}_zL^p_{xy}$-setting. This solution is constructed in critical spaces.