Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise

TL;DR

This paper proves the global existence and regularity of strong solutions for the 3D stochastic Primitive Equations with multiplicative noise, advancing the mathematical understanding of climate models under uncertainty.

Contribution

It establishes the first global strong solution results for the 3D stochastic Primitive Equations with nonlinear multiplicative noise.

Findings

Proved global existence of strong solutions in 3D

Developed anisotropic and pressure estimates for stochastic PDEs

Addressed nonlinear multiplicative noise in climate models

Abstract

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. 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 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.