The stochastic primitive equations with transport noise and turbulent pressure

TL;DR

This paper proves the global existence and uniqueness of solutions for a stochastic primitive equation modeling geophysical flows with transport noise and turbulent pressure, allowing very rough noise terms without smallness assumptions.

Contribution

It introduces a novel approach to handle rough transport noise in stochastic primitive equations without requiring smallness conditions, expanding the scope of solvable models.

Findings

Proved global existence and uniqueness of solutions.

Allowed very rough noise terms without smallness assumptions.

Discussed extensions to Stratonovich noise and variable viscosity.

Abstract

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 $L^2$-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.