Local Martingale Solutions and Pathwise Uniqueness for the Three-dimensional Stochastic Inviscid Primitive Equations

TL;DR

This paper investigates the existence and uniqueness of solutions for three-dimensional stochastic primitive equations, including cases with added vertical viscosity and complex noise, demonstrating solutions become analytic over time.

Contribution

It extends the analysis of stochastic primitive equations by considering broader noise types and establishing solution regularity and uniqueness in analytic function spaces.

Findings

Proved local existence of martingale solutions under stochastic effects.

Established pathwise uniqueness for solutions.

Showed solutions become instantaneously analytic in the vertical variable.

Abstract

We study the stochastic effect on the three-dimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale solutions and pathwise uniqueness. By adding vertical viscosity, i.e., considering the hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and allow the transport noise in the vertical direction. We establish the local existence of martingale solutions and pathwise uniqueness, and show that the solutions become analytic in the vertical variable instantaneously as $t>0$ and the vertical analytic radius increases as long as the solutions exist.