Existence and Uniqueness of Energy Solutions to the Stochastic Diffusive Surface Quasi-Geostrophic Equation with Additive Noise

TL;DR

This paper proves the existence of global energy solutions for a stochastic diffusive surface quasi-geostrophic equation with additive noise, and establishes pathwise uniqueness when the diffusion parameter exceeds 1.5.

Contribution

It extends the theory of stochastic SQG equations by constructing solutions for any positive diffusion and proving uniqueness for sufficiently high diffusion levels.

Findings

Global energy solutions exist for all elta > 0.

Pathwise uniqueness holds for elta > 1.5.

Small diffusion enables solution construction.

Abstract

We continue our study of the dynamics of a nearly inviscid periodic surface quasi-geostrophic equation. Here we consider a slightly diffusive stochastic SQG equation of the form \begin{equation*} \begin{cases} d\theta_t + |D|^{2\delta}\theta_t\,dx + (u_t \cdot \nabla)\theta_t\,dx + |D|^{\delta}dW_t = 0 \\ u_t = \nabla^\perp|D|^{-1}\theta_t. \end{cases} \end{equation*} We construct global energy solutions as introduced by P. Goncalves and M. Jara (2014) for any $\delta > 0$, so that any small amount of diffusion permits us to construct solutions. We show moreover that pathwise uniqueness of these energy solutions holds in the presence of sufficiently high diffusion $\delta > \frac32$.