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

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} dθ_t + |D|^{2δ}θ_t\,dx + (u_t \cdot \nabla)θ_t\,dx + |D|^δdW_t = 0 \\ u_t = \nabla^\perp|D|^{-1}θ_t. \end{cases} \end{equation*} We construct global energy solutions as introduced by P. Goncalves and M. Jara (2014) for any $δ> 0$, 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 $δ> \frac32$.

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