Invariant Measures and Global Well Posedness for the SQG Equation

We construct an invariant measure $$\mu $$ for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of $$\mu $$ are initial conditions of global, unique solutions of SQG that depend continuously on the initial data. In addition, we show that the support of $$\mu $$ is infinite dimensional, meaning that it is not locally a subset of any compact set with finite Hausdorff dimension. Also, there are global solutions that have arbitrarily large initial condition. The measure a $$\mu $$ is obtained via fluctuation–dissipation method, that is, as a limit of invariant measures for stochastic SQG with a carefully chosen dissipation and random forcing.