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We construct an invariant measure $\mu$ for the Surface Quasi-Geostrophic(SQG) equation and show that almost all functions in the support of $\mu$ areinitial conditions of global, unique solutions of SQG, that depend continuouslyon the initial data. In addition, we show that the support of $\mu$ is infinitedimensional, meaning that it is not locally a subset of any compact set withfinite Hausdorff dimension. Also, there are global solutions that havearbitrarily large initial condition. The measures $\mu$ is obtained viafluctuation-dissipation method, that is, as a limit of invariant measures forstochastic SQG with a carefully chosen dissipation and random forcing.

Invariant measures and global well posedness for SQG equation