Estimate on the dimension of the singular set of the supercritical surface quasigeostrophic equation

Abstract

We consider the SQG equation with dissipation given by a fractional Laplacian of order $\hat{I}\pm <\frac{1}{2}$. We introduce a notion of suitable weak solution, which exists for every $L^2$ initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most $\frac{1}{2\hat{I}\pm }\left(\frac{1+\hat{I}\pm }{\hat{I}}\pm (1-2\hat{I}\pm )+2\right)$.