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 \(\alpha <\frac{1}{2}\). 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 \(\frac{1}{2\alpha } \left( \frac{1+\alpha }{\alpha } (1-2\alpha ) + 2\right) \).