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

TL;DR

This paper investigates the supercritical surface quasigeostrophic (SQG) equation with fractional dissipation, establishing an upper bound on the Hausdorff dimension of the singular set for suitable weak solutions.

Contribution

It introduces a notion of suitable weak solutions for the supercritical SQG equation and provides a bound on the Hausdorff dimension of their singular set.

Findings

Singular set is contained in a set of Hausdorff dimension at most rac{1}{2eta} imes ( rac{1+eta}{eta} (1-2eta) + 2)

Suitable weak solutions exist for all initial data in L^2

Bound on the size of the singular set depends explicitly on the dissipation parameter 0

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