Improved Hausdorff dimension estimate of the singular set of the supercritical surface quasigeostrophic equation

TL;DR

This paper improves the upper bound on the Hausdorff dimension of the singular set for solutions of the supercritical surface quasigeostrophic equation, advancing understanding of singularity structure in these fluid models.

Contribution

It provides a sharper estimate of the singular set dimension by refining the local energy inequality analysis for the supercritical SQG equation.

Findings

Hausdorff dimension of singular set at most 1/(2α^2)

Improved upon previous estimates

Utilizes enhanced local energy inequality iteration

Abstract

We prove that the spacetime singular set of any suitable Leray-Hopf solution of the surface quasigeostrophic equation with fractional dissipation of order $0< \alpha < \frac{1}{2}$ has Hausdorff dimension at most $\frac{1}{2\alpha^2}\,.$ This result improves previously known dimension estimate established in [6] and builds on the excess decay result and the control on the particle flow already developed there. The improvement lies in the initial iteration of the local energy inequality in analogy with the celebrated result of Caffarelli-Kohn-Nirenberg [2] for the Navier-Stokes equations.