Eventual regularization for the slightly supercritical quasi-geostrophic equation

TL;DR

This paper demonstrates that weak solutions to a slightly supercritical quasi-geostrophic equation become smooth over time, using a De Giorgi type argument inspired by recent advances.

Contribution

It introduces a novel regularization result for the supercritical quasi-geostrophic equation employing a De Giorgi method.

Findings

Weak solutions become smooth for large times.

The method adapts De Giorgi techniques to supercritical regimes.

Provides a new approach to regularization in fluid dynamics equations.

Abstract

We prove that weak solutions of a slightly supercritical quasi-geostrophic equation become smooth for large time. We prove it using a De Giorgi type argument using ideas from a recent paper by Caffarelli and Vasseur.