Eventual Regularity of the Solutions to the Supercritical Dissipative Quasi-Geostrophic Equation

TL;DR

This paper proves that solutions to the supercritical dissipative quasi-geostrophic equation, which are smooth up to a certain time, remain smooth indefinitely, extending previous results on regularity.

Contribution

It introduces a modified class of test functions to demonstrate the eternal smoothness of solutions beyond initial regularity in the supercritical SQG equation.

Findings

Solutions become smooth after initial regularity

Solutions remain smooth forever if initially smooth

Extension of regularity results to supercritical regime

Abstract

Recently, Silvestre proved that certain weak solutions of the slightly supercritical surface quasi-geostrophic equation eventually become smooth. To prove this, he employed a De Giorgi type argument originated in the work of Caffarelli and Vasseur. Kiselev and Nazarov proved a variation of the result of Caffarelli and Vasseur by introducing a class of test functions. Motivated by the results of Silvestre, we will modify the class of test functions from the work of Kiselev and Nazarov and use this modified class to show that a solution to the supercritical SQG that is smooth up to a certain time must remain smooth forever.