Regularity results for a class of generalized surface quasi-geostrophic equations

TL;DR

This paper establishes global existence and regularity results for a class of generalized surface quasi-geostrophic equations, including supercritical cases, using commutator estimates and modulus of continuity methods.

Contribution

It provides the first global regularity results for these equations with supercritical dissipation, extending understanding of their solution behavior.

Findings

Global existence of weak solutions in the inviscid case

Global regularity for equations with supercritical dissipation

Eventual regularity in supercritical regimes

Abstract

We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical dissipation, which turns out to correspond to a logarithmically supercritical diffusion due to the singular nature of the velocity. Our last result is the eventual regularity in the supercritical cases for such weak solutions. The main idea in the proof of the existence part is based on suitable commutator estimates along with a careful cutting into low/high frequencies and inner/outer spatial scales to pass to the limit; while the proof of both the global regularity result and the eventual regularity for the supercritical diffusion are essentially based on the use of the so-called modulus of continuity method.