Continuity of weak solutions of the critical quasigeostrophic equation on the sphere

TL;DR

This paper establishes regularity results for weak solutions of the critical surface quasigeostrophic equation on the sphere, advancing understanding of active scalar equations in curved geometries.

Contribution

It provides the first regularity results for weak solutions of the critical quasigeostrophic equation on the sphere, including anisotropic and non-homogeneous media.

Findings

Regularity results for weak solutions on the sphere

Extension of active scalar theory to curved surfaces

Analysis applicable to anisotropic media

Abstract

In this paper we provide regularity results for active scalars that are weak solutions of almost critical drift-diffusion equations in general surfaces. This includes models of anisotropic non-homogeneous media and the physically motivated case of the two-dimensional sphere. Our finest result deals with the critical surface quasigeostrophic equation on the round sphere.