Global well-posedness of slightly supercritical active scalar equations

TL;DR

This paper proves global regularity for certain slightly supercritical active scalar equations, including SQG, Burgers, and 2D Euler, using a nonlocal maximum principle to control solution regularity.

Contribution

It establishes the first global regularity results for these equations in the supercritical regime with logarithmic diffusion behavior.

Findings

Global regularity for supercritical SQG and Burgers equations.

Sharpness of results demonstrated for Burgers equation.

Global regularity proven for a supercritical 2D Euler equation.

Abstract

The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasi-geostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions.