Global well-posedness for a slightly supercritical surface quasi-geostrophic equation

TL;DR

This paper proves the global existence of smooth solutions for a slightly supercritical surface quasi-geostrophic equation using a nonlocal maximum principle, extending understanding of such equations in fluid dynamics.

Contribution

It introduces a novel approach employing a nonlocal maximum principle to establish global well-posedness for a supercritical regime of the surface quasi-geostrophic equation.

Findings

Global smooth solutions exist for the supercritical SQG equation.

The velocity field is derived from the scalar via a Fourier multiplier with controlled growth.

The method extends previous results to a broader class of supercritical equations.

Abstract

We use a nonlocal maximum principle to prove the global existence of smooth solutions for a slightly supercritical surface quasi-geostrophic equation. By this we mean that the velocity field $u$ is obtained from the active scalar $\theta$ by a Fourier multiplier with symbol $i k^\perp |k|^{-1} m(k|)$, where $m$ is a smooth increasing function that grows slower than $\log \log |k|$ as $|k|\rightarrow \infty$.