Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations

TL;DR

This paper introduces a family of active scalar equations generalizing the 2D Euler and SQG equations, establishing global regularity results for specific cases and developing tools to analyze the regularity of solutions.

Contribution

It develops new analytical tools to bound velocity gradients and proves global regularity for a novel Loglog-Euler model, extending understanding of active scalar equations.

Findings

Proved global regularity for the Loglog-Euler equation with specific operator P.

Established a regularity criterion for models with P(Λ)=Λ^β.

Developed bounds for the gradient of velocity in a general class of operators.

Abstract

Any classical solution of the 2D incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component $u_j$ of the velocity field $u$ is determined by the scalar $\theta$ through $u_j =\mathcal{R} \Lambda^{-1} P(\Lambda) \theta$ where $\mathcal{R}$ is a Riesz transform and $\Lambda=(-\Delta)^{1/2}$. The 2D Euler vorticity equation corresponds to the special case $P(\Lambda)=I$ while the SQG equation to the case $P(\Lambda) =\Lambda$. We develop tools to bound $\|\nabla u||_{L^\infty}$ for a general class of operators $P$ and establish the global regularity for the Loglog-Euler equation for which $P(\Lambda)= (\log(I+\log(I-\Delta)))^\gamma$ with $0\le \gamma\le 1$. In addition, a regularity criterion for the model corresponding to $P(\Lambda)=\Lambda^\beta$ with $0\le \beta\le 1$ is also obtained.