Global regularity for critical SQG in bounded domains

TL;DR

This paper establishes the existence and uniqueness of smooth solutions for the critical dissipative SQG equation in bounded domains, solving an open problem using a novel transformation and maximum principle approach.

Contribution

It introduces a new method transforming the SQG equation into an extended system in the whole space, enabling the proof of global regularity in bounded domains.

Findings

Proves global regularity for critical SQG in bounded domains.

Develops a new transformation technique for nonlocal equations.

Utilizes a nonlinear maximum principle for nonlocal operators.

Abstract

We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in $\mathbb R^2$. This solves an open problem. We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.