Unique existence of solutions to the inviscid SQG equation in a critical space

TL;DR

This paper proves the unique existence of strong solutions to the inviscid SQG equation in a critical Besov space within a bounded domain, using spectral localization and uniform estimates.

Contribution

It establishes the first proof of unique strong solutions in a critical space for the SQG equation with Dirichlet boundary conditions.

Findings

Unique solutions exist in the critical Besov space $ ext{dot} B^2_{2,1}$.

Spectral localization techniques are effective for this problem.

Uniform estimates are key to constructing solutions.

Abstract

We study the Cauchy problem for the surface quasi-geostrophic (SQG) equations in a two-dimensional bounded domain with the homogeneous Dirichlet boundary condition. We establish the unique existence of strong solutions in the critical Besov space $\dot B^2_{2,1}$, which is embedded in $C^1$. The proof is based on spectral localization using dyadic decomposition associated with the Dirichlet Laplacian. We obtain the solution by establishing uniform estimates for a sequence of solutions to the equation with a regularized nonlinear term.