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

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 ${\dot{B}}_{2,1}^2$, 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.