On the uniqueness of the quasi-geostrophic equation with the fractional Laplacian
Tsukasa Iwabuchi, Taiki Okazaki
TL;DR
This work establishes uniqueness for the surface quasi-geostrophic equation with a fractional Laplacian in non-homogeneous Besov spaces, without imposing extra constructive assumptions. The authors employ mild-solution theory, dyadic (frequency) decompositions, and a detailed network of bilinear, product, and commutator estimates in Besov spaces to treat a full range of fractional powers $0<\alpha\leq 2$. They prove endpoint uniqueness results, notably in $C([0,T];B_{p,p/(p-2)}^{-1/2})$ with $p=4/(2\alpha-3)$ for $\tfrac{3}{2}<\alpha\leq 2$, and in $C([0,T];B_{\infty,1}^{-1/2})$ when $\alpha=3/2$ under a bounded Riesz-transform condition on the initial data; for $0<\alpha<3/2$, uniqueness is shown in $C([0,T];B_{\infty,\infty}^{1-\alpha})$ under similar initial-data regularity. The results advance the understanding of well-posedness at rough, near-critical regularities and provide a refined landscape for SQG uniqueness in Besov spaces. They also illuminate how the nonlinear term can be handled at minimal regularity through a second-derivative representation and careful frequency analysis. The findings have implications for the broader study of geophysical-fluid models and related PDEs in low-regularity function spaces.
Abstract
We consider the uniqueness of the solution of the surface quasi-geostrophic equation with fractional Laplacian. We show that the uniqueness holds in non-homogeneous Besov spaces without any additional assumption which is supposed to constract solutions. When the power of the fractional Laplacian is close to 2, we prove that the uniqueness with the regularity index $s=-1/2$. We extract the least regularity $s=-1/2$ for the well-definedness of the nonlinear term of the equation.