Remarks on weak-strong uniqueness of the generalized quasi-geostrophic equation in critical Besov spaces

This paper is devoted to studying the improved weak-strong uniqueness of the 2D generalized surface quasi-geostrophic equation with the singular velocity $u=\Lambda ^{1-\beta}R^{\bot}\theta $. Based on the continuity of the trilinear form in critical Besov spaces and energy methods, we proved that the solution $\theta (x, t)$ is unique in the class of weak solutions, if $\theta (x, t)$ lies in the following critical Besov spaces $$ \theta (x, t)\in L^{q}(0,T;B^{s}_{p,\infty}(\mathbb{R}^{2}))\ \ ~~\text{with} ~~\frac{2}{p}+\frac{\alpha }{q}=\alpha +\beta +s-2~~and~~ \frac{2}{\alpha +\beta +s-2}< p< \infty . $$ This result extends the uniqueness criterion from Lebesgue space to critical Besov spaces.