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=Λ 1-βR^θ. Based on the continuity of the trilinear form in critical Besov spaces and energy methods, we proved that the solution θ (x, t) is unique in the class of weak solutions, if θ (x, t) lies in the following critical Besov spaces θ (x, t)∈ Lq(0,T;Bˢp,∞(R²))\ \ ~~with ~~2/p+α /q=α +β +s-2~~and~~ 2/α +β +s-2< p< ∞ . This result extends the uniqueness criterion from Lebesgue space to critical Besov spaces.