Uniqueness criterion of weak solutions for the dissipative quasi-geostrophic equations in Orlicz-Morrey spaces

Consider the quasi-geostrophic equations with the initial data theta(0) is an element of L-2 (R-n). Let theta and (theta) over tilde be two weak solutions with the same initial value theta(0). If del theta is an element of L2 alpha/2 alpha-r ([0, T], M-L2LogPL (n/r)) with 0 <= r <= alpha, where M-L2LogPL (n/r) (R-n) is the OrliczMorrey space (for a definition of this space, see Definition 2.1), then we have theta = (theta) over tilde. In view of the embedding L-n/r subset of M-p(n/r) subset of M-L2LogPL (n/r) with 2 < p <= n/r and p > 1, we see that our result improves the previous result of Dong and Chen. [Asymptotic stability of the critical and super-critical dissipative quasigeostrophic equation. Nonlinearity. 19 (2006), 2919-2928]. This is an extension of earlier regularity results in the Serrins type space L-q ([0, T], L-p (R-2)).