Weak-strong uniqueness criteria for the critical quasi-geostrophic equation

We give two weak-strong uniqueness results for the weak solutions to the critical dissipative quasi-geostrophic equation when the initial data belongs to Ḣ−1/2. The first one shows that we can construct a unique Ḣ−1/2-solution when the initial data belongs moreover to L∞ with a small L∞ norm. The other one gives the uniqueness of a Ḣ−1/2-solution which belongs to C([0,T),CMO).