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 H ̇ − 1 / 2 . The first one shows that we can construct a unique H ̇ − 1 / 2 -solution when the initial data belongs moreover to L ∞ with a small L ∞ norm. The other one gives the uniqueness of a H ̇ − 1 / 2 -solution which belongs to C ( [ 0 , T ) , CMO ) .