Note on the weak–strong uniqueness criterion for the β-QG in Morrey–Campanato space

Consider the β  -quasi-geostrophic equations with the initial data θ0∈L2(R2)θ0∈L2(R2). Let θ   and θ˜ be two weak solutions with the same initial value θ0. If θ satisfies the energy inequality and if ∇θ∈Lr0,T;M.p,q(R2)with2p+αr=α+β−1and2α+β−1