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

Consider the β-quasi-geostrophic equations with the initial data 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<3r,where M.p,q(R2) denotes the homogeneous Morrey–Campanato space, then we have θ=θ˜. Due to the embedding relation Lp(R2)⊂M.p,q(R2) with 1 < p < q < ∞, we see that our result is an improvement of the corresponding result given by Zhao and Liu (2013). © 2016 Elsevier Inc.