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

Consider the beta-quasi-geostrophic equations with the initial data theta(0) epsilon L-2(R-2). Let theta and (theta) over tilde be two weak solutions with the same initial value theta(0). If theta satisfies the energy inequality and if del(theta) epsilon L(r)0, T; (M) over dot(p,q) (R-2) with 2/p + alpha/r = alpha + beta - 1 and 2/alpha + beta - 1 < p< 3/r, where (M) over dot(p,g)(R-2) denotes the homogeneous Morrey-Campanato space, then we have theta = (theta) over tilde. Due to the embedding relation L-p( R-2) subset of (M) over dot(p,q) with 1 < p < q < infinity, we see that our result is an improvement of the corresponding result given by Zhao and Liu (2013). (C) 2016 Elsevier Inc. All rights reserved.