Consider the β -quasi-geostrophic equations with the initial data θ0∈L2(R2). Let θ and be two weak solutions with the same initial value θ0. If θ satisfies the energy inequality and if
where denotes the homogeneous Morrey–Campanato space, then we have . Due to the embedding relation with 1 < p < q < ∞, we see that our result is an improvement of the corresponding result given by Zhao and Liu (2013).