Uniqueness criterion of weak solutions for the dissipative quasi-geostrophic equations in Orlicz-Morrey spaces

Consider the quasi-geostrophic equations with the initial data. Let and be two weak solutions with the same initial value. If where is the Orlicz-Morrey space (for a definition of this space, see Definition), then we have. In view of the embedding with and, we see that our result improves the previous result of Dong and Chen. This is an extension of earlier regularity results in the Serrin's type space.