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

Abstract

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 .

Keywords:

• uniqueness
• Orlicz–Morrey spaces
• quasi-geostrophic equations

AMS Subject Classifications:

• 76U05
• 76B03
• 35Q35

Acknowledgments

The authors want to express their sincere thanks to the referee for his/her invaluable comments and suggestions which helped improve the paper greatly. This work is partially supported by Grant-in-Aid for Young Scientists (B) No. 21740104, Japan Society for the Promotion of Science, by the Global COE program at the Graduate School of Mathematical Sciences, the University of Tokyo, and by Grant-in-Aid for Scientific Research (C) (No. 23540187), the Japan Society for the Promotion of Science. We are indebted to Professor Akihiko Miyachi for his pointing () out.