Abstract
The stability of smooth solutions of an anisotropic surface quasi-geostrophic equation with horizontal dissipation remains an open problem. In this work, we present a partial answer to this problem in a rougher function space. More precisely, if the initial data θ0 belong to anisotropic Sobolev space H0,s with 12<s<1, then there exists a global small solution, provided that the initial data are small in H0,s. The stability problem in the space H0,1 or more regular spaces remains open. The main tools used are the Littlewood–Paley theory and standard techniques.
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