On Gevrey Regularity of the Supercritical Sqg Equation in Critical Besov Spaces

In this paper, we show that the solution of the supercrtical surface quasi-geostrophic (SQG) equation, with initial data in a critical Besov space, belongs to a subanalytic Gevrey class. In order to prove this, a suitable estimate on the nonlinear term, in the form of a commutator, is required. We express the commutator as a bilinear multiplier operator and obtain single-scale estimates for its symbol. In particular, we show that the localized symbol is of Marcinkiewicz type, and show that due to the localizations inherited from working in the Besov spaces, this condition implies the requisite boundedness of the corresponding operator. This result strengthens previous ones which showed that solutions starting from initial data in critical Besov spaces are classical. As a direct consequence of our method, decay estimates of higher-order derivatives are easily deduced.