GEOMETRIC CRITERIA FOR INVISCID 2D SURFACE QUASIGEOSTROPHIC EQUATIONS

It remains an open question if all classical solutions of the inviscid surface quasigeostrophic (SQG) equation are global in time or not. In this article, this issue is addressed through a geometric approach. This article contains three sections. The first section introduces the SQG equation, and presents existing results along with open problems. The second section presents local uniqueness and existence results of the SQG equations. Finally, the third section presents several geometric criteria under which the solutions of the SQG equation become regular for all time. The relation between the geometry of the level curves and the regularity of the solutions is the central focus of this part.