Singularity Formation and Energy Dissipation in Supercritical Quasi-Geostrophic Equations

This monograph investigates the analytical structure of the supercritical Surface Quasi-Geostrophic (SQG) equation, focusing on the interplay between fractional dissipation and nonlinear advection. We analyze the formation of finite-time singularities and the non-uniqueness of weak solutions in the regime where the dissipative operator (with < 1) fails to control the transport term. By employing convex integration techniques and De Giorgi-Nash-Moser iteration schemes, we delineate the threshold for global regularity versus blow-up. Furthermore, we examine the phenomenon of anomalous energy dissipation, establishing a connection between the Onsager conjecture for the 3D Euler equations and the Hölder regularity of the active scalar field in the SQG context. The results presented herein synthesize recent advances in the construction of "wild" weak solutions and provide rigorous criteria for the eventual regularization of large data solutions.