Global regularity in the supercritical generalized SQG regime

Prove global regularity of smooth solutions to the dissipative generalized surface quasi-geostrophic equations in the supercritical regime, where the dissipation parameter satisfies 0<α<1/2, for general smooth initial data.

Background

The paper discusses the dissipative generalized surface quasi-geostrophic (gSQG) family, which includes the SQG equation and is divided into subcritical, critical, and supercritical regimes according to the dissipation parameter α. Global regularity is known in the subcritical and critical regimes, but the corresponding problem for smooth solutions in the supercritical regime remains unresolved. The paper instead establishes global existence of weak solutions in Sobolev spaces and does not address smooth-solution regularity.