Generalized surface quasi-geostrophic equations with singular velocities

Abstract

This paper establishes several existence and uniqueness results for two fam- ilies of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasi- geostrophic (SQG) equation with the velocity field u related to the scalar  by u D r ? ƒ ˇ 2 , where 1 < ˇ  2 and ƒ D ./ 1=2 is the Zygmund operator. The borderline case ˇ D 1 corresponds to the SQG equation and the situation is more singular for ˇ > 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local exis- tence of patch-type solutions. The second family is a dissipative active scalar equation with u D r ? .log.I  //   for  > 0, which is at least logarithmi- cally more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step to- wards resolving the global regularity issue recently proposed by K. Ohkitani.