This paper establishes several existence and uniqueness results for two
families 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 θ by u = ∇⊥Λ β−2 θ, where 1 1. We obtain the local existence and uniqueness of classical solutions, the global existence of
weak solutions and the local existence of patch type solutions. The second family is a dissipative active scalar equation with u = ∇⊥(log(I − ∆))µθ for µ > 0, which is at least logarithmically 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 towards resolving the global regularity issue recently proposed by K. Ohkitani.National Research Foundation of KoreaNational Science FoundationMinisterio de Ciencia e InnovaciónEuropean Research Counci
