Generalized surface quasi‐geostrophic equations with singular velocities

Abstract

AbstractThis 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 quasigeostrophic (SQG) equation with the velocity fieldurelated to the scalarθby$u=\nabla^\perp\Lambda^{\beta-2}\theta$, where$1<\beta\le 2$and$\Lambda=(-\Delta)^{1/2}$is the Zygmund operator. The borderline caseβ= 1 corresponds to the SQG equation and the situation is more singular forβ> 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$u=\nabla^\perp (\log(I-\Delta))^\mu\theta\ {\rm for}\ \mu>0$, 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. © 2012 Wiley Periodicals, Inc.