Generalized surface quasi-geostrophic equations with singular velocities

Abstract

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 $\theta$ by $u=\nabla^\perp\Lambda^{\beta-2}\theta$, where $1<\beta\le 2$ and $\Lambda=(-\Delta)^{1/2}$ is the Zygmund operator. The borderline case $\beta=1$ corresponds to the SQG equation and the situation is more singular for $\beta>1$. 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 $u=\nabla^\perp (\log(I-\Delta))^\mu\theta$ 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 \cite{Oh}.