Well-posedness for Ohkitani model and long-time existence for surface quasi-geostrophic equations

TL;DR

This paper proves local well-posedness and long-time existence results for a logarithmically singular surface quasi-geostrophic equation, improving prior results and analyzing the effects of dissipation on solution behavior.

Contribution

It establishes local existence and uniqueness of smooth solutions for the Ohkitani model with decreasing Sobolev exponents and extends results to long-time dynamics of δ-SQG equations.

Findings

Local well-posedness in Sobolev spaces with decreasing exponent

Ill-posedness in fixed Sobolev spaces shown in companion paper

Global well-posedness for dissipative δ-SQG equations with small δ

Abstract

We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, $$\partial_t \theta - \nabla^\perp \log(10+(-\Delta)^{\frac12})\theta \cdot \nabla \theta = 0 ,$$ and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae, Constantin, C\'{o}rdoba, Gancedo, and Wu in \cite{CCCGW}. This well-posedness result can be applied to describe the long-time dynamics of the $\delta$-SQG equations, defined by $$\partial_t \theta + \nabla^\perp (10+(-\Delta)^{\frac12})^{-\delta}\theta \cdot \nabla \theta = 0,$$ for all sufficiently small $\delta>0$ depending on the size of the initial data. For the same range of $\delta$, we establish global well-posedness of smooth solutions to the logarithmically dissipative counterpart: $$\partial_t \theta + \nabla^\perp (10+(-\Delta)^{\frac12})^{-\delta}\theta \cdot \nabla \theta + \log(10+(-\Delta)^{\frac12})\theta = 0.$$