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

Abstract

We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, $${\partial }_t\theta -{\nabla }^{\perp }\log (10+(-\Delta {)}^{\frac{1}{2}})\theta \cdot \nabla \theta =0,$$ 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, Córdoba, 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 $${\partial }_t\theta +{\nabla }^{\perp }(10+(-\Delta {)}^{\frac{1}{2}}{)}^{-\delta }\theta \cdot \nabla \theta =0,$$ 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: $${\partial }_t\theta +{\nabla }^{\perp }(10+(-\Delta {)}^{\frac{1}{2}}{)}^{-\delta }\theta \cdot \nabla \theta +\log (10+(-\Delta {)}^{\frac{1}{2}})\theta =0.$$

21 pages, to appear in Comm. Math. Phys