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, $$\partial_t θ- \nabla^\perp \log(10+(-Δ)^{\frac12})θ\cdot \nabla θ= 0 ,$$ 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 $δ$-SQG equations, defined by $$\partial_t θ+ \nabla^\perp (10+(-Δ)^{\frac12})^{-δ}θ\cdot \nabla θ= 0,$$ for all sufficiently small $δ>0$ depending on the size of the initial data. For the same range of $δ$, we establish global well-posedness of smooth solutions to the logarithmically dissipative counterpart: $$\partial_t θ+ \nabla^\perp (10+(-Δ)^{\frac12})^{-δ}θ\cdot \nabla θ+ \log(10+(-Δ)^{\frac12})θ= 0.$$