Remarks on well-posedness of the generalized surface quasi-geostrophic equation

Abstract

In this paper, we are concerned with the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation in which the velocity field is expressed as $u=K\ast \ddot{I}‰$, where $\ddot{I}‰=\ddot{I}‰(x,t)$ is an unknown function and $K(x)=\frac{x^{\perp }}{|x{|}^{2+2\hat{I}\pm }},0\le \hat{I}\pm \le \frac{1}{2}.$ When $\hat{I}\pm =0$, it is the two-dimensional Euler equations. When $\hat{I}\pm =\frac{1}{2}$, it corresponds to the inviscid SQG. We will prove that if the existence interval of the smooth solution to the generalized SQG for some $0<\hat{I}{\pm }_0\le \frac{1}{2}$ is $[0,T]$, then under the same initial data, the existence interval of the generalized SQG with $\hat{I}\pm$ which is close to $\hat{I}{\pm }_0$ will keep on $[0,T]$. As a byproduct, our result implies that the construction of the possible singularity of the smooth solution of the Cauchy problem to the generalized SQG with $\hat{I}\pm >0$ will be subtle, in comparison with the singularity presented in [Kiselev et al 2016]. To prove our main results, the difference between the two solutions and meanwhile the approximation of the singular integrals will be dealt with. Some new uniform estimates with respect to $\hat{I}\pm$ on the singular integrals and commutator estimates will be shown in this paper.

Accepted by Archive for Rational Mechanics and Analysis