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

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 $u=K\ast\omega$, where $\omega=\omega(x,t)$ is an unknown function and $K(x)=\frac{x^\perp}{|x|^{2+2\alpha}}, 0\le\alpha\le \frac12.$ When $\alpha=0$, it is the two-dimensional Euler equations. When $\alpha=\frac 12$, it corresponds to the inviscid SQG. We will prove that if the existence interval of the smooth solution to the generalized SQG for some $0<\alpha_0\le\frac12$ is $[0,T]$, then under the same initial data, the existence interval of the generalized SQG with $\alpha$ which is close to $\alpha_0$ 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 $\alpha>0$ 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 $\alpha$ on the singular integrals and commutator estimates will be shown in this paper.