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 ${u=K\ast\omega}$, where ${\omega=\omega(x,t)}$ is an unknown function and ${K(x)=\frac{x^\perp}{|x|^{2+2\alpha}}, 0 \leqq \alpha \leqq \frac12.}$ When ${\alpha=0}$, the equation involves 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 ${0 \leqq \alpha_0 \leqq \frac12}$ 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 remain on [0, T]. As a byproduct, our results imply 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. (Ann Math 184(3):909–948, 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.