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

TL;DR

This paper investigates the well-posedness of the generalized surface quasi-geostrophic (SQG) equation, showing stability of existence intervals with respect to the parameter alpha and highlighting the subtlety of potential singularities for solutions with alpha > 0.

Contribution

It establishes the continuous dependence of solution existence intervals on alpha and introduces new uniform estimates for singular integrals in the generalized SQG context.

Findings

Existence interval stability for solutions as alpha varies.

New uniform estimates for singular integrals and commutator estimates.

Highlighting the subtlety of singularity formation for alpha > 0.

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 $K(x)=\frac{x^\perp}{|x|^{2+2\alpha}}, 0\le\alpha\le \frac12.$ 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.