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* ω, where ω= ω(x, t) is an unknown function and K(x)=x⊥|x|2+2α,0≦α≦12. When α= 0 , the equation involves the two-dimensional Euler equations. When α=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≦α0≦12 is [0,T], then under the same initial data, the existence interval of the generalized SQG with α which is close to α 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 α> 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 α on the singular integrals and commutator estimates will be shown in this paper.