On the locally self-similar blowup for the generalized SQG equation

Abstract

We analyze finite-time blowup scenarios of locally self-similar type for the inviscid generalized surface quasi-geostrophic equation (gSQG) in $\mathbb{R}^2$. Under an $L^r$ growth assumption on the self-similar profile and its gradient, we identify appropriate ranges of the self-similar parameter where the profile is either identically zero, and hence blowup cannot occur, or its $L^p$ asymptotic behavior can be characterized, for suitable $r, p$. Our results extend the work [Xue; Journal of Differential Equations, 2016] regarding the SQG equation, and also partially recover the results proved in [Cannone, Xue; Proceedings of the American Mathematical Society, 2015] concerning globally self-similar solutions of the gSQG equation.