Existence of hyperbolic blow-up to the generalized quasi-geostrophic equation

Abstract

In this work, we investigate the blow-up of solutions to the generalized surface quasi-geostrophic (gSQG) equation in ${\mathbb{R}}^2$, within the more singular range $\beta \in (1,2)$ for the coupling of the velocity field. This behavior is studied under a hyperbolic setting based on the framework originally introduced by Córdoba (1998, Annals of Math. 148, 1135--52) for the classical SQG equation. Assuming that the level sets of the solution contains a hyperbolic saddle, and under suitable conditions on the solution at the origin, we obtain the existence of a time $T^{\ast }\in {\mathbb{R}}^{+}\cup \{\infty \}$ at which the opening angle of the saddle collapses. Moreover, we derive a lower bound for the blow-up time $T^{\ast }$. This geometric degeneration leads to the blow-up of the Hölder norm $\Vert \theta (t){\Vert }_{C^{\sigma }}$ as $t\to T^{\ast }$, for $\sigma \in (0,\beta -1)$, showing the formation of singularity in the Hölder space at time $T^{\ast }$. To the best of our knowledge, these are the first results in the literature to rigorously prove the formation of a singularity, whether in finite or infinite time, for a class of smooth solutions to the gSQG equation.

We have identified technical issues in a key lemma of the preprint and have therefore decided to withdraw it while we work on an attempt to resolve them