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

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 C\'{o}rdoba (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 H\"{o}lder norm $\Vert\theta(t)\Vert_{C^{\sigma}}$ as $t\rightarrow T^{\ast}$, for $\sigma\in(0, \beta -1)$, showing the formation of singularity in the H\"{o}lder 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.