标题： 广义准地转方程的双曲爆破解的存在性 显示英文标题

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

摘要： 在本工作中，我们研究了广义表面准地转（gSQG）方程在$\mathbb{R}^{2}$中解的爆破，在速度场耦合的更奇异范围$\beta\in(1,2)$内。 这种行为是在基于 Córdoba（1998, Annals of Math. 148, 1135--52）最初为经典 SQG 方程引入的框架下的双曲设定中进行研究的。 假设解的等值线包含一个双曲鞍点，并且在原点处的解满足适当的条件，我们得到了一个时间$T^{\ast}\in\mathbb{R}^{+}\cup\{\infty\}$的存在，此时鞍点的开角会坍塌。 此外，我们推导了爆破时间$T^\ast$的下界。 这种几何退化导致霍尔德范数$\Vert\theta(t)\Vert_{C^{\sigma}}$随$t\rightarrow T^{\ast}$而爆破，对于$\sigma\in(0, \beta -1)$，表明在霍尔德空间中时间$T^{\ast}$时奇点的形成。据我们所知，这些是文献中首次严格证明了gSQG方程光滑解类在有限或无限时间内形成奇点的结果。 显示英文摘要

摘要： 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.