发表机构
Instituto de Ciencias Matemáticas（数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
证明二维环面上无外力不可压缩多孔介质方程从光滑初值发生有限时间梯度爆破，通过恒等式将层梯度对数增长与负压力曲率关联，利用近垂直母层旋转放大新相位，非对称周期轮廓控制压力Hessian，经尺度归纳实现双导数分量爆破并收敛至终端密度。
AI 中文摘要
我们证明在二维环面上，无外力的不可压缩多孔介质方程会发生有限时间的梯度爆破，从而去除了Alpöge、Buckmaster和Coiculescu \cite{lit:ABC}以及Córdoba和Martínez-Zoroa \cite{lit:CMIPM}的IPM爆破构造中所使用的外力。关键在于一个恒等式，该恒等式将层的梯度的对数增长表示为垂直于其输运相位的负压力曲率。一个近乎垂直的母层旋转并放大一个新的相位，其增益随母层倾斜角的倒数呈指数增长。一个非对称周期轮廓为压力Hessian提供一致的上界，从而防止梯度幅度的快速衰减。每一层在零时刻被播种；其相位均值由终端位移问题选择，并且有限阶展开被修正为精确的无外力解。尺度归纳随后表明，在原点处两个导数分量发生爆破，并在每个$C^\eta$（$\eta<1$）中收敛到终端密度。
英文摘要
We prove finite-time gradient blow-up for the unforced incompressible porous media equation on the two-dimensional torus, removing the forcing used in the IPM blow-up constructions of Alpöge, Buckmaster, and Coiculescu \cite{lit:ABC} and Córdoba and Martínez-Zoroa \cite{lit:CMIPM}. The key is an identity that expresses the logarithmic growth of a layer's gradient as the negative pressure curvature transverse to its transported phase. A nearly vertical parent layer rotates and amplifies a new phase, with a gain exponential in the reciprocal parent tilt. An asymmetric periodic profile gives a uniform upper bound for the pressure Hessian, preventing rapid decay of the gradient magnitude. Each layer is seeded at time zero; its phase means are selected by a terminal displacement problem, and a finite order expansion is corrected to an exact unforced solution. The scale induction then shows blow-up of two derivative components at the origin and convergence to a terminal density in every $C^η$, $η<1$.
Comments74 pages
