发表机构
Friedrich–Alexander–Universität Erlangen–Nürnberg; Max Planck Institute for Mathematics in the Sciences（埃尔朗根-纽伦堡弗里德里希·亚历山大大学; 马克斯·普朗克科学促进研究所数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带非线性保守噪声的随机多孔介质方程，证明其动力学解的有限传播速度，提出新颖迭代技术推导Stampacchia型不等式，确定初值平坦性条件以保证等待时间现象发生，结果与预期行为一致。
AI 中文摘要
从局域能量估计出发，我们证明了带非线性保守噪声的随机多孔介质方程的动力学解具有有限传播速度，这类解的存在性与唯一性近期已得到确立。特别地，我们提出一种新颖的迭代技术，尽管多孔介质项与噪声项可能具有不同的标度行为，该技术仍能让我们得到仅含一个积分量的Stampacchia型不等式。这使我们可以应用针对线性源型噪声情况开发的随机滤波论证。利用相关思路，我们确定了初值的平坦性条件，该条件保证解的支集的正向传播在局域上出现等待时间现象，即其支集的正向传播起始在局域上被延迟。在噪声为临界非线性的情况下，该条件与确定性多孔介质方程的对应条件一致，仅需一个对数修正项；但当非线性趋向线性保守噪声时，该条件要求初值具有越来越强的平坦性。这与预期行为相符：在线性保守噪声情况下，由于随机输运效应，无论初值剖面多平坦，解都可能发生瞬时正向运动。
英文摘要
Starting from localized energy estimates, we prove finite speed of propagation for kinetic solutions to stochastic porous media equations with nonlinear conservative noise, the existence and uniqueness of which has recently been established. In particular, we propose a novel iteration technique which allows us to obtain a Stampacchia-type inequality involving one single integral quantity, despite the possibly different scaling behaviors of the porous media and the noise term. This allows us to apply stochastic filtering arguments developed for the case of linear source-type noise. Using related ideas, we identify flatness conditions on initial data which guarantee locally the occurrence of a waiting time phenomenon, i.e., the onset of forward propagation of the solution's support is locally delayed. The condition for the latter matches the one for the deterministic porous media equation up to a logarithmic correction in the case of critical nonlinearity in the noise, but it requires more and more flatness of the initial data as the nonlinearity tends towards linear conservative noise. This is in line with the expected behavior: In the case of linear conservative noise, instantaneous forward motion is possible due to the effects of stochastic transport, no matter how flat the initial profile is.
