Various degenerate diffusion equations exhibit a waiting time phenomenon: Dependening on the "flatness" of the compactly supported initial datum at the boundary of the support, the support of the solution may not expand for a certain amount of time. We show that this phenomenon is captured by particular Lagrangian discretizations of the porous medium and the thin-film equations, and we obtain suffcient criteria for the occurrence of waiting times that are consistent with the known ones for the original PDEs. Our proof is based on estimates on the fluid velocity in Lagrangian coordinates. Combining weighted entropy estimates with an iteration technique a la Stampacchia leads to upper bounds on free boundary propagation. Numerical simulations show that the phenomenon is already clearly visible for relatively coarse discretizations.

中文翻译：

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空间离散多孔介质和薄膜方程中的等待时间现象

各种退化扩散方程都表现出等待时间现象：依赖于支撑边界处紧支撑初始数据的“平坦度”，解的支撑在一定时间内可能不会膨胀。我们表明这种现象是由多孔介质的特定拉格朗日离散和薄膜方程捕获的，并且我们获得了足够的等待时间发生的标准，这些标准与原始 PDE 的已知等待时间一致。我们的证明是基于对拉格朗日坐标中流体速度的估计。将加权熵估计与 a la Stampacchia 的迭代技术相结合，可以得出自由边界传播的上限。数值模拟表明，对于相对粗糙的离散化，这种现象已经清晰可见。