The waiting time phenomenon in spatially discretized porous medium and thin film equations

TL;DR

This paper investigates the waiting time phenomenon in degenerate diffusion equations, demonstrating that certain Lagrangian discretizations can accurately capture this behavior and providing criteria for its occurrence.

Contribution

It introduces criteria for the waiting time phenomenon in discretized porous medium and thin-film equations, aligning with the continuous PDE behavior.

Findings

Waiting times are captured by specific Lagrangian discretizations.

Criteria for waiting times are consistent with continuous PDEs.

Numerical simulations confirm the phenomenon appears even with coarse discretizations.

Abstract

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.