Focusing of an elongated hole in porous medium flow

In the focusing problem, we study solutions to the porous medium equation ∂tu=Δ(um) whose initial distributions are positive in the exterior of a compact 2D region and zero inside. We assume that the initial interface is elongated (i.e., has an aspect ratio >1), and possesses reflectional symmetry with respect to both the x- and y-axes. We implement a numerical scheme that adapts the numerical grid around the interface so as to maintain a high resolution as the interface shrinks to a point. We find that as t tends to the focusing time T, the interface becomes oval-like with lengths of the major and minor axes O(T−t) and O(T−t), respectively. Thus the aspect ratio is O(1/T−t). By scaling and formal asymptotic arguments we derive an approximate solution which is valid for all m. This approximation indicates that the numerically observed power law behavior for the major and minor axes is universal for all m>1.