Focusing of an elongated hole in porous medium flow

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Focusing of an elongated hole in porous medium flow

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In the focusing problem we study solutions to the porous medium equation $u_t=\Delta u^m$ whose initial distributions are positive in the exterior of a compact two-dimensional region and zero inside. We assume that the initial interface is elongated 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 the lengths of the major and minor axes $O(\sqrt{T-t})$ and $O(T-t)$ respectively. Thus, the aspect ratio is $O(1/\sqrt{T-t})$. 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 behavior for the major and minor axes is universal for all m>1.





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Angenent, Sigurd; Aronson, D.G.; Betelu, Santiago; Lowengrub, J.S.. (2001). Focusing of an elongated hole in porous medium flow. Retrieved from the University Digital Conservancy,

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