Formation of discontinuities in flux-saturated degenerate parabolic equations

Alina Chertock, Alexander Kurganov, Philip Rosenau

Research output: Contribution to journalArticlepeer-review

Abstract

We endow the nonlinear degenerate parabolic equation used to describe propagation of thermal waves in plasma or in a porous medium, with a mechanism for flux saturation intended to correct the nonphysical gradient-flux relations at high gradients. We study both analytically and numerically the resulting equation: ut = [un Q(g(u)x)]x, n > 0, where Q is a bounded increasing function. This model reveals that for n > 1 the motion of the front is controlled by the saturation mechanism and instead of the typical infinite gradients resulting from the linear flux-gradients relations, Q ∼ ux, we obtain a sharp, shock-like front, typically associated with nonlinear hyperbolic phenomena. We prove that if the initial support is compact, independently of the smoothness of the initial datum inside the support, a sharp front discontinuity forms in a finite time, and until then the front does not expand.

Original languageEnglish
Pages (from-to)1875-1898
Number of pages24
JournalNonlinearity
Volume16
Issue number6
DOIs
StatePublished - Nov 2003

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