TY - JOUR

T1 - Formation of discontinuities in flux-saturated degenerate parabolic equations

AU - Chertock, Alina

AU - Kurganov, Alexander

AU - Rosenau, Philip

PY - 2003/11

Y1 - 2003/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0242510858&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/16/6/301

DO - 10.1088/0951-7715/16/6/301

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AN - SCOPUS:0242510858

SN - 0951-7715

VL - 16

SP - 1875

EP - 1898

JO - Nonlinearity

JF - Nonlinearity

IS - 6

ER -