Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Front motion in multi-dimensional viscous conservation laws with stiff source terms driven by mean curvature and variation of front thickness


Authors: Haitao Fan and Shi Jin
Journal: Quart. Appl. Math. 61 (2003), 701-721
MSC: Primary 35K57; Secondary 35B25, 35L65
DOI: https://doi.org/10.1090/qam/2019619
MathSciNet review: MR2019619
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Abstract: The bistable reaction-diffusion-convection equation

$\displaystyle {\partial _t}u + \nabla \cdot f\left( u \right) = {- \frac{1}{\ep... ...a u}, \qquad {x \in \mathbb{R}{^n}}, {u \in \mathbb{R}} \qquad \left( 1 \right)$

is considered. Stationary traveling waves of the above equation are proved to exist when $ f\left( u \right)$ is symmetric and $ g\left( u \right)$ is antisymmetric about $ u = 0$. Solutions of initial value problems tend to almost piecewise constant functions within $ O\left( 1 \right)\epsilon $ time. The almost constant pieces are separated by sharp interior layers, called fronts. The motion of these fronts is studied by asymptotic expansion. The equation for the motion of the front is obtained. In the case of $ f = b{u^2}$ and $ g\left( u \right) = au\left( 1 - {u^2} \right)$, where $ b \in {\mathbb{R}^{n}}$ and $ 0 < a \in \mathbb{R}$ are constants, the front motion equation takes a more explicit form, showing that the front's speed is

$\displaystyle \epsilon \left( {k + \frac{{\nabla \mu }}{\mu } \cdot T} \right)$

, where $ \kappa $ is the mean curvature of the front, $ \mu $ is the width of the planar traveling of (1) in the normal direction n of the front, and T is a vector tangential to the front. Both $ \kappa $ and $ \nabla \mu /\mu \cdot T$ T are elliptic operators, contributing to the shrinkage of closed curves. An ellipse in $ {\mathbb{R}^{2}}$ is found to preserve its shape while shrinking.

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DOI: https://doi.org/10.1090/qam/2019619
Article copyright: © Copyright 2003 American Mathematical Society


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