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

Haitao Fan, Shi Jin
2003 Quarterly of Applied Mathematics  
The bistable reaction-diffusion-convection equation dtu + V ■ f(u): = -g(u) + eAu, x G Rn, u £ R (1) is considered. Stationary traveling waves of the above equation are proved to exist when f(u) is symmetric and g(u) is antisymmetric about u = 0. Solutions of initial value problems tend to almost piecewise constant functions within 0{l)e time. The almost constant pieces are separated by sharp interior layers, called fronts. The motion of these fronts is studied by asymptotic expansion. The
more » ... expansion. The equation for the motion of the front is obtained. In the case of f = bu2 and g(u) = au(l -u2), where b G R™ and 0 < a G R are constants, the front motion equation takes a more explicit form, showing that the front's speed is V/U _ K+ -■ T where k is the mean curvature of the front, /i 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 k and V/x/// • T are elliptic operators, contributing to the shrinkage of closed curves. An ellipse in R2 is found to preserve its shape while shrinking.
doi:10.1090/qam/2019619 fatcat:mc2byk5qxrexlpyzcns2jthgem