Equations with singular diffusivity
Ryo Kobayashi, Yoshikazu Giga · EPrints - Department of Mathematics, Hokkaido University · 1998
Recently the models of the faceted crystal growth and of the grain boundary were proposed based on the gradient system with nondifferentiable energy. In this article, we study their most basic forms given by the equations ut = (ux/|ux|)x and ut = a/1(a * ux / ux)x where both of the related energy include luxl term of power one which is nondiffer entiable at Ux = 0. The first equation is spatially homogeneous while the second one is spatially inhomogeneous when a depends on x. These equations naturally express non-local interactions through their singular diffusivities ( Ux = 0) which make the profiles of the solutions completely flat. Mathematical basis for justifying and analyzing these equations will be explained, and also it will be shown how the solutions of the equations evolve by theoretical and numerical approach.