Equations with Singular Diffusivity

摘要

Recently models of faceted crystal growth and of grain boundaries were proposed based on the gradient system with nondifferentiable energy. In this article, we study their most basic forms given by the equations u_t=(u_x/|u_x|)_x and u_t=(1/a)(au_x/|u_x|)_x, where both of the related energies include a |u_x| term of power one which is nondifferentiable at u_x=0. The first equation is spatially homogeneous, while the second one is spatially inhomogeneous when a depends on x. These equations naturally express nonlocal interactions through their singular diffusivities (infinitely large diffusion constant), which make the profiles of the solutions completely flat. The mathematical basis for justifying and analyzing these equations is explained, and theoretical and numerical approaches show how the solutions of the equations evolve.