Analysis of solutions to a model parabolic equation with very singular diffusion

TL;DR

This paper analyzes a singular parabolic equation with competing smoothing and facet-creating nonlinearities, describing the evolution of solutions via free boundary problems and providing laws for facet endpoints.

Contribution

It introduces a detailed analysis of the qualitative behavior and regularity of solutions, including a system of free boundary problems and evolution laws for facets.

Findings

Solutions exhibit smooth evolution between facet merging events.

The paper derives laws governing the motion of facet endpoints.

Local smoothness of the solution's unfaceted regions is established.

Abstract

We consider a singular parabolic equation of form \[ u_t = u_{xx} + \frac{\alpha}{2}(\mathrm{sgn}\,u_x)_x \] with periodic boundary conditions. Solutions to this kind of equations exhibit competition between smoothing due to one-dimensional Laplace operator and tendency to create flat facets due to strongly nonlinear operator $(\mathrm{sgn}\,u_x)_x$ coming from the total variation flow. We present results concerning analysis of qualitative behaviour and regularity of the solutions. Our main result states that locally (between moments when facets merge), the evolution is described by a system of free boundary problems for $u$ in intervals between facets coupled with equations of evolution of facets. In particular, we provide a proper law governing evolution of endpoints of facets. This leads to local smoothness of the motion of endpoints and the unfaceted part of the solution.