A caricature of a singular curvature flow in the plane

TL;DR

This paper investigates a simplified singular curvature flow in one dimension, establishing existence, uniqueness, and detailed analysis of facet formation and interactions through semi-discretization methods.

Contribution

It introduces a new approach to analyze singular curvature flows, proving existence and uniqueness of solutions and detailing facet dynamics and asymptotic behavior.

Findings

Facets must form during the flow.

Existence and uniqueness of weak solutions are established.

Detailed analysis of facet interactions and asymptotics.

Abstract

We study a singular parabolic equation of the total variation type in one dimension. The problem is a simplification of the singular curvature flow. We show existence and uniqueness of weak solutions. We also prove existence of weak solutions to the semi-discretization of the problem as well as convergence of the approximating sequences. The semi-discretization shows that facets must form. For a class of initial data we are able to study in details the facet formation and interactions and their asymptotic behavior. We notice that our qualitative results may be interpreted with the help of a special composition of multivalued operators.