Uniqueness and existence of spirals moving by forced mean curvature motion

TL;DR

This paper investigates the existence and uniqueness of spiral motions driven by forced mean curvature in the plane, addressing singularities and establishing well-posedness for the associated parabolic equations.

Contribution

It introduces a comparison principle for a quasi-linear PDE with singularities and proves well-posedness of the Cauchy problem for spiral motions in the plane.

Findings

Comparison principle for the PDE with singularities

Well-posedness of the Cauchy problem in viscosity solutions

Existence of smooth solutions under compatibility conditions

Abstract

In this paper, we study the motion of spirals by mean curvature type motion in the (two dimensional) plane. Our motivation comes from dislocation dynamics; in this context, spirals appear when a screw dislocation line reaches the surface of a crystal. The first main result of this paper is a comparison principle for the corresponding parabolic quasi-linear equation. As far as motion of spirals are concerned, the novelty and originality of our setting and results come from the fact that, first, the singularity generated by the attached end point of spirals is taken into account for the first time, and second, spirals are studied in the whole space. Our second main result states that the Cauchy problem is well-posed in the class of sub-linear weak (viscosity) solutions. We also explain how to get the existence of smooth solutions when initial data satisfy an additional compatibility condition.