Viscosity solutions for a polymer crystal growth model

TL;DR

This paper establishes the existence and regularity of solutions for a polymer crystal growth model involving a nonlocal velocity linked to a heat equation, advancing mathematical understanding of such growth processes.

Contribution

It introduces new regularity results for the eikonal equation with measurable velocities, enabling the proof of existence and regularity for the crystal growth model.

Findings

Existence of solutions for the polymer crystal growth model.

New regularity results for the eikonal equation with non-smooth velocities.

Regularity estimates for the associated heat equation solutions.

Abstract

We prove existence of a solution for a polymer crystal growth model describing the movement of a front $(\Gamma(t))$ evolving with a nonlocal velocity. In this model the nonlocal velocity is linked to the solution of a heat equation with source $\delta_\Gamma$. The proof relies on new regularity results for the eikonal equation, in which the velocity is positive but merely measurable in time and with H\"{o}lder bounds in space. From this result, we deduce \textit{a priori} regularity for the front. On the other hand, under this regularity assumption, we prove bounds and regularity estimates for the solution of the heat equation.