The radius of analyticity for solutions to a problem in epitaxial growth on the torus

TL;DR

This paper proves that solutions to a specific epitaxial growth model on the torus become analytic over time, with the radius of analyticity increasing linearly, under a smallness condition on the initial data.

Contribution

It establishes the analyticity and growth rate of the radius of solutions to the epitaxial growth model on the torus, with explicit smallness conditions on initial data.

Findings

Solutions become analytic at positive times

Radius of analyticity grows linearly over time

Smallness condition on initial data in Wiener algebra

Abstract

A certain model for epitaxial film growth has recently attracted attention, with the existence of small global solutions having being proved in both the case of the n-dimensional torus and free space. We address a regularity question for these solutions, showing that in the case of the torus, the solutions become analytic at any positive time, with the radius of analyticity growing linearly for all time. As other authors have, we take the Laplacian of the initial data to be in the Wiener algebra, and we find an explicit smallness condition on the size of the data. Our particular condition on the torus is that the Laplacian of the initial data should have norm less than 1/4 in the Wiener algebra.