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

摘要

A certain model for epitaxial film growth has recently attracted attention, with the existence of small global solutions having been 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 (in fact, the result holds for data a bit larger than this).