Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions

TL;DR

This paper proves the existence of weak solutions for a complex PDE modeling crystal surface evolution in multiple dimensions, highlighting its connection to a related nonlinear surface model.

Contribution

It provides the first analytical validation of a continuum PDE model for crystal surface evolution in multiple space dimensions.

Findings

Existence of weak solutions with non-negative Laplacian established.

Demonstrates the connection between two different crystal surface models.

Provides a mathematical foundation for the PDE model in crystal surface dynamics.

Abstract

In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation $\frac{\partial u}{\partial t} = \Delta\left(\Delta u\right)^{-3}$. This problem arises in the mathematical modeling of the evolution of a crystal surface. Existence of a weak solution $u$ with $\Delta u\geq 0$ is obtained via a suitable substitution. Our investigations reveal the close connection between this problem and the equation $\partial_t\rho+\rho^2\Delta^2\rho^3=0$, another crystal surface model first proposed by H. Al Hajj Shehadeh, R. V. Kohn and J. Weare in Physica D: Nonlinear Phenomena, bf 240 (2011), no. 21, 1771-1784.