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

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 $\Delta u\ge 0$ 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.

References in corpus (1)

• The relaxation of a family of broken bond crystal surface models