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

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Abstract

In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation $$\partial_t u= \Delta(\Delta u)^{-3}$$. 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 \cite{AKW}.

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Journal
2016-12-13
1612.04269