Vol. 3, No. 4, 2021

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Analysis of a fourth-order exponential PDE arising from a crystal surface jump process with Metropolis-type transition rates

Yuan Gao, Anya E. Katsevich, Jian-Guo Liu, Jianfeng Lu and Jeremy L. Marzuola

Vol. 3 (2021), No. 4, 595–612
Abstract

We analytically and numerically study a fourth-order PDE modeling rough crystal surface diffusion on the macroscopic level. We discuss existence of solutions globally in time and long-time dynamics for the PDE model. The PDE, originally derived by Katsevich is the continuum limit of a microscopic model of the surface dynamics, given by a Markov jump process with Metropolis-type transition rates. We outline the convergence argument, which depends on a simplifying assumption on the local equilibrium measure that is valid in the high-temperature regime. We provide numerical evidence for the convergence of the microscopic model to the PDE in this regime.

Keywords
crystal surface relaxation, gradient flows, metropolis rates
Mathematical Subject Classification 2010
Primary: 35Q70, 82D25
Milestones
Received: 16 March 2020
Revised: 19 November 2020
Accepted: 29 April 2021
Published: 12 February 2022
Authors
Yuan Gao
Department of Mathematics
Purdue University
West Lafayette, IN
United States
Anya E. Katsevich
Courant Institute of Mathematical Sciences
New York University
New York City, NY
United States
Jian-Guo Liu
Departments of Mathematics and Physics
Duke University
Durham, NC
United States
Jianfeng Lu
Departments of Mathematics, Physics and Chemistry
Duke University
Durham, NC
United States
Jeremy L. Marzuola
Department of Mathematics
University of North Carolina
Chapel Hill, NC
United States