TY - JOUR

T1 - The evolution of a crystal surface

T2 - Analysis of a one-dimensional step train connecting two facets in the ADL regime

AU - Shehadeh, Hala Al Hajj

AU - Kohn, Robert V.

AU - Weare, Jonathan

N1 - Funding Information:
We thank John Ball for the observation that the “energy in similarity variables” is convex when viewed as a function of . The work of Al Hajj Shehadeh and Kohn was supported in part by NSF through grant DMS-0807347 and that of Weare was supported in part by the Applied Mathematical Sciences Program of the US Department of Energy under Contract DEFG0200ER25053.

PY - 2011/10/15

Y1 - 2011/10/15

N2 - We study the evolution of a monotone step train separating two facets of a crystal surface. The model is one-dimensional and we consider only the attachmentdetachment-limited regime. Starting with the well-known ODEs for the velocities of the steps, we consider the system of ODEs giving the evolution of the "discrete slopes." It is the l2-steepest-descent of a certain functional. Using this structure, we prove that the solution exists for all time and is asymptotically self-similar. We also discuss the continuum limit of the discrete self-similar solution, characterizing it variationally, identifying its regularity, and discussing its qualitative behavior. Our approach suggests a PDE for the slope as a function of height and time in the continuum setting. However, existence, uniqueness, and asymptotic self-similarity remain open for the continuum version of the problem.

AB - We study the evolution of a monotone step train separating two facets of a crystal surface. The model is one-dimensional and we consider only the attachmentdetachment-limited regime. Starting with the well-known ODEs for the velocities of the steps, we consider the system of ODEs giving the evolution of the "discrete slopes." It is the l2-steepest-descent of a certain functional. Using this structure, we prove that the solution exists for all time and is asymptotically self-similar. We also discuss the continuum limit of the discrete self-similar solution, characterizing it variationally, identifying its regularity, and discussing its qualitative behavior. Our approach suggests a PDE for the slope as a function of height and time in the continuum setting. However, existence, uniqueness, and asymptotic self-similarity remain open for the continuum version of the problem.

KW - Epitaxial relaxation

KW - Facet

KW - Self similar solution

KW - Steepest descent

UR - http://www.scopus.com/inward/record.url?scp=80052936913&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052936913&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2011.07.016

DO - 10.1016/j.physd.2011.07.016

M3 - Article

AN - SCOPUS:80052936913

SN - 0167-2789

VL - 240

SP - 1771

EP - 1784

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 21

ER -