{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:29:03Z","timestamp":1787228943899,"version":"build-2736575974"},"reference-count":52,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Multiscale Model. Simul."],"published-print":{"date-parts":[[2013,1]]},"abstract":"<jats:p>We study linkages of two scales in the relaxation of an axisymmetric crystal with a facet in evaporation-condensation kinetics. The macroscale evolution is driven by the motion of concentric circular, repulsively interacting line defects (steps) which exchange atoms with the vapor. At the microscale, the step velocity is proportional to the variation of the total step free energy, leading to large systems of differential equations for the step radii. We focus on two step flow models. In one model (called M1) the discrete mobility is simply proportional to the upper-terrace width; in another model (M2) the mobility is altered by an extra geometric factor. By invoking self-similarity at long time, we numerically demonstrate that (i) in M1, discrete slopes follow closely a continuum thermodynamics approach with \u201cnatural boundary conditions\u201d at the facet edge; (ii) in contrast, predictions of M2 deviate from results of the above continuum approach; and (iii) this discrepancy can be eliminated via a continuum boundary condition with a geometry-induced jump for top-step collapses. At the macroscale, both step models give rise to free-boundary problems for a second-order, parabolic partial differential equation, which we study via the subgradient formalism. We discuss the interpretation of the facet height as shock and prove convergence of the solution of each discrete scheme to the (weak) entropy solution of a conservation law if steps do not interact.<\/jats:p>","DOI":"10.1137\/110849687","type":"journal-article","created":{"date-parts":[[2013,1,29]],"date-time":"2013-01-29T11:01:31Z","timestamp":1359457291000},"page":"244-281","source":"Crossref","is-referenced-by-count":2,"title":["Discrete and Continuum Relaxation Dynamics of Faceted Crystal Surface in Evaporation Models"],"prefix":"10.1137","volume":"11","author":[{"given":"Kanna","family":"Nakamura","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dionisios","family":"Margetis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2013,1,29]]},"reference":[{"key":"atypb1","unstructured":"H. Al Hajj Shehadeh,\n                      The Evolution of a Crystal Surface: Step ODEs, PDEs and Self-similarity\n                      , Ph.D. thesis, Courant Institute, New York University, New York, 2010."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2011.07.016"},{"key":"atypb3","unstructured":"G. Birkhoff and S. MacLane,\n                      A Survey of Modern Algebra\n                      , MacMillan, New York, 1959."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/BF00620292"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1951.0006"},{"key":"atypb6","first-page":"398","volume":"29","author":"Chame A.","year":"1996","journal-title":"Bulgarian Chem. Commun."},{"key":"atypb7","first-page":"13","author":"Coddington E. A.","year":"1955","journal-title":"Boston"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02124750"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1023\/A:1010361711825"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1063\/1.1726787"},{"key":"atypb11","unstructured":"P.W. Fok,\n                      Simulation of Axisymmetric Stepped Surfaces with a Facet\n                      , Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 2006."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.76.033408"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.78.235401"},{"key":"atypb14","unstructured":"Y. Giga,\n                      Shocks and very strong vertical diffusion\n                      , in Lectures on Partial Differential Equations: Proceedings in Honor of Louis Nirenberg's 75th Birthday, New Studies in Advance Mathematics, Vol. 2, S.Y. A. Chang, C.S. Lim, and H.T. Yau, eds., International Press, Somerville, MA, 2003, pp. 95-102."},{"key":"atypb15","first-page":"121","author":"Giga Y.","year":"2003","journal-title":"Philadelphia"},{"key":"atypb16","first-page":"73","author":"Giga M.-H.","year":"2003","journal-title":"RI"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.2969\/aspm\/03110093"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/0022-3697(67)90017-0"},{"key":"atypb19","first-page":"12","author":"Hale J. K.","year":"1969","journal-title":"RI"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1063\/1.1699658"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.82.87"},{"key":"atypb22","unstructured":"M. H. Holmes,\n                      Introduction to Perturbation Methods\n                      , Springer, New York, 1998."},{"key":"atypb23","unstructured":"B. R. Hunt, R. L. Lipsman, J. E. Osborn, and J. Rosenberg,\n                      Differential Equations with Matlab\n                      , Wiley, New York, 2005."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.60.5946"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.62.13707"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-5729(98)00010-7"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1023\/A:1004570921372"},{"key":"atypb28","unstructured":"R. V. Kohn, T. S. Lo, and N. K. Yip,\n                      Continuum limit of a step flow model of epitaxial growth\n                      , in Statistical Mechanical Modeling in Materials Science, MRS Symposia Proceedings 701, M. C. Bartelt, J. W. Evans, A. S. Karma, S. Torquato, and D. E. Wolf, eds., Materials Research Society, Warrendale, PA, 2002, pp. T1.7.1-T1.7.7."},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"P. Lax,\n                      Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves\n                      , SIAM, Philadelphia, 1973.","DOI":"10.1137\/1.9781611970562"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160130205"},{"key":"atypb31","unstructured":"R. J. LeVeque,\n                      Numerical Methods for Conservation Laws\n                      , 2nd ed., Birkha\u0308user, Basel, 2005."},{"key":"atypb32","first-page":"129","volume":"52","author":"Marchenko V. I.","year":"1980","journal-title":"Sov. Phys. JETP"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.71.165432"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.97.096102"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/06065297X"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2011.03.007"},{"key":"atypb37","doi-asserted-by":"crossref","unstructured":"T. Michely and J. Krug,\n                      Islands, Mounds and Atoms: Patterns and Processes in Crystal Growth Far From Equilibrium\n                      , Springer, Berlin, 2004.","DOI":"10.1007\/978-3-642-18672-1"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.82.981"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1016\/0001-6160(58)90020-8"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1016\/0039-6028(94)90269-0"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1051\/jphys:0198700480100160500"},{"key":"atypb42","unstructured":"I. V. Odisharia,\n                      Simulation and Analysis of the Relaxation of a Crystalline Surface\n                      , Ph.D. thesis, Courant Institute, New York University, New York, 2006."},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.42.5013"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.88.206103"},{"key":"atypb45","doi-asserted-by":"crossref","unstructured":"A. Pimpinelli and J. 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