CNA Seminar

Yoshikazu Giga
University of Tokyo
Title: On some macroscopic PDE models for evolving phase boundaries

Abstract: When studying crystal growth one observes the occurrence of flat surfaces called facets. Such behaviour happens especially at low temperatures. From a macroscopic thermodynamical point of view one explains this by a singular interfacial energy. This leads us to a model equation describing the motion of a crystal surface. It is a very singular diffusion equation where the speed is determined by a nonlocal quantity, namely the area of a facet. Such models for crystalline flow in materials science also describe total variation flow (which is well-known in image processing). Mathematical analysis on these equations is highly nontrivial. There are several approaches including a viscosity and a variational approach. There are also fourth order models describing relaxation dynamics of a crystal surface. Compared with second order problems its analysis becomes is even harder because comparison principle does not apply. In this talk we survey mathematical analysis of both second and fourth models with emphasis on recent developments. This lecture is based on my joint work with M.-H. Giga, R. V. Kohn, P. Rybka.

References: (1) M.-H. Giga and Y. Giga, Very singular diffusion equations - second and fourth order problems, Japanese J. Ind. Appl. Math., 27 (2010), 323-345. (2) Y. Giga and R. V. Kohn, Scale-invariant extinction time estimates for some singular diffusion equations, Discrete and Continuous Dynamical Systems, 30 (2011), no. 2, 509-535. (3) M.-H. Giga, Y. Giga and P. Rybka, P. A comparison principle for singular diffusion equations with spatially inhomogeneous driving force for graphs, Hokkaido University Preprint Series in Math. #981 http://eprints3.math.sci.hokudai.ac.jp/2160/

Date: Tuesday, April 30, 2013
Time: 1:30 pm
Location: Wean 7128
Submitted by:  David Kinderlehrer