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第25回PDE実解析研究会 (北大数学COE協賛)

PDE Real Analysis Seminar

Contents

Program

組織委員：
新井仁之（東大），儀我美一（東大／北大）
幹事：
石井仁司（早大），河添　健（慶大），剣持信幸（千葉大），酒井　良（都立大），柴田良弘（早大），望月　清（中央大），宮地晶彦（東女大），山崎昌男（早大）
日　　時：
2006年7月12日(水) 10:30-11:30
場　　所：
東京大学大学院　数理科学研究科056号室
※会場へのアクセスは下記にてご確認下さい。
駒場アクセスマップ
http://www.u-tokyo.ac.jp/campusmap/map02_02_j.html
駒場キャンパス数理科学研究科棟
http://www.u-tokyo.ac.jp/campusmap/cam02_01_27_j.html
講 演 者：
Piotr Rybka 氏（Warsaw University）
演　　題：
Analysis of a crystal growth model
ABSTRACT:
We are concerned with mathematical model of a single crystal growing from vapor. Mathematically this is an exterior, one-phase Stefan-type problem with Gibbs-Thomson law. We restrict our attention to an idealization of a ice crystal, i.e. our evolving free boundary is a circular cylinder. The system under consideration consists of an equation for the motion of the free boundary (the crystal surface) coupled to the quasi-steady approximation of the diffusion equation for the supersaturation of vapor. We present analysis of the system, we show well-posedness and draw the phase portrait, we use here the fact that we need just to variable to describe evolution of a cylinder.

We are mostly concerned with the shape-persitency problem of the evolution. The problem is, the Gibbs-Thomson relation is in fact a driven, weighted, mean, singular curvature flow and it is not obvious that the shape of the initial interface will persists throughout the evolution or even for some time. In order to solve this problem we show existence of the region in the phase plane which is a neighborhood of a unique steady state, such that in this region the shape of the cylinder is preserved. However, this set is not invariant with respect to dynamics of the problem.

It is a very interesting question what happens to surface of our crystal at the boundary of the shape-persitency (or shape stability) region. This problem in its full generality is open. However, we give some insight by studying the Gibbs-Thomson relation with a given driving, which inherits properties of the coupling to the diffusion field. We study the resulting driven weighted mean curvature flow for graphs and some special closed Lipschitz curves. We show well-posedness of the problem, but mainly we exhibit the phenomenon of bending flat parts of the curve, which grow "too big".