学术报告（偏微分方程讨论班）

题目：Existence and speed determining of traveling waves in an undulating cylinder (joint work with Hiroshi Matano)
报告人：娄本东（同济大学教授，研究方向：曲率流方程和周期行波解）
时 间：11月25日（周五）16：00－17：00
地 点：理科大楼A1414
摘 要：
We study a curvature-dependent motion of curves in a two-dimensional cylinder in $mathbb{R}^2$ with periodically undulating boundaries. The law of motion is given by $V=kappa + A$, where $V$ is the normal velocity of the curve, $kappa$ is the curvature, and $A$ is a positive constant. We first establish a necessary and sufficient condition for the existence of periodic traveling waves, then we study how the average speed of the periodic traveling wave depends on the geometry of the domain boundaries. More specifically, we consider the homogenization problem as the period of the boundary undulation, denoted by $varepsilon$, tends to zero, and determine the homogenization limit of the avarage speed of periodic traveling waves. Quite surprisingly, this homogenized speed depends only on the maximum opening angle of the domain boundaries. Our analysis also shows that, for any small $varepsilon>0$, the average speed of the traveling wave is smaller than $A$, which equals the speed of the planar front. This implies that boundary undulation always lowers the speed of the traveling waves.