[00607] Analysis and computation of interface evolution equation and related topics
Session Time & Room : 5C (Aug.25, 13:20-15:00) @F411
Type : Proposal of Minisymposium
Abstract : Analyzing and computational methods for interfacial motion including some singularities or topological changes has been continued to develop and applied to various fields. Recently, these methods are extended to the problems with strong singularity and constraint, nonlocal evolution law, or coupling system and other phenomena. In these developments, there has been high demand for fast and accurate computing, and rigorous mathematical analysis of parametric or non-parametric interface motion. This minisymposium will feature the recent developments on modelling, computation and analysis for interface evolution equation involving the above motivation and related topics.
[03814] A minimizing movement approach to surface constrained interfacial motions
Format : Talk at Waseda University
Author(s) :
Elliott Ginder (Meiji University)
Abstract : By extending the applicability of minimizing movements to the surface PDE setting, we will develop threshold dynamics for surface-constrained interfacial motions. In particular, we will show how our approach enables one to approximate multiphase, volume-preserving, curvature flows on surfaces via generalized MBO and HMBO algorithms.
[02801] Geometric Sobolev gradient flows on spaces of curves
Format : Talk at Waseda University
Author(s) :
Philip Schrader (Murdoch University)
Abstract : The curve-shortening flow, which deforms a closed planar curve by moving its points perpendicular to the curve with velocity proportional to curvature, was proposed by Mullins as a model for the motion of grain boundaries in the process of annealing. It can be characterised as the gradient descent of the length functional on curves, when the gradient is taken with respect to a parametrisation invariant $L^2$ inner product. In this talk I will describe some of the gradient flows which result when taking instead the gradient with respect to some Sobolev parametrisation invariant inner products. I will discuss the different kinds of asymptotic behaviour that are possible and also the numerical advantages of $H^1$ products.
[03938] A Simple Algorithm for the Monge-Ampere Equation on a Sphere
Format : Talk at Waseda University
Author(s) :
Richard Tsai (The University of Texas at Austin)
Axel Turnquist (University of Texas at Austin)
Abstract : In this talk, we present a novel approach for solving the Monge-Ampere (MA) equation defined on a sphere. Specifically, we extend the MA equation on a sphere to a narrowband around the sphere by formulating an equivalent optimal transport problem. We demonstrate that the extended MA equation can be solved using existing algorithms developed for the MA equation on Euclidean space, making the resulting algorithm simple and easy to implement. Our approach provides a useful tool for solving problems that involve the MA equation defined on or near a sphere, which has a wide range of applications in fields such as computer graphics, image processing, and fluid dynamics.
[02029] Waiting time effects for the wearing process of a non-convex stone
Format : Talk at Waseda University
Author(s) :
Nao Hamamuki (Hokkaido University)
Ryosuke Takahashi (Hokkaido University)
Abstract : We investigate evolution of a non-convex stone by the wearing process. Following the formulation introduced by Ishii and Mikami 2001, 2004, we study the unique viscosity solution of a nonlocal Gauss curvature flow equation describing the wearing process and prove that waiting time effects occur on an appropriate subset in a cavity of the stone, that is, any point on the set does not move at all for some positive time.
