Mathematical Analysis on Change of Patterns by Anisotropy and Diffusion

各向异性和扩散引起的图案变化的数学分析

基本信息

批准号：
14204011
负责人：
GIGA Yoshikazu
金额：
$ 31.28万
依托单位：
University of Tokyo (2004-2005)Hokkaido University (2002-2003)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

项目摘要

Several mathematical foundation is established to investigate the mechanism of variation of shape of solutions due to anisotropy and diffusion effect for nonlinear diffusion equations. We just give two typical topics.(i)Equation describing motion of a crystal surface. When anisotropy of surface energy is strong like snow crystal, there appears a plat surface called a facet in their equilibrium shape. The equations governing its evolution is formally a curvature flow equation. However, the diffusion is so strong that it cannot be viewed as a partial differential equation. We studied a Stefan type free boundary problem describing snow crystal growth assuming that its equilibrium shape is a cylinder. We established (a)Local solvability under the assumption that a facet stays as a facet ; (b)Berg's effect concerning behavior diffusion field near on a facet ; (c)a necessary and sufficient condition for facet bending ; (d)a sufficient condition for existence of a self-similar solution and size condition on facet bending ; (e)a rigorous proof that a facet preserved near equilibrium. A further problem is whether one can track evolution after facet bending.(ii)Equations in fluid mechanics : An invicid Burgers equation is a typical coarse model to describe motion of compressible fluid. The solution develops jump discontinuity in finite time even if the initial data is smooth for the Burgers equation. Just before this proposal, the principal inverstigator introduced a notion of a proper viscosity solution to describe a solution with jumps which is also useful for equations with nondivergence type. In this project we established a notion of singular vertical diffusivity which is useful to calculate the solution numerically in the graph space. In particular we are successful to calculate a proper viscosity solution by a level set method numerically.

建立了几个数学基础，以研究由于各向异性和非线性扩散方程的扩散效应而引起的溶液形状变化的机理。我们只提供了两个典型的主题。（i）描述晶体表面运动的方程式。当表面能的各向异性像雪晶体一样强，出现一个平衡的平面表面，其平衡形状。控制其演变的方程式是正式的曲率流程方程。但是，扩散是如此之强，以至于无法将其视为部分微分方程。我们研究了Stefan型自由边界问题，描述了雪晶的生长，假设其平衡形状是圆柱体。我们建立了（a）局部解决性，假设一个方面是一个方面；（b）Berg在一个方面附近的行为扩散场的影响；（c）方面弯曲的必要条件；（d）有足够的条件，可以在弯曲弯曲方面存在自相似的解决方案和尺寸条件；（e）严格的证据表明，一个方面保存在平衡附近。另一个问题是一个人是否可以跟踪方面弯曲后的进化。（ii）流体力学中的方程：Invicid Burgers方程是描述可压缩流体运动的典型粗制模型。即使对汉堡方程的初始数据平滑，解决方案也会在有限的时间内发展不连续性。就在此提案之前，主要的InverStigator引入了适当的粘度解决方案的概念，以描述具有跳跃的解决方案，该解决方案对于非词类型的方程式也很有用。在这个项目中，我们建立了一个单数垂直扩散率的概念，该概念对于在图形空间中数值计算解决方案很有用。特别是，我们成功地通过数值级别设置方法来计算适当的粘度解决方案。