Mathematical Analysis of partial differential equations related to a variational problem via the discrete Morse Semiflows

通过离散莫尔斯半流对与变分问题相关的偏微分方程进行数学分析

基本信息

批准号：
11640159
负责人：
OMATA Seiro
金额：
$ 2.3万
依托单位：
Kanazawa University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

We mainly investigated partial differential equations related to a variational problem via the discrete Morse semiflows. Our main interest is on sets of singular points of a solutions. Such sets has sometimes big energy concentrate on it. So, we can cosider that our purpose is on treating the energy concentration phenomena on the singularity of solutions. In this stand point of view, we treated the following type of problems :(1) Develop a prallel machine for solving mininizing problems,(2) Develop a Numerical method via a minimization process,(3) Develop a method to solve both parabolic and hyperbolic equations via minimizing.For these problems, we have developped a 8-CPU parallel computer for solving minimizing problems. By use of this, we did a numerical copmutations to catch the structure of singularities for eikonal equation, Ginzburg-Landau system, and smestics liquid crystal problems. Basic method due to discrete Morse semiflow for parabolic and hyperbolic problems.We also solved the asymptotic behavior of solitary wave solutions for BBM-Burgers equations.Moreover we developped a software to solve hyperbolic free boundary problems. This is based on the smoothing method of a equation and we can get good results even when the free boundary changes its topology.We summed up these results into 8 papers (appeared or in press) and 2 preprint (submitted).

我们主要通过离散莫尔斯半流研究与变分问题相关的偏微分方程。我们的主要兴趣是解决方案的奇异点集。这样的组合有时会有很大的精力集中在上面。因此，我们可以认为我们的目的是处理解奇点上的能量集中现象。从这个角度来看，我们处理了以下类型的问题：（1）开发一个并行机来解决最小化问题，（2）通过最小化过程开发一种数值方法，（3）开发一种解决抛物线和通过最小化双曲方程。针对这些问题，我们开发了一个8-CPU并行计算机来解决最小化问题。利用这一点，我们进行了数值计算，以捕获 eikonal 方程、Ginzburg-Landau 系统和 smestics 液晶问题的奇点结构。基于离散莫尔斯半流的抛物线和双曲问题的基本方法。我们还解决了 BBM-Burgers 方程的孤立波解的渐近行为。此外，我们开发了一个软件来解决双曲自由边界问题。这是基于方程的平滑方法，即使自由边界改变其拓扑结构，我们也能得到很好的结果。我们将这些结果总结为8篇论文（已发表或正在出版）和2篇预印本（已提交）。