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

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640159/
• 关键词: Variational problem; Nonlinear partial differential equations; Numerical Analysis; Minimizing methods; Free boundary problem

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).

本文主要研究与变分问题有关的偏微分方程的离散莫尔斯半流问题。我们的主要兴趣是对集的奇点的解决方案。这样的集合有时会有很大的能量集中在它上面，所以我们可以认为我们的目的是处理解的奇异性上的能量集中现象。本文从这一观点出发，研究了以下几类问题：（1）发展一个求解极小化问题的并行机，（2）发展一个通过极小化过程的数值方法，（3）发展一个通过极小化求解抛物型和双曲型方程的方法，针对这些问题，我们发展了一个求解极小化问题的8-CPU并行机。利用这一点，我们对程函方程、Ginzburg-Landau系统和液晶问题的奇异性结构进行了数值变换。利用离散的莫尔斯半流方法，给出了抛物型和双曲型方程的基本方法，并解决了BBM-Burgers方程孤立波解的渐近性态，开发了求解双曲型自由边界问题的软件。这是基于方程的光滑化方法，即使自由边界拓扑发生变化，也能得到很好的结果，我们将这些结果归纳为8篇论文（已发表或出版）和2篇预印本（已提交）。

项目成果

K.Kikuchi,S.Omata: "A free boundary problem for a one dimensional hyperbolic equation"Adv.Math.Sci.Appl.. 10 No.1. 775-786 (1999)

K.Kikuchi，S.Omata：“一维双曲方程的自由边界问题”Adv.Math.Sci.Appl.. 10 No.1。

H.Imai,K.Kikuchi,K.Nakane,S.Omata,T.Tachikawa: "A Numerical Approach to the Asymptotic Bbehavior of Solutions of a One-Dimensional Free Boundary Problem of Hyperbolic Type"Japan Journal of Industrial and Applied Mathematics. 18(1). 43-58 (2001)

H.Imai，K.Kikuchi，K.Nakane，S.Omata，T.Tachikawa：“双曲型一维自由边界问题解的渐近行为的数值方法”日本工业与应用数学杂志。

T.Nagasawa K.Nakane S.Omata: "Hyperbolic Ginzburg Landau system" Nonliear Analysis. to appear.

T.Nagasawa K.Nakane S.Omata：“双曲 Ginzburg Landau 系统”非线性分析。

S.Kinami, M.Mei and S.Omata: "Asymptotic Toward Diffusion Waves of the Solutions for Benjamin-Bona-Mahony-Burgers Equations"Applicable Analysis. 75, (3-4). 317-340 (2000)

S.Kinami、M.Mei 和 S.Omata：“Benjamin-Bona-Mahony-Burgers 方程解的渐近扩散波”应用分析。

S.Omata, T.Okamura and K.Nakane: "Numerical analysis for the discrete Morse semiflow related to the Ginzburg Landau functional"Nonlinear Analysis. 37, No.5. 589-602 (1999)

S.Omata、T.Okamura 和 K.Nakane：“与 Ginzburg Landau 泛函相关的离散莫尔斯半流的数值分析”非线性分析。