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Study of evolution equations from the aspects of the theory of minimizing movements

从最小化运动理论研究演化方程

基本信息

批准号：
16540186
负责人：
KIKUCHI Koji
金额：
$ 2.05万
依托单位：
Shizuoka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

Minimizing movement is proposed by De Giorgi which is based on theories used in the studies of partial differential equations and mean curvature flows. This research was projected in order to investigate the problems which appear in the following studies: (A) Study of minimizing movements in mathematical physics and geometry, (B) Study of minimizing movements associated with second order quasilinear hyperbolic partial differential equations. In the first year Seminar on Partial Differential Equations and Its Applications was held at Pukyong National University, Pusan, Korea, and, head investigator attended this conference, announced his recent result and gathered information. Besides, during the term of the project the head investigator and other investigators attended vari-ous conferences and discussed with specialists in related research areas. Thereby following research results are obtained. The most progresses are obtained in Study (B). It has been expected that ap-plication by minimizing movement method is equivalent to Yosida approximation, however the head investigator presents an example and shows that they are different from each other. Furthermore this example is also an example of a minimizing movement which does not satisfy energy conser-vation law. The equation which appears in the example does not satisfy uniqueness of a solution and hence it also turns out that uniqueness is not important in existence of a minimizing movement. Namely, this research has obtained important but negative facts. Probably structures of minimiz-ing movements associated with second order quasilinear hyperbolic partial differential equations are much more complicated than one expects at first. Some facts related to Study (A) are also obtained. However, the results seem to be the halfway stage and the future investigations are expected.

最小化运动由 De Giorgi 提出，它基于偏微分方程和平均曲率流研究中使用的理论。本研究旨在调查以下研究中出现的问题：（A）数学物理和几何中最小化运动的研究，（B）最小化与二阶拟线性双曲偏微分方程相关的运动的研究。第一年偏微分方程及其应用研讨会在韩国釜山国立釜庆大学举行，首席研究员出席了这次会议，宣布了他的最新成果并收集了信息。此外，项目期间，首席研究员和其他研究人员参加了各种会议，并与相关研究领域的专家进行了讨论。从而得到以下研究结果。研究（B）取得了最大的进展。人们预计最小化运动方法的应用与 Yosida 近似等效，但首席研究员举了一个例子并表明它们彼此不同。此外，这个例子也是不满足能量守恒定律的最小化运动的例子。示例中出现的方程不满足解的唯一性，因此也证明唯一性在最小化运动的存在中并不重要。也就是说，这项研究获得了重要但负面的事实。与二阶拟线性双曲偏微分方程相关的最小化运动的结构可能比人们一开始预期的要复杂得多。还获得了与研究（A）相关的一些事实。然而，结果似乎已经进行了一半，未来的调查值得期待。

项目成果

期刊论文数量（82）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Global solvability for systems of nonlinear wave equations with multiple speeds

多速度非线性波动方程组的全局可解性

DOI：
发表时间：
2005
期刊：
Differential and Integral Equations 17巻
影响因子：
0
作者：
Edited by Toshiaki;Shoji;Masaki Kashiwara;Noriaki Kawanaka 他;Y. Shibata and S. Shimizu;Akira Hoshiga;Akira Hoshiga and Hideo Kubo
通讯作者：
Akira Hoshiga and Hideo Kubo

Decay properties of the Stokes semigroup in exterior domains with Neumann boundary condition

具有诺依曼边界条件的外域斯托克斯半群的衰变性质