Mathematical analysis of interface problems in mathematical physics

数学物理中界面问题的数学分析

• 批准号: 14540171
• 负责人: SHIMIZU Senjo
• 金额: $ 2.24万
• 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: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540171/
• 关键词: Stokes Equations; Neumann boundary condition; Resolvent problems; L_p estimates; Analytic semigroups; L_p-L_q estimates; Free boundary problems; Local energy decay; 最大正則性; 自由境界値問題; 有界領域; 外部領域; 界面問題; L_p評価; 非有界領域

In this research, we consider the Stokes equation with Neumann boundary condition which is obtained as a linearized equation of the free boundary problem for the Navier-Stokes equation. We analyzed this problem by the following procedure : (1) Analysis of the resolvent problem (2) Generation of Analytic semigroups (3) L_p-L_q estimates(1)Obtained is the L_p estimate of solutions to the resolvent problem for Stokes system with Neumann type boundary condition in a bounded or exterior domain in R^n. The result has been obtained by Grubb and Solonnikov by the systematic use of theory of pseudo-differential operators. In this paper, we give an essentially different proof from theirs. The core of my approach is to estimate the solutions in the whole space and half-space case. We apply the Fourier multiplier theorem to solution of the model problems.(2)First we introduce the Helmholtz decomposition. Then we delete pressure trem and reduce to the problem only including velocity vector. Then we generated analytic semigroup to this reduced Stokes equation.(3)We obtained local energy decay estimates and L_p-L_q estimates of the solutions to the Stokes equation with Neumann boudary condition. Comparing with the non-slip (Dirichlet) boundary condition case, we have a better decay estimate for the gradient of the semigroup because of the null net force at the boundary.

在本研究中，我们考虑了带有Neumann边界条件的Stokes方程，它是Navier-Stokes方程自由边界问题的线性化方程。我们通过以下步骤分析了这个问题：（1）预解问题的分析（2）解析半群的生成（3）L_p-L_q估计（1）获得了R^n中有界或外部区域上具有Neumann型边界条件的Stokes系统的预解问题解的L_p估计。Grubb和Solonnikov系统地利用伪微分算子理论得到了这个结果。本文给出了一个与他们的证明有本质区别的证明。我的方法的核心是估计全空间和半空间情况下的解。我们应用傅立叶乘子定理来解决模型问题。（2）首先介绍Helmholtz分解。然后去掉压力项，将问题简化为只包含速度矢量的问题。然后对这类约化的Stokes方程生成解析半群。(3)We得到了具有Neumann-bouquet条件的Stokes方程解的局部能量衰减估计和L_p-L_q估计。与无滑移（Dirichlet）边界条件的情形相比，由于边界净力为零，我们对半群的梯度有了更好的衰减估计。

项目成果

Yoshihiro Shibata, Senjo Shimizu: "Some resolvent estimates for the Stokes system in bounded and exterior domains."Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis (Hirosaki, 2001). 451-461 (2003)

Yoshihiro Shibata、Senjo Shimizu：“对有界域和外部域中的斯托克斯系统的一些解析估计。”非线性分析和凸分析国际会议记录（Hirosaki，2001）。

Existence and blowing up of solutions to systems of quasilinear wave equations

拟线性波动方程组解的存在性与爆炸

• 发表时间: 2005
• 期刊: Adv.Math.Sci.Appl. 15(掲載予定)
• 作者: Akira Hoshiga;Hideo Kubo;Akira Hoshiga
• 通讯作者: Akira Hoshiga

Yoshihiro Shibata, Senjo Shimizu: "On a resolvent estimate for the Stokes system with Neumann boundary condition"Differential Integral Equations. 発表予定(to appear). (2003)

Yoshihiro Shibata，Senjo Shimizu：“关于具有诺伊曼边界条件的斯托克斯系统的解析估计”微分积分方程出现（2003）。

Wave front sets of the Riemann function of elastic interface problems

弹性界面问题的黎曼函数波前集

• 发表时间: 2004
• 期刊: Math. Meth. Appl. Sci. 27巻
• 作者: Akira Hoshiga;Hideo Kubo;Akira Hoshiga;Shinji Adachi;Shinji Adachi;Senjo Shimizu
• 通讯作者: Senjo Shimizu

Local energy decay of solutions to the Oseen equation in the exterior domains

外部域 Oseen 方程解的局部能量衰减

• 发表时间: 2004
• 期刊: Indiana Univ.Math.J. 53・5
• 作者: Yoshihiro Shibata;Y.Enomoto
• 通讯作者: Y.Enomoto