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; 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.

在这项研究中，我们考虑了具有Neumann边界条件的Stokes方程，该方程是作为Navier-Stokes方程的自由边界问题的线性化方程。我们通过以下步骤分析了此问题：（1）分析问题（2）分析性分析的分析（3）获得的L_P-L_Q估计值（1）是在r^n中使用noumann类型边界结构域中使用Neumann类型边界条件的Stokes System解决方案解决方案的L_P估计值。结果是通过Grubb和Solonnikov通过系统使用伪差异算子的系统来获得的。在本文中，我们给出了与他们的基本不同的证据。我的方法的核心是估计整个空间和半空间情况下的解决方案。我们将傅立叶乘数定理应用于模型问题的解决方案。（2）首先，我们引入了Helmholtz分解。然后，我们删除压力trem并减少到包括速度向量在内的问题。然后，我们生成了该减少的Stokes方程的分析半群。（3）我们获得了具有Neumann Boudary条件的Stokes方程的溶液的局部能量衰减估计值和L_P-L_Q估计值。与非滑动（Dirichlet）边界条件情况相比，由于边界处的无效净力，我们对Semigroup的梯度有更好的衰减估计值。