The approach of an analytic semigroup for free boundary problems of viscous compressible fluids

粘性可压缩流体自由边界问题的解析半群方法

• 批准号: 17540156
• 负责人: SHIMIZU Senjo
• 金额: $ 2.11万
• 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: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540156/
• 关键词: Free boundary problems; Analytic semigroups; Lp-maximal regularity; Stokes equations; Navier-Stokes equations; R-boundedness; Fourier multiplier theorem; 非圧縮性粘性流体; 最大正則性; ストークス方程式; 準線形方程式; 2相流体; ノイマン境界条件

In this research, we consider a free boundary problem for the Navier-Stokes equation which describes the motion of an isolated finite volume of viscous incompressible fluid without taking surface tension into account By using the Lagrange coordinates, free boundary problem is written by the quasi-linear equations on the fixed boundary. Our purpose is to prove a local in time unique existence theorem for any initial data and a global in time unique existence theorem for some small initial data.To treat quasi-linear equations, first we prove the Lp-Lq maximal regularity of solutions to the linearized problem, which is described by the Neumann problem for the Stokes equation. We consider this problem by analytic semigroup approach. Our main issues is to use R-boundedness and operator valued Fourier multiplier theorem which are recently developed by Weis ('01, Math.Ann.) and Denk-Hieber-Pruss ('03, Mem.AMS).Based on the Lp-Lq maximal regularity result of the linearized problem, by using the contraction mapping principle, we proved a local in time unique existence theorem for any initial data and external force and a global in time unique existence theorem for some small initial data which are orthogonal to the rigid space in the case where external force vanishes.

在这项研究中，我们考虑纳维-斯托克斯方程的自由边界问题，该方程描述了孤立的有限体积的粘性不可压缩流体的运动，而不考虑表面张力。通过使用拉格朗日坐标，自由边界问题由拟固定边界上的线性方程。我们的目的是证明任意初始数据的局部时间唯一存在定理和一些小的初始数据的全局时间唯一存在定理。为了处理拟线性方程，首先我们证明解的 Lp-Lq 最大正则性线性化问题，由斯托克斯方程的诺伊曼问题描述。我们通过解析半群方法来考虑这个问题。我们的主要问题是使用最近由 Weis ('01, Math.Ann.) 和 Denk-Hieber-Pruss ('03, Mem.AMS) 开发的 R 有界性和算子值傅里叶乘数定理。基于 Lp- Lq最大正则性结果的线性化问题，利用收缩映射原理，证明了任意初始数据和外力的局部时间唯一存在性定理和全局时间唯一存在性外力消失情况下与刚性空间正交的一些小初始数据的定理。

项目成果

On a free boundary problem for the Navier-Stokes equations

• DOI: 10.57262/die/1356039501
• 发表时间: 2007-01
• 期刊: Differential and Integral Equations
• 影响因子: 1.4
• 作者: Y. Shibata;Senjo Shimizu

On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier-Stokes equation

外域Oseen半群的衰变率及其在Navier-Stokes方程中的应用

• 发表时间: 2005
• 期刊: J.Math.Fluid Mech. 7・no.3
• 作者: Y.Enomoto;Yoshihiro Shibata
• 通讯作者: Yoshihiro Shibata

On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

• DOI: 10.1515/crelle.2008.013
• 发表时间: 2008-02
• 作者: Y. Shibata;Senjo Shimizu

Free boundary problems for a viscous incompressible fluid

• 发表时间: 2018
• 期刊: 2021 IEEE/CVF International Conference on Computer Vision Workshops (ICCVW)
• 作者: Y. Shibata

Globally in time existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle

旋转障碍物外部纳维-斯托克斯流的全局时间存在定理

• 发表时间: 2006
• 期刊: WSEAS Trans. Math. 5
• 作者: Toshiaki Hishida;Yoshihiro Shihata
• 通讯作者: Yoshihiro Shihata