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，我们考虑了Navier-Stokes方程的一个自由边界问题，该方程描述了孤立的有限体积粘性不可压缩的流体的运动，而无需使用Lagrange坐标，而无需考虑表面张力，而自由边界问题由固定边界上的准线性方程编写。我们的目的是证明任何初始数据的局部时间定理和全球范围的某些初始数据的独特存在定理。要处理准线性方程，首先，我们证明了线性化问题的LP-LQ最大规律性，这是由Stokes方程的Neumann问题描述的。我们通过分析的分析方法来考虑这个问题。 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存在与刚性空间正交的一些小初始数据的存在定理，如果外力消失。

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