Study on the Navier-Stokes equations on unbounded domains by way of real analysis

无界域纳维-斯托克斯方程的实分析研究

• 批准号: 21540202
• 负责人: YAMAZAKI Masao
• 金额: $ 2.75万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21540202/
• 关键词: 関数方程式; 偏微分方程式論; 安定性; Navier-Stokes方程式; 外部領域; 減衰; 対称性; 平行平板; Sobolev空間; Besov空間; Poiseuille流; 定常解

This research is concerned with the Navier-Stokes equations on either the whole plane or two-dimensional exterior domains. It was shown that, if there exists a small stationary external force with strong symmetry, the equation has a small stationary solution decaying rapidly at infinity. It was also shown that, if the stationary solution above is sufficiently small, it is stable under initial perturbation without restriction on the size. The Navier-Stokes equaions in an infinite layer is also studied. It is shown that, if the equation is treated in the Besov spaces, nontrivial solutions with no external forces exist if p is infinite, and that these solutions correspond to the Poiseuille flows.

本文研究了全平面或二维外部区域上的Navier-Stokes方程。结果表明，如果存在一个具有强对称性的小的定常外力，则方程有一个小的定常解在无穷远处迅速衰减。结果还表明，如果上述定常解足够小，它在初始摄动下是稳定的，而不受大小的限制。研究了无限大层中的Navier-Stokes方程。证明了如果在Besov空间中处理该方程，当p是无穷大时，存在无外力的非平凡解，并且这些解对应于Poiseuille流。

项目成果

On a resolvent estimate of the Stokes system in a half space arising from a free boundary problem for the Navier-Stokes equations

纳维-斯托克斯方程自由边界问题引起的半空间斯托克斯系统的求解估计

• DOI: 10.1002/mana.200710749
• 发表时间: 2009
• 期刊: Math. Nachr.
• 作者: Robert Denk;Reinhard Racke and Yoshihiro Shibata;Yoshihiro Shibata and Senjo Shimizu
• 通讯作者: Yoshihiro Shibata and Senjo Shimizu

The stationary Navier-Stokes equation on the whole plane with external force with antisymmetry

反对称外力作用下全平面平稳Navier-Stokes方程

• 发表时间: 2009
• 期刊: Annali dell'universita de Ferrara Sez.7, Scienze Matematiche 55
• 作者: Robert Denk;Reinhard Racke and Yoshihiro Shibata;Yoshihiro Shibata and Senjo Shimizu;Masao Yamazaki
• 通讯作者: Masao Yamazaki

The Fujita–Kato approach to the Navier–Stokes equations in the rotational framework

• DOI: 10.1007/s00209-009-0525-8
• 发表时间: 2010-06
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Matthias Hieber;Y. Shibata

On some decay properties of Stokes semigroup of compressible viscous fluid flow in a 2-dimensional exterior domain

二维外域可压缩粘性流体斯托克斯半群的一些衰变性质

• 发表时间: 2012
• 期刊: J. Differential Equations
• 作者: Y.Shibata;S. Shimizu;Yuko Enomoto and Yoshihiro Shibata
• 通讯作者: Yuko Enomoto and Yoshihiro Shibata

Lp-Lq Estimate of the Stokes Operator and Navier–Stokes Flows in the Exterior of a Rotating Obstacle

• DOI: 10.1007/s00205-008-0130-8
• 发表时间: 2009-08
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: T. Hishida;Y. Shibata