约束诱导的限制和扩张算子及其应用

基金项目: 上海市重点学科建设资助项目(Y0103);国家自然科学基金资助项目(10772103)

Constraint Induced Restriction and Extension Operators With Applications

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P. R. China

摘要: Stokes方程是由动量方程和不可压缩约束耦合而成的方程组，Stokes算子是由Stokes方程诱导所得到的微分积分算子．该文试从Helmholtz最小耗散原理的角度，采用对零散度矢量场进行Hodge正交分解的方法，对Stokes算子的性质进行分析．结果指出Stokes算子是Helmholtz耗散泛函的Fréchet导算子，零散度约束通过Hodge正交分解诱导出一对有界线性算子，即限制算子R和扩张算子ε．作为结果的应用，利用它计算Stokes算子的特征值
- Stokes算子 /
- 诱导算子 /
- Hodge分解 /
- 变分方法 /
- 特征值问题
Abstract: The Stokes operator is a differential-integral operator induced by the Stokes equations. From the point of view of the Helm holtzminimum dissipation principle the Stokes operator was analyzed. It's shown that, th rough the Hodge orthogonal decomposition, a pair of bounded linear operators, namely, a restriction operator and an extension operator, are induced from the divergence-free constraint. As a consequence of the observation, it's utilized to calculate the eigenvalues of the Stokes operator.
- the Stokes operator /
- induced operators /
- restriction and extension /
- variational method /
- Hodge decomposition

[1]	Constantin P,Foias C.The Navier-Stokes Equations[M].Chicago:University of Chicago Press,1992.
[2]	Temam R.Navier-Stokes Equations and Nonlinear Functional Analysis[M].2nd Ed.Philadelphia:SIAM,1995.
[3]	Temam R.Navier-Stokes Equations:Theory and Numerical Analysis[M].3rd Ed.Amsterdam:North-Holland,1984.
[4]	Abels H,Teresawa Y.On Stokes operators with variable viscosity in bounded and unbounded domains[J].Math Ann,2009,344(2):381-429. doi: 10.1007/s00208-008-0311-7
[5]	Bae Hyeon-Ohk.Analyticity of the Stokes operator in Besov spaces[J].J Korean Math Soc,2003,40(6):1061-1074. doi: 10.4134/JKMS.2003.40.6.1061
[6]	Lee D-S,Rummler B.Then eigenfunctions of the Stokes operator in special domains[J].ZAMM,2002,82(6):399-407. doi: 10.1002/1521-4001(200206)82:6<399::AID-ZAMM399>3.0.CO;2-6
[7]	Turinsky V V.A lower bound for the principle eigenvalue of the Stokes operator in a random domain[J].Ann Inst Henri Poincaré Prob Stat,2008,44(1)：1-18. doi: 10.1214/07-AIHP136
[8]	王立周，李东升，李开泰.球间隙区域上的Stokes算子的特征值问题及应用[J]. 高校应用数学学报,A辑，2001，16(2):87-96.
[9]	陈波，李孝伟, 刘高联. Hodge分解在最小耗散原理中的应用[J].数学的实践和认识，2009，39(8):197-200
[10]	陆文端.微分方程中的变分方法[M].北京：科学出版社，2003.