平面平行管道中的MHD方程组的边界层

数学物理学报 ›› 2019, Vol. 39 ›› Issue (4): 738-760.

平面平行管道中的MHD方程组的边界层

王娜^1,*(),王术²

¹ 河北师范大学数学与信息科学学院河北石家庄 050024
² 北京工业大学应用数理学院北京 100124

收稿日期:2018-04-04 出版日期:2019-08-26 发布日期:2019-09-11
通讯作者: 王娜 E-mail:wangna1989@hebtu.edu.cn
基金资助:
国家自然科学基金(11371042);河北师范大学科技类博士（后）基金(L2019B03)

The Boundary Layer for MHD Equations in a Plane-Parallel Channel

Na Wang^1,*(),Shu Wang²

¹ College of Mathematics and Information Science, Hebei Normal University, Hebei Shijiazhuang 050024
² College of Applied Sciences, Beijing University of Technology, Beijing 100124

Received:2018-04-04 Online:2019-08-26 Published:2019-09-11
Contact: Na Wang E-mail:wangna1989@hebtu.edu.cn
Supported by:
the NSFC(11371042);the (Post) Doctor Fund of Science and Technology of Hebei Normal University(L2019B03)

摘要/Abstract

摘要：

该文研究平面平行管道中不可压缩MHD方程组的边界层问题.利用多尺度分析和精细的能量方法，证明了当粘性系数与磁耗散系数趋近于0时，粘性与磁耗散MHD方程组的解收敛到理想MHD方程组的解.

关键词: 不可压缩MHD方程组, 边界层, 平面平行管道

Abstract:

In this paper, we study the boundary layer problem for the incompressible MHD equations in a plane-parallel channel. Using the multiscale analysis and the careful energy method, we prove the convergence of the solution of viscous and diffuse MHD equations to that of the ideal MHD equations as the viscosity and magnetic diffusion coefficient tend to zero.

Key words: Incompressible MHD equations, Boundary layer, Plane-parallel channel

中图分类号:

O175.29

王娜,王术. 平面平行管道中的MHD方程组的边界层[J]. 数学物理学报, 2019, 39(4): 738-760.

Na Wang,Shu Wang. The Boundary Layer for MHD Equations in a Plane-Parallel Channel[J]. Acta mathematica scientia,Series A, 2019, 39(4): 738-760.

参考文献 25

[1]	Mazzucato A L, Niu D J, Wang X M. Boundary layer associated with a class of 3D nonlinear plane parallel channel flows. Indiana Univ Math J, 2011, 60:1113-1136<br />
[2]	Biskamp D. Nonlinear Magnetohydrodynamics. Cambridge:Cambridge University Press, 1993<br />
[3]	E W N. Boundary layer theory and the zero viscosity limit of the Navier-Stokes equations. Acta Math Sin (English Series), 2000, 16:207-218<br />
[4]	Han D Z, Mazzucato A L, Niu D J, Wang X M. Boundary layer for a class of nonlinear pipe flow. J Differential Equations, 2012, 252:6387-6413<br />
[5]	Duvaut G, Lions J L. Inéquation en themoélasticite et magnétohydrodynamique. Arch Ration Mech Anal, 1972, 46:241-279<br />
[6]	Gie G M, Kelliher J P, Lopes Filho M C, et al. The vanishing viscosity limit for some symmetric flows. 2017, arXiv:1706.06039<br />
[7]	He C, Xin Z P. On the regularity of weak solutions to the magnetohydrodynamic equations. J Differential Equations, 2005, 213:235-254<br />
[8]	He C, Xin Z P. Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations. J Funct Anal, 2005, 227:113-152<br />
[9]	Masmoudi N. The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary. Arch Ration Mech Anal, 1998, 142:375-394<br />
[10]	Liu C J, Xie F, Yang T. MHD boundary layers in Sobolev spaces without monotonicity Ⅱ:convergence theory. 2017, arXiv:1704.00523[math.AP]<br />
[11]	Masmoudi N. Remarks about the inviscid limit of the Navier-Stokes system. Comm Math Phys, 2007, 270(3):777-788<br />
[12]	Sammartino M, Caflisch R E. Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half space I:Existence for Euler and Prandtl equations. Comm Math Phys, 1998, 192:433-461<br />
[13]	Sammartino M, Caflisch R E. Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half space Ⅱ:Construction of Navier-Stokes solution. Comm Math Phys, 1998, 192:463-491<br />
[14]	Sermange M, Temam R. Some mathematical questions related to the MHD equations. Comm Pure Appl Math, 1983, 36:635-664<br />
[15]	Temam R, Wang X M. Asymptotic analysis of Oseen type equations in a channel at small viscosity. Indiana Univ Math J, 1996, 45:863-916<br />
[16]	Temam R, Wang X M. Remarks on the Prandtl equation for a permeable wall. Z Angew Math Mech, 2000, 80:835-843<br />
[17]	Temam R, Wang X M. Boundary layer associated with incompressible Navier-Stokes equations:The noncharacteristic boundary case. J Differential Equations, 2002, 179:647-686<br />
[18]	Wu J H. Regularity criteria for the generalized MHD equations. Comm Partial Differential Equations, 2008, 33:285-306<br />
[19]	Wang S, Wang B Y, Liu C D, Wang N. Boundary layer problem and zero viscosity-diffusion vanishing limit of the incompressible Magnetohydrodynamic system with no-slip boundary conditions. J Differential Equations, 2017, 263:4723-4749<br />
[20]	王术, 王娜. 不可压缩MHD方程组的边界层问题. 北京工业大学学报, 2017, 43(10):1596-1603 Wang S, Wang N. The boundary layer problem for the incompressible MHD equations. J Beijing Univ Technol, 2017, 43(10):1596-1603<br />
[21]	Wang S, Wang N. Boundary layer problem of MHD system with noncharacteristic perfect conducting wall. Applicable Analysis, 2019, 98(3):516-535<br />
[22]	Wang N, Wang S. Vanishing vertical limit of the incompressible combined viscosity and magnetic diffusion magnetohydrodynamic system. Math Meth Appl Sci, 2018, 41:5015-5049<br />
[23]	Nguyen T T, Sueur F. Boundary-layer interactions in the plane-parallel incompressible flows. Nonlinearity, 2012, 25(12):3327-3342<br />
[24]	Wang X M. A kato type theorem on zero viscosity limit of Navier-Stokes flows. Indiana Univ Math J, 2001, 50:223-241<br />
[25]	谢晓强, 罗琳, 李常敏. 非特征边界的MHD方程的边界层, 数学年刊,2014, 35A(2):171-192 Xie X Q, Luo L, Li C M. Boundary layer for MHD equations with the noncharacteristic boundary conditions. Chin Ann Math, 2014, 35A(2):171-192