Acta mathematica scientia, Series B >
A STUDY ON THE BOUNDARY LAYER FOR THE PLANAR MAGNETOHYDRODYNAMICS SYSTEM
Received date: 2015-01-27
Online published: 2015-07-01
Supported by
Xulong Qin and Zheng-an Yao's research was supported in part by NNSFC (11271381 and 11431015). Tong Yang's research was supported in part by the Joint NSFC-RGC Research Fund, N-CityU 102/12. Wenshu Zhou's research was supported in part by the Program for Liaoning Excellent Talents in University (LJQ2013124) and the Fundamental Research Fund for the Central Universities.
The paper aims to estimate the thickness of the boundary layer for the planar MHD system with vanishing shear viscosity μ. Under some conditions on the initial and boundary data, we show that the thickness is of the order √μ|lnμ|. Note that this estimate holds also for the Navier-Stokes system so that it extends the previous works even without the magnetic effect.
Xulong QIN , Tong YANG , Zheng-an YAO , Wenshu ZHGOU . A STUDY ON THE BOUNDARY LAYER FOR THE PLANAR MAGNETOHYDRODYNAMICS SYSTEM[J]. Acta mathematica scientia, Series B, 2015 , 35(4) : 787 -806 . DOI: 10.1016/S0252-9602(15)30022-9
[1] Balescu R. Transport Processes in Plasmas I: Classical Transport Theory. New York: North-Holland, 1988
[2] Chen G,Wang D. Global solutions of nonlinear magnetohydrodynamics with large initial data. J Differential Equations, 2002, 182: 344-376
[3] Chen G, Wang D. Existence and continuous dependence of large solutions for the magnetohydrodynamic equations. Z Angew Math Phys, 2003, 54: 608-632
[4] Clemmow P, Dougherty J. Electrodynamics of Particles and Plasmas. New York: Addison-Wesley, 1990
[5] Fan J, Jiang S, Nakamura G. Vanishing shear viscosity limit in the magnetohydrodynamic equations. Commun Math Phys, 2007, 270: 691-708
[6] Fan J, Huang S, Li F. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vaccum. arXiv: 1206.3624v3[math.AP]19 Jul 2012
[7] Fan J, Jiang S. Zero shear viscosity limit of the Navier-Stokes equations of compressible isentropic fluids with cylindrical symmetry. Rend Sem Mat Univ Politec Torino, 2007, 65: 35-52
[8] Frid H, Shelukhin V V. Boundary layer for the Navier-Stokes equations of compressible fluids. Commun Math Phys, 1999, 208: 309-330
[9] Frid H, Shelukhin V V. Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry. SIAM J Math Anal, 2000, 31: 1144-1156
[10] Hoff D, Tsyganov E. Uniqueness and continuous dependence of weak solutions in compressible magneto- hydrodynamics. Z Angew Math Phys, 2005, 56: 791-804
[11] Hu X, Wang D. Global existence and large-time behavior of solutions to the threedimensional equations of compressible magnetohydrodynamic flows. Arch Ration Mech Anal, 2010, 197: 203-238
[12] Jiang S, Zhang J W. Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry. SIAM J Math Anal, 2009, 41: 237-268
[13] Kazhikhov A, Shelukhin V V. Unique global solutions in time of initial boundary value problems for one- dimensional equations of a viscous gas. J Appl Math Mech, 1977, 41: 273-283
[14] Kawashima S, Okada M. Smooth global solutions for the one-dimensional equations in magnetohydrody- namics. Proc Japan Acad Ser A, Math Sci, 1982, 58 384-387
[15] Laudau L, Lifshitz E. Electrodynamics of Continuous Media. 2nd ed. New York: Pergamon, 1984
[16] Lieberman G M. Second Order Parabolic Diffrential Equations. Singapore: World Scientific, 1996
[17] Liu T, Zeng Y. Large-time behavior of solutions for general quasilinear hyperbolicparabolic systems of conservation laws. Mem Amer Math Soc, 1997, 125(599)
[18] Qin X, Yang T, Yao Z, Zhou W. Vanishing shear viscosity and boundary layer for the Navier-Stokes equations with cylindrical symmetry. Arch Rational Mech Anal, 2015, 216(3): 1049-1086
[19] Shelukhin V V. The limit of zero shear viscosity for compressible fluids. Arch Rational Mech Anal, 1998, 143: 357-374
[20] Tsyganov E, Hoff D. Systems of partial differential equations of mixed hyperbolic-parabolic type. J Differ- ential Equations, 2004, 204: 163-201
[21] Wang D. Large solutions to the initial boundary value problem for planar magnetohydrodynamics. SIAM J Appl Math, 2003, 23: 1424-1441
[22] Woods L. Principles of Magnetoplasma Dynamics. New York: Oxford University Press, 1987
[23] Wu C. Formation, structure, and stability of MHD intermediate shocks. J Geophys Res, 1990, 95: 8149- 8175
[24] Vol'pert A, Hudjaev S. On the Cauchy problem for composite systems of nonlinear differential equations. Math USSR-Sb, 1972, 16: 517-544
[25] Yao L, Zhang T, Zhu C J. Boundary layers for compressible Navier-Stokes equations with density dependent viscosity and cylindrical symmetry. Ann I H Poincaré, 2011, 28: 677-709
/
| 〈 |
|
〉 |