Abstract
We study an initial boundary value problem for the equations of plane magnetohydrodynamic compressible flows, and prove that as the shear viscosity goes to zero, global weak solutions converge to a solution of the original equations with zero shear viscosity. As a by-product, this paper improves the related results obtained by Frid and Shelukhin for the case when the magnetic effect is neglected.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cabannes H. (1970). Theoretical Magnetofluiddynamics. Academic Press, New York-London
Chen G.Q. and Wang D. (2003). Existence and continuous dependence of large solutions for the magnetohydrodynamics equations. Z. angew. Math. Phys. 54: 608–632
Chen G.Q. and Wang D. (2002). Global solutions of nonlinear magnetohydrodynamics with large initial data. J. Diff. Eqs. 182: 344–376
Fan, J., Jiang, S.: Zero shear viscosity limit for the Navier–Stokes equations of compressible isentropic fluids with cylindric symmetry. Rend. Sem. Mat. Univ. Pol. Torino (in press)
Fan, J., Jiang, S., Nakamura, G.: Stability of weak solutions to equations of magnetohydrodynamics with Lebesgue initial data. Preprint, 2005
Fan, J.: Stability of spherically symmetric weak solutions to the Navier–Stokes equations of compressible heat conductive fluids in bounded annular domains. Submitted, 2005
Feireisl E. (2001). On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable. Comment. Math. Univ. Carlolinae 42: 83–98
Feireisl E. (2004). Dynamics of Viscous Compressible Fluids. Oxford Univ. Press, Oxford
Frid H. and Shelukhin V.V. (1999). Boundary layers for the Navier–Stokes equations of compressible fluids. Comm. Math. Phys. 208: 309–330
Frid H. and Shelukhin V.V. (2000). Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry. SIAM J. Math. Anal. 31: 1144–1156
Gajewski H., Gröger K. and Zacharias K. (1974). Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademil Verlag, Berlin
Hoff D. (1995). Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Diff. Eqns. 120: 215–254
Hoff D. and Tsyganov E. (2005). Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics. Z. angew. Math. Phys. 56: 791–804
Jiang S. and Zhang P. (2001). On spherically symmetric solutions of the compressible isentropic Navier–Stokes equations. Commun. Math. Phys. 215: 559–581
Jiang S. and Zlotnik A. (2004). Global well-posedness of the Cauchy problem for the equations of a one-dimensional viscous heat-conducting gas with Lebesgue initial data. Proc. Royal Soc. Edinburgh 134A: 939–960
Kazhikhov A.V. and Smagulov Sh.S. (1986). Well-posedness and approximation methods for a model of magnetogasdynamics. Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat. 5: 17–19
Kawashima S. and Okada M. (1982). Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc Japan Acad. Ser. A, Math. Sci. 58: 384–387
Landau L.D., Lifshitz E.M. and Pitaevskii L.P. (1999). Electrodynamics of Continuous Media, 2nd ed. Butterworth-Heinemann, London
Li, T.-T., Qin, T.: Physics and Partial Differential Equations, 2nd. Volume 1, Beijing: Higher Education Press, 2005 (in Chinese)
Lions P.-L. (1996). Mathematical Topics in Fluid Dynamics. Volume 1, Incompressible Models. Oxford Science Publication, Oxford
Lions P.-L. (1998). Mathematical Topics in Fluid Dynamics. Volume 2, Compressible Models. Oxford Science Publication, Oxford
Liu, T.-P., Zeng, Y.: Long-time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Mem. Amer. Math. Soc. 599, Providence, RI: Amer. Math. Soc. 1997
Moreau R. (1990). Magnetohydrodynamics. Klumer Academic Publishers, Dordrecht
Polovin R.V. and Demutskii V.P. (1990). Fundamentals of Magnetohydrodynamics. Consultants. Bureau, New York
Rakotoson J.M. (1992). A compactness lemma for quasilinear problem: application to parabolic equations. J. Funct. Anal. 106: 358–374
Serre D. (1986). Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible. C. R. Acad. Sci. Paris. 303: 639–642
Shelukhin V.V. (1998). The limit of zero shear viscosity for compressible fluids. Arch. Rat. Mech. Appl. 143: 357–374
Shelukhin V.V. (1998). A shear flow problem for the compressible Navier-Stokes equations. Int. J. Non-linear Mech. 33: 247–257
Shelukhin V.V. (2000). Vanishing shear viscosity in a free-boundary problem for the equations of compressible fluids. J. Diff. Eqs. 167: 73–86
Simon J. (1987). Compact sets and the space L p(0, T; B). Ann. Mat. Pura Appl. 146: 65–96
Smagulov Sh.S., Durmagambetov A.A. and Iskenderova D.A. (1993). The Cauchy problem for the equations of magnetogasdynamics. Diff. Eqs. 29: 278–288
Ströhmer G. (1990). About compressible viscous fluid flow in a bounded region. Pacific J. Math. 143: 359–375
Vol’pert A.I. and Hudjaev S.I. (1972). On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR-Sb. 16: 517–544
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Constantin
Supported by NSFC (Grant No. 10301014, 10225105) and the National Basic Research Program (Grant No. 2005CB321700) of China.
Rights and permissions
About this article
Cite this article
Fan, J., Jiang, S. & Nakamura, G. Vanishing Shear Viscosity Limit in the Magnetohydrodynamic Equations. Commun. Math. Phys. 270, 691–708 (2007). https://doi.org/10.1007/s00220-006-0167-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-0167-1