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On existence, uniqueness and Lr-exponential stability for stationary solutions to the MHD equations in three-dimensional domains

Published online by Cambridge University Press:  17 February 2009

Chunshan Zhao  and
Kaitai Li
Chunshan Zhao
Affiliation:
Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA; e-mail: chuzhao@math.uiowa.edu.
Kaitai Li
Affiliation:
School of Science, Xi'an Jiaotong University, Shaanxi, 710049, China; e-mail: ktli@mail.xjtu.edu.cn.
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Abstract

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The existence of stationary solutions to the MHD equations in three-dimensional bounded domains will be proved. At the same time if the assumption of smallness is made on external forces, uniqueness of the stationary solutions can be guaranteed and it can be shown that any Lr (r > 3) global bounded non-stationary solution to the MHD equations approaches the stationary solution under both L2 and Lr norms exponentially as time goes to infinity.

Type
Research Article
Information
The ANZIAM Journal , Volume 46 , Issue 1 , July 2004 , pp. 95 - 109
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]da Veiga, H. Beirao, “Existence and asymptotic behavior for the strong solutions of the Navier-Stokes equations in the whole space”, Indiana Univ. Math. J. 36 (1987) 149–166.CrossRefGoogle Scholar
[2]Chizhonkov, E. V., “On a system of equations of magneto-hydrodynamic type”, Soviet Math. Dokl. 30 (1984) 542–545.Google Scholar
[3]Cowling, T. G., Magnetohydrodynamics, Interscience Tracts on Physics and Astronomy 4 (Inter-science Publishers, New York, 1957).Google Scholar
[4]Evans, L., Partial differential equations (American Mathematical Society, Providence, RI, 1998).Google Scholar
[5]Guo, B. and Zhang, L., “Decay of solutions to magnetohydrodynamics equations in two space dimensions”, Proc. Roy. Soc. London Ser A 449 (1995) 79–91.Google Scholar
[6]Lassener, G., “Uber ein Rand-anfangswert-problem der Magnetohydrodinamik”, Arch. Rational Mech. Anal. 25 (1967) 388–405.CrossRefGoogle Scholar
[7]Qu, C. and Wang, P., “L p exponential stability for the equilibrium solutions of the Navier-Stokes equations”, J. Math. Anal. Appl. 190 (1995) 419–427.CrossRefGoogle Scholar
[8]Rojas-Medar, M. and Boldrini, J., “Global strong solutions of equations of magneto-hydrodynamic type”, J. Austral. Math. Soc. Ser B 38 (1997) 291–306.CrossRefGoogle Scholar
[9]Schonbek, M., “The long time behaviors of solutions to the MHD equations”, Math. Ann. 304 (1996) 717–756.CrossRefGoogle Scholar
[10]Stein, E. M., Singular integrals and differentiability properties of functions (Princeton University Press, Princeton, NJ, 1970).Google Scholar
[11]Temam, R., Navier-Stokes equations (North-Holland, Amsterdam, 1979).Google Scholar
[12]Temam, R., Navier-Stokes equations and nonlinear functional analysis (SIAM, Philadelphia, PA, 1983).Google Scholar