Abstract
The equations of an electrically conducting compressible fluid in electro-magneto-fluid dynamics are studied. It is proved that in a certain case of two-dimensional flow, the equations of the fluid become a symmetric hyperbolic-parabolic system in both of the viscous and non-viscous cases. Therefore, the initial value problem is well posed in the Sobolev spaces at least for short time interval. Furthermore, in the viscous case, the solution exists globally in time and tends to the constant state as time goes to infinity, provided the initial data are closed to the constant state. The proof is based on a technical energy method, which makes use of a quadratic function associated with the total energy of the fluid.
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Reference
K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. U.S.A.,68 (1971), 1686–1688.
I. Imai, General priciples of magneto-fluid dynamics, in “Magneto-Fluid Dynamics,” Suppl. Prog. Theor. Phys. No. 24 (ed. H. Yukawa), Chap. I, 1–34, RIFP Kyoto Univ., 1962.
T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal.,58, 3 (1975), 181–205.
S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A,58 (1982), 384–387.
L. D. Landau and E. M. Lifshitz, “Electrodynamics of Continuous Media”, Pergamon, New York, 1960.
A. Matsumura, An energy method for the equations of motion of compressible viscous and heatconductive fluids, Univ. of Wisconsin-Madison, MRC Technical Summary Report, No. 2194, 1981.
A. I. Vol’pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR.-Sb.,16 (1972), 517–544.
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Kawashima, S. Smooth global solutions for two-dimensional equations of electro-magneto-fluid dynamics. Japan J. Appl. Math. 1, 207–222 (1984). https://doi.org/10.1007/BF03167869
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DOI: https://doi.org/10.1007/BF03167869