Abstract
The linearized equations of the electrically conducting compressible viscous fluids are studied. It is shown that the decay estimate (1+t)−3/4 inL 2(R 3) holds for solutions of the above equations, provided that the initial data are inL 2(R 3)∩L 1(R 3). Since the systems of equations are not rotationally invariant, the perturbation theory for one parameter family of matrices is not useful enough to derive the above result. Therefore, by exploiting an energy method, we show that the decay estimate holds for the solutions of a general class of equations of symmetric hyperbolic-parabolic type, which contains the linearized equations in both electro-magneto-fluid dynamics and magnetohydrodynamics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. Imai, General principles of magneto-fluid dynamics, “Magneto-Fluid Dynamics”. Suppl. Progr. Theoret. Phys. No. 24 (ed. H. Yukawa). RIFP Kyoto Univ., 1962, Chapter I, 1–34.
T. Kato, “Perturbation Theory for Linear Operators” (Second Ed.) Springer-Verlag, New York, 1976.
S. Kawashima, Smooth global solutions for two-dimensional equations of electro-magneto-fluid dynamics. (preprint).
S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc. Japan Acad. Ser. A,58 (1982), 384–387.
S. Klainerman and G. Ponce, Global, small amplitude solutions to nonlinear evolution equations. Comm. Pure Appl. Math.,36 (1983), 133–141.
A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations. Publ. RIMS, Kyoto Univ.,12 (1976), 169–189.
A. Matsumura, An energy method for the equations of motion of compressible viscous and heat-conductive fluids. MRC Technical Summary Report, No. 2194, Univ. Wisconsin, Madison, 1981.
A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A,55 (1979), 337–342.
T. Nishida and K. Imai, Global solutions to the initial value problem for the nonlinear Boltzmann equation. Publ. RIMS, Kyoto Univ.,12 (1976), 229–239.
J. Shatah, Global existence of small solutions to non-linear evolution equations, to appear.
Y. Shibata, On the global existence of classical solutions of second order fully non-linear hyperbolic equations with first order dissipation in the exterior domain, to appear in Tsukuba J. Math.
S. Ukai, Les solutions globales de l’équation de Boltzmann dans l’espace tout entier et dans le demispace. C. R. Acad. Sci. Paris, Sér. A,282 (1976), 317–320.
Author information
Authors and Affiliations
About this article
Cite this article
Umeda, T., Kawashima, S. & Shizuta, Y. On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics. Japan J. Appl. Math. 1, 435–457 (1984). https://doi.org/10.1007/BF03167068
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF03167068