Periodic strong solutions of the magnetohydrodynamic type equations
DOI:
https://doi.org/10.4067/S0716-09172002000300001Keywords:
Magnetohydrodynamic type equations, periodic strong solutions, Galerkin method, ecuaciones de tipo magnetohidrodinámico, soluciones periódicas fuertes, método de Galerkin.Abstract
We obtain, using the spectral Galerkin method together with compactness arguments, existence and uniqueness of periodic strong solutions for the magnetohydrodynamic type equations.References
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[16] Notte-Cuello, E. A. and Rojas-Medar, M. A., On a system of evolution equations of magnetohydrodynamic type: an iterational approach. Proyecciones 17, pp. 133-165, (1998).
[17] Rojas-Medar, M.A. and Beltrán-Barrios, R., The initial value problem for the equations of magnetohydrodynamic type in noncylindrical domains, Rev. Mat. Univ. Compl. Madrid, 8, pp. 229-251, (1995).
[18] Rojas-Medar, M.A. and Boldrini, J. L., Global strong solutions of equations of magnetohydrodynamic type, J. Australian Math. Soc., Serie B, Applied Math 38, pp. 291-306, (1997).
[19] Serrin, J., A note on the existence of periodic solutions of the NavierStokes equations, Arch. Rat. Mech. Anal. 3, pp. 120-122, (1959).
[20] Simon J., Nonhomogeneous viscous incompressible fluids: existence of velocity, density and pressure, SIAM J. Math. Anal. , V. 21, N 5, pp. 1093-1117, (1990).
[21] Schlüter, A., Dynamic des Plasmas, I and II. Z. Naturforsch. 5a, pp. 72-78, (1950); 6a, pp. 73-79, (1951).
[22] Temam, R., Navier-Stokes equations, North-Holland, Amsterdam, Rev. Edit., (1979).
[2] Amrouche, G. and Girault, V., On the existence and regularity of the solutions of Stokes problem in arbitrary dimension, Proc. Japan Acad., 67, Ser. A. (1991), 171-175.
[3] Berger, M.S. Nonlinearity and Functional Analysis, Acad. Press,1987.
[4] Berselli, L.C. and Ferreira, J., On the magnetohydrodynamic type equations in a new class of non-cylindrical domains, To appear in B.U.M.I..
[5] Boldrini, J.L. And Rojas-Medar, M.A., On a system of evolution equations of Magnetohydrodinamic type, Mat. Cont., 8, pp. 1-19, 1995.
[6] Burton, T.A. Stability and periodic solutions of ordinary and functional differential equations, Academic Press, (1985).
[7] Constantin, P. and Foias C., Navier-Stokes equations, The Univ. Chicago Press, Chicago and London, (1988).
[8] Damásio, P. and Rojas-Medar, M.A., On some questions of the weak solutions of evolution equations for magnetohydrodynamic type, Proyecciones 16, pp. 83-97, (1997).
[9] Fujita, H. and Kato, T., On the Navier-Stokes initial value problem I, Arch. Rat. Mech. Anal. 16, pp. 269-315, (1964).
[10] Giga, Y. and Miyakawa, T., Solutions in Lr of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89, pp. 267-281, (1985).
[11] Heywood, J. G. - The Navier-Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29, pp. 639-681, (1980).
[12] Kato, H., Existence of periodic solutions of the Navier-Stokes equations, J. Math. Anal. and Appl. 208, pp. 141-157, (1997).
[13] Lassner, G., Uber Ein Rand-Anfangswert - Problem der Magnetohydrodinamik, Arch. Rat. Mech. Anal. 25, pp. 388-405, (1967).
[14] Lions, J. L., Quelques Méthods de Résolution des Problémes aux Limits Non Linéares, Dunod, Paris, (1969).
[15] Pikelner, S.B., Grundlangen der Kosmischen Elektrohydrodynamik, Moscou, (1966).
[16] Notte-Cuello, E. A. and Rojas-Medar, M. A., On a system of evolution equations of magnetohydrodynamic type: an iterational approach. Proyecciones 17, pp. 133-165, (1998).
[17] Rojas-Medar, M.A. and Beltrán-Barrios, R., The initial value problem for the equations of magnetohydrodynamic type in noncylindrical domains, Rev. Mat. Univ. Compl. Madrid, 8, pp. 229-251, (1995).
[18] Rojas-Medar, M.A. and Boldrini, J. L., Global strong solutions of equations of magnetohydrodynamic type, J. Australian Math. Soc., Serie B, Applied Math 38, pp. 291-306, (1997).
[19] Serrin, J., A note on the existence of periodic solutions of the NavierStokes equations, Arch. Rat. Mech. Anal. 3, pp. 120-122, (1959).
[20] Simon J., Nonhomogeneous viscous incompressible fluids: existence of velocity, density and pressure, SIAM J. Math. Anal. , V. 21, N 5, pp. 1093-1117, (1990).
[21] Schlüter, A., Dynamic des Plasmas, I and II. Z. Naturforsch. 5a, pp. 72-78, (1950); 6a, pp. 73-79, (1951).
[22] Temam, R., Navier-Stokes equations, North-Holland, Amsterdam, Rev. Edit., (1979).
Published
2017-05-22
How to Cite
[1]
E. A. Notte Cuello, M. D. Rojas Medar, and M. A. Rojas Medar, “Periodic strong solutions of the magnetohydrodynamic type equations”, Proyecciones (Antofagasta, On line), vol. 21, no. 3, pp. 199-224, May 2017.
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