On some questions of the weak solutions of evolution equations for magnetohydrodynamic type
DOI:
https://doi.org/10.22199/S07160917.1997.0002.00001Abstract
We prove that the weak solution of the equations for magneto-hydrodynamic type posses fractional derivatives in time of any order less that 1/2 if n = 2 and that it is true conditionally in the three and four-dimensional cases. Also, we give some results of uniqueness of weak solutions similar to the Navier-Stokes equations for n > 3. Thus, we reach the same level of knoweledge as the one in the case of the classical Navier-Stokes.
References
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[17] Temam, R., Navier-Stokes equations, revised ed., Amsterdam, North-Holland, 1979.
[18] Zhang, K., On Shinbrot's conjecture for the Navier-Stokes equations, Proc. R. Soc. Lond. A, 440, 537-540, 1993.
[19] Zygmund, A., Trigonometric series, 2nd. Ed., vol 1, Cambridge University Press, 1959.
[2] Boldrini, J.L. and Rojas-Medar, M.A., On a system of evolution equations of magnetohydrodynamic type: on the existence, regularity and approximations of solutions, Actas 2° Congreso de Matemática Capricornio, Arica, Chile, 23-28, 1992.
[3] Boldrini, J.L. and Rojas-Medar, M.A., On a system of evolution equations of magnetohydrodynamic type. Mat. Cont. 8, 1-19, 1995.
[4] Chizhonkov, E. V., On a system of equation of magnetohydrodynamic type, Soviet Math. Dokl., 30, 542-545, 1984.
[5] Fujita, H. and Kato, T., On the Navier-Stokes initial value problem, I, Arch. Rational Mech. Anal., 16, 269-315, 1964.
[6] Lassner, G., Übereir Randanfargswert Problem der Magnetohydrodynamik, Arch. Rational Mech. Anal., 25, 388-405, 1967.
[7] Lions, J.L., Quelques résultats d'existence dans les équations aux dérivées partielles non linéaires, Bull Soc. Math. Fr., 87, 245-273, 1959.
[8] Lions, J.L., Quelques méthodes de résolution des problemes aux limites non linéaires. Paris-Dunod, 1969.
[9] Pikelner, S.B., Grundlanger der Kosmischen Elektrodynamik, Moscou, 1966.
[10] Rojas-Medar, M.A. and Beltrán-Barrios, R., The initial value problema for the equations of magnetohydrodynamic type in non-cylindrical domains, Rev. Mat. Univ. Complutense de Madrid, 8 (1), 229-251, 1995.
[11] Rojas-Medar, M.A. and Boldrini, J.L., The weak solutions and reproductive property for a system of evolution equations of magnetohydrodynamics type, Proyecciones, 13 (2), 85-97, 1994.
[12] Rojas-Medar, M.A. and Boldrini, J.L., Global strong solutions of equations of magnetohydrodynamic type, to appear in J. Aust. Math. Sci. Série B, Applied Math.
[13] Schlüter, A., Dynamik des Plasmas, 1 and 11, Z. Naturforsch. 5a, 72-78, 1950, 6a, 73-79, 1951.
[14] Shinbrot, M., Fractional derivatives of solutions of the Navier-Stokes equations. Arch. Ration. Mech. Analysis, 40, 139-154, 1971.
[15] Shinbrot, M., Lectures on Fluid Mechanics. New York, Gordon and
Breach, 1973.
[16] Simon, J., Sobolev, Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on interval, Annali Mat. Pura Appl. 157, 117-148, 1990.
[17] Temam, R., Navier-Stokes equations, revised ed., Amsterdam, North-Holland, 1979.
[18] Zhang, K., On Shinbrot's conjecture for the Navier-Stokes equations, Proc. R. Soc. Lond. A, 440, 537-540, 1993.
[19] Zygmund, A., Trigonometric series, 2nd. Ed., vol 1, Cambridge University Press, 1959.
Published
2018-04-04
How to Cite
[1]
P. D. Damázio and M. A. Rojas-Medar, “On some questions of the weak solutions of evolution equations for magnetohydrodynamic type”, Proyecciones (Antofagasta, On line), vol. 16, no. 2, pp. 83-97, Apr. 2018.
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