A~nonlinear hydrodynamic stability criterion derived by a generalized energy method
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2005), pp. 85-90.

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By applying a new variant of the A. Georgescu – L. Palese – A. Redaelli (G-P-R) method [8], based on the symmetrization of a linear operator, we deduce a nonlinear stability criterion of a state of thermal conduction of a horizontal fluid layer subject to a vertical upwards uniform magnetic field and a vertical upwards constant temperature gradient. The Boussinesq approximation is used. The upper and lower surfaces of the layer are two rigid walls. It is assumed that the magnetic Prandtl number is strictly greater than unity.
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Cătălin Liviu Bichir; Adelina Georgescu; Lidia Palese. A~nonlinear hydrodynamic stability criterion derived by a generalized energy method. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2005), pp. 85-90. http://geodesic.mathdoc.fr/item/BASM_2005_1_a6/

[1] Chandrasekhar S., Hydrodynamic and hydromagnetic stability, Clarendon, Oxford, 1961 | MR | Zbl

[2] Chandrasekhar S., “On the inhibition of convection by a magnetic field”, Phil. Mag. Sc. Ser., 43:7 (1952), 501–532 | MR | Zbl

[3] Chandrasekhar S., “On the inhibition of convection by a magnetic field, II”, Phil. Mag. Sc. Ser., 45:7 (1954), 1177–1191 | MR

[4] Galdi G. P., “Nonlinear stability analysis of the magnetic Benard problem via a generalized method”, Arch. Rational Mech. Anal., 87 (1985), 167–186 | DOI | MR | Zbl

[5] Georgescu A., Hydrodynamic stability theory, Kluwer, Dordrecht, 1985 | MR | Zbl

[6] Georgescu A., Variational formulation of some non-selfadjoint problems occuring in Benard instablity theory, I, Preprint Series in Mathematics, 35, Institute of Mathematics, Bucharest, Romania, 1977

[7] Georgescu A., Palese L., “Extension of a Joseph criterion to the non linear stability of mechanical equilibria in the presence of thermodiffusive conductivity”, Theoretical and Computational Fluid Dynamics, 8 (1996), 403–413 | Zbl

[8] Georgescu A., Palese L., Redaelli A., “On a new method in hydrodynamic stability theory”, Math. Sci. Res. Hot-Line, 4:7 (2000), 1–16 | MR | Zbl

[9] Georgescu A., Palese L., Redaelli A., “The complete form of the Joseph extended criterion”, Ann. Univ. Ferrara, Sez. VII, Sc. Mat., XLVIII (2001), 9–12 | MR

[10] Girault V., Raviart P.-A., Finite element methods for Navier-Stokes equations: Theory and algorithms, Springer, Berlin, 1986 | MR | Zbl

[11] Joseph D. D., “Global stability of the conduction-diffusion solution”, ARMA, 36:4 (1970), 285–292 | DOI | MR | Zbl

[12] Joseph D. D., Stability of Fluid Motions, I, II, Springer, Berlin, 1976 | MR

[13] Mulone G., Rionero S., “Necessary and sufficient conditions for nonlinear stability in the magnetic Benard problem”, ARMA, 166 (2003), 197–218 | DOI | MR | Zbl

[14] Nakagawa Y., “An experiment on the inhibition of thermal convection by a magnetic field”, Nature, 175 (1955), 417–419 | DOI

[15] Nakagawa Y., “Experiments on the inhibition of thermal convection by a magnetic field”, Proc. Royal Soc. (London), A 240 (1957), 108–113 | DOI

[16] Rionero S., Mulone G., “A non-linear stability analysis of the magnetic Benard problem through the Lyapunov direct method”, ARMA, 103:4 (1988), 347–368 | DOI | MR | Zbl

[17] Rionero S., “Sulla stabilità asintotica in media in magnetoidrodinamica”, Ann. Mat. Pura Appl., 76 (1967), 75–92 | DOI | MR | Zbl

[18] Rionero S., “Sulla stabilità asintotica in media in magnetoidrodinamica non isoterma”, Ricerche Mat., 16 (1967), 250–263 | MR

[19] Rionero S., “Metodi variazionali per la stabilità asintotica in media in magnetoidrodinamica”, Ann. Mat. Pura Appl., 78 (1968), 339–364 | DOI | MR | Zbl

[20] Rionero S., “Sulla stabilità magnetoidrodinamica non lineare asintotica in media con vari tipi di condizioni al contorno”, Ricerche di Mat., 17 (1968), 64–78 | MR | Zbl

[21] Rionero S., “On magnetohydrodynamic stability”, Quaderni di Matematica, 1 (1997), 347–376 | MR | Zbl

[22] Straughan B., The Energy Method, Stability, and Nonlinear Convection, Springer, New-York, 2004 | MR

[23] Temam R., Navier-Stokes equations. Theory and numerical analysis, North-Holland, Amsterdam, 1979 | MR | Zbl

[24] Thompson W. B., “Thermal convection in a magnetic field”, Phil. Mag. Sc. Ser., 42:7 (1951), 1417–1432 | MR | Zbl