Digital Object Identifier (DOI) 10.1007/s00205-002-0230-9 Arch. Rational Mech. Anal. 166 (2003) 197–218 Necessary and Sufficient Conditions for Nonlinear Stability in the Magnetic Bénard Problem G. Mulone & S. Rionero Communicated by C. M. Dafermos Abstract We study the magnetic Bénard problem with the Lyapunov direct method and obtain necessary and sufficient conditions of conditional nonlinear stability. By introducing a suitable change of fields and a generalized energy functional, we show that, whenever the magnetic Prandtl number is less than the usual Prandtl number, the critical linear and nonlinear stability Rayleigh numbers coincide for any Chandrasekhar number. 1. Introduction The questions of the stability and instability of the magnetic Bénard problem are concerned with the study of the onset of convection of a horizontal layer of a homogeneous, viscous, and electrically conducting fluid, permeated by an imposed uniform magnetic field H, normal to the layer, and heated from below. The experi- ments show that the magnetic field has a stabilizing effect, i.e., it inhibits the onset of convection. For this reason and for the applications in many physical situations, this problem has been studied by many writers, Thompson [33], Chandrasekhar [1–3] Nakagawa [20, 21], Rionero [22–27], Galdi, [6], Rionero &Mulone [29], Galdi &Padula [7], Mulone &Rionero, [17] (see also the references in the last paper). The inhibiting effect of the magnetic field on the onset of convection has been proved in the linear case, by using the normal-modes analysis, by Thompson, [33], and Chandrasekhar [1–3], who obtained a critical Rayleigh number (which is a non-dimensional parameter that describes the onset of convection) and showed that it is an increasing function of the magnetic field. The stabilizing effect of the magnetic field has also been confirmed by the experiments of Nakagawa [20, 21]. Rionero, [22–25], studied the problem in the nonlinear context. By applying the Lyapunov method, and using the classical energy (the L 2 norm of perturbations