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We investigate the stability and instability of the magneticRayleigh--B\'enard problem with zero resistivity. An stability criterion isestablished, under which the magnetic B\'enard problem is stable. The proofmainly is based on a three-layers energy method and an idea of magneticinhibition mechanism. The stable result first mathematically verifiesChandrasekhar's assertion in 1955 that the thermal instability can be inhibitedby strong magnetic field in magnetohydrodynamic (MHD) fluid with zeroresistivity (based on a linearized steady magnetic B\'enard equations). Inaddition, we also provide an instability criterion, under which the magneticRayleigh--B\'enard problem is unstable. The proof mainly is based on thebootstrap instability method by further developing new analysis technique. Ourinstability result presents that the thermal instability occurs for a smallmagnetic field.

On the Inhibition of Thermal Convection by a Magnetic Field under Zero Resistivity