Large solutions of the magnetic Bénard problem on the infinite layer (Q5958916)

The author considers the magnetic Bénard problem in a Hilbert space setting on the infinite plate admitting arbitrarily large initial values which give raise to exponentially decaying strong solutions. The underlaying assumption is that the Rayleigh parameter \(\lambda\) (encoding temperature) satisfies \(\lambda < \lambda_C\) where \(\lambda_C\) is a critical Rayleigh parameter related to the Bénard problem without magnetic field. The magnetic parameter however, which measures the strenght of the imposed perpendicular field, may be arbitrary large. In the subspace \(E^g\), characterised by certain symmetry requirements, the zero solution is asymptotically stable and the basin of attraction is unbounded. An initial condition induces a solution which either decays exponentially or explodes after finite time. If the magnetic parameter is small, then the Lyapunov stability can be asserted. Both the statement on the unbounded basin of attraction and the dichotomy for trajectories in \(E^g\) admit counterparts on \(E\). The method used here apply to Navier-Stokes and Bénard problems.