Equilibrium solutions of the Bénard equations with an exterior force

Abstract

In this paper we investigate questions of existence and uniqueness of equilibrium solutions of the inhomogeneous Bénard equations with exterior force $f$ which may be generated by a magnetic field which is not too strong. Two types of results are obtained. The first, based on a priori estimates and degree arguments states that there is a certain range $\lambda\leq \lambda_c+\delta_1$ for the Rayleigh parameter $\lambda$, for which existence can be asserted for any given force $f$, while the second result says that for $\lambda <\lambda_c$, $\lambda_c-\lambda$ small, there are forces $f$ which give rise to three different solutions. Here, $\lambda^2_c=R_c$ is the critical Rayleigh number which enters basically into our considerations.