Equilibrium solutions of the Bénard equations with an exterior force

Abstract

We study the inhomogeneous B\'enard equations on the infinite layer, $\Omega={\mathbb R}^2\times (-\frac12,\frac12)$, provided with an exterior force $f=f(z)$, depending only on the bounded variable $z\in[-\frac12,\frac12]$. There is a unique equilibrium solution $v=v(z)$ depending only on $z$. We study the stability of small $v(z)$, once under $L$-periodic perturbations, and once under spatially localized perturbations, i.e., perturbations in ${\mathcal L}^2(\Omega)$. Loss of stability may occur in the neighbourhood of the critical Rayleigh numbers $\lambda_L$ and $\lambda_\omega$, where $\lambda_L$ refers to the $L$-periodic setting, $\lambda_\omega$ to the ${\mathcal L}^2(\Omega)$ setting. Among others we give a characterization of $\lambda_\omega$ in terms of Orr-Sommerfeld theory. It is shown that if $\lambda_L\ne\lambda_\omega$ then $v(z)$ may be stable under $L$-periodic perturbations, but is necessarily unstable under ${\mathcal L}^2(\Omega)$ perturbations. The proofs are based on energy methods and on Bloch space theory.