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.
Citation
B. Scarpellini. "Equilibrium solutions of the Bénard equations with an exterior force." Adv. Differential Equations 10 (12) 1321 - 1344, 2005. https://doi.org/10.57262/ade/1355867737
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