Advances in Differential Equations

Equilibrium solutions of the Bénard equations with an exterior force

B. Scarpellini

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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.

Article information

Adv. Differential Equations Volume 10, Number 12 (2005), 1321-1344.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 76D05: Navier-Stokes equations [See also 35Q30] 76E06: Convection 76E25: Stability and instability of magnetohydrodynamic and electrohydrodynamic flows


Scarpellini, B. Equilibrium solutions of the Bénard equations with an exterior force. Adv. Differential Equations 10 (2005), no. 12, 1321--1344.

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