Advances in Differential Equations

Equilibrium solutions of the Bénard equations with an exterior force

B. Scarpellini

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

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

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867737

Mathematical Reviews number (MathSciNet)
MR2175008

Subjects
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

Citation

Scarpellini, B. Equilibrium solutions of the Bénard equations with an exterior force. Adv. Differential Equations 10 (2005), no. 12, 1321--1344. https://projecteuclid.org/euclid.ade/1355867737.


Export citation