Equilibrium solutions of the Bénard equations with an exterior force
type: Item
instance of: scholarly article
title: Equilibrium solutions of the Bénard equations with an exterior force
zbMATH Open document ID: 1109.35093
published in: Advances in Differential Equations
publication date: 2006-09-22T00:00:00Z
review text: The author investigates the nonhomogeneous Bénard equations in the infinite layer \(\Omega=\mathbb{R}^2\times (-1/2,1/2)\) under the action of the exterior force \(f=f(z)\) \[ \begin{aligned} &\partial_t v=\nu\Delta v+\lambda kV-(v,\nabla)v+\widetilde{f}-\nabla p,\quad \text{div}(v)=0,\\ &\partial_tV=\kappa\Delta V+\lambda v_3-(v,\nabla) V+f_4=0,\quad v=V=0\; \text{ at }\; \pm\frac12, \end{aligned} \] where \(k=(0,0,1)^t\), \(f=(\tilde{f},f_4)\), \(\tilde{f}=(f_1,f_2,f_3)\). Stability of small \(v(z)\) under \(L\)-periodic perturbations belonging to \(\mathcal{L}^2(\Omega)\) is studied. Also the loss of stability arises in the neighbourhood of critical Rayleigh numbers \(\lambda_L\) (\(\lambda_\omega\)) for \(L\)-periodic (\(\mathcal{L}^2(\Omega)\)) setting, where \(\lambda_\omega\) is characterized in terms of Orr-Sommerfeld theory. Stability of solutions under \(L\)-periodic and \(\mathcal{L}^2(\Omega)\) perturbations when \(\lambda_L\neq\lambda_\omega\) is studied. In the proofs energy methods and Bloch space theory are used.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76E06
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 5056137
zbMATH Keywords: inhomogeneous Bénard equations
zbMATH Keywords: infinite layer
zbMATH Keywords: exterior force action
zbMATH Keywords: stability of solutions
author: Bruno Scarpellini
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
description: scientific article