Equilibrium solutions of the Bénard equations with an exterior force
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.
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