Dynamic of thermo-MHD flows via a new approach

Abstract

Via a new approach, the dynamic of thermo-MHD flows in horizontal layers heated from below, in the Boussinesq scheme, is investigated. Denoting by $m_0$, $E$, $P_r$, $P_m$ and ${\bar{R}}_C$ the thermal conduction solution, the $L^2$-energy of the nonlinear perturbations to $m_0$, the Prandtl number, the magnetic Prandtl number and the steady convection onset critical value of the Rayleigh number $R^2$, first of all we obtain, for any initial data, the ultimately boundedness of $E$, via the existence of $L^2-$absorbing sets. Successively we introduce a new approach for the E-longtime behaviour associated to the $R^2-$increasing. This new approach is based on an $L^2$-energy linearization principle and on a new way of analyzing and using the linear stability. As concerns the linearization principle, denoting by $\mathcal{E}$ the $L^2$-energy of the linear perturbations to $m_0$, it is shown that: "${\left(\frac{d\mathcal{E}}{dt}\right)}_{(t=0)}<0$ for any initial data, implies $\frac{dE}{dt}<0$ $\forall t\in {\mathbb{R}}^{+}$“.

In order to obtain ${\left(\frac{d\mathcal{E}}{dt}\right)}_{(t=0)}<0$ for any initial data, denoting by ${\text{L}}_n$ the linear operator governing the nth-Fourier component of perturbations, we introduce the characteristic values ${\text{I}}_{jn},(j=1,2,3)$ of ${\text{L}}_n$ via the ${\text{L}}_n-$entries and obtain the equation ${\lambda }^3-{\text{I}}_{1n}{\lambda }^2+{\text{I}}_{2n}\lambda -{\text{I}}_{3n}=0$, governing the ${\text{L}}_n-$eigenvalues. To this equation we apply the Hurwitz's Criterion guaranteeing that all the eigenvalues have negative real part. As matter of fact, the Hurwitz's Criterion, applied for each $n\in \mathbb{N}$, allows to obtain conditions necessary and sufficient for being ${\left(\frac{d\mathcal{E}}{dt}\right)}_{(t=0)}<0$ for any initial data. Following this new approach, we show that the unconditional nonlinear asymptotic stability of $m_0$, with respect to the $L^2-$energy norm, is guaranteed by the linear stability and obtain – among other things – two conditions, in a very simple closed forms, guaranteeing the onset of oscillatory convection (overstability laws).

All the results, first obtained for the free-free boundary case, are successively generalized to the rigid-rigid, rigid-free and free-rigid boundary cases.