# Hopf bifurcations and global nonlinear $L^2$-energy stability in thermal MHD

### Salvatore Rionero

Università degli Studi di Napoli Federico II, Italy

## Abstract

The transfer of heat and mass by convection in a fluid horizontal layer $\texttt L$, heated from below – in the past as nowadays – has attracted the attention of many scientists since it can be driven by different factors and has a very relevant influence on the behaviour of many phenomena of the real world concerning meteorology, sun and stars physics, oceanography, heat insulation, air and water pollution, ... (see [1–12, 25–29] and references therein). When $\texttt L$ is filled by a plasma and is embedded in a transverse constant magnetic field, in the non-relativistic scheme of magneto-hydrodynamic (MHD), in the early of 1950 ([1–2]), the Nobel Prize laureate (1983) S. Chandrasekhar – in *linear* thermal MHD – obtained a relevant inhibition of convection by a magnetic field, verified experimentally [9] and appeared in 1961 in the celebrated monograph [3]. Successively, during the years, many efforts have been done in order to recover this relevant stabilizing effect in the nonlinear thermal MHD theory (see [6, 10, 11, 13, 16]). This goal has been reached partially in 1988 in [13] and totally in [16] but *only under very severe restrictions on the initial data* (of the order $<10^{-6}$). Recently in [17], via a non standard approach, assuming the verticality of the gradient pressure perturbations, it has been totally recovered in the nonlinear thermal MHD theory, the linear inhibition of convection by magnetic field *for any admissible initial data* (*Linearization Principle*). In the present paper, we return to the problem and *show that, in thermal MHD, the linear asymptotic stability implies the global exponential nonlinear $L^2-$energy stability*, without requiring the verticality of the perturbations to the pressure gradient. As concerns the onset of instabilities in the free-free case, since $P_r \geq P_m$ (with $P_r,P_m$ Prandtl and Prandlt magnetic numbers), implies the onset of steady bifurcation for any value of the Chandrasekhar number $Q^2$, we analyze the case $P_r<P_m$ via the introduction of two critical numbers $R_{C_2}, R_{C_3}$ coming from the application of the Hurwitz criterion to the spectral equation of the nth-component of the perturbations to the thermal conduction $m_0$ and obtain that if and only if $R_{C_2}<R_{C_3}$ the Hopf bifurcation occurs. Named *Hopf bifurcation number* the threshold $Q_c$ that the Chandrasekhar number has to cross for the occurring of Hopf bifurcations, we obtain that $Q_c=\frac{1+P_r}{P_m-P_r}\pi^2$. This formula – new in the existing literature – removes the difficulties (mentioned in page 184 of [3]) on finding a "simpler formula which gives $Q_c$ as function of $P_r$, $P_m$".

## Cite this article

Salvatore Rionero, Hopf bifurcations and global nonlinear $L^2$-energy stability in thermal MHD. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 4, pp. 881–905

DOI 10.4171/RLM/874