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Open AccessArticle

On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection

by Dongfen Bian 1,2
1
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
2
Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China
Mathematics 2020, 8(7), 1049; https://doi.org/10.3390/math8071049
Received: 26 May 2020 / Revised: 17 June 2020 / Accepted: 26 June 2020 / Published: 30 June 2020
(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)
This paper is concerned with the nonlinear stability and instability of the two-dimensional (2D) Boussinesq-MHD equations around the equilibrium state ( u ¯ = 0 , B ¯ = 0 , θ ¯ = θ 0 ( y ) ) with the temperature-dependent fluid viscosity, thermal diffusivity and electrical conductivity in a channel. We prove that if a + a , and d 2 d y 2 κ ( θ 0 ( y ) ) 0 or 0 < d 2 d y 2 κ ( θ 0 ( y ) ) β 0 , with β 0 > 0 small enough constant, and then this equilibrium state is nonlinearly asymptotically stable, and if a + < a , this equilibrium state is nonlinearly unstable. Here, a + and a are the values of the equilibrium temperature θ 0 ( y ) on the upper and lower boundary.
Keywords: Boussinesq-MHD system; asymptotic stability; nonlinear instability Boussinesq-MHD system; asymptotic stability; nonlinear instability
MDPI and ACS Style

Bian, D. On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection. Mathematics 2020, 8, 1049.

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