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

Dongfen Bian
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Dongfen Bian: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Mathematics, 2020, vol. 8, issue 7, 1-23

Abstract: 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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