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arXiv · 2012.11875

Stability of Couette flow for 2D Boussinesq system in a uniform magnetic field

Dongfen Bian·Shouyi Dai·Jingjing Mao
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Abstract

In this paper, we consider the Boussinesq equations with magnetohydrodynamics convection in the domain $\mathbb{T} \times \mathbb{R}$ and establishes the nonlinear stability of the Couette flow $(\mathbf{u}_{sh} = (y,0), \mathbf{b}_{sh} = (1,0), p_{sh} = 0, \theta_{sh} = 0$). The novelty in this paper is that we design a new Fourier multiplier operator by using the properties of the enhanced dissipation to overcome the difficult term $\partial_{xy}(-\Delta)^{-1}j$ in the linearized and nonlinear system. Then, we prove the asymptotic stability for the linearized system. Finally, we establish the nonlinear stability for the full system by bootstrap principle.

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Dongfen Bian, Shouyi Dai, Jingjing Mao. 2020-12-22. Stability of Couette flow for 2D Boussinesq system in a uniform magnetic field. https://arxiv.org/abs/2012.11875

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