The resistive magnetohydrodynamic equation near an equilibrium

This work intends to understand the stability and large-time behavior of perturbations near a stationary solution of the 2D resistive magnetohydrodynamic (MHD) equation. The stationary solution is taken to be a background magnetic field parallel to the horizontal axis. We obtain three main results....

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Bibliographic Details
Published in:Journal of Differential Equations 2020-02, Vol.268 (4), p.1854-1871
Main Authors: Ji, Ruihong, Wu, Jiahong
Format: Article
Language:English
Subjects:
Background magnetic field
Large-time behavior
Resistive magnetohydrodynamic equation
Stability
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Summary:This work intends to understand the stability and large-time behavior of perturbations near a stationary solution of the 2D resistive magnetohydrodynamic (MHD) equation. The stationary solution is taken to be a background magnetic field parallel to the horizontal axis. We obtain three main results. The first result assesses the stability and the precise large-time asymptotic behavior for solutions to the linearzied system satisfied by the perturbation. Due to the lack of viscosity, the standard energy estimates do not work and the proof is achieved by constructing a suitable Lyapunov function. The second result makes use of the special wave structure of the linearization to establish the linear stability and decay rates. The third result obtains the H1-stability for the full nonlinear system and shows that the Lq-norm (q∈(2,∞)) of the magnetic field perturbation b˜ and the L2-norm of the gradient of b˜ approach zero as t→∞.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2019.09.027