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Sobolev Stability for the 2D MHD Equations in the Non-Resistive Limit
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abstract
In this article, we consider the stability of the 2D magnetohydrodynamics (MHD) equations close to a combination of Couette flow and a constant magnetic field. We consider the ideal conductor limit for the case when viscosity $\nu$ is larger than resistivity $\kappa$, $\nu\ge \kappa>0$. For this regime, we establish a bound on the Sobolev stability threshold. Furthermore, for $\kappa\le \nu^3$ this system exhibits instability, which leads to norm inflation of size $\nu \kappa^{-\frac 1 3 }$.
Forward citations
Cited by 2 Pith papers
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The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field
For 3D MHD with a rationally aligned background magnetic field, the sharp stability threshold around Couette flow is gamma=1, with inviscid damping and a nu^{-1/3} magnetic amplification.
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Suppression of Fluid Echoes and Sobolev Stability Threshold for 2D Dissipative Fluid Equations Around Couette Flow
A unified nonlinear estimate suppresses fluid echoes and reduces the Sobolev stability threshold for 2D Boussinesq and MHD Couette flow from 1/2 to 1/3, with a logarithmic correction for MHD.
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