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On the Sobolev stability threshold for shear flows near Couette in 2D MHD equations
Published online by Cambridge University Press: 12 February 2024
Abstract
In this work, we study the Sobolev stability of shear flows near Couette in the 2D incompressible magnetohydrodynamics (MHD) equations with background magnetic field $(\alpha,0 )^\top$ on $\mathbb {T}\times \mathbb {R}$
. More precisely, for sufficiently large $\alpha$
, we show that when the initial datum of the shear flow satisfies $\left \| U(y)-y\right \|_{H^{N+6}}\ll 1$
, with $N>1$
, and the initial perturbations ${u}_{\mathrm {in}}$
and ${b}_{\mathrm {in}}$
satisfy $\left \| ( {u}_{\mathrm {in}},{b}_{\mathrm {in}}) \right \| _{H^{N+1}}=\epsilon \ll \nu ^{\frac 56+\tilde \delta }$
for any fixed $\tilde \delta >0$
, then the solution of the 2D MHD equations remains $\nu ^{-(\frac {1}{3}+\frac {\tilde \delta }{2})}\epsilon$
-close to $( e^{\nu t \partial _{yy}}U(y),0)^\top$
for all $t>0$
.
Keywords
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- Research Article
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- Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
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