On the Sobolev stability threshold for shear flows near Couette in 2D MHD equations

In this work, we study the Sobolev stability of shear flows near Couette in the 2D incompressible magnetohydrodynamics (MHD) equations with background magneticfield (alpha,0)(T)on TxR. More precisely, for sufficiently large alpha, we show that when the initial datum of the shear flow satisfies & Vert;U(y)-y & Vert;HN+6 << 1, with N>1, and the initial perturbationsu in and b(in) satisfy & Vert;(u(in),b(in))& Vert;HN+1=<<nu 5/(6)+ delta for any fixed delta>0, then the solution of the 2D MHD equations remains nu-((1)/(3)+ (delta)/(2))is an element of-close to(e(nu t partial derivative yy)U(y),0)(T)for allt>0.