2026/07/28 by Roberto Feola, Luca Franzoi, Riccardo Montalto +1
Mathematics · #math.AP #math-ph #math.MP
We investigate the long-wave linear stability and instability of the two-dimensional viscous, resistive Magnetohydrodynamic (MHD) equations, in vorticity-current formulation, on the periodic domain \mathbb Tα× \mathbb T = ( \mathbb R/(\frac2 πα \mathbb Z) × \mathbb R/(2 π\mathbb Z) ), around a periodic shear flow (U(y),0) coupled with a constant background magnetic field \bf b=(\rm b1,\rm b2). It is a non-trivial extension of a recent paper for the Navier-Stokes equations by Colombo, Dolce, Montalto & Ventura to the MHD setting in the spirit of the classical works of Kolmogorov, Meshalkin, Sinai and Yudovich. We establish explicit conditions on the shear flow profile U(y) involving the viscosity ν, the resistivity η and the components of the background magnetic field \bf b to obtain linear long-wave stability and instability in the regime α≪ 1. The proof combines a non-perturbative normal form transformation decoupling the zero Fourier mode from the non-zero modes with sharp asymptotic expansions of the eigenvalues bifurcating from the zero unperturbed eigenvalue with respect to the parameter α. As a dynamical consequence, we obtain a splitting of the phase space into unstable and stable subspaces, on which solutions grow or decay exponentially in Sobolev norm.