APDE seminar@EHU: Long-wave instability of periodic shear flows with constant magnetic field for the 2D resistive MHD equations

Abstract

The purpose of this talk is to present the article [1] in which we establish two results:

On the one hand, we study the long-wave linear (in)stability of the two-dimensional viscous, resistive Magnetohydrodynamic (MHD) equations on the periodic domain T_\alpha × T = R/(2π\alpha Z) × R/(2πZ) in the regime α ≪ 1, around a periodic shear flow (U(y), 0) coupled with a constant background magnetic field b = (b1, b2). It is a non-trivial extension of the recent paper [2] for the Navier-Stokes equations to the MHD setting. 

On the other hand, as a dynamical consequence of the spectral analysis carried out along the proof, we obtain a splitting of the phase space into unstable and stable subspaces, on which solutions grow or decay exponentially in Sobolev norm.

[1] Feola, R., Franzoi, L., Montalto, R., Peña, C. Long-wave instability of periodic shear flows with constant magnetic field for the 2D resistive MHD equations. arXiv:2607.25836.

[2] Colombo, M., Dolce, M., Montalto, R. and Ventura, P.: Long-wave instability of periodic shear flows for the 2D Navier-Stokes equations. arXiv:2509.18070.