On the localization of the magnetic and the velocity fields in the equations of magnetohydrodynamics

Abstract

We study the behaviour at infinity, with respect to the spatial variable, of solutions to the magnetohydrodynamics equations in $\mathbb{R}^d$. We prove that if the initial magnetic field decays sufficiently fast, then the plasma flow behaves as a solution of the free non-stationary Navier–Stokes equations when $|x|\to\infty$, and that the magnetic field will govern the decay of the plasma, if it is poorly localized at the beginning of the evolution. Our main tools are new boundedness criteria for convolution operators in weighted spaces.