A dyadic model for ideal MHD

Mimi Dai, Susan Friedlander

Published 2021-04-19Version 1

We study a dyadic model for incompressible ideal magnetohydrodynamics. Besides establishing global existence of weak solutions and local well-posedness, a set of fixed points are found. Solutions starting from positive data near a positive fixed point are shown to remain positive for all the time. Solutions with positive data are also shown to develop blow-up at finite time. In addition, we prove that the pure fluid steady state (with zero magnetic field) attracts solutions with stronger perturbation in the fluid component than that of the magnetic field; vice versa, the non-fluid steady state attracts solutions with stronger perturbation in the magnetic field. The solutions attracted by a fixed point dissipate energy.