On Moffatt's magnetic relaxation equations

Abstract

We investigate the stability properties for a family of equations introduced by Moffatt to model magnetic relaxation. These models preserve the topology of magnetic streamlines, contain a cubic nonlinearity, and yet have a favorable L 2 energy structure. We consider the local and global in time well-posedness of these models and establish a difference between the behavior as t → ∞ with respect to weak and strong norms. CONTENTS 1. Introduction 2. Local existence in Sobolev spaces for all γ ≥ 0 3. Global existence for γ > d/2 + 1 4. Convergence as t → ∞ for γ > d/2 + 1 5. 2D stability of the state B = e 1 and u = 0 6. Nonlinear instabilities in 3D 7. Open problems References