Effective Dynamics and Blow Up in a Model of Magnetic Relaxation

Abstract

In this article we study a one dimensional model for Magnetic Relaxation. This model was introduced by Moffatt and describes a low resistivity viscous plasma, in which the pressure and the inercia are much smaller than the magnetic pressure. In the limit of resistivity $\epsilon \to 0$, we prove the existence of two time scales for the evolution of the magnetic field: a fast one for times of order $\log ({\epsilon }^{-1})$ in which the resistivity plays no role and the energy is dissipated only via viscosity; and a slow one for times of order ${\epsilon }^{-1}$ characterized by the influence of the resistivity. We show that in this second time scale, as $\epsilon \to 0$, the modulus of magnetic field approaches a function that depends only on time. We also prove that, in this regime, the magnetic field $b_{\epsilon }(t,x)$ can be approximated as $\epsilon \to 0$ by the solution of a PDE whose solutions exhibit blow up for some choices of initial data.

52 pages, 6 figures