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