Evidence for singularity formation in a class of stretched solutions of the equations for ideal MHD

Abstract

A class of stretched solutions of the equations for incompressible, ideal 3D-MHD are studied using Elsasser variables V ^±=U±B. This class takes the form V^±=(v^±, v ^±₃ ) where v ^±=v^±(x,y,t) and v ^±₃ (x,y,z,t)=zγ ^±(x,y,t)+β^±(x,y,t). The chosen domain is of a tubular form which is infinite in the z-direction with periodic cross-section. This follows a previous study by the authors on this same class of solutions for the 3D Euler equations. In both cases the systems are of infinite energy. Strong numerical evidence for a finite time singularity in the Euler case was subsequently confirmed by a rigorous analytical proof by Constantin. In the MHD case, pseudo-spectral computations of the 2D partial differential equations for & γ ^±, v ^± and β ^± valid on the cross-sectional domain provide evidence for a finite time blow-up in both the fluid and magnetic variables although an analytical proof for the existence of this singularity remains elusive