Tubes, Sheets and Singularities in Fluid Dynamics pp 295-304 | Cite as

# 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 3*D*-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 3*D* 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 2*D* 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*

## Keywords

Singularity Formation Vortex Tube Incompressible Euler Equation Magnetic Variable Infinite Energy## Preview

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