Singularity formation in a class of stretched solutions of the equations for ideal magneto-hydrodynamics (Q2757158)

The authors study a class of stretched solutions of the equations of three-dimensional incompressible ideal magnetohydrodynamics. Two-dimensional partial differential equations are obtained that are valid in a tubular domain which is infinite in the \(z\)-direction with periodic cross section. Pseudo-spectral computations of these equations provide evidence for a blow-up in a finite time in the above variables. This apparent blow up is an infinite energy process that gives rise to certain subtleties; while all the variables appear to blowup simultaneously, the two-dimensional part of the magnetic field blows up at a very late stage. This singularity in the magnetic field is hard to detect numerically, but a supporting analytical evidence of Lagrangian nature is provided for its existence. In three dimensions these solutions correspond to magnetic vortices developing along the axis of the tube prior to breakdown.