1D Model for the 3D Magnetohydrodynamics

We propose a one-dimensional (1D) model for the three-dimensional (3D) incompressible ideal magnetohydrodynamics. For this 1D model, local well-posedness is established, and a regularity criterion of the Beale-Kato-Majda type is obtained. Without the stretching effect, the model with only transport effect is shown to have global in time strong solution. Some numerical simulations suggest that solutions of the model with certain smooth periodic initial data are not likely to develop singularities in finite time, while solutions starting from other initial data have the tendency to form singularities.

Automated summary

We propose a one-dimensional (1D) model for the three-dimensional (3D) incompressible ideal magnetohydrodynamics. Local well-posedness is established for this 1D model, and a regularity criterion of the Beale-Kato-Majda type is obtained. The model with only transport effect, without the stretching effect, is shown to have a global in time strong solution. Numerical simulations suggest that solutions of the model with certain smooth periodic initial data are unlikely to develop singularities in finite time, while solutions starting from other initial data have the tendency to form singularities.