Symmetry breaking in ideal magnetohydrodynamics: the role of the velocity

TL;DR

This paper investigates the lifespan of solutions to ideal magnetohydrodynamics equations, emphasizing the role of velocity and providing refined lifespan bounds under less restrictive initial magnetic field conditions.

Contribution

It establishes a continuation criterion based solely on the velocity field and improves the lower lifespan bound for 2D flows by relaxing initial magnetic field regularity.

Findings

Lifespan of solutions can be continued based on velocity criteria alone.

Refined lower bounds for 2D flow lifespan with weaker initial magnetic field assumptions.

The equations behave similarly to incompressible Euler equations in the small magnetic field regime.

Abstract

The ideal magnetohydrodynamic equations are, roughly speaking, a quasi-linear symmetric hyperbolic system of PDEs, but not all the unknowns play the same role in this system. Indeed, in the regime of small magnetic fields, the equations are close to the incompressible Euler equations. In the present paper, we adopt this point of view to study questions linked with the lifespan of strong solutions to the ideal magnetohydrodynamic equations. First of all, we prove a continuation criterion in terms of the velocity field only. Secondly, we refine the explicit lower bound for the lifespan of $2$-D flows found in [11], by relaxing the regularity assumptions on the initial magnetic field.