Asymptotic results for pressureless magneto--hydrodynamics

TL;DR

This paper studies the lifespan and asymptotic behavior of solutions to a pressureless magneto-hydrodynamics system under a strong magnetic field, revealing uniform existence time and phase oscillation properties as the magnetic parameter tends to zero.

Contribution

It establishes uniform existence of smooth solutions and characterizes the asymptotic phase behavior of the velocity field in a pressureless gas influenced by a strong, inhomogeneous magnetic field.

Findings

Solutions exist on a uniform time interval independent of the magnetic field strength.

The phase of oscillation of the velocity is an order one perturbation of pure rotation.

As the magnetic parameter tends to zero, the density's asymptotics are characterized.

Abstract

We are interested in the life span and the asymptotic behaviour of the solutions to a system governing the motion of a pressureless gas, submitted to a strong, inhomogeneous magnetic field $ \e^{-1} B(x)$, of variable amplitude but fixed direction -- this is a first step in the direction of the study of rotating Euler equations. This leads to the study of a multi--dimensional Burgers type system on the velocity field $ u_\e$, penalized by a rotating term $ \e^{-1} u_\e \wedge B(x)$. We prove that the unique, smooth solution of this Burgers system exists on a uniform time interval $ [0,T]$. We also prove that the phase of oscillation of $ u_\e$ is an order one perturbation of the phase obtained in the case of a pure rotation (with no nonlinear transport term), $ \e^{-1}B(x)t$. Finally going back to the pressureless gas system, we obtain the asymptotics of the density as $ \e$ goes to zero.