Asymptotic results for pressureless magneto--hydrodynamics

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.