Asymptotic results for pressureless magneto--hydrodynamics

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.