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 ${\text{e}}^{-1}B(x)$, 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 $u_{\text{e}}$, penalized by a rotating term ${\text{e}}^{-1}{u}_{\text{e}}\wedge B(x)$. 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 $u_{\text{e}}$ is an order one perturbation of the phase obtained in the case of a pure rotation (with no nonlinear transport term), ${\text{e}}^{-1}B(x)t$. Finally going back to the pressureless gas system, we obtain the asymptotics of the density as $\text{e}$ goes to zero.