On Pressureless Gases Driven by a Strong Inhomogeneous Magnetic Field

Abstract

We are interested in the life span and the asymptotic behavior of the solutions to a system governing the motion of a pressureless gas that is submitted to a strong, inhomogeneous magnetic field ${\varepsilon}^{-1} B(x)$ 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 multidimensional Burgers-type system on the velocity field $u_{\varepsilon}$, penalized by a rotating term ${\varepsilon}^{-1} u_{\varepsilon}\wedge B(x)$. 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 $u_{\varepsilon}$ is an order one perturbation of the phase obtained in the case of a pure rotation (with no nonlinear transport term), ${\varepsilon}^{-1}B(x)t$. Finally, going back to the pressureless gas system, we obtain the asymptotics of the density as ${\varepsilon}$ goes to zero.