On pressureless gases driven by a strong inhomogeneous magnetic field

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 epsilon(-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(epsilon), penalized by a rotating term epsilon(-1) u(epsilon) boolean AND 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(epsilon) is an order one perturbation of the phase obtained in the case of a pure rotation (with no nonlinear transport term), epsilon(-1) B(x)t. Finally, going back to the pressureless gas system, we obtain the asymptotics of the density as epsilon goes to zero.