Exponential convergence to the Maxwell distribution of solutions of spatially inhomogeneous Boltzmann equations

We consider the rate of convergence of solutions of spatially inhomogeneous Boltzmann equations, with hard-sphere potentials, to some equilibriums, called Maxwellians. Maxwellians are spatially homogeneous static Maxwell velocity distributions with different temperatures and mean velocities. We study solutions in weighted space L¹ (R³×T³). The result is that, assuming the solution is sufficiently localized and sufficiently smooth, then the solution, in L¹-space, converges to a Maxwellian, exponentially fast in time