  We consider the rate of convergence of solutions of spatially inhomogenous
Boltzmann equations, with hard sphere potentials, to some equilibriums, called
Maxwellians. Maxwellians are spatially homogenous static Maxwell velocity
distributions with different temperatures and mean velocities. We study
solutions in weighted space $L^{1}(\mathbb{R}^{3}\times \mathbb{T}^3)$. We
prove a conjecture of C. Villani: assume the solution is sufficiently localized
and sufficiently smooth, then the solution, in $L^{1}$-space, converges to a
Maxwellian, exponentially fast in time.
