Optimal time-decay rates of the Boltzmann equation

We consider the optimal time-convergence rates of the global solution to the Cauchy problem for the Boltzmann equation in a"e(3). We show that the global solution tends to the global Maxwellian at the optimal time-decay rate (1+t)(-3/4), where the macroscopic density, momentum and energy decay at the optimal rate (1 + t)(-3/4) and the microscopic part decays at the optimal rate (1 + t)(-5/4). We also show that the solution tends to the Maxwellian at the optimal time-decay rate (1 + t)(-5/4) in the case of the macroscopic part of the initial data is zero, where the macroscopic density, momentum and energy decay at the optimal rate (1+t)(-5/4) and the microscopic part decays at the optimal rate (1+t)(-7/4). These convergence rates are shown to be optimal for the Boltzmann equation.