  We study the Boltzmann equation with hard sphere in a near-equilibrium
setting. The initial data is compactly supported in the space variable and has
a polynomial tail in the microscopic velocity. We show that the solution can be
decomposed into a particle-like part (polynomial tail) and a fluid-like part
(Gaussian tail). The particle-like part decays exponentially in both space and
time, while the fluid-like part corresponds to the behavior of the compressible
Navier-Stokes equation, which dominates the long time behavior and exhibits
rich wave motion. The nonlinear wave interactions in the fluid-like part are
precisely characterized and therefore we are able to distinguish the linear and
nonlinear wave of the solution. It is notable that although the solution has
polynomial tail in the velocity initially, the transition process from the
polynomial to the Gaussian tail can be quantitatively revealed due to the
collision with the background global Maxwellian.
