  In this work, we are concerned with the regularities of the solutions to
Boltzmann equation with the physical collision kernels for the full range of
intermolecular repulsive potentials, $r^{-(p-1)}$ with $p>2$. We
  give the new and constructive upper and lower bounds for the collision
operator in terms of standard fractional Sobolev norm. As an application, we
prove that the strong solutions obtained by Desvillettes \& Mouhot \cite{dm} to
homogeneous Boltzmann equation and classical solutions obtained by
Gressman-Strain \cite{gs1,gs2} or Alexandre-Morimoto-Ukai-Xu-Yang
\cite{amuxy3,amuxy5} for the inhomogeneous Boltzmann equation become
immediately smooth with respect to all variables. And as another application,
we obtain the global entropy dissipation estimate which is a little stronger
than the one of Alexandre-Desvillettes-Villani-Wennberg \cite{advw}.

Analysis of PDEs