Macroscopic regularity for the Boltzmann equation

The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e. the density, momentum and total energy are continuous functions of $(x,t)$ in the region $\mathbb{R}^3\times(0,+\infty)$. More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see \cite{Hoff}. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.