Macroscopic regularity for the Boltzmann equation

摘要

The regularity of solutions to the Boltzmann equation is a fundamentalproblem in the kinetic theory. In this paper, the case with angular cut-off isinvestigated. It is shown that the macroscopic parts of solutions to theBoltzmann equation, i.e. the density, momentum and total energy are continuousfunctions of $(x,t)$ in the region $\mathbb{R}^3\times(0,+\infty)$. Moreprecisely, these macroscopic quantities immediately become continuous in anypositive time even though they are initially discontinuous and thediscontinuities of solutions propagate only in the microscopic level. It shouldbe noted that such kind of phenomenon can not happen for the compressibleNavier-Stokes equations in which the initial discontinuities of the densitynever vanish in any finite time, see \cite{Hoff}. This hints that the Boltzmannequation has better regularity effect in the macroscopic level thancompressible Navier-Stokes equations.