  The Boltzmann equation is a nonlinear partial differential equation that
plays a central role in statistical mechanics. From the mathematical point of
view, the existence of global smooth solutions for arbitrary initial data is an
outstanding open problem. In the present article, we review a program focused
on the type of particle interactions known as non-cutoff. It is dedicated to
the derivation of a priori estimates in $C^\infty$, depending only on
physically meaningful conditions. We prove that the solution will stay
uniformly smooth provided that its mass, energy and entropy densities remain
bounded, and away from vacuum.
