Compactness in kinetic transport equations and hypoellipticity

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L1fλ(x,v)∈L1 satisfying some appropriate transport relationv⋅∇xfλ=(1−Δx)β2(1−Δv)α2gλ may be inferred solely from additional integrability and compactness with respect to v. In a forthcoming work, the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the Boltzmann equation with a rough force field (Arsénio and Saint-Raymond, in preparation [4]).