Hypocoercivity in cascade for linearized Boltzmann type equations with external potentials

In this talk we present a very recent results on the return to the equilibrium for inhomogeneous linearized  kinetic equations when the collision kernel has 5 conserved moments (mass momentum and energy), in the case when there is an external potential acting on the system of particles.
In the case when there is no axisymetry of the potential, there is a unique equilibrium state (for a given initial mass and energy), and the return to the equilibrium is the result of a cascade of hypocoercive and damping effects. We shall try to explain the general scheme of the proof and the main mathematical and physical ideas behind.
This a joint work with  K. Carrapatoso, J. Dolbeault, S. Mischler, C. Schmeiser and C. Mouhot.