Hypocoercivity in Phi-entropy for the linear relaxation Boltzmann equation on the Torus

In this article we have proved that the solution of the linear transport equation converges in the long time limit to its global equilibrium state at an exponential rate if and only

In the clas- sical di↵usion approximation of the linear Boltzmann equation, the di↵usion coeffi- cient is proportional to the reciprocal absorption rate (see for instance formula

A key feature of our approach is that it clearly distinguishes the mechanisms of relaxation at microscopic level (convergence towards a local equilibrium, in velocity space)

A second part of the paper is devoted to the time discretization of the Boltzmann equation, the main results being estimates of the rate of convergence for the

0 (the Boltzmann-Grad limit), the prob- ability density of a test particle converges to a solution of the linear (uncutoed) Boltzmann equation with the cross section relative to

Keywords: Boltzmann equation with external force, Hydrodynamical limit, In- compressible Navier-Stokes equation, Hypocoercivity, Knudsen

After this discussion on the theory concerning the Boltzmann equation for phonons and its solution in the relaxation time approximation, in a next paper, we will address how

Keywords: Boltzmann equation; spatially homogeneous; hard potentials; measure solutions; equilibrium; exponential rate of convergence; eternal solution; global-in-time