Hypocoercivity in Phi-entropy for the Linear Relaxation Boltzmann Equation on the Torus

Abstract: This paper studies convergence to equilibrium for the spatially inhomogeneous linear relaxation Boltzmann equation in Boltzmann entropy and related entropy functionals the $p$-entropies. Villani proved in \cite{V09} entropic hypocoercivity for a class of PDEs in a H\"{o}rmander sum of squares form. It was an open question to prove such a result for an operator which does not share this form. We show exponentially fast convergence to equilibrium with explicit rate in entropy for a linear relaxation Boltzmann equation. The key new idea appearing in our proof is the use of a total derivative of the entropy of a projection of our solution to compensate for additional error term which appear when using non-linear entropies. We also extend the proofs for hypocoercivity of both the linear relaxation Boltzmann and kinetic Fokker-Planck to the case of $p$-entropy functionals.