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

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 [Hypocoercivity, in Memoirs of the American Mathematical Society, Vol. 202, American Mathematical Society, Providence, RI, 2006] proved entropic hypocoercivity for a class of PDEs in a Hörmander sum-of-squares form. It was an open question to prove such a result for an operator which does not share this form. We prove a closed entropy-entropy production inequality à la Villani which implies exponentially fast convergence to equilibrium for the linear Boltzmann equation with a quantitative rate. 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 an error term which appears when using nonlinear entropies. We also extend the proofs for hypocoercivity for the linear relaxation Boltzmann to the case of Φ-entropy functionals.