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 entropy and related functionals, $p$-entropies. Villani [Hypocoercivity, Memoirs of American Mathematical Society, Vol. 202, Providence, RI, 2006] proved entropic hypocoercivity a class PDEs Hörmander sum-of-squares form. It was an open question prove such result operator which does not share this We closed entropy-entropy production inequality à la implies exponentially fast with quantitative rate. The key new idea appearing our proof is use total derivative projection solution compensate error term appears when using nonlinear entropies. also extend proofs case $\Phi$-entropy functionals.