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. In [29], Villani 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 inequalityà 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 non-linear entropies. We also extend the proofs for hypocoercivity for the linear relaxation Boltzmann to the case of Φ-entropy functionals.Contents 8 2. Proofs for General Φ-entropy 9 2.1. Proof of the main theorem 16 Appendix A.

Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces

“…(5) In this case, other methods have been developed in [2,3,12,20] to solve this question in Hilbert space with Gaussian weight. Furthermore, the exponential decay rate toward to the global equilibrium µ of the equation ( 5) with an external confining potential is also obtained by [9,10,11,14].…”

Hypocoercivity of linear kinetic equations via Harris's Theorem

“…and we denote by (X t (x 0 , v 0 ), V t (x 0 , v 0 )) the solution at time t to (22) with initial data x(0) = x 0 , v(0) = v 0 . Performing time integration twice, it clearly satisfies…”

“…For the potentials of interest we will have that the solutions to these ODEs will exist for infinite time. We prove bounds on the solutions and ∇Φ(X t ) for any potential: (22) is defined (at least) for |t| ≤ T , with…”

“…For subquadratic potentials we give algebraic convergence rates, again in the same kind of weighted L 1 norms. Some results were already available for these equations [11,21,38,18,29,22]. We will give a more detailed account of them after we describe them more precisely.…”

Hypocoercivity with Schur complements

“…Recently, it was shown how to remove the assumption that the Hessian of the potential is bounded [21]. Entropic hypocoercivity for the linear Boltzmann equation was studied in [31,52], and for linearized BGK models (that generalize the linear Boltzmann equation) in dimension one in [1].…”