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arXiv · 1702.04168

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

Josephine Evans
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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.

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Josephine Evans. 2017-02-14. Hypocoercivity in Phi-entropy for the Linear Relaxation Boltzmann Equation on the Torus. https://arxiv.org/abs/1702.04168

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