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Hypocoercivity in Phi-Entropy for the linear relaxation Boltzmann Equation on the Torus
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Evans, Josphine (2021) Hypocoercivity in Phi-Entropy for the linear relaxation Boltzmann Equation on the Torus. SIAM Journal of Mathematical Analysis, 53 (2). pp. 1357-1378. doi:10.1137/19M1277631 ISSN 0036-1410.
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Official URL: https://doi.org/10.1137/19M1277631
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 [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 $\Phi$-entropy functionals.
Item Type: | Journal Article | ||||||
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Alternative Title: | |||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Journal or Publication Title: | SIAM Journal of Mathematical Analysis | ||||||
Publisher: | Society for Industrial and Applied Mathematics | ||||||
ISSN: | 0036-1410 | ||||||
Official Date: | 8 March 2021 | ||||||
Dates: |
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Volume: | 53 | ||||||
Number: | 2 | ||||||
Page Range: | pp. 1357-1378 | ||||||
DOI: | 10.1137/19M1277631 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Reuse Statement (publisher, data, author rights): | First Published in SIAM Journal of Mathematical Analysis in 53(2) 2021, published by the Society for Industrial and Applied Mathematics (SIAM) Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 17 November 2020 | ||||||
Date of first compliant Open Access: | 31 March 2021 | ||||||
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