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

2017/02/14 by Josephine Evans, Evans, Josephine · 1 citation
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Mathematical Physics (math-ph) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1702.04168

openalex publication_date 2017/02/14 · openalex created_date 2022/08/05 · openalex updated_date 2026/07/28

Abstract

This paper studies convergence to equilibrium for the spatially inhomogeneous\nlinear relaxation Boltzmann equation in Boltzmann entropy and related entropy\nfunctionals the p-entropies. Villani proved in citeV09 entropic\nhypocoercivity for a class of PDEs in a H "ormander sum of squares form. It\nwas an open question to prove such a result for an operator which does not\nshare this form. We show exponentially fast convergence to equilibrium with\nexplicit rate in entropy for a linear relaxation Boltzmann equation. The key\nnew idea appearing in our proof is the use of a total derivative of the entropy\nof a projection of our solution to compensate for additional error term which\nappear when using non-linear entropies. We also extend the proofs for\nhypocoercivity of both the linear relaxation Boltzmann and kinetic\nFokker-Planck to the case of p-entropy functionals.\n

Citations

• Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential
• Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation
• Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus

Cited by

• Discrete H-theorem for a finite volume discretization of a nonlinear kinetic system: application to hypocoercivity

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