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A semigroup approach to the convergence rate of a collisionless gas
Analysis of PDEs · Submitted 8 Nov 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim By semigroup arguments,we prove that in the $L^1$ norm, the polynomial rate of convergence$\frac{1}{(t+1)^{n-}}$ given by Tsuji et al [2010] and Kuo et al[2013,2014,2015] can be extended to any $C^2$ domain, with standard assumptionson the initial data.
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We study the rate of convergence to equilibrium for a collisionless (Knudsen)gas enclosed in a vessel in dimension $n \in \{2,3\}$. By semigroup arguments,we prove that in the $L^1$ norm, the polynomial rate of convergence$\frac{1}{(t+1)^{n-}}$ given by Tsuji et al [2010] and Kuo et al[2013,2014,2015] can be extended to any $C^2$ domain, with standard assumptionson the initial data. This is to our knowledge, the first quantitative result incollisionless kinetic theory in dimension equal to or larger than 2 relying ondeterministic arguments that does not require any symmetry of the domain, nor amonokinetic regime. The dependency of the rate with respect to the initialdistribution is detailed. Our study includes the case where the temperature atthe boundary varies. The demonstrations are adapted from a deterministicversion of a subgeometric Harris' theorem recently established by Ca\~nizo andMischler. We also compare our model with a free-transport equation withabsorbing boundary.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the rate of convergence to equilibrium for a collisionless (Knudsen)gas enclosed in a vessel in dimension $n \in \{2,3\}$. | contribution 0.66 | 0.79 | 0.06 | asserted 0.74 |
| s1 | By semigroup arguments,we prove that in the $L^1$ norm, the polynomial rate of convergence$\frac{1}{(t+1)^{n-}}$ given by Tsuji et al [2010] and Kuo et al[2013,2014,2015] can be extended to any $C^2$ domain, with standard assumptionson the initial data. | result 0.81 | 0.88 | 0.84 | asserted 1.00 |
| s2 | This is to our knowledge, the first quantitative result incollisionless kinetic theory in dimension equal to or larger than 2 relying ondeterministic arguments that does not require any symmetry of the domain, nor amonokinetic regime. | contribution 0.48 | 0.83 | 0.30 | asserted 0.97 |
| s3 | The dependency of the rate with respect to the initialdistribution is detailed. | contribution 0.73 | 0.89 | 0.05 | asserted 0.94 |
| s4 | Our study includes the case where the temperature atthe boundary varies. | contribution 0.95 | 0.87 | 0.04 | asserted 1.00 |
| s5 | The demonstrations are adapted from a deterministicversion of a subgeometric Harris' theorem recently established by Ca\~nizo andMischler. | method 0.98 | 0.34 | 0.04 | asserted 1.00 |
| s6 | We also compare our model with a free-transport equation withabsorbing boundary. | contribution 0.68 | 0.87 | 0.03 | asserted 0.97 |
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Original abstract
We study the rate of convergence to equilibrium for a collisionless (Knudsen)gas enclosed in a vessel in dimension $n \in \{2,3\}$. By semigroup arguments,we prove that in the $L^1$ norm, the polynomial rate of convergence$\frac{1}{(t+1)^{n-}}$ given by Tsuji et al [2010] and Kuo et al[2013,2014,2015] can be extended to any $C^2$ domain, with standard assumptionson the initial data. This is to our knowledge, the first quantitative result incollisionless kinetic theory in dimension equal to or larger than 2 relying ondeterministic arguments that does not require any symmetry of the domain, nor amonokinetic regime. The dependency of the rate with respect to the initialdistribution is detailed. Our study includes the case where the temperature atthe boundary varies. The demonstrations are adapted from a deterministicversion of a subgeometric Harris' theorem recently established by Ca\~nizo andMischler. We also compare our model with a free-transport equation withabsorbing boundary.
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We also mention the preprint [ref] which shows exponential convergence towards non-equilibrium steady states for the free transport equation in a domain with Maxwell boundary conditions.
Existence of a Non-Equilibrium Steady State for the Non-linear BGK equation on an interval · State of the Art · 0.96
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In [ref] , the authors study a similar model with probabilistic tools, more precisely they use a coupling of two Markov processes to derive a rate similar (up to logarithmic factors) to the one of Corollary [ref].
A coupling approach for the convergence to equilibrium for a collisionless gas · Introduction · 0.99
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