Hypocoercivity of linear kinetic equations via Harris's Theorem

Cañizo, José A.; Cao, Chuqi; Evans, Josephine; Yoldaş, Havva

doi:10.3934/krm.2020004

Cited by 18 publications

(15 citation statements)

References 37 publications

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“…The linear Boltzmann equation was first shown to converge to equilibrium by Hérau in [31] and also falls under the scope of the powerful general theorem in [22]. In [16], written by the authors and others, we show that Harris's theorem from Markov process theory provides an alternative way of showing convergence to equilibrium for the linear Boltzmann equation amongst other equations. The run and tumble equation differs from the linear Boltzmann, and similar hypocoercive equations, in two key ways.…”

Section: Motivation Methodology and Noveltymentioning

confidence: 77%

“…This fact was exploited by the first author in [26] where we used Harris's theorem to find existence of a steady state for a non-linear kinetic equation with nonequilibrium steady states. Moreover, in [16], we showed that Harris's theorem can be applied efficiently to kinetic equations with nonlocal collision operators to obtain quantitative hypocoercivity results. In conclusion, the classical tools from hypocoercivity are difficult to apply on the run and tumble equation but Harris's approach gives promising results.…”

Section: Motivation Methodology and Noveltymentioning

confidence: 99%

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On the asymptotic behaviour of a run and tumble equation for bacterial chemotaxis

Evans¹,

Yoldaş²

2021

Preprint

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We study the asymptotic behaviour of the run and tumble model for bacteria movement. Experiments show that under the effect of a chemical stimulus, the movement of bacteria is a combination of a transport with a constant velocity, "run", and a random change in the direction of the movement, "tumble". This so-called velocity jump process can be described by a kinetic-transport equation. We focus on the situation for bacteria called E. Coli where the tumbling rate depends on a chemical stimulus but the post tumbling velocities do not.In this paper, we show that the linear run and tumble equation converges to a unique steady state solution with an exponential rate in a weighted total variation distance in dimension d ≥ 1. We provide a constructive and quantitative proof by using Harris's Theorem from ergodic theory of Markov processes. The result is an improvement of a recent paper by Mischler and Weng [40], since we are able to remove the radial symmetry assumption on the chemoattractant concentration. We also consider a weakly non-linear run and tumble equation by coupling it with a nonlocal equation on the chemoattractant concentration. We construct a unique stationary solution for the weakly nonlinear equation and show the exponential convergence towards it. The novelty of our paper consist in our generalisation of the spectral gap result to dimension d ≥ 1 under relaxed assumptions and the methods we used in the linear setting; and, all the results in the non-linear setting. Contents

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Section: Motivation Methodology and Noveltymentioning

confidence: 77%

Section: Motivation Methodology and Noveltymentioning

confidence: 99%

On the asymptotic behaviour of a run and tumble equation for bacterial chemotaxis

Evans¹,

Yoldaş²

2021

Preprint

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“…This is true for many distributions which decay only polynomially at infinity. For the linear relaxation Boltzmann equation convergence to equilibrium results in a polynomially weighted L 1 space exist in the paper [12] due to the author and collaborators. This polynomially weighted L 1 space allows for an even wider class of initial data than is considered in this paper.…”

Section: Previous Workmentioning

confidence: 99%

“…This polynomially weighted L 1 space allows for an even wider class of initial data than is considered in this paper. The work [12] uses Harris's theorem from Markov process theory rather than the entropy method used here which results in a less explicit rate of decay. • If we want to eventually study non-linear equations then it is often the case that strong spaces like f ∈ L 2 (µ −1 )/h ∈ L 2 (µ) will not be a natural space for the equation.…”

Section: Previous Workmentioning

confidence: 99%

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

Evans¹

2021

SIAM J. Math. Anal.

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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.

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“…Concerning Fokker-Planck equations and kinetic Fokker-Planck equations we shall quote [73,59] and [23], as well as the references therein. We also mention the results concerning degenerate linear transport equations [31,12,20], as well as degenerate linear Boltzmann equations [53]. Finally, the free transport equation with diffusive or Maxwell boundary condition has been tackled in [3,62,13] for instance.…”

Section: Weakly Coercive Operatorsmentioning

confidence: 99%

Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

Bernou¹,

Carrapatoso²,

Mischler³

et al. 2021

Preprint

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We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator, in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all type of boundary conditions in a unified framework. In particular, our result includes the case of vanishing accommodation coefficient and thus the specific case of the specular reflection boundary condition.

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Hypocoercivity of linear kinetic equations via Harris's Theorem

Cited by 18 publications

References 37 publications

On the asymptotic behaviour of a run and tumble equation for bacterial chemotaxis

On the asymptotic behaviour of a run and tumble equation for bacterial chemotaxis

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

Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

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