The kinetic Fokker–Planck equation with general force

Cao, Chuqi

doi:10.1007/s00028-021-00684-4

Cited by 11 publications

(17 citation statements)

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“…Moreover in a later work [20], the author considers the kinetic Fokker-Planck equation with general force term, i.e. (1.1) with (3.42) with β ≥ 2 and γ ≥ 1 and shows the exponential convergence to a unique equilibrium in a quantitative manner by using Harris-type arguments.…”

Section: The Kinetic Fokker-planck Equationmentioning

confidence: 99%

“…Then we prove the minorisation condition by iterating this and applying Duhamel's formula. The details can be found in Theorem 5.2 of [20].…”

Section: The Kinetic Fokker-planck Equationmentioning

confidence: 99%

“…Particularly they are used in the study of asymptotic behaviour of some models in population dynamics (e.g., [3,4,14,15,37,63]), kinetic theory (e.g. [8,13,19,20,32]) and extended to non-conservative cases, i.e. not conserving mass (e.g.…”

Section: Introductionmentioning

confidence: 99%

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On quantitative hypocoercivity estimates based on Harris-type theorems

Yoldaş¹

2022

Preprint

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This review concerns recent results on the quantitative study of convergence towards equilibrium for spatially in-homogeneous linear kinetic equations. We focus on analytical results obtained by means of certain probabilistic techniques from the ergodic theory of Markov processes. These techniques are sometimes referred to as Harris-type theorems. They provide constructive proofs for convergence results in L 1 (or total variation) setting for a wide class of initial data. The convergence rates can be made explicit (both for geometric and sub-geometric rates) by tracking the constants appearing in the hypotheses. Harris-type theorems are particularly well-adapted for equations exhibiting non-explicit and non-equilibrium steady states since they do not require prior information on the existence of stationary states. This allows for significant improvements of some already-existing results by relaxing assumptions and providing explicit convergence rates.We aim at presenting Harris-type theorems, providing a guideline on how to apply these techniques on kinetic equations at hand, presenting an exposition of recent quantitative results obtained for kinetic equations in gas theory and mathematical biology, as well as giving some perspectives on potential extensions to non-linear equations. Contents

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Section: The Kinetic Fokker-planck Equationmentioning

confidence: 99%

“…Then we prove the minorisation condition by iterating this and applying Duhamel's formula. The details can be found in Theorem 5.2 of [20].…”

Section: The Kinetic Fokker-planck Equationmentioning

confidence: 99%

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On quantitative hypocoercivity estimates based on Harris-type theorems

Yoldaş¹

2022

Preprint

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“…Moreover, hypocoercivity can be measured in many different ways, such as entropy, ϕentropies, total variation, Wasserstein distance, L 2 norms, H k norms, or more general weighted Sobolev norms, see for instance [5], [9], [3], [41], [16], [21], [7], [24](see also related work [22] and [23]), [25] and the references therein. We also call attentions to [39], [18], [10], [11], [12], [15], [14] and [30] for some recent results concerning hypocoercivity with various kinds of potentials. In particular, we highlight [8] and [37] for some results concerning singular potentials.…”

Section: Introductionmentioning

confidence: 99%

Hypocoercivity and global hypoellipticity for the kinetic Fokker-Planck equation in $H^k$ spaces

Zhang

2020

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The purpose of this paper is to extend the hypocoercivity results for the kinetic Fokker-Planck equation in H 1 space in Villani's memoir [45] to higher order Sobolev spaces. As in the L 2 and H 1 setting, there is lack of coercivity in H k for the associated operator. To remedy this issue, we shall modify the usual H k norm with certain well-chosen mixed terms and with suitable coefficients which are constructed by induction on k. In parallel, a similar strategy but with coefficients depending on time (c.f. [34]), usually referred as Hérau's method, can be employed to prove global hypoellipticity in H k . The exponents in our regularity estimates are optimal in short time. Moreover, as in our recent work [30], the general results here can be applied in the mean-field setting to get estimates independent of the dimension; in particular, an application to the Curie-Weiss model is presented.

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“…Their H 1 -convergence holds under the same assumptions. Note that is has triggered quite a lot of results for kinetic equations [16,26,8,9,10,20]. However Both Villani's and DMS's approach on the exponential convergence rate depends highly on the number N of particles.…”

Section: Introductionmentioning

confidence: 99%

The Kinetic Fokker-planck Equation With Mean Field Interaction

Guillin

Liu

et al. 2019

Preprint

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We study the long time behaviour of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fkker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H 1 (µ) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.

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The kinetic Fokker–Planck equation with general force

Cited by 11 publications

References 17 publications

On quantitative hypocoercivity estimates based on Harris-type theorems

On quantitative hypocoercivity estimates based on Harris-type theorems

Hypocoercivity and global hypoellipticity for the kinetic Fokker-Planck equation in $H^k$ spaces

The Kinetic Fokker-planck Equation With Mean Field Interaction

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