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

Bernou, Armand; Carrapatoso, Kleber; Mischler, Stéphane; Tristani, Isabelle

doi:10.48550/arxiv.2102.07709

Cited by 2 publications

(2 citation statements)

References 70 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also mention a recent paper (cf. [10]) where the authors studied the hypocoercivity for various linear kinetic equations with Maxwell boundary conditions. After this general introduction for the nonlinear Boltzmann equation, its properties and some boundary conditions, we look at long-time behaviour of some kinetic models for interacting gas particles.…”

Section: Kinetic Equations Arising In Mathematical Physicsmentioning

confidence: 99%

On quantitative hypocoercivity estimates based on Harris-type theorems

Yoldaş¹

2022

Preprint

View full text Add to dashboard Cite

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

show abstract

Section: Kinetic Equations Arising In Mathematical Physicsmentioning

confidence: 99%

On quantitative hypocoercivity estimates based on Harris-type theorems

Yoldaş¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…x,v . Such a norm is defined in Subsection 3.1 and is inspired by [33] (see also [7]) in which the more complex case of bounded domains with various boundary conditions is treated. Due to the fact that derivatives in x commute with Λ ε , it is easy to deduce a similar result on the space H 3 x L 2 v .…”

Section: Theorem 11mentioning

confidence: 99%

Regularization estimates and hydrodynamical limit for the Landau equation

Carrapatoso¹,

Rachid²,

Tristani³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we study the Landau equation under the Navier-Stokes scaling in the torus for hard and moderately soft potentials. More precisely, we investigate the Cauchy theory in a perturbative framework and establish some new short time regularization estimates for our rescaled nonlinear Landau equation. These estimates are quantified in time and optimal, indeed, we obtain the instantaneous expected anisotropic gain of regularity (see [53] for the corresponding hypoelliptic estimates on the linearized Landau collision operator). Moreover, the estimates giving the gain of regularity in the velocity variable are uniform in the Knudsen number. Intertwining these new estimates on the Landau equation with estimates on the Navier-Stokes-Fourier system, we are then able to obtain a result of strong convergence towards this fluid system.

show abstract

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

Cited by 2 publications

References 70 publications

On quantitative hypocoercivity estimates based on Harris-type theorems

On quantitative hypocoercivity estimates based on Harris-type theorems

Regularization estimates and hydrodynamical limit for the Landau equation

Contact Info

Product

Resources

About