A kinetic eikonal equation

Emeric Bouin; Vincent Calvez

doi:10.1016/j.crma.2012.03.009

Partial Differential Equations

A kinetic eikonal equation
[Une équation eikonale cinétique]

Présenté par : Pierre-Louis Lions

Emeric Bouin ¹ ; Vincent Calvez ¹

¹ UMR CNRS 5669 ‘UMPA’ and INRIA project ‘NUMED’, École normale supérieure de Lyon, 46, allée dʼItalie, 69364 Lyon cedex 07, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 243-248.

Résumé
Abstract

Nous analysons une équation cinétique linéaire de transport avec un opérateur de relaxation BGK. Nous étudions la limite hyperbolique de grande échelle $(t, x) \to (t / ε, x / ε)$ . Nous obtenons à la limite une nouvelle équation de Hamilton–Jacobi, qui est lʼanalogue de lʼéquation eikonale classique obtenue à partir de lʼéquation de la chaleur avec petite diffusion. Il est alors intéressant de constater que la limite hydrodynamique ne commute pas avec lʼasymptotique des grandes déviations. Nous démontrons le caractère bien posé de lʼéquation vérifiée par la phase, ainsi que la convergence vers une solution de viscosité de lʼéquation de Hamilton–Jacobi. Ceci est un travail préliminaire en vue dʼanalyser la propagation de fronts de réaction pour des équations cinétiques.

We analyse the linear kinetic transport equation with a BGK relaxation operator. We study the large scale hyperbolic limit $(t, x) \to (t / ε, x / ε)$ . We derive a new type of limiting Hamilton–Jacobi equation, which is analogous to the classical eikonal equation derived from the heat equation with small diffusivity. Interestingly, the hydrodynamic limit and the large deviation approach do not commute. We prove well-posedness of the phase problem and convergence towards the viscosity solution of the Hamilton–Jacobi equation. This is a preliminary work before analyzing the propagation of reaction fronts in kinetic equations.

Reçu le : 2012-02-10
Accepté le : 2012-03-07
Publié le : 2012-03-01

DOI : 10.1016/j.crma.2012.03.009

Affiliations des auteurs :

Emeric Bouin ¹ ; Vincent Calvez ¹

¹ UMR CNRS 5669 ‘UMPA’ and INRIA project ‘NUMED’, École normale supérieure de Lyon, 46, allée dʼItalie, 69364 Lyon cedex 07, France

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     title = {A kinetic eikonal equation},
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     pages = {243--248},
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     number = {5-6},
     year = {2012},
     doi = {10.1016/j.crma.2012.03.009},
     language = {en},
}

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Emeric Bouin; Vincent Calvez. A kinetic eikonal equation. Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 243-248. doi : 10.1016/j.crma.2012.03.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.03.009/

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Vincent Calvez; Pierre Gabriel; Álvaro Mateos González Limiting Hamilton–Jacobi equation for the large scale asymptotics of a subdiffusion jump-renewal equation, Asymptotic Analysis, Volume 115 (2019) no. 1-2, p. 63 | DOI:10.3233/asy-191528
EMERIC BOUIN; NILS CAILLERIE Spreading in kinetic reaction–transport equations in higher velocity dimensions, European Journal of Applied Mathematics, Volume 30 (2019) no. 2, p. 219 | DOI:10.1017/s0956792518000037
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Hélène Hivert A first-order asymptotic preserving scheme for front propagation in a one-dimensional kinetic reaction–transport equation, Journal of Computational Physics, Volume 367 (2018), p. 253 | DOI:10.1016/j.jcp.2018.04.036
Songting Luo; Nicholas Payne Properties-preserving high order numerical methods for a kinetic eikonal equation, Journal of Computational Physics, Volume 331 (2017), p. 73 | DOI:10.1016/j.jcp.2016.11.040
Songting Luo; Nicholas Payne An asymptotic method based on a Hopf–Cole transformation for a kinetic BGK equation in the hyperbolic limit, Journal of Computational Physics, Volume 341 (2017), p. 295 | DOI:10.1016/j.jcp.2017.04.028
Christopher Henderson; Panagiotis E. Souganidis The reactive-telegraph equation and a related kinetic model, Nonlinear Differential Equations and Applications NoDEA, Volume 24 (2017) no. 6 | DOI:10.1007/s00030-017-0488-0
Emeric Bouin; Vincent Calvez; Grégoire Nadin Propagation in a Kinetic Reaction-Transport Equation: Travelling Waves And Accelerating Fronts, Archive for Rational Mechanics and Analysis, Volume 217 (2015) no. 2, p. 571 | DOI:10.1007/s00205-014-0837-7
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