A kinetic eikonal equation

Bouin, Emeric; Calvez, Vincent

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

Nous analysons une équation cinétique linéaire de transport avec un opérateur de relaxation BGK. Nous étudions la limite hyperbolique de grande échelle (t,x)→(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)→(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.

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