A kinetic eikonal equation

We analyse the linear kinetic transport equation with a BGK relaxation operator. We study the large scale hyperbolic limit (t,x) -> (t/epsilon, x/epsilon). 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. (C) 2012 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.