A kinetic eikonal equation

Emeric Bouin (1, 2) , Vincent Calvez (1, 2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 NUMED - Numerical Medicine

We analyse the linear kinetic transport equation with a BGK relaxation operator. We study the large scale hyperbolic limit $(t,x)\to (t/\eps,x/\eps)$. 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. 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 analysing the propagation of reaction fronts in kinetic equations.

Mots clés: Kinetic equations; front propagation; hyperbolic limit

Domaines: Equations aux dérivées partielles [math.AP]

Emeric Bouin, Vincent Calvez. A kinetic eikonal equation. 2012. ⟨hal-00668874⟩