A kinetic eikonal equation

TL;DR

This paper derives a new Hamilton-Jacobi equation as a large-scale limit of a kinetic transport equation with BGK relaxation, establishing well-posedness and convergence to viscosity solutions, paving the way for reaction front analysis.

Contribution

It introduces a novel limiting Hamilton-Jacobi equation from kinetic equations, extending classical eikonal concepts to kinetic transport with BGK relaxation.

Findings

Derived a new Hamilton-Jacobi equation as a limit of kinetic transport.

Proved well-posedness of the phase problem.

Established convergence to viscosity solutions.

Abstract

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.