A Hamilton-Jacobi approach for front propagation in kinetic equations

TL;DR

This paper applies viscosity solutions of Hamilton-Jacobi equations to analyze front propagation in kinetic models with particles undergoing velocity jumps and reactions, revealing conditions for convergence and phenomena like front acceleration.

Contribution

It introduces a Hamilton-Jacobi framework for kinetic front propagation, including convergence results and analysis of acceleration phenomena in unbounded velocity cases.

Findings

Convergence of phase to viscosity solution under bounded velocities

Identification of front acceleration in unbounded velocity scenarios

Effective Hamiltonian obtained via eigenvalue problem in velocity space

Abstract

In this paper we use the theory of viscosity solutions for Hamilton-Jacobi equations to study propagation phenomena in kinetic equations. We perform the hydrodynamic limit of some kinetic models thanks to an adapted WKB ansatz. Our models describe particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The scattering operator is supposed to satisfy a maximum principle. When the velocity space is bounded, we show, under suitable hypotheses, that the phase converges towards the viscosity solution of some constrained Hamilton-Jacobi equation which effective Hamiltonian is obtained solving a suitable eigenvalue problem in the velocity space. In the case of unbounded velocities, the non-solvability of the spectral problem can lead to different behavior. In particular, a front acceleration phenomena can occur. Nevertheless, we expect that when the spectral problem is solvable one can extend the convergence result.