A Hamilton-Jacobi approach for front propagation in kinetic equations

Emeric Bouin (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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.

Mots clés: Kinetic equations; Front propagation; Hyperbolic limit; Hopf-Cole transformation; Spectral problem; Geometric optics approximation.; Geometric optics approximation

Domaines: Equations aux dérivées partielles [math.AP]

Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. 2014. ⟨hal-01002918⟩