A Hamilton–Jacobi approach to nonlocal kinetic equations

Abstract

Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, kinetic model with BGK type operators implementing a velocity-jump process for cell migration, directed by the nonlocal sensing of either an external signal or the cell population density itself. We describe, in an asymptotic regime, the precise profile of these concentrations which, at the macroscale, are Dirac masses. Because Dirac concentrations look like Gaussian potentials, we use the Hopf–Cole transform to calculate the potential adapted to the problem. This potential, as in other similar situations, is obtained through the viscosity solutions of a Hamilton–Jacobi equation. We begin with the linear case, when the heterogeneous external signal is given, and we show that the concentration profile obtained after the diffusion approximation is not correct and is a simple eikonal approximation of the true H–J equation. Its heterogeneous nature leads us to develop a new analysis of the implicit equation defining the Hamiltonian and a new condition to circumvent the ‘dimensionality problem’. In the nonlinear case, when the signal occurs from the cell density itself, it is shown that the already observed linear instability (pattern formation) occurs when the Hamiltonian is convex-concave, a striking new feature of our approach.