Existence of Solutions of the Hyperbolic Keller-segel Model

We are concerned with the hyperbolic Keller-Segel model with quorum sensing, a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities. This is a first order quasilinear equation, its flux depends on space and time via the solution to an elliptic PDE in which the right-hand side is the solution to the hyperbolic equation. This model lacks strong compactness or contraction properties. Our purpose is to prove the existence of an entropy solution obtained, as usual, in passing to the limit in a sequence of solutions to the parabolic approximation. The method consists in the derivation of a kinetic formulation for the weak limit. The specific structure of the limiting kinetic equation allows for a ‘rigidity theorem’, which identifies some property of the solution (which might be nonunique) to this kinetic equation. This is enough to deduce a posteriori the strong convergence of a subsequence.