Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller--Segel type models

TL;DR

This paper proves the convergence of nonlinear hyperbolic chemotaxis models with volume filling effects to Keller--Segel type systems, using compensated compactness and existence results for small perturbations.

Contribution

It establishes the singular limit of hyperbolic chemotaxis systems to Keller--Segel models, including existence of solutions near constant states and convergence analysis.

Findings

Convergence of hyperbolic models to Keller--Segel systems proven.

Existence of solutions for small perturbations established.

Results applicable in two-dimensional periodic settings.

Abstract

In this paper we deal with diffusive relaxation limits of nonlinear systems of Euler type modeling chemotactic movement of cells toward Keller--Segel type systems. The approximating systems are either hyperbolic--parabolic or hyperbolic--elliptic. They all feature a nonlinear pressure term arising from a \emph{volume filling effect} which takes into account the fact that cells do not interpenetrate. The main convergence result relies on compensated compactness tools and is obtained for large initial data under suitable assumptions on the approximating solutions. In order to justify such assumptions, we also prove an existence result for initial data which are small perturbation of a constant state. Such result is proven via classical Friedrichs's symmetrization and linearization. In order to simplify the coverage, we restrict to the two--dimensional case with periodical boundary conditions.