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 volume filling effect which takes into account the fact that cells do not interpenetrate. The main convergence result relies on energy methods and 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.
Highlights
This paper deals with diffusive relaxation limits for the nonlinear hyperbolic model describing chemotactic movement of cells, known as the persistence and chemotaxis model, ∂τ ρ + ∇ · = 0
In order to produce a nontrivial class of solutions to the nonlinear hyperbolic system (1) which relax toward a Keller–Segel type model after a proper rescaling, we shall provide an existence theorem for the approximating system (4) and prove the uniform estimates needed to justify the assumptions (1) in case of initial densities ρ0 which are small perturbation of an arbitrary non zero constant state
The initial datum is not concentrated enough in order to see the appearance of a concentration in a finite time
Summary
In order to produce a nontrivial class of solutions to the nonlinear hyperbolic system (1) which relax toward a Keller–Segel type model after a proper rescaling, we shall provide an existence theorem for the approximating system (4) and prove the uniform estimates needed to justify the assumptions (1) in case of initial densities ρ0 which are small perturbation of an arbitrary non zero constant state (see Theorem 5.1). This result is achieved by means of the classical Friedrichs’ symmetrization technique and by a linearization argument, see [21, 30]. We shall deal with three different asymptotic regimes for (6), corresponding to small parameter limits of three different types of scaling
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