Shock-type singularity of the hyperbolic-parabolic chemotaxis system

TL;DR

This paper analytically investigates shock-type singularity formation in a hyperbolic-parabolic chemotaxis model, revealing detailed blow-up profiles and regularity properties of solutions near singularities.

Contribution

It constructs explicit blow-up profiles for the 1D HPC system, demonstrating stability and detailed regularity behavior of solutions at the singularity.

Findings

Density and velocity gradients blow up at the singularity

Density and velocity are bounded but develop cusp singularities

Chemoattractant concentration remains smooth with $C^2$ regularity

Abstract

This paper deals with the hyperbolic-parabolic chemotaxis (HPC) model, which is a hydrodynamic model describing vascular network formation at the early stage of the vasculature. We study analytically the singularity formation associated with the shock-type structure, which was numerically observed by Filbet, Lauren{\c{c}}ot, and Perthame \cite{filbet2005derivation} and Filbet and Shu \cite{filbet2005approximation}. We construct the blow-up profile in a 1D HPC system on $\mathbb{R}$ as follows: The blow-up profile is stable in the sense of $H^m$ topology ($m\geq 5$) prior to the occurrence of the singularity. For the first singularity, while the density and velocity $(\rho, u)$ of endothelial cells themselves remain bounded, the gradients of the density and velocity blow up. The chemoattractant concentration $\phi$ has $C^2$ regularity. However, the density and velocity with $C^ {\frac{1}{3}}$ regularity exhibit a cusp singularity at a unique blow-up point, the location and time of which are explicitly estimated. Furthermore, the HPC system is $C^1$ differentiable except in any neighborhood of the blow-up point.