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Shock-type singularity of the hyperbolic-parabolic chemotaxis system

Woojae Lee

TL;DR

The paper tackles finite-time shock-type singularity formation in a 1D hyperbolic-parabolic chemotaxis system modeling vasculogenesis. It innovates via a self-similar modulation framework around a Burgers-type profile, deriving a detailed blow-up construction that yields a unique blow-up point, a cusp with exponent 1/3 in the self-similar variable, and divergence of density- and velocity-gradients while keeping the chemoattractant regular. The main contributions include proving $H^m$ stability up to blow-up, explicit blow-up time and location estimates, and a comprehensive bootstrap that controls both hyperbolic and parabolic components. The results illuminate how sharp density-gradient accumulation drives singularity formation in HPC, with implications for understanding early vascular network formation and related nonlocal hyperbolic-parabolic PDEs. The approach combines Riemann-type reformulations, self-similar variables, and modulation methods to deliver a rigorous, quantitative blow-up scenario with cusp structure and preserved regularity of the chemoattractant field.

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{ç}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 $(ρ, u)$ of endothelial cells themselves remain bounded, the gradients of the density and velocity blow up. The chemoattractant concentration $φ$ 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.

Shock-type singularity of the hyperbolic-parabolic chemotaxis system

TL;DR

The paper tackles finite-time shock-type singularity formation in a 1D hyperbolic-parabolic chemotaxis system modeling vasculogenesis. It innovates via a self-similar modulation framework around a Burgers-type profile, deriving a detailed blow-up construction that yields a unique blow-up point, a cusp with exponent 1/3 in the self-similar variable, and divergence of density- and velocity-gradients while keeping the chemoattractant regular. The main contributions include proving stability up to blow-up, explicit blow-up time and location estimates, and a comprehensive bootstrap that controls both hyperbolic and parabolic components. The results illuminate how sharp density-gradient accumulation drives singularity formation in HPC, with implications for understanding early vascular network formation and related nonlocal hyperbolic-parabolic PDEs. The approach combines Riemann-type reformulations, self-similar variables, and modulation methods to deliver a rigorous, quantitative blow-up scenario with cusp structure and preserved regularity of the chemoattractant field.

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{ç}ot, and Perthame \cite{filbet2005derivation} and Filbet and Shu \cite{filbet2005approximation}. We construct the blow-up profile in a 1D HPC system on as follows: The blow-up profile is stable in the sense of topology () prior to the occurrence of the singularity. For the first singularity, while the density and velocity of endothelial cells themselves remain bounded, the gradients of the density and velocity blow up. The chemoattractant concentration has regularity. However, the density and velocity with 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 differentiable except in any neighborhood of the blow-up point.
Paper Structure (25 sections, 27 theorems, 300 equations)

This paper contains 25 sections, 27 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stable self-similar Burgers profile
  4. Reformulation
  5. Main result
  6. Assumptions on the initial data
  7. Statement of the main result
  8. Local well-posedness and continuation criterion
  9. Constraints of $W$ and the system of modulation variables
  10. Bootstrap assumptions
  11. Closing bootstrap argument I
  12. Modulation variables $\tau,\xi$ estimates
  13. $\Phi$ estiamtes
  14. Bounds for particle trajectories
  15. Z estimates
  16. ...and 10 more sections

Key Result

Theorem 1.1

For some smooth initial data $(q_0,u_0,\phi_0)$ with $\mathcal{O}(1)$ amplitude and with the maximally negative slope of $(q_0,u_0)$ equal to $-1/\epsilon$, for $\epsilon>0$ taken sufficiently small, there exist a smooth solution of the HPC system (simple model) such that the following hold true:

Theorems & Definitions (56)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Theorem 3.1: Shock-type singularity
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • Remark 4.1
  • ...and 46 more