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Analysis of steady-state solutions to a vasculogenesis model in dimension two

Sinchita Lahiri, Kun Zhao

TL;DR

This work analyzes steady-state solutions of a chemotaxis-driven vasculogenesis model in a bounded two-dimensional domain and proves their local, and under favorable diffusion conditions, global stability. By constructing explicit non-constant steady states with $P(\rho)=A_0\rho^2$ on $\Omega=(0,1)^2$ via eigenfunction expansion and enforcing mass conservation, the authors reveal how a large diffusion coefficient $d$ ensures positivity and controllability of the equilibria. In the stability analysis, they focus on the fast-relaxation regime $\tau=0$ and develop a multi-step energy method that reduces the energy to temporal derivatives, recovers density dissipation, and couples curl and velocity estimates to establish exponential decay toward the steady state. A key finding is the quantitative link between stability thresholds and $d$, providing theoretical insight into pattern formation and self-organization in vasculogenesis under chemotactic coupling.

Abstract

We investigate the steady state solutions of a vasculogenesis model governed by coupled partial differential equations in a bounded two dimensional domain. Explicit steady state solutions are analytically constructed, and their stability is rigorously analyzed under prescribed initial and boundary conditions. By employing the energy method, we prove that these solutions exhibit local asymptotic stability when specific parametric criteria are satisfied. The analysis establishes a direct connection between the stability thresholds and the system's diffusion coefficient, offering quantitative insights into the mechanisms governing pattern formation. These results provide foundational theoretical advances for understanding self organization in chemotaxis-driven biological systems, particularly vasculogenesis.

Analysis of steady-state solutions to a vasculogenesis model in dimension two

TL;DR

This work analyzes steady-state solutions of a chemotaxis-driven vasculogenesis model in a bounded two-dimensional domain and proves their local, and under favorable diffusion conditions, global stability. By constructing explicit non-constant steady states with on via eigenfunction expansion and enforcing mass conservation, the authors reveal how a large diffusion coefficient ensures positivity and controllability of the equilibria. In the stability analysis, they focus on the fast-relaxation regime and develop a multi-step energy method that reduces the energy to temporal derivatives, recovers density dissipation, and couples curl and velocity estimates to establish exponential decay toward the steady state. A key finding is the quantitative link between stability thresholds and , providing theoretical insight into pattern formation and self-organization in vasculogenesis under chemotactic coupling.

Abstract

We investigate the steady state solutions of a vasculogenesis model governed by coupled partial differential equations in a bounded two dimensional domain. Explicit steady state solutions are analytically constructed, and their stability is rigorously analyzed under prescribed initial and boundary conditions. By employing the energy method, we prove that these solutions exhibit local asymptotic stability when specific parametric criteria are satisfied. The analysis establishes a direct connection between the stability thresholds and the system's diffusion coefficient, offering quantitative insights into the mechanisms governing pattern formation. These results provide foundational theoretical advances for understanding self organization in chemotaxis-driven biological systems, particularly vasculogenesis.

Paper Structure

This paper contains 12 sections, 2 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Literature Review
  4. Open question and current work
  5. Finding Steady state solution
  6. Stability of steady-state solution
  7. Preparation
  8. Reduction of total energy
  9. Energy estimates of temporal derivatives
  10. Recovery of density dissipation
  11. Estimate of curl
  12. Coupling of energy estimates

Key Result

Theorem 3.1

Consider system AA1 with initial data $(\rho,\mathbf{u})(\mathbf{x},0)=(\rho_0,\mathbf{u}_0)(\mathbf{x})$ and boundary conditions eqn:ourmodel, where $\alpha, d, a, b>0$ and $\beta\in\mathbb{R}$. Suppose that the initial data are compatible with the boundary condition, i.e., $\partial_t^k\mathbf{u}_ where $\tilde{\rho}=\rho-\hat{\rho}$, $\tilde{\Phi}=\Phi-\hat{\Phi}$, and $\hat{\Phi}$ denotes the

Theorems & Definitions (2)

  • Theorem 3.1
  • Lemma 3.2