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General Stability Estimates in NonLocal Traffic Models for Several Populations

Rinaldo M. Colombo, Mauro Garavello, Claudia Nocita

TL;DR

This work analyzes a nonlocal, multiclass macroscopic traffic model with densities $\rho_1,\dots,\rho_n$ and nonlocal fluxes driven by kernels $\eta_{ij}$. The authors establish global existence, uniqueness, and $L^1$ stability by a fixed-point approach on $\mathbf{C^0}([0,T]; \mathbf{L^1}(\mathbb{R};\mathbb{R}^n))$, representing the solution via characteristics and nonlocal renewals, and then extend globally using BV estimates. They also prove continuous dependence of solutions on the initial data, speed laws $v_i$, and kernels $\eta_{ij}$, with explicit Lipschitz-type bounds, and provide detailed BV control and positivity. Complementing the theory, numerical simulations using a Lax--Friedrichs scheme demonstrate how horizon length, maximal speeds, and space inhomogeneities affect wave propagation, overtaking, and bottleneck passage, highlighting the practical impact of nonlocal terms in traffic dynamics.

Abstract

We prove global existence, uniqueness and $\L1$ stability of solutions to general systems of nonlocal conservation laws modeling multiclass vehicular traffic. Each class follows its own speed law and has specific effects on the other classes' speeds. Moreover, general explicit dependencies of the speed laws on space and time are allowed. Solutions are proved to depend continuously -- in suitable norms -- on all terms appearing in the equations, as well as on the initial data. Numerical simulations show the relevance and the effects of the nonlocal terms.

General Stability Estimates in NonLocal Traffic Models for Several Populations

TL;DR

This work analyzes a nonlocal, multiclass macroscopic traffic model with densities and nonlocal fluxes driven by kernels . The authors establish global existence, uniqueness, and stability by a fixed-point approach on , representing the solution via characteristics and nonlocal renewals, and then extend globally using BV estimates. They also prove continuous dependence of solutions on the initial data, speed laws , and kernels , with explicit Lipschitz-type bounds, and provide detailed BV control and positivity. Complementing the theory, numerical simulations using a Lax--Friedrichs scheme demonstrate how horizon length, maximal speeds, and space inhomogeneities affect wave propagation, overtaking, and bottleneck passage, highlighting the practical impact of nonlocal terms in traffic dynamics.

Abstract

We prove global existence, uniqueness and stability of solutions to general systems of nonlocal conservation laws modeling multiclass vehicular traffic. Each class follows its own speed law and has specific effects on the other classes' speeds. Moreover, general explicit dependencies of the speed laws on space and time are allowed. Solutions are proved to depend continuously -- in suitable norms -- on all terms appearing in the equations, as well as on the initial data. Numerical simulations show the relevance and the effects of the nonlocal terms.

Paper Structure

This paper contains 20 sections, 10 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Analytical Results
  3. Numerical Integrations
  4. The Role of the Horizon
  5. Different Maximal Speeds
  6. Space Inhomogeneity and Forward Horizon
  7. Analytical Proofs
  8. $\mathbf{L^1}$-norm of $\Sigma_\rho w$.
  9. $\mathop\mathrm{TV}$ estimate.
  10. $\Sigma_{\rho_o}$ is well defined.
  11. The map $\Sigma_{\rho_o}$ in \ref{['def:Sigma']} is Lipschitz continuous.
  12. The map $\mathcal{T}$ is well defined and Lipschitz continuous.
  13. Properties of the function $\mathcal{T}\rho.$
  14. The map $\mathcal{T}$ is a contraction for sufficiently small times.
  15. Proof of \ref{['item:4']}:
  16. ...and 5 more sections

Key Result

Theorem 2.3

Let item:1 and item:2 hold. Then, problem eq:9 generates a unique map with the following properties:

Figures (3)

  • Figure 3.1: Solutions to \ref{['eq:9']} for $n=2$ with the parameters as in § \ref{['subsec: horizon']}. The two populations have the same speed law, but the forward horizon of the first population is $5$ times that of the second one. As a result, the first population moves faster and its front vehicles share almost the same speed. On the contrary, the rightmost part of the graph of $\rho_2$ is less steep.
  • Figure 3.2: Solutions to \ref{['eq:9']}--\ref{['eq:30']}--\ref{['eq:29']} with $n=3$ populations on the domain $[0,\, 100]$ with the parameters given in Subsection \ref{['subsec:overtaking']}. Note that faster populations overtake slower populations and towards the end of the integration the populations appear ordered according to their maximal speed. Moreover, the graph at $t=28.7$ shows that a density may reach values higher than those attained at the initial time.
  • Figure 3.3: Solution $\rho$ to the nonlocal equation \ref{['eq:9']}--\ref{['eq:22']}--\ref{['eq:23']}--\ref{['eq:27']} compared to $r$, solving the LWR model \ref{['eq:28']}, with the same initial datum \ref{['eq:27']} and speed law \ref{['eq:22']}--\ref{['eq:23']}. The effect of the bottleneck in $[5,\,10]$ is clearly seen since time $t=4.0$. Note that already at time $t=37.5$ the solution to the nonlocal problem is supported on the right of the bottleneck, while the same happens only after $t=43.3$ for the solution to the LWR model.

Theorems & Definitions (12)

  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 4.1
  • Proposition 4.2
  • Lemma 4.3
  • Proposition 4.4
  • Lemma 4.5
  • Lemma 4.6
  • Proposition 4.7
  • ...and 2 more