Stability estimates for scalar conservation laws with moving flux constraints

Maria Laura Delle Monache; Paola Goatin

doi:10.3934/nhm.2017010

Article Contents

2017, Volume 12, Issue 2: 245-258. Doi: 10.3934/nhm.2017010

This issue Previous Article Numerical approximation of a coagulation-fragmentation model for animal group size statistics Next Article Existence of solutions to a boundary value problem for a phase transition traffic model

Stability estimates for scalar conservation laws with moving flux constraints

Maria Laura Delle Monache^1,2, and
Paola Goatin^3, ,

1.
Department of mathematical sciences, Rutgers University -Camden, 311 N. 5th Street, Camden, NJ 08102, USA
2.
Inria, Univ. Grenoble Alpes, CNRS, GIPSA-lab, F-38000 Grenoble, France
3.
Inria Sophia Antipolis -Méditerranée, Université Côte d'Azur, Inria, CNRS, LJAD, 2004, route des Lucioles -BP 93, 06902 Sophia Antipolis Cedex, France

* Corresponding author: Paola Goatin
* Corresponding author: Paola Goatin

Received: October 2016

Revised: January 2017

Published: October 2017

This research was supported by the Inria Associated Team "Optimal REroute Strategies for Traffic managEment" (ORESTE).

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Abstract

We study well-posedness of scalar conservation laws with moving flux constraints. In particular, we show the Lipschitz continuous dependence of BV solutions with respect to the initial data and the constraint trajectory. Applications to traffic flow theory are detailed.

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Mathematics Subject Classification: Primary: 35L65; Secondary: 35B35.

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Figure 1. Graphical representation of the constraint action in the fixed (left) and moving (right) reference frames

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Figure 2. The set $\mathcal{G}_\alpha (\dot{y})$ (thick lines) in the case of a flux function of the form $f(\rho)=V\rho(1-\rho/R)$, as in Section 3

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Figure 3. Bus and cars speed

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Figure 4. Different solutions of the Riemann problem (17). Each subfigure illustrates a point of the Definition 3.2: fundamental diagram representation (left) and space-time diagram (right)

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Figure 5. Case 1

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Figure 6. Case 2

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Figure 7. Case 3

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References

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