Global Weak Solutions to the Cauchy Problem for a Two-Phase Model at a Node

Garavello, Mauro; Marcellini, Francesca

doi:10.1137/19m1265041

Cited by 2 publications

(3 citation statements)

References 29 publications

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“…We can now state our well-posedness result. We explicitly point out that it is an existence and uniqueness result, whereas several other results concerning traffic models on road networks only establish existence, see for instance [15,24,25].…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

“…On the other hand, networks only containing T-junctions are the only ones where one can reasonably hope for propagation of the condition that the traffic is not congested. See also [24] for another recent work where the authors restrict to T-junctions. In the present work we also establish propagation of BV regularity and stability, and these results are again interesting in view of counterexamples in [12] that we discuss in the following.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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New regularity results for scalar conservation laws, and applications to a source-destination model for traffic flows on networks

Dovetta¹,

Marconi²,

Spinolo³

2021

Preprint

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We focus on entropy admissible solutions of scalar conservation laws in one space dimension and establish new regularity results with respect to time. First, we assume that the flux function f is strictly convex and show that, for every x ∈ R, the total variation of the composite function f • u(•, x) is controlled by the total variation of the initial datum. Next, we assume that f is monotone and, under no convexity assumption, we show that, for every x, the total variation of the left and right trace u(•, x ± ) is controlled by the total variation of the initial datum. We also exhibit a counter-example showing that in the first result the total variation bound does not extend to the function u, or equivalently that in the second result we cannot drop the monotonicity assumption. We then discuss applications to a source-destination model for traffic flows on road networks. We introduce a new approach, based on the analysis of transport equations with irregular coefficients, and, under the assumption that the network only contains so-called T-junctions, we establish existence and uniqueness results for merely bounded data in the class of solutions where the traffic is not congested. Our assumptions on the network and the traffic congestion are basically necessary to obtain wellposedness in view of a counter-example due to Bressan and Yu. We also establish stability and propagation of BV regularity, and this is again interesting in view of recent counter-examples.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 87%

Section: Introduction and Main Resultsmentioning

confidence: 99%

New regularity results for scalar conservation laws, and applications to a source-destination model for traffic flows on networks

Dovetta¹,

Marconi²,

Spinolo³

2021

Preprint

View full text Add to dashboard Cite

show abstract

mentioning

confidence: 89%

New Regularity Results for Scalar Conservation Laws, and Applications to a Source-Destination Model for Traffic Flows on Networks

Dovetta¹,

Marconi²,

Spinolo³

2022

SIAM J. Math. Anal.

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We focus on entropy admissible solutions of scalar conservation laws in one space dimension and establish new regularity results with respect to time. First, we assume that the flux function f is strictly convex and show that, for every x \in \BbbR , the total variation of the composite function f \circ u(\cdot , x) is controlled by the total variation of the initial datum. Next, we assume that f is monotone and, under no convexity assumption, we show that, for every x, the total variation of the left and the right trace u(\cdot , x \pm ) is controlled by the total variation of the initial datum. We also exhibit a counterexample showing that in the first result the total variation bound does not extend to the function u, or equivalently that in the second result we cannot drop the monotonicity assumption. We then discuss applications to a source-destination model for traffic flows on road networks. We introduce a new approach, based on the analysis of transport equations with irregular coefficients, and, under the assumption that the network only contains so-called T-junctions, we establish existence and uniqueness results for merely bounded data in the class of solutions where the traffic is not congested. Our assumptions on the network and the traffic congestion are basically necessary to obtain well-posedness in view of a counterexample due to Bressan and Yu. We also establish stability and propagation of BV regularity, and this is again interesting in view of recent counterexamples.

show abstract

Global Weak Solutions to the Cauchy Problem for a Two-Phase Model at a Node

Cited by 2 publications

References 29 publications

New regularity results for scalar conservation laws, and applications to a source-destination model for traffic flows on networks

New regularity results for scalar conservation laws, and applications to a source-destination model for traffic flows on networks

New Regularity Results for Scalar Conservation Laws, and Applications to a Source-Destination Model for Traffic Flows on Networks

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