Existence of BV solutions for a non-conservative constrained Aw–Rascle–Zhang model for vehicular traffic

Dymski, Nikodem; Goatin, Paola; Rosini, Massimiliano D.

doi:10.1016/j.jmaa.2018.07.025

Cited by 9 publications

(10 citation statements)

References 20 publications

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“…Every weak solution of , is a coupling solution, but the converse does not necessarily hold true. This is in the same spirit of the solutions considered in [].…”

Section: The Mathematical Modeling Of the Flow Through A Valvesupporting

confidence: 54%

Coherence and chattering of a one‐way valve

Corli

Rosini

2019

Z Angew Math Mech

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In this paper we analyze the effects of a one-way valve on the isothermal gas flow through a pipe. The valve keeps the flow at a constant value * > 0, if possible; otherwise it is closed. First, for fixed * , we define a Riemann solver and characterize the coherence of its initial data; coherence is a necessary condition for the construction of solutions to a general initial-value problem based on a wave-front tracking scheme. We also give an example of an invariant and coherent domain where the valve can be either open or closed. Second, for suitable compact sets of initial data we make precise the range of values * that guarantee the coherence. At last, in the case of a real valve with finite reaction time, we show the chattering (rapid switch on and off) of the valve in correspondence with incoherent initial data. K E Y W O R D Schattering, coupling conditions, gas flow, Riemann problem, systems of conservation laws, valve 2 0 1 0 A

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“…Every weak solution of , is a coupling solution, but the converse does not necessarily hold true. This is in the same spirit of the solutions considered in [].…”

Section: The Mathematical Modeling Of the Flow Through A Valvesupporting

confidence: 54%

Coherence and chattering of a one‐way valve

Corli

Rosini

2019

Z Angew Math Mech

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“…For this reason in (7) (and then also in (8)) we consider test functions φ such that φ(·, 0) ≡ 0. This is in the same spirit of the solutions considered in [6,[12][13][14][15] for traffic through locations with reduced capacity. However, with this choice for the test functions in (7) and (8) we loose the possibility to better characterize the (density) flux at x = 0 associated to non-classical shocks.…”

Section: The Constrained Cauchy Problemmentioning

confidence: 63%

“…Choose T > 0 such that φ(t, x) = 0 whenever t ≥ T . By (14), the a.e. convergence of u n to u and the uniform continuity of E k and Q k , it is sufficient to prove that lim inf…”

Section: 4mentioning

confidence: 99%

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An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic

Benyahia

Donadello

Dymski

et al. 2018

Nonlinear Differ. Equ. Appl.

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In this paper we study a phase transition model for vehicular traffic flows. Two phases are taken into account, according to whether the traffic is light or heavy. We assume that the two phases have a non-empty intersection, the so called metastable phase. The model is given by the Lighthill-Whitham-Richards model in the free-flow phase and by the Aw-Rascle-Zhang model in the congested phase. In particular, we study the existence of solutions to Cauchy problems satisfying a local point constraint on the density flux. We prove that if the constraint F is higher than the minimal flux f − c of the metastable phase, then constrained Cauchy problems with initial data of bounded total variation admit globally defined solutions. We also provide sufficient conditions on the initial data that guarantee the global existence of solutions also in the case F < f − c . These results are obtained by applying the wave-front tracking technique.2010 Mathematical Subject Classification. Primary: 35L65, 90B20, 35L45

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“…As a consequence, a solution to (2) taking values in Ω ex c is not necessarily a weak solution to ARZ model. This is in the same spirit of the solutions considered in [9,[20][21][22][23] for traffic through locations with reduced capacity.…”

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confidence: 63%

Lack of BV bounds for approximate solutions to a two‐phase transition model arising from vehicular traffic

Benyahia

Rosini

2020

Math Methods in App Sciences

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We consider wave-front tracking approximate solutions to a two-phase transition model for vehicular traffic. We construct an explicit example showing that the total variation in space of the solution blows up in finite time even for an initial datum with bounded total variation. 2010 Mathematical Subject Classification. Primary: 35L65, 90B20, 35L45 Key words. Conservation laws, phase transitions, Lighthill-Whitham-Richards model, Aw-Rascle-Zhang model, point constraint on the density flux, wave-front tracking, blow up in finite time.

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Existence of BV solutions for a non-conservative constrained Aw–Rascle–Zhang model for vehicular traffic

Cited by 9 publications

References 20 publications

Coherence and chattering of a one‐way valve

Coherence and chattering of a one‐way valve

An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic

Lack of BV bounds for approximate solutions to a two‐phase transition model arising from vehicular traffic

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