Abstract
We rigorously prove the convergence of the micro–macro limit for particle approximations of the constrained pressureless gas dynamics system. The lack of BV bounds on the density variable is supplied by a compensated compactness argument.
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Aw, A., Klar, A., Materne, T., Rascle, M.: Derivation of continuum traffic flow models from microscopic follow-the-leader models. SIAM J. Appl. Math. 63(1), 259–278 (2002)
Aw, A., Rascle, M.: Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60, 916–938 (2000)
Berthelin, F.: Existence and weak stability for a pressureless model with unilateral constraint. Math. Models Methods Appl. Sci. 12(2), 249–272 (2002)
Berthelin, F., Broizat, D.: A model for the evolution of traffic jams in multi-lane. Kinet. Relat. Models 5(4), 697–728 (2012)
Berthelin, F., Degond, P., Delitala, M., Rascle, M.: A model for the formation and evolution of traffic jams. Arch. Ration. Mech. Anal. 187(2), 185–220 (2008)
Colombo, R.M., Marcellini, F., Rascle, M.: A 2-phase traffic model based on a speed bound. SIAM J. Appl. Math. 70(7), 2652–2666 (2010)
Colombo, R.M., Rossi, E.: On the micro–macro limit in traffic flow. Rend. Semin. Mat. Univ. Padova 131, 217–235 (2014)
Di Francesco, M., Fagioli, S., Rosini, M.D.: Deterministic particle approximation of scalar conservation laws. Boll. Unione Mat. Ital. (2017)
Di Francesco, M., Rosini, M.D.: Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit. Arch. Ration. Mech. Anal. 217(3), 831–871 (2015)
Francesco, M.D., Fagioli, S., Rosini, M.D.: Many particle approximation of the Aw–Rascle–Zhang second order model for vehicular traffic. Math. Biosci. Eng. 14(1), 127–141 (2017)
Francesco, M.D., Fagioli, S., Rosini, M.D., Russo, G.: Deterministic particle approximation of the Hughes model in one space dimension. Kinet. Relat. Models 10(1), 215–237 (2017)
Goatin, P., Rossi, F.: A traffic flow model with non-smooth metric interaction: well-posedness and micro–macro limit. Commun. Math. Sci. 15(1), 261–287 (2017)
Holden, H., Risebro, N.H.: Continuum Limit of Follow-the-Leader Models. ArXiv e-prints (2017)
Zhang, H.M.: A non-equilibrium traffic model devoid of gas-like behavior. Transp. Res. B: Methodol. 36(3), 275–290 (2002)
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To Prof. Alberto Bressan.
This article is part of the topical collection “Hyperbolic PDEs, Fluids, Transport and Applications: Dedicated to Alberto Bressan for his 60th birthday” guest edited by Fabio Ancona, Stefano Bianchini, Pierangelo Marcati, Andrea Marson.
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Berthelin, F., Goatin, P. Particle approximation of a constrained model for traffic flow. Nonlinear Differ. Equ. Appl. 24, 55 (2017). https://doi.org/10.1007/s00030-017-0480-8
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DOI: https://doi.org/10.1007/s00030-017-0480-8