{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T06:36:00Z","timestamp":1764570960478},"reference-count":40,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["NHM"],"published-print":{"date-parts":[[2021]]},"abstract":"<p style='text-indent:20px;'>We prove the convergence of the vanishing viscosity approximation for a class of <inline-formula><tex-math id=\"M2\">\\begin{document}$ 2\\times2 $\\end{document}<\/tex-math><\/inline-formula> systems of conservation laws, which includes a model of traffic flow in congested regimes. The structure of the system allows us to avoid the typical constraints on the total variation and the <inline-formula><tex-math id=\"M3\">\\begin{document}$ L^1 $\\end{document}<\/tex-math><\/inline-formula> norm of the initial data. The key tool is the compensated compactness technique, introduced by Murat and Tartar, used here in the framework developed by Panov. The structure of the Riemann invariants is used to obtain the compactness estimates.<\/p><\/jats:p>","DOI":"10.3934\/nhm.2021011","type":"journal-article","created":{"date-parts":[[2021,6,4]],"date-time":"2021-06-04T04:12:03Z","timestamp":1622779923000},"page":"413","source":"Crossref","is-referenced-by-count":1,"title":["Vanishing viscosity for a $ 2\\times 2 $ system modeling congested vehicular traffic"],"prefix":"10.3934","volume":"16","author":[{"given":"Giuseppe Maria","family":"Coclite","sequence":"first","affiliation":[]},{"given":"Nicola De","family":"Nitti","sequence":"additional","affiliation":[]},{"given":"Mauro","family":"Garavello","sequence":"additional","affiliation":[]},{"given":"Francesca","family":"Marcellini","sequence":"additional","affiliation":[]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2021011-1","doi-asserted-by":"publisher","unstructured":"A. Aw, M. Rascle.Resurrection of \"second order\" models of traffic flow, SIAM J. Appl. Math.<\/i>, 60<\/b> (2000), 916-938.","DOI":"10.1137\/S0036139997332099"},{"key":"key-10.3934\/nhm.2021011-2","doi-asserted-by":"publisher","unstructured":"C. Bardos, A. Y. le Roux, J.-C. N\u00e9d\u00e9lec.First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations<\/i>, 4<\/b> (1979), 1017-1034.","DOI":"10.1080\/03605307908820117"},{"key":"key-10.3934\/nhm.2021011-3","doi-asserted-by":"publisher","unstructured":"S. Bianchini, A. Bressan.Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2)<\/i>, 161<\/b> (2005), 223-342.","DOI":"10.4007\/annals.2005.161.223"},{"key":"key-10.3934\/nhm.2021011-4","doi-asserted-by":"publisher","unstructured":"S. Blandin, D. Work, P. Goatin, B. Piccoli, A. Bayen.A general phase transition model for vehicular traffic, SIAM J. Appl. Math.<\/i>, 71<\/b> (2011), 107-127.","DOI":"10.1137\/090754467"},{"key":"key-10.3934\/nhm.2021011-5","doi-asserted-by":"crossref","unstructured":"A. Bressan., Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem<\/i>, ${ref.volume}<\/b> (2000).","DOI":"10.1093\/oso\/9780198507000.001.0001"},{"key":"key-10.3934\/nhm.2021011-6","doi-asserted-by":"publisher","unstructured":"A. Bressan, R. M. Colombo.The semigroup generated by $2\\times 2$ conservation laws, Arch. Rational Mech. Anal.<\/i>, 133<\/b> (1995), 1-75.","DOI":"10.1007\/BF00375350"},{"key":"key-10.3934\/nhm.2021011-7","doi-asserted-by":"publisher","unstructured":"A. Bressan, G. Crasta and B. Piccoli, Well-posedness of the Cauchy problem for $n\\times n$ systems of conservation laws, Mem. Amer. Math. Soc.<\/i>, 146<\/b> (2000).","DOI":"10.1090\/memo\/0694"},{"key":"key-10.3934\/nhm.2021011-8","doi-asserted-by":"publisher","unstructured":"A. Bressan, T.-P. Liu, T. Yang.$L^1$ stability estimates for $n\\times n$ conservation laws, Arch. Ration. Mech. Anal.<\/i>, 149<\/b> (1999), 1-22.","DOI":"10.1007\/s002050050165"},{"key":"key-10.3934\/nhm.2021011-9","doi-asserted-by":"publisher","unstructured":"G.-Q. Chen.Remarks on R. J. DiPerna's paper: \"Convergence of the viscosity method for isentropic gas dynamics\", Proc. Amer. Math. Soc.<\/i>, 125<\/b> (1997), 2981-2986.","DOI":"10.1090\/S0002-9939-97-03946-4"},{"key":"key-10.3934\/nhm.2021011-10","doi-asserted-by":"publisher","unstructured":"G. -Q. Chen and H. Frid, Vanishing viscosity limit for initial-boundary value problems for conservation laws, in Nonlinear Partial Differential Equations<\/i>, Contemp. Math., Vol. 238, Amer. Math. Soc., Providence, RI, 1999, 35\u201351.","DOI":"10.1090\/conm\/238\/03538"},{"key":"key-10.3934\/nhm.2021011-11","unstructured":"G. M. Coclite, K. H. Karlsen, S. Mishra, N. H. Risebro.Convergence of vanishing viscosity approximations of $2\\times2$ triangular systems of multi-dimensional conservation laws, Boll. Unione Mat. Ital. (9)<\/i>, 2<\/b> (2009), 275-284."},{"key":"key-10.3934\/nhm.2021011-12","doi-asserted-by":"publisher","unstructured":"R. M. Colombo.Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math.<\/i>, 63<\/b> (2002), 708-721.","DOI":"10.1137\/S0036139901393184"},{"key":"key-10.3934\/nhm.2021011-13","doi-asserted-by":"publisher","unstructured":"R. M. Colombo, F. Marcellini, M. Rascle.A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math.<\/i>, 70<\/b> (2010), 2652-2666.","DOI":"10.1137\/090752468"},{"key":"key-10.3934\/nhm.2021011-14","doi-asserted-by":"publisher","unstructured":"C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics<\/i>, 4th edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 325, Springer-Verlag, Berlin, 2016.","DOI":"10.1007\/978-3-662-49451-6"},{"key":"key-10.3934\/nhm.2021011-15","doi-asserted-by":"publisher","unstructured":"R. J. DiPerna.Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys.<\/i>, 91<\/b> (1983), 1-30.","DOI":"10.1007\/BF01206047"},{"key":"key-10.3934\/nhm.2021011-16","doi-asserted-by":"publisher","unstructured":"L. C. Evans, Weak convergence methods for nonlinear partial differential equations<\/i>, CBMS Regional Conference Series in Mathematics, Vol. 74, Conference Board of the Mathematical Sciences, American Mathematical Society, Providence, RI, 1990.","DOI":"10.1090\/cbms\/074"},{"key":"key-10.3934\/nhm.2021011-17","doi-asserted-by":"publisher","unstructured":"S. Fan, M. Herty, B. Seibold.Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media<\/i>, 9<\/b> (2014), 239-268.","DOI":"10.3934\/nhm.2014.9.239"},{"key":"key-10.3934\/nhm.2021011-18","unstructured":"A. Friedman, Partial Differential Equations of Parabolic Type<\/i>, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964."},{"key":"key-10.3934\/nhm.2021011-19","doi-asserted-by":"publisher","unstructured":"M. Garavello, F. Marcellini.The Riemann problem at a junction for a phase transition traffic model, Discrete Contin. Dyn. Syst.<\/i>, 37<\/b> (2017), 5191-5209.","DOI":"10.3934\/dcds.2017225"},{"key":"key-10.3934\/nhm.2021011-20","doi-asserted-by":"publisher","unstructured":"P. Goatin.The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling<\/i>, 44<\/b> (2006), 287-303.","DOI":"10.1016\/j.mcm.2006.01.016"},{"key":"key-10.3934\/nhm.2021011-21","doi-asserted-by":"publisher","unstructured":"J. M. Greenberg, A. Klar, M. Rascle.Congestion on multilane highways, SIAM J. Appl. Math.<\/i>, 63<\/b> (2003), 818-833.","DOI":"10.1137\/S0036139901396309"},{"key":"key-10.3934\/nhm.2021011-22","doi-asserted-by":"publisher","unstructured":"F. Gu, Y.-g. Lu, Q. Zhang.Global solutions to one-dimensional shallow water magnetohydrodynamic equations, J. Math. Anal. Appl.<\/i>, 401<\/b> (2013), 714-723.","DOI":"10.1016\/j.jmaa.2012.12.042"},{"key":"key-10.3934\/nhm.2021011-23","doi-asserted-by":"publisher","unstructured":"H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws<\/i>, 2nd edition, Applied Mathematical Sciences, Vol. 152, Springer, Heidelberg, 2015.","DOI":"10.1007\/978-3-662-47507-2"},{"key":"key-10.3934\/nhm.2021011-24","unstructured":"B. S. Kerner, The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory<\/i>, Springer, Berlin, New York, 2004."},{"key":"key-10.3934\/nhm.2021011-25","doi-asserted-by":"publisher","unstructured":"C. Klingenberg, Y.-g. Lu.The vacuum case in Diperna's paper, J. Math. Anal. Appl.<\/i>, 225<\/b> (1998), 679-684.","DOI":"10.1006\/jmaa.1998.6050"},{"key":"key-10.3934\/nhm.2021011-26","unstructured":"S. N. Kru\u017ekov.First order quasilinear equations with several independent variables, Mat. Sb. (N.S.)<\/i>, 81<\/b> (1970), 228-255."},{"key":"key-10.3934\/nhm.2021011-27","doi-asserted-by":"publisher","unstructured":"J. P. Lebacque, X. Louis, S. Mammar, B. Schnetzlera, H. Haj-Salem.Mod\u00e9lisation du trafic autoroutier au second ordre, C. R. Math. Acad. Sci. Paris<\/i>, 346<\/b> (2008), 1203-1206.","DOI":"10.1016\/j.crma.2008.09.024"},{"key":"key-10.3934\/nhm.2021011-28","doi-asserted-by":"publisher","unstructured":"M. J. Lighthill, G. B. Whitham.On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A.<\/i>, 229<\/b> (1955), 317-345.","DOI":"10.1098\/rspa.1955.0089"},{"key":"key-10.3934\/nhm.2021011-29","doi-asserted-by":"publisher","unstructured":"T.-P. Liu, T. Yang.$L_1$ stability for $2\\times 2$ systems of hyperbolic conservation laws, J. Amer. Math. Soc.<\/i>, 12<\/b> (1999), 729-774.","DOI":"10.1090\/S0894-0347-99-00292-1"},{"key":"key-10.3934\/nhm.2021011-30","doi-asserted-by":"publisher","unstructured":"T.-P. Liu, T. Yang.$L_1$ stability of conservation laws with coinciding Hugoniot and characteristic curves, Indiana Univ. Math. J.<\/i>, 48<\/b> (1999), 237-247.","DOI":"10.1512\/iumj.1999.48.1601"},{"key":"key-10.3934\/nhm.2021011-31","unstructured":"Y. Lu, Hyperbolic Conservation Laws and the Compensated Compactness Method<\/i>, Chapman & Hall\/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 128, Chapman & Hall\/CRC, Boca Raton, FL, 2003."},{"key":"key-10.3934\/nhm.2021011-32","unstructured":"F. Murat.L'injection du c\u00f4ne positif de $H^{-1}$ dans $W^{-1, q}$ est compacte pour tout $q < 2$, J. Math. Pures Appl. (9)<\/i>, 60<\/b> (1981), 309-322."},{"key":"key-10.3934\/nhm.2021011-33","doi-asserted-by":"publisher","unstructured":"E. Panov.On weak completeness of the set of entropy solutions to a scalar conservation law, SIAM J. Math. Anal.<\/i>, 41<\/b> (2009), 26-36.","DOI":"10.1137\/080724587"},{"key":"key-10.3934\/nhm.2021011-34","doi-asserted-by":"publisher","unstructured":"P. I. Richards.Shock waves on the highway, Operations Res.<\/i>, 4<\/b> (1956), 42-51.","DOI":"10.1287\/opre.4.1.42"},{"key":"key-10.3934\/nhm.2021011-35","doi-asserted-by":"publisher","unstructured":"D. Serre.Solutions \u00e0 variations born\u00e9es pour certains syst\u00e8mes hyperboliques de lois de conservation, J. Differential Equations<\/i>, 68<\/b> (1987), 137-168.","DOI":"10.1016\/0022-0396(87)90189-6"},{"key":"key-10.3934\/nhm.2021011-36","unstructured":"L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV<\/i>, Res. Notes in Math., Vol. 39, Pitman, Boston, MA, London, 1979, 136\u2013212."},{"key":"key-10.3934\/nhm.2021011-37","doi-asserted-by":"publisher","unstructured":"M. E. Taylor, Partial Differential Equations I. Basic Theory<\/i>, 2nd edition, Applied Mathematical Sciences, Vol. 115, Springer, New York, 2011.","DOI":"10.1007\/978-1-4419-7055-8"},{"key":"key-10.3934\/nhm.2021011-38","doi-asserted-by":"publisher","unstructured":"B. Temple.Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc.<\/i>, 280<\/b> (1983), 781-795.","DOI":"10.1090\/S0002-9947-1983-0716850-2"},{"key":"key-10.3934\/nhm.2021011-39","doi-asserted-by":"publisher","unstructured":"G. Wong, S. Wong.A multi-class traffic flow model - an extension of LWR model with heterogeneous drivers, Transportation Research Part A: Policy and Practice<\/i>, 36<\/b> (2002), 827-841.","DOI":"10.1016\/S0965-8564(01)00042-8"},{"key":"key-10.3934\/nhm.2021011-40","doi-asserted-by":"publisher","unstructured":"D. -y. Zheng, Y. -g. Lu, G. -q. Song and X. -z. Lu, Global existence of solutions for a nonstrictly hyperbolic system, Abstr. Appl. Anal.<\/i> (2014), Art. ID 691429, 7 pp.","DOI":"10.1155\/2014\/691429"}],"container-title":["Networks & Heterogeneous Media"],"original-title":[],"deposited":{"date-parts":[[2023,11,4]],"date-time":"2023-11-04T13:52:48Z","timestamp":1699105968000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/nhm.2021011"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021]]},"references-count":40,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021]]}},"alternative-id":["1556-1801_2021_3_413"],"URL":"https:\/\/doi.org\/10.3934\/nhm.2021011","relation":{},"ISSN":["1556-181X"],"issn-type":[{"value":"1556-181X","type":"print"}],"subject":[],"published":{"date-parts":[[2021]]}}}