September  2021, 16(3): 413-426. doi: 10.3934/nhm.2021011

Vanishing viscosity for a $ 2\times 2 $ system modeling congested vehicular traffic

1. 

Department of Mechanics, Mathematics and Management, Polytechnic University of Bari, Via E. Orabona 4, 70125 Bari, Italy

2. 

Department of Data Science, Chair in Applied Analysis (Alexander von Humboldt Professorship), Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany

3. 

Department of Mathematics and its Applications, University of Milano Bicocca, via R. Cozzi 55, 20125 Milano, Italy

4. 

Department of Information Engineering, University of Brescia, via Branze 38, 25123 Brescia, Italy

* Corresponding author: Nicola De Nitti

Received  October 2020 Revised  March 2021 Published  September 2021 Early access  May 2021

We prove the convergence of the vanishing viscosity approximation for a class of $ 2\times2 $ 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 $ L^1 $ 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.

Citation: Giuseppe Maria Coclite, Nicola De Nitti, Mauro Garavello, Francesca Marcellini. Vanishing viscosity for a $ 2\times 2 $ system modeling congested vehicular traffic. Networks and Heterogeneous Media, 2021, 16 (3) : 413-426. doi: 10.3934/nhm.2021011
References:
[1]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.  doi: 10.1137/S0036139997332099.

[2]

C. BardosA. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034.  doi: 10.1080/03605307908820117.

[3]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2), 161 (2005), 223-342.  doi: 10.4007/annals.2005.161.223.

[4]

S. BlandinD. WorkP. GoatinB. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127.  doi: 10.1137/090754467.

[5] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, Vol. 20, Oxford University Press, Oxford, 2000. 
[6]

A. Bressan and R. M. Colombo, The semigroup generated by $2\times 2$ conservation laws, Arch. Rational Mech. Anal., 133 (1995), 1-75.  doi: 10.1007/BF00375350.

[7]

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., 146 (2000). doi: 10.1090/memo/0694.

[8]

A. BressanT.-P. Liu and T. Yang, $L^1$ stability estimates for $n\times n$ conservation laws, Arch. Ration. Mech. Anal., 149 (1999), 1-22.  doi: 10.1007/s002050050165.

[9]

G.-Q. Chen, Remarks on R. J. DiPerna's paper: "Convergence of the viscosity method for isentropic gas dynamics", Proc. Amer. Math. Soc., 125 (1997), 2981-2986.  doi: 10.1090/S0002-9939-97-03946-4.

[10]

G. -Q. Chen and H. Frid, Vanishing viscosity limit for initial-boundary value problems for conservation laws, in Nonlinear Partial Differential Equations, Contemp. Math., Vol. 238, Amer. Math. Soc., Providence, RI, 1999, 35–51. doi: 10.1090/conm/238/03538.

[11]

G. M. CocliteK. H. KarlsenS. Mishra and N. H. Risebro, Convergence of vanishing viscosity approximations of $2\times2$ triangular systems of multi-dimensional conservation laws, Boll. Unione Mat. Ital. (9), 2 (2009), 275-284. 

[12]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.  doi: 10.1137/S0036139901393184.

[13]

R. M. ColomboF. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666.  doi: 10.1137/090752468.

[14]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 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.

[15]

R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys., 91 (1983), 1-30.  doi: 10.1007/BF01206047.

[16]

L. C. Evans, Weak convergence methods for nonlinear partial differential equations, 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.

[17]

S. FanM. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2014), 239-268.  doi: 10.3934/nhm.2014.9.239.

[18]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[19]

M. Garavello and F. Marcellini, The Riemann problem at a junction for a phase transition traffic model, Discrete Contin. Dyn. Syst., 37 (2017), 5191-5209.  doi: 10.3934/dcds.2017225.

[20]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303.  doi: 10.1016/j.mcm.2006.01.016.

[21]

J. M. GreenbergA. Klar and M. Rascle, Congestion on multilane highways, SIAM J. Appl. Math., 63 (2003), 818-833.  doi: 10.1137/S0036139901396309.

[22]

F. GuY.-g. Lu and Q. Zhang, Global solutions to one-dimensional shallow water magnetohydrodynamic equations, J. Math. Anal. Appl., 401 (2013), 714-723.  doi: 10.1016/j.jmaa.2012.12.042.

[23]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2nd edition, Applied Mathematical Sciences, Vol. 152, Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2.

[24]

B. S. Kerner, The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory, Springer, Berlin, New York, 2004.

[25]

C. Klingenberg and Y.-g. Lu, The vacuum case in Diperna's paper, J. Math. Anal. Appl., 225 (1998), 679-684.  doi: 10.1006/jmaa.1998.6050.

[26]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. 

[27]

J. P. LebacqueX. LouisS. MammarB. Schnetzlera and H. Haj-Salem, Modélisation du trafic autoroutier au second ordre, C. R. Math. Acad. Sci. Paris, 346 (2008), 1203-1206.  doi: 10.1016/j.crma.2008.09.024.

[28]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.

[29]

T.-P. Liu and T. Yang, $L_1$ stability for $2\times 2$ systems of hyperbolic conservation laws, J. Amer. Math. Soc., 12 (1999), 729-774.  doi: 10.1090/S0894-0347-99-00292-1.

[30]

T.-P. Liu and T. Yang, $L_1$ stability of conservation laws with coinciding Hugoniot and characteristic curves, Indiana Univ. Math. J., 48 (1999), 237-247.  doi: 10.1512/iumj.1999.48.1601.

[31]

Y. Lu, Hyperbolic Conservation Laws and the Compensated Compactness Method, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 128, Chapman & Hall/CRC, Boca Raton, FL, 2003.

[32]

F. Murat, L'injection du cône positif de $H^{-1}$ dans $W^{-1, q}$ est compacte pour tout $q < 2$, J. Math. Pures Appl. (9), 60 (1981), 309-322. 

[33]

E. Panov, On weak completeness of the set of entropy solutions to a scalar conservation law, SIAM J. Math. Anal., 41 (2009), 26-36.  doi: 10.1137/080724587.

[34]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.

[35]

D. Serre, Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservation, J. Differential Equations, 68 (1987), 137-168.  doi: 10.1016/0022-0396(87)90189-6.

[36]

L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., Vol. 39, Pitman, Boston, MA, London, 1979, 136–212.

[37]

M. E. Taylor, Partial Differential Equations I. Basic Theory, 2nd edition, Applied Mathematical Sciences, Vol. 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[38]

B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795.  doi: 10.1090/S0002-9947-1983-0716850-2.

[39]

G. Wong and S. Wong, A multi-class traffic flow model - an extension of LWR model with heterogeneous drivers, Transportation Research Part A: Policy and Practice, 36 (2002), 827-841.  doi: 10.1016/S0965-8564(01)00042-8.

[40]

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. (2014), Art. ID 691429, 7 pp. doi: 10.1155/2014/691429.

show all references

References:
[1]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.  doi: 10.1137/S0036139997332099.

[2]

C. BardosA. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034.  doi: 10.1080/03605307908820117.

[3]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2), 161 (2005), 223-342.  doi: 10.4007/annals.2005.161.223.

[4]

S. BlandinD. WorkP. GoatinB. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127.  doi: 10.1137/090754467.

[5] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and its Applications, Vol. 20, Oxford University Press, Oxford, 2000. 
[6]

A. Bressan and R. M. Colombo, The semigroup generated by $2\times 2$ conservation laws, Arch. Rational Mech. Anal., 133 (1995), 1-75.  doi: 10.1007/BF00375350.

[7]

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., 146 (2000). doi: 10.1090/memo/0694.

[8]

A. BressanT.-P. Liu and T. Yang, $L^1$ stability estimates for $n\times n$ conservation laws, Arch. Ration. Mech. Anal., 149 (1999), 1-22.  doi: 10.1007/s002050050165.

[9]

G.-Q. Chen, Remarks on R. J. DiPerna's paper: "Convergence of the viscosity method for isentropic gas dynamics", Proc. Amer. Math. Soc., 125 (1997), 2981-2986.  doi: 10.1090/S0002-9939-97-03946-4.

[10]

G. -Q. Chen and H. Frid, Vanishing viscosity limit for initial-boundary value problems for conservation laws, in Nonlinear Partial Differential Equations, Contemp. Math., Vol. 238, Amer. Math. Soc., Providence, RI, 1999, 35–51. doi: 10.1090/conm/238/03538.

[11]

G. M. CocliteK. H. KarlsenS. Mishra and N. H. Risebro, Convergence of vanishing viscosity approximations of $2\times2$ triangular systems of multi-dimensional conservation laws, Boll. Unione Mat. Ital. (9), 2 (2009), 275-284. 

[12]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.  doi: 10.1137/S0036139901393184.

[13]

R. M. ColomboF. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666.  doi: 10.1137/090752468.

[14]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 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.

[15]

R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys., 91 (1983), 1-30.  doi: 10.1007/BF01206047.

[16]

L. C. Evans, Weak convergence methods for nonlinear partial differential equations, 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.

[17]

S. FanM. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2014), 239-268.  doi: 10.3934/nhm.2014.9.239.

[18]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[19]

M. Garavello and F. Marcellini, The Riemann problem at a junction for a phase transition traffic model, Discrete Contin. Dyn. Syst., 37 (2017), 5191-5209.  doi: 10.3934/dcds.2017225.

[20]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303.  doi: 10.1016/j.mcm.2006.01.016.

[21]

J. M. GreenbergA. Klar and M. Rascle, Congestion on multilane highways, SIAM J. Appl. Math., 63 (2003), 818-833.  doi: 10.1137/S0036139901396309.

[22]

F. GuY.-g. Lu and Q. Zhang, Global solutions to one-dimensional shallow water magnetohydrodynamic equations, J. Math. Anal. Appl., 401 (2013), 714-723.  doi: 10.1016/j.jmaa.2012.12.042.

[23]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2nd edition, Applied Mathematical Sciences, Vol. 152, Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2.

[24]

B. S. Kerner, The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory, Springer, Berlin, New York, 2004.

[25]

C. Klingenberg and Y.-g. Lu, The vacuum case in Diperna's paper, J. Math. Anal. Appl., 225 (1998), 679-684.  doi: 10.1006/jmaa.1998.6050.

[26]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. 

[27]

J. P. LebacqueX. LouisS. MammarB. Schnetzlera and H. Haj-Salem, Modélisation du trafic autoroutier au second ordre, C. R. Math. Acad. Sci. Paris, 346 (2008), 1203-1206.  doi: 10.1016/j.crma.2008.09.024.

[28]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.

[29]

T.-P. Liu and T. Yang, $L_1$ stability for $2\times 2$ systems of hyperbolic conservation laws, J. Amer. Math. Soc., 12 (1999), 729-774.  doi: 10.1090/S0894-0347-99-00292-1.

[30]

T.-P. Liu and T. Yang, $L_1$ stability of conservation laws with coinciding Hugoniot and characteristic curves, Indiana Univ. Math. J., 48 (1999), 237-247.  doi: 10.1512/iumj.1999.48.1601.

[31]

Y. Lu, Hyperbolic Conservation Laws and the Compensated Compactness Method, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 128, Chapman & Hall/CRC, Boca Raton, FL, 2003.

[32]

F. Murat, L'injection du cône positif de $H^{-1}$ dans $W^{-1, q}$ est compacte pour tout $q < 2$, J. Math. Pures Appl. (9), 60 (1981), 309-322. 

[33]

E. Panov, On weak completeness of the set of entropy solutions to a scalar conservation law, SIAM J. Math. Anal., 41 (2009), 26-36.  doi: 10.1137/080724587.

[34]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.

[35]

D. Serre, Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservation, J. Differential Equations, 68 (1987), 137-168.  doi: 10.1016/0022-0396(87)90189-6.

[36]

L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., Vol. 39, Pitman, Boston, MA, London, 1979, 136–212.

[37]

M. E. Taylor, Partial Differential Equations I. Basic Theory, 2nd edition, Applied Mathematical Sciences, Vol. 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[38]

B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795.  doi: 10.1090/S0002-9947-1983-0716850-2.

[39]

G. Wong and S. Wong, A multi-class traffic flow model - an extension of LWR model with heterogeneous drivers, Transportation Research Part A: Policy and Practice, 36 (2002), 827-841.  doi: 10.1016/S0965-8564(01)00042-8.

[40]

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. (2014), Art. ID 691429, 7 pp. doi: 10.1155/2014/691429.

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