American Institute of Mathematical Sciences

November  2017, 37(11): 5913-5942. doi: 10.3934/dcds.2017257

Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network

 1 Laboratoire de Mathématiques et Physique Théorique CNRS UMR7350, Université de Tours, Parc de Grandmont, 37200 Tours, France 2 Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4, 70125 Bari, Italy 3 Laboratoire de Mathématiques CNRS UMR6623, Université de Bourgogne-Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France

* Corresponding author: G. M. Coclite

Received  November 2016 Revised  June 2017 Published  July 2017

Fund Project: The second author is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
The work on this paper was supported by the French ANR-11-JS01-0006 project CoToCoLa

We provide a complete study of the model investigated in [Coclite, Garavello, SIAM J. Math. Anal., 2010]. We prove well-posedness of solutions obtained as vanishing viscosity limits for the Cauchy problem for scalar conservation laws $ρ_{h, t} + f_h(ρ_h)_x = 0$, for $h∈ \{1, ..., m+n\}$, on a junction where $m$ incoming and $n$ outgoing edges meet. Our analysis and the definition of the admissible solution rely upon the complete description of the set of edge-wise constant solutions and its properties, which is of some interest on its own. The Riemann solver at the junction is characterized. In order to prove uniqueness, we introduce a family of Kruzhkov-type adapted entropies at the junction. Existence is justified both by the vanishing viscosity method and via the proof of convergence of a monotone well-balanced finite volume discretization. Beyond the classical vanishing viscosity framework, the numerical procedure and the uniqueness argument can be applied to general junction solvers enjoying the crucial order-preservation property.

Citation: Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello. Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5913-5942. doi: 10.3934/dcds.2017257
References:
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Sci., 1 (2014), 47-111. doi: 10.4171/EMSS/2. Google Scholar [10] R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425. doi: 10.1007/s10665-007-9148-4. Google Scholar [11] R. Bürger, K. H. Karlsen and J. D. Towers, An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47 (2009), 1684-1712. doi: 10.1137/07069314X. Google Scholar [12] G. M. Coclite and M. Garavello, Vanishing viscosity for traffic on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783. doi: 10.1137/090771417. Google Scholar [13] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. Google Scholar [14] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. 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Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289. Google Scholar [20] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, 2nd edition, Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2. Google Scholar [21] C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. éc. Norm. Supér., (4) 50 (2017), 357-448, URL https: //hal.archives-ouvertes.fr/hal-00832545. doi: 10.24033/asens.2323. Google Scholar [22] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002. Google Scholar [23] S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. Google Scholar [24] J.-P. Lebacque, The Godunov scheme and what it means for first order traffic flow models, in Internaional symposium on transportation and traffic theory, (1996), 647-677. Google Scholar [25] E. Y. Panov, On sequences of measure-valued solutions of a first-order quasilinear equation, Mat. Sb., 185 (1994), 87-106. doi: 10.1070/SM1995v081n01ABEH003621. Google Scholar [26] E. Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770. doi: 10.1142/S0219891607001343. Google Scholar [27] L. Tartar, Nonlinear analysis and mechanics: Heriot-watt symposium, in Compensated Compactness and Applications to Partial Differential Equations, vol. IV, Pitman, Boston, (1979), 317-345. Google Scholar

show all references

References:
 [1] B. Andreianov and C. Cancés, On interface transmission conditions for conservation laws with discontinuous flux of general shape, J. Hyperbolic Differ. Equ., 12 (2015), 343-384. doi: 10.1142/S0219891615500101. Google Scholar [2] B. Andreianov, K. H. Karlsen and N. H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux, Netw. Heterog. Media, 5 (2010), 617-633. doi: 10.3934/nhm.2010.5.617. Google Scholar [3] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86. doi: 10.1007/s00205-010-0389-4. Google Scholar [4] B. Andreianov and D. Mitrović, Entropy conditions for scalar conservation laws with discontinuous flux revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1307-1335. doi: 10.1016/j.anihpc.2014.08.002. Google Scholar [5] B. Andreianov and K. Sbihi, Well-posedness of general boundary-value problems for scalar conservation laws, Trans. Amer. Math. Soc., 367 (2015), 3763-3806. doi: 10.1090/S0002-9947-2015-05988-1. Google Scholar [6] E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265. doi: 10.1017/S0308210500003863. Google Scholar [7] P. Baiti and H. K. Jenssen, Well-posedness for a class of $2×2$ conservation laws with $L^∞$ data, J. Differential Equations, 140 (1997), 161-185. doi: 10.1006/jdeq.1997.3308. Google Scholar [8] C. Bardos, A. Y. Leroux and J.-C. Nedelec, First order quasilinear equations with boundary conditions, Communications in partial differential equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117. Google Scholar [9] A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111. doi: 10.4171/EMSS/2. Google Scholar [10] R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425. doi: 10.1007/s10665-007-9148-4. Google Scholar [11] R. Bürger, K. H. Karlsen and J. D. Towers, An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47 (2009), 1684-1712. doi: 10.1137/07069314X. Google Scholar [12] G. M. Coclite and M. Garavello, Vanishing viscosity for traffic on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783. doi: 10.1137/090771417. Google Scholar [13] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. Google Scholar [14] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1-21, URL http://dx.doi.org/10.2307/2006218. doi: 10.1090/S0025-5718-1980-0551288-3. Google Scholar [15] S. Diehl, A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients, J. Hyperbolic Differ. Equ., 6 (2009), 127-159. doi: 10.1142/S0219891609001794. Google Scholar [16] F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122. doi: 10.1016/0022-0396(88)90040-X. Google Scholar [17] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis, Vol. Ⅶ, Handb. Numer. Anal., Ⅶ, North-Holland, Amsterdam, (2000), 713-1020. Google Scholar [18] M. Garavello and B. Piccoli, Traffic Flow on Networks, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. Google Scholar [19] H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289. Google Scholar [20] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, 2nd edition, Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2. Google Scholar [21] C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. éc. Norm. Supér., (4) 50 (2017), 357-448, URL https: //hal.archives-ouvertes.fr/hal-00832545. doi: 10.24033/asens.2323. Google Scholar [22] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002. Google Scholar [23] S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. Google Scholar [24] J.-P. Lebacque, The Godunov scheme and what it means for first order traffic flow models, in Internaional symposium on transportation and traffic theory, (1996), 647-677. Google Scholar [25] E. Y. Panov, On sequences of measure-valued solutions of a first-order quasilinear equation, Mat. Sb., 185 (1994), 87-106. doi: 10.1070/SM1995v081n01ABEH003621. Google Scholar [26] E. Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770. doi: 10.1142/S0219891607001343. Google Scholar [27] L. Tartar, Nonlinear analysis and mechanics: Heriot-watt symposium, in Compensated Compactness and Applications to Partial Differential Equations, vol. IV, Pitman, Boston, (1979), 317-345. Google Scholar
A junction consisting of $m$ incoming and $n$ outgoing roads.
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