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June  2020, 15(2): 197-213. doi: 10.3934/nhm.2020009

Vanishing viscosity on a star-shaped graph under general transmission conditions at the node

1. 

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

2. 

Laboratoire de Mathématiques CNRS UMR6623, Université de Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France

* Corresponding author: Giuseppe Maria Coclite

Received  May 2019 Revised  November 2019 Published  June 2020 Early access  June 2020

In this paper we consider a family of scalar conservation laws defined on an oriented star shaped graph and we study their vanishing viscosity approximations subject to general matching conditions at the node. In particular, we prove the existence of converging subsequence and we show that the limit is a weak solution of the original problem.

Citation: Giuseppe Maria Coclite, Carlotta Donadello. Vanishing viscosity on a star-shaped graph under general transmission conditions at the node. Networks and Heterogeneous Media, 2020, 15 (2) : 197-213. doi: 10.3934/nhm.2020009
References:
[1]

Ad imurthiS. Mishra and G. D. V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837.  doi: 10.1142/S0219891605000622.

[2]

B. Andreianov and C. Cancès, 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.

[3]

B. P. AndreianovG. M. Coclite and C. Donadello, Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network, Discrete Contin. Dyn. Syst., 37 (2017), 5913-5942.  doi: 10.3934/dcds.2017257.

[4]

B. AndreianovK. 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.

[5]

B. AndreianovK. 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.

[6]

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

[7]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.

[8]

G. BrettiR. Natalini and M. Ribot, A hyperbolic model of chemotaxis on a network: A numerical study, ESAIM Math. Model. Numer. Anal., 48 (2014), 231-258.  doi: 10.1051/m2an/2013098.

[9]

R. BürgerK. 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.

[10]

G. M. Coclite and M. Garavello, Vanishing viscosity for traffic on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783.  doi: 10.1137/090771417.

[11]

G. M. Coclite and L. di Ruvo, Vanishing viscosity for traffic on networks with degenerate diffusivity, Mediterr. J. Math., 16 (2019), Art. 110, 21 pp. doi: 10.1007/s00009-019-1391-1.

[12]

R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234 (2007), 654-675.  doi: 10.1016/j.jde.2006.10.014.

[13]

M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[14]

F. R. Guarguaglini and R. Natalini, Global smooth solutions for a hyperbolic chemotaxis model on a network, SIAM J. Math. Anal., 47 (2015), 4652-4671.  doi: 10.1137/140997099.

[15]

E. F. Kaasschieter, Solving the buckley-leverett equation with gravity in a heterogeneous porous medium, Comput. Geosci., 3 (1999), 23-48.  doi: 10.1023/A:1011574824970.

[16]

O. Kedem and A. Katchalsky, Thermodynamic analysis of permeability of biological membranes to non-electrolytes, Biochimica et Biophysica Acta, 27 (1958), 229-246.  doi: 10.1016/0006-3002(58)90330-5.

[17]

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

[18]

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

show all references

References:
[1]

Ad imurthiS. Mishra and G. D. V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837.  doi: 10.1142/S0219891605000622.

[2]

B. Andreianov and C. Cancès, 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.

[3]

B. P. AndreianovG. M. Coclite and C. Donadello, Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network, Discrete Contin. Dyn. Syst., 37 (2017), 5913-5942.  doi: 10.3934/dcds.2017257.

[4]

B. AndreianovK. 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.

[5]

B. AndreianovK. 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.

[6]

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

[7]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.

[8]

G. BrettiR. Natalini and M. Ribot, A hyperbolic model of chemotaxis on a network: A numerical study, ESAIM Math. Model. Numer. Anal., 48 (2014), 231-258.  doi: 10.1051/m2an/2013098.

[9]

R. BürgerK. 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.

[10]

G. M. Coclite and M. Garavello, Vanishing viscosity for traffic on networks, SIAM J. Math. Anal., 42 (2010), 1761-1783.  doi: 10.1137/090771417.

[11]

G. M. Coclite and L. di Ruvo, Vanishing viscosity for traffic on networks with degenerate diffusivity, Mediterr. J. Math., 16 (2019), Art. 110, 21 pp. doi: 10.1007/s00009-019-1391-1.

[12]

R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234 (2007), 654-675.  doi: 10.1016/j.jde.2006.10.014.

[13]

M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[14]

F. R. Guarguaglini and R. Natalini, Global smooth solutions for a hyperbolic chemotaxis model on a network, SIAM J. Math. Anal., 47 (2015), 4652-4671.  doi: 10.1137/140997099.

[15]

E. F. Kaasschieter, Solving the buckley-leverett equation with gravity in a heterogeneous porous medium, Comput. Geosci., 3 (1999), 23-48.  doi: 10.1023/A:1011574824970.

[16]

O. Kedem and A. Katchalsky, Thermodynamic analysis of permeability of biological membranes to non-electrolytes, Biochimica et Biophysica Acta, 27 (1958), 229-246.  doi: 10.1016/0006-3002(58)90330-5.

[17]

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

[18]

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

Figure 1.  A junction consisting of $ m $ incoming and $ n $ outgoing edges
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