No Access

Existence and uniqueness results for a class of nonlocal conservation laws by means of a Lax–Hopf-type solution formula

Alexander Keimer

Institute of Transportation Studies (ITS), University of California, Berkeley CA-94720, USA

E-mail Address: keimer@berkeley.edu

Search for more papers by this author

Manish Singh

TIFR Centre For Applicable Mathematics, Sharada Nagar, Chikkabommsandra, Bangalore 560065, India

E-mail Address: manish@math.tifrbng.res.in

Search for more papers by this author

, and

Tanya Veeravalli

Department of Electrical Engineering and Computer Science, University of California, Berkeley CA-94720, USA

E-mail Address: tveeravalli19@berkeley.edu

Search for more papers by this author

https://doi.org/10.1142/S0219891620500204Cited by:6 (Source: Crossref)

Abstract

We study the initial value problem and the initial boundary value problem for nonlocal conservation laws. The nonlocal term is realized via a spatial integration of the solution between specified boundaries and affects the flux function of a given “local” conservation law in a multiplicative way. For a strictly convex flux function and strictly positive nonlocal impact we prove existence and uniqueness of weak entropy solutions relying on a fixed-point argument for the nonlocal term and an explicit Lax–Hopf-type solution formula for the corresponding Hamilton–Jacobi (HJ) equation. Using the developed theory for HJ equations, we obtain a semi-explicit Lax–Hopf-type formula for the solution of the corresponding nonlocal HJ equation and a semi-explicit Lax–Oleinik-type formula for the nonlocal conservation law.

Communicated by The Editors

Keywords:

AMSC: 35L65, 35F21, 35F31, 49L99, 35L04

References

1. A. Aggarwal, R. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal. 53(2) (2015) 963–983. Crossref, Web of Science, Google Scholar
2. A. Aggarwal and P. Goatin, Crowd dynamics through non-local conservation laws, Bull. Brazil. Math. Soc. New Ser. 47(1) (2016) 37–50. Crossref, Google Scholar
3. D. Armbruster, D. E. Marthaler, C. A. Ringhofer, K. G. Kempf and T.-C. Jo, A continuum model for a re-entrant factory, Oper. Res. 54(5) (2006) 933–950. Crossref, Web of Science, Google Scholar
4. C. Bardos, A. Y. Leroux and J. C. Nedelec, First order quasilinear equations with boundary conditions, Commun. Partial Differential Equations 4(9) (1979) 1017–1034. Crossref, Google Scholar
5. S. Benzoni-Gavage and R. M. Colombo, An $n$ $n$ -populations model for traffic flow, Eur. J. Appl. Math. 14(5) (2003) 587–612. Crossref, Web of Science, Google Scholar
6. S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math. 132 (2016) 217–241. Crossref, Web of Science, Google Scholar
7. A. Bressan, Hyperbolic Systems of Conservation Laws, Vol. 20 (Oxford University Press, Oxford, 2000). Crossref, Google Scholar
8. A. Bressan and W. Shen, Entropy Admissibility of the Limit Solution for a Nonlocal Model of Traffic Flow, arXiv:2011.05430 (2020). Google Scholar
9. A. Bressan and W. Shen, On traffic flow with nonlocal flux: A relaxation representation, Arch. Ration. Mech. Anal. 237(3) (2020) 1213–1236. Crossref, Web of Science, Google Scholar
10. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext (Springer, New York, 2011). Crossref, Google Scholar
11. G. M. Coclite, J.-M. Coron, N. De Nitti, A. Keimer and L. Pflug, A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels, arXiv e-prints, 2020. Google Scholar
12. M. Colombo, G. Crippa, E. Marconi and L. V. Spinolo, Local limit of nonlocal traffic models: Convergence results and total variation blow-up, preprint (2019), arXiv:1808.03529. Google Scholar
13. M. Colombo, G. Crippa and L. V. Spinolo, On the singular local limit for conservation laws with nonlocal fluxes, Arch. Ration. Mech. Anal. 233(3) (2019) 1131–1167. Crossref, Web of Science, Google Scholar
14. R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci. 22(4) (2012) 1150023. Link, Web of Science, Google Scholar
15. R. M. Colombo and G. Guerra, Hyperbolic balance laws with a non-local source, Commun. Partial Differential Equations 32(12) (2007) 1917–1939. Crossref, Web of Science, Google Scholar
16. R. M. Colombo and G. Guerra, On general balance laws with boundary, J. Differential Equations 248(5) (2010) 1017–1043. Crossref, Web of Science, Google Scholar
17. R. M. Colombo and M. Lécureux-Mercier, Nonlocal crowd dynamics models for several populations, Acta Math. Sci. 32(1) (2012) 177–196. Crossref, Web of Science, Google Scholar
18. J.-M. Coron, M. Kawski and Z. Wang, Analysis of a conservation law modeling a highly re-entrant manufacturing system, Discrete Contin. Dyn. Syst. Ser. B 14(4) (2010) 1337–1359. Crossref, Web of Science, Google Scholar
19. G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, Nonlinear Differential Equations Appl. NoDEA 20(3) (2013) 523–537. Crossref, Web of Science, Google Scholar
20. C. De Filippis and P. Goatin, The initial–boundary value problem for general non-local scalar conservation laws in one space dimension, Nonlinear Anal. 161 (2017) 131–156. Crossref, Web of Science, Google Scholar
21. L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19 (American Mathematical Society, Providence RI, 2007). Google Scholar
22. P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the lwr traffic flow model with non-local velocity, Netw. Hetereog. Media 11(1) (2016) 107–121. Crossref, Web of Science, Google Scholar
23. M. Gugat, A. Keimer, W. Leugering and Z. Wang, Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks, Netw. Heterog. Media 10(4) (2015) 749–785. Crossref, Web of Science, Google Scholar
24. M. Haderlein, D. Segets, M. Gröschel, L. Pflug, G. Leugering and W. Peukert, FIMOR: An efficient simulation for ZnO quantum dot ripening applied to the optimization of nanoparticle synthesis, Chem. Eng. J. 260 (2015) 706–715. Crossref, Web of Science, Google Scholar
25. K. T. Joseph and G. D. V. Gowda, Explicit formula for the solution of convex conservation laws with boundary condition, Duke Math. J. 62(2) (1991) 401–416. Crossref, Web of Science, Google Scholar
26. A. Keimer, N. Laurent-Brouty, F. Farokhi, H. Signargout, V. Cvetkovic, A. M. Bayen and K. H. Johansson, Information patterns in the modeling and design of mobility management services, Proc. IEEE 106(4) (2018) 554–576. Crossref, Web of Science, Google Scholar
27. A. Keimer and L. Pflug, Existence, uniqueness and regularity results on nonlocal balance laws, J. Differential Equations 263 (2017) 4023–4069. Crossref, Web of Science, Google Scholar
28. A. Keimer and L. Pflug, On approximation of local conservation laws by nonlocal conservation laws, J. Math. Anal. Appl. 475(2) (2019) 1927–1955. Crossref, Web of Science, Google Scholar
29. A. Keimer, L. Pflug and M. Spinola, Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping, J. Math. Anal. Appl. 466(1) (2018) 18–55. Crossref, Web of Science, Google Scholar
30. A. Keimer, L. Pflug and M. Spinola, Nonlocal scalar conservation laws on bounded domains and applications in traffic flow, SIAM J. Math. Anal. 50(6) (2018) 6271–6306. Crossref, Web of Science, Google Scholar
31. B. Perthame, Transport Equations in Biology (Springer Science & Business Media, 2006). Google Scholar
32. W. Peukert, D. Segets, L. Pflug and G. Leugering, Unified design strategies for particulate products, Adv. Chem. Eng. 46 (2015) 1–81. Crossref, Google Scholar
33. B. Piccoli, N. P. Duteil and E. Trélat, Sparse control of Hegselmann–Krause models: Black hole and declustering, SIAM J. Control Optim. 57(4) (2018) 2628–2659. Crossref, Web of Science, Google Scholar
34. R. T. Rockafellar, Conjugate Duality and Optimization, Vol. 16 (Siam, 1974). Crossref, Google Scholar
35. E. Rossi and R. M. Colombo, Non-local conservation laws in bounded domains, SIAM J. Math. Anal. 50(4) (2018) 4041–4065. Crossref, Web of Science, Google Scholar
36. D. Segets, M. A. J. Hartig, J. Gradl and W. Peukert, A population balance model of quantum dot formation: Oriented growth and ripening of ZnO, Chem. Eng. Sci. 70 (2012) 4–13. Crossref, Web of Science, Google Scholar
37. P. Shang and Z. Wang, Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system, J. Differential Equations 250(2) (2011) 949–982. Crossref, Web of Science, Google Scholar
38. E. Zeidler, Nonlinear Functional Analysis and its Applications, I. Fixed-Point Theorems (Springer-Verlag, New York, 1986). Google Scholar
39. K. Zumbrun, On a nonlocal dispersive equation modeling particle suspensions, Quarterly Appl. Math. 57 (1999) 573–600. Crossref, Web of Science, Google Scholar