No Access

Oleinik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit

Giuseppe Maria Coclite

https://orcid.org/0000-0001-6019-4757

Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4, 70125 Bari, Italy

E-mail Address: giuseppemaria.coclite@poliba.it

Search for more papers by this author

Maria Colombo

https://orcid.org/0000-0003-3294-472X

Institut de Mathématiques, EPFL, Station 8, 1015 Lausanne, Switzerland

E-mail Address: maria.colombo@epfl.ch

Search for more papers by this author

Gianluca Crippa

https://orcid.org/0000-0001-6825-4275

Departement Mathematik und Informatik, Universität, Basel, Spiegelgasse 1, 4051 Basel, Switzerland

E-mail Address: gianluca.crippa@unibas.ch

Search for more papers by this author

Nicola De Nitti

https://orcid.org/0000-0003-0402-7502

Institut de Mathématiques, EPFL, Station 8, 1015 Lausanne, Switzerland

E-mail Address: nicola.denitti@epfl.ch

Search for more papers by this author

Alexander Keimer

https://orcid.org/0000-0003-3825-5853

Department of Mathematics, Friedrich-Alexander-Universität, Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany

E-mail Address: alexander.keimer@fau.de

Search for more papers by this author

Elio Marconi

https://orcid.org/0000-0003-2989-2594

Dipartimento di Matematica Tullio Levi-Civita, Università, degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy

E-mail Address: elio.marconi@unipd.it

Search for more papers by this author

Lukas Pflug

https://orcid.org/0000-0001-8001-5832

Friedrich-Alexander-Universität Erlangen-Nürnberg, Competence Unit for Scientific Computing, Martensstr. 5a, 91058 Erlangen, Germany

Department of Mathematics, Chair of Applied Mathematics, (Continuous Optimization), Cauerstr. 11, 91058 Erlangen, Germany

E-mail Address: lukas.pflug@fau.de

Search for more papers by this author

, and

Laura V. Spinolo

https://orcid.org/0000-0002-7696-8873

IMATI-CNR, Via Ferrata 5, 27100 Pavia, Italy

E-mail Address: spinolo@imati.cnr.it

Search for more papers by this author

https://doi.org/10.1142/S021989162440006XCited by:0 (Source: Crossref)

This article is part of the issue:

Special Issue: Recent Advances in Nonlinear Hyperbolic PDEs and Fluid Dynamics
Special Issue Editors: Fabio Ancona (University of Padua, Italy), Stefano Bianchini (SISSA, Italy)

Abstract

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W : = 𝟙_{(- \infty, 0]} (\cdot) exp (\cdot) * ρ$ satisfy an Oleĭnik-type entropy condition. More precisely, under different sets of assumptions on the velocity function $V$ , we prove that $W$ satisfies a one-sided Lipschitz condition and that $V^{'} (W) W \partial_{x} W$ satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.

Communicated by F. Ancona and S. Bianchini

Keywords:

AMSC: 35L65

References

1. F. Ancona, O. Glass and K. T. Nguyen , Lower compactness estimates for scalar balance laws, Comm. Pure Appl. Math. 65(9) (2012) 1303–1329. Crossref, Web of Science, Google Scholar
2. S. Blandin and P. Goatin , Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math. 132(2) (2016) 217–241. Crossref, Web of Science, Google Scholar
3. F. Bouchut and F. James , One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal. 32(7) (1998) 891–933. Crossref, Web of Science, Google Scholar
4. F. Bouchut, F. James and S. Mancini , Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4(1) (2005) 1–25. Google Scholar
5. A. Bressan and R. M. Colombo , Decay of positive waves in nonlinear systems of conservation laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26(1) (1998) 133–160. Google Scholar
6. 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
7. A. Bressan and W. Shen , Entropy admissibility of the limit solution for a nonlocal model of traffic flow, Commun. Math. Sci. 19(5) (2021) 1447–1450. Crossref, Web of Science, Google Scholar
8. K. S. Cheng , A regularity theorem for a nonconvex scalar conservation law, J. Differential Equations 61(1) (1986) 79–127. Crossref, Web of Science, Google Scholar
9. 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, Ann. Inst. H. Poincaré C Anal. Non Linéaire 40(5) (2023) 1205–1223. Crossref, Web of Science, Google Scholar
10. G. M. Coclite, N. De Nitti, A. Keimer and L. Pflug , Singular limits with vanishing viscosity for nonlocal conservation laws, Nonlinear Anal. 211(12) (2021) 112370. Crossref, Google Scholar
11. G. M. Coclite, N. De Nitti, A. Keimer and L. Pflug , On existence and uniqueness of weak solutions to nonlocal conservation laws with BV kernels, Z. Angew. Math. Phys. 73(6) (2022) 241. Crossref, Web of Science, Google Scholar
12. G. M. Coclite, N. De Nitti, A. Keimer, L. Pflug and E. Zuazua , Long-time convergence of a nonlocal Burgers’ equation towards the local $N$ -wave, Nonlinearity 36(11) (2023) 5998–6019. Crossref, Google Scholar
13. M. Colombo, G. Crippa, M. Graff and L. V. Spinolo , On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws, ESAIM Math. Model. Numer. Anal. 55(6) (2021) 2705–2723. Crossref, Web of Science, Google Scholar
14. M. Colombo, G. Crippa, E. Marconi and L. V. Spinolo , Local limit of nonlocal traffic models: Convergence results and total variation blow-up, Ann. Inst. H. Poincaré C Anal. Non Linéaire 38(5) (2021) 1653–1666. Crossref, Web of Science, Google Scholar
15. M. Colombo, G. Crippa, E. Marconi and L. V. Spinolo , Nonlocal traffic models with general kernels: Singular limit, entropy admissibility, and convergence rate, Arch. Ration. Mech. Anal. 247(2) (2023) 18. Crossref, Web of Science, Google Scholar
16. 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
17. A. Constantin and J. Escher , Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181(2) (1998) 229–243. Crossref, Web of Science, Google Scholar
18. J.-M. Coron, A. Keimer and L. Pflug , Nonlocal transport equations — existence and uniqueness of solutions and relation to the corresponding conservation laws, SIAM J. Math. Anal. 52(6) (2020) 5500–5532. Crossref, Web of Science, Google Scholar
19. C. M. Dafermos , Characteristics in hyperbolic conservation laws. A study of the structure and the asymptotic behaviour of solutions, in Nonlinear Analysis and Mechanics: Heriot–Watt Symp. (Edinburgh, 1976), Vol. I, Research Notes in Mathematics, Vol. 17 (Pitman, London, 1977), pp. 1–58. Google Scholar
20. C. M. Dafermos , Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 325, 4th edn. (Springer-Verlag, Berlin, 2016). Crossref, Google Scholar
21. C. De Lellis and F. Golse , A quantitative compactness estimate for scalar conservation laws, Comm. Pure Appl. Math. 58(7) (2005) 989–998. Crossref, Web of Science, Google Scholar
22. M. Garavello and B. Piccoli , Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, Vol. 1 (American Institute of Mathematical Sciences, Springfield, MO, 2006). Google Scholar
23. F. Ghiraldin and X. Lamy , Optimal Besov differentiability for entropy solutions of the eikonal equation, Comm. Pure Appl. Math. 73(2) (2020) 317–349. Crossref, Web of Science, Google Scholar
24. O. Glass , An extension of Oleinik’s inequality for general 1D scalar conservation laws, J. Hyperbol. Differ. Equ. 5(1) (2008) 113–165. Link, Web of Science, Google Scholar
25. P. Goatin and S. Scialanga , Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media 11(1) (2016) 107–121. Crossref, Web of Science, Google Scholar
26. B. D. Greenshields, J. Bibbins, W. Channing and H. Miller , A study of traffic capacity, in Highway Research Board Proc., Vol. 14 (Washington, DC, 1935), pp. 448–477. Google Scholar
27. E. Hopf , The partial differential equation $u_{t} + u u_{x} = μ u_{x x}$ , Comm. Pure Appl. Math. 3 (1950) 201–230. Crossref, Web of Science, Google Scholar
28. D. Hoff , The sharp form of Oleĭnik’s entropy condition in several space variables, Trans. Amer. Math. Soc. 276(2) (1983) 707–714. Google Scholar
29. H. K. Jenssen and C. Sinestrari , On the spreading of characteristics for non-convex conservation laws, Proc. Roy. Soc. Edinburgh Sect. A 131(4) (2001) 909–925. Crossref, Google Scholar
30. A. Keimer and L. Pflug , Existence, uniqueness and regularity results on nonlocal balance laws, J. Differential Equations 263(7) (2017) 4023–4069. Crossref, Web of Science, Google Scholar
31. 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
32. 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
33. 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
34. A. Keimer, L. Pflug and M. Spinola , Nonlocal balance laws: Theory of convergence for nondissipative numerical schemes, Submitted (2020). Google Scholar
35. O. A. Ladyženskaya , On the construction of discontinuous solutions of quasi-linear hyperbolic equations as limits of solutions of the corresponding parabolic equations when the “coefficient of viscosity” tends toward zero, Trudy Moskov. Mat. Obšč. 6 (1957) 465–480. Google Scholar
36. P. D. Lax , Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7 (1954) 159–193. Crossref, Web of Science, Google Scholar
37. P. D. Lax , Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957) 537–566. Crossref, Web of Science, Google Scholar
38. P. G. LeFloch and K. Trivisa , Continuous Glimm-type functionals and spreading of rarefaction waves, Commun. Math. Sci. 2(2) (2004) 213–236. Crossref, Google Scholar
39. E. Marconi , The rectifiability of the entropy defect measure for Burgers equation, J. Funct. Anal. 283(6) (2022) 109568. Crossref, Web of Science, Google Scholar
40. O. A. Oleĭnik , Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963) 95–172. Crossref, Google Scholar
41. L. Tartar , Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot–Watt Symp., Vol. IV, Research Notes in Mathematics, Vol. 39 (Pitman, Boston, Mass.-London, 1979), pp. 136–212. Google Scholar

Vol. 21, No. 03

Metrics

Downloaded 9 times

History

Received 9 February 2024

Accepted 11 April 2024

Published: 24 January 2025

Keywords

PDF download

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Oleinik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit

Abstract

References

Degenerate Cahn–Hilliard systems: From nonlocal to local

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

CONVERGENCE OF THIN FILM APPROXIMATION FOR A SCALAR CONSERVATION LAW

Applications of local linking to nonlocal Neumann problems

CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX

One-dimensional hyperbolic conservation laws: past and future

Preface: Recent advances in nonlinear hyperbolic PDEs and fluid dynamics

Oscillations in compressible Navier–Stokes and homogenization in phase transition problems

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Oleinik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit

Abstract

Recommended

Degenerate Cahn–Hilliard systems: From nonlocal to local

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

CONVERGENCE OF THIN FILM APPROXIMATION FOR A SCALAR CONSERVATION LAW

Applications of local linking to nonlocal Neumann problems

CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX

One-dimensional hyperbolic conservation laws: past and future

Preface: Recent advances in nonlinear hyperbolic PDEs and fluid dynamics

Oscillations in compressible Navier–Stokes and homogenization in phase transition problems