Abstract
We give an answer to a question posed in Amorim et al. (ESAIM Math Model Numer Anal 49(1):19–37, 2015), which can loosely speaking, be formulated as follows: consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law. Can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one does not in general have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ambrosio, L.: Transport equation and Cauchy problem for \(BV\) vector fields. Invent. Math. 158(2), 227–260 (2004)
Amorim, P., Colombo, R.M., Teixeira, A.: On the numerical integration of scalar nonlocal conservation laws. ESAIM Math. Model. Numer. Anal. 49(1), 19–37 (2015)
Betancourt, F., Bürger, R., Karlsen, K.H., Tory, E.M.: On nonlocal conservation laws modelling sedimentation. Nonlinearity 24(3), 855–885 (2011)
Bhat, H.S., Fetecau, R.C.: A Hamiltonian regularization of the Burgers equation. J. Nonlinear Sci. 16(6), 615–638 (2006)
Blandin, S., Goatin, P.: Well-posedness of a conservation law with non-local flux arising in traffic flow modeling. Numer. Math. 132(2), 217–241 (2016)
Calderoni, P., Pulvirenti, M.: Propagation of chaos for Burgers' equation. Ann. Inst. H. Poincaré Sect. A (N.S.) 39(1), 85–97, 1983
Colombo, M., Crippa, G., Graff, M., Spinolo, L.V.: On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws. Preprint arXiv:1902.07513
Colombo, M., Crippa, G., Spinolo, L.V.: Blow-up of the total variation in the local limit of a nonlocal traffic model. Preprint arXiv:1808.03529
Colombo, R.M., Garavello, M., Lécureux-Mercier, M.: A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci. 22(4), 1150023 (2012)
Colombo, R.M., Herty, M., Mercier, M.: Control of the continuity equation with a non local flow. ESAIM Control Optim. Calc. Var. 17(2), 353–379 (2011)
Crippa, G., Donadello, C., Spinolo, L.V.: A note on the initial–boundary value problem for continuity equations with rough coefficients. In: Hyperbolic Problems: Theory, Numerics and Applications, Volume 8 of AIMS Series on Applied Mathematics, pp. 957–966, 2014
Crippa, G., Lécureux-Mercier, M.: Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow. NoDEA Nonlinear Differ. Equ. Appl. 20(3), 523–537 (2013)
Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 4th edn, vol. 325. Springer, Berlin (2016)
De Lellis, C.: Notes on hyperbolic systems of conservation laws and transport equations. In: Handbook of Differential Equations: Evolutionary Equations. Vol. III, Handbook of Differential Equations, pp. 277–382. Elsevier/North-Holland, Amsterdam, 2007
DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, 2nd edn, vol. 19. American Mathematical Society, Providence, RI (2010)
Keimer, A., Pflug, L.: Existence, uniqueness and regularity results on nonlocal balance laws. J. Differ. Equ. 263(7), 4023–4069 (2017)
Kružkov, S.N.: First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (123), 228–255, 1970
LeVeque, R.J.: Finite volume methods for hyperbolic problems. In: Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002
Zumbrun, K.: On a nonlocal dispersive equation modeling particle suspensions. Quart. Appl. Math. 57(3), 573–600 (1999)
Acknowledgements
The authors wish to thank Stefano Bianchini, Rinaldo Colombo, ElioMarconi,Mario Pulvirenti, Giuseppe Savaré and Giuseppe Toscani for valuable discussions. In particular, Lemma 5.4 is due to Elio Marconi and the equality (5.35) was pointed out to us by Giuseppe Savaré. GC is partially supported by the Swiss National Science Foundation Grant 200020_156112 and by the ERC Starting Grant 676675 FLIRT. LVS is a member of the GNAMPA group of INDAM and of the PRIN National Project “Hyperbolic Systems of Conservation Laws and Fluid Dynamics: Analysis and Applications”. Part of this work was done when LVS andMC were visiting the University of Basel, and its kind hospitality is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Saint-Raymond
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Colombo, M., Crippa, G. & Spinolo, L.V. On the Singular Local Limit for Conservation Laws with Nonlocal Fluxes. Arch Rational Mech Anal 233, 1131–1167 (2019). https://doi.org/10.1007/s00205-019-01375-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-019-01375-8