Nonlocal conservation laws with time delay

  • Alexander KeimerEmail author
  • Lukas Pflug


This study considers nonlocal conservation laws in which the velocity depends nonlocally on the solution not in real time but in a time-delayed manner. Nonlocal refers to the fact that the velocity of the conservation law depends on the solution integrated over a specific area in space. In every model modelling human’s behavior a time delay as reaction/response time is crucial. We distinguish a so called nonlocal classical delay model where only the velocity of the conservation law is delayed from a more realistic nonlocal delay model where also a shift backwards in space is considered. For both models we show existence and uniqueness of the solutions and study their analytical properties. We also present a direct application in traffic flow modelling. We also show that for delay approaching zero, the solutions of the considered delayed models converge to the solutions of the non-delayed models in the proper topology. Finally, a comprehensive numerical study illustrating exemplary the impacts of delay and nonlocality are presented and compared with nonlocal models without delay, as well as the corresponding local models.


Nonlocal conservation laws Nonlocal balance laws Delay Traffic flow modelling with delay LWR PDE with delay Convergence for vanishing delay Existence Uniqueness 

Mathematics Subject Classification

35L65 35L03 65M25 



The authors would like to thank the referees for their careful and detailed proof reading and suggestions for improvements.

Travel funding provided by the “Bavaria California Technology Center” (BaCaTeC) is thankfully acknowledged.


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Authors and Affiliations

  1. 1.Institute of Transportation Studies (ITS)University of California, BerkeleyBerkeleyUSA
  2. 2.Continuous OptimizationFriedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

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