On the Singular Local Limit for Conservation Laws with Nonlocal Fluxes
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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.
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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.
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