Calculus of Variations and Geometric Measure Theory
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G. M. Coclite - N. De Nitti - A. Keimer - L. Pflug

Singular limits with vanishing viscosity for nonlocal conservation laws

created by denitti on 05 Oct 2020

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Inserted: 5 oct 2020
Last Updated: 5 oct 2020

Year: 2020

Abstract:

We consider a class of nonlocal conservation laws with a second-order viscous regularization term which finds an application in modelling macroscopic traffic flow. The velocity function depends on a weighted average of the density ahead, where the averaging kernel is of exponential type. We show that, as the nonlocal reach and the viscosity parameter simultaneously tend to zero (under a suitable balance condition), the solution of the nonlocal problem converges to the entropy solution of the corresponding local conservation law. The key idea of our proof is to observe that the nonlocal term satisfies a third-order equation with diffusive and dispersive effects and to deduce a suitable energy estimate on the nonlocal term. The convergence result is then based on the compensated compactness theory.


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