Calculus of Variations and Geometric Measure Theory
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M. Colombo - G. Crippa - L. V. Spinolo

On the singular local limit for conservation laws with nonlocal fluxes

created by crippa on 13 Oct 2017

[BibTeX]

Preprint

Inserted: 13 oct 2017
Last Updated: 13 oct 2017

Year: 2017

Abstract:

We give an answer to a question posed in $[$P. Amorim, R. Colombo, A. Teixeira, ESAIM Math. Model. Numerics. Anal. 2015$]$, which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity field is given by the convolution of the solution with 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 counter-examples showing that, despite numerical evidence suggesting a positive answer, one in general does not 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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