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We give an answer to a question posed in [P. Amorim, R. Colombo, and A.Teixeira, ESAIM Math. Model. Numerics. Anal. 2015], which can be looselyspeaking formulated as follows. Consider a family of continuity equations wherethe velocity depends on the solution via the convolution by a regular kernel.In the singular limit where the convolution kernel is replaced by a Diracdelta, one formally recovers a conservation law: can we rigorously justify thisformal limit? We exhibit counterexamples showing that, despite numericalevidence suggesting a positive answer, one in general does not have convergenceof the solutions. We also show that the answer is positive if we considerviscous perturbations of the nonlocal equations. In this case, in the singularlocal limit the solutions converge to the solution of the viscous conservationlaw.

On the singular local limit for conservation laws with nonlocal fluxes