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 loosely
speaking 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 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.Comment: 26 page
