Data Entry: Please note that the research database will be replaced by UNIverse by the end of October 2023. Please enter your data into the system https://universe-intern.unibas.ch. Thanks
On the singular local limit for conservation laws with nonlocal fluxes
Journal
Archive for Rational Mechanics and Analysis
Volume
233
Number
3
Pages / Article-Number
1131-1167
Abstract
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.
