We give an answer to a question posed in Amorim et al. ESAIM Math Model Numer Anal 49 1 โ37, , 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. This is a preview of subscription content, log in via an institution to check access. Rent this article via DeepDyve. Institutional subscriptions. Ambrosio, L. Amorim, P. Betancourt, F. Nonlinearity 24 3 , โ Bhat, H. Nonlinear Sci. Blandin, S. Calderoni, P.
Colombo, M. Preprint arXiv Colombo, R. Models Methods Appl. Crippa, G. Dafermos, C. Springer, Berlin De Lellis, C. DiPerna, R. Evans, L. Graduate Studies in Mathematics, 2nd edn, vol. Keimer, A. LeVeque, R. In: Cambridge Texts in Applied Mathematics.
Cambridge University Press, Cambridge, Zumbrun, K. Download references. In particular, Lemma 5. You can also search for this author in PubMed Google Scholar. Correspondence to Gianluca Crippa. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Reprints and permissions.
