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On the singular local limit for conservation laws with nonlocal fluxes

Colombo, Maria and Crippa, Gianluca and Spinolo, Laura Valentina. (2019) On the singular local limit for conservation laws with nonlocal fluxes. Archive for Rational Mechanics and Analysis, 233 (3). pp. 1131-1167.

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Official URL: https://edoc.unibas.ch/74298/

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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.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa)
UniBasel Contributors:Crippa, Gianluca
Item Type:Article, refereed
Article Subtype:Research Article
Publisher:Springer
ISSN:0003-9527
e-ISSN:1432-0673
Note:Publication type according to Uni Basel Research Database: Journal article
Language:English
Identification Number:
Last Modified:13 Jan 2020 11:30
Deposited On:13 Jan 2020 11:30

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