Conservation Laws with Nonlocal Velocity: The Singular Limit Problem

SIAM Journal on Applied Mathematics, vol. 84, pp. 497–522

Abstract

Abstract. We consider conservation laws with nonlocal velocity and show, for nonlocal weights of exponential type, that the unique solutions converge in a weak or strong sense (dependent on the regularity of the velocity) to the entropy solution of the local conservation law when the nonlocal weight approaches a Dirac distribution. To this end, we first establish a uniform total variation bound on the nonlocal velocity, which can be used to pass to the limit in the weak solution. For the required entropy admissibility, we use a tailored entropy-flux pair and take advantage of a well-known result that a single strictly convex entropy-flux pair is sufficient for uniqueness, given some additional constraints on the velocity. For general weights, we show that the monotonicity of the initial datum is preserved over time, which enables us to prove convergence to the local entropy solution for rather general kernels if the initial datum is monotone. This case covers the archetypes of local conservation laws: shock waves and rarefactions. These results suggest that a “nonlocal in the velocity” approximation might be better suited to approximating local conservation laws than a nonlocal in the solution approximation, in which such monotonicity only holds for specific velocities.

Authors 4

1. Jan O. Friedrich Aachen Institute of Applied Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute of Applied Mathematics, RWTH Aachen University, 52064 Aachen, Germany

2. Simone Göttlich

   University of Mannheim

   Affiliation as printed

   Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany

3. Alexander Keimer

   Friedrich-Alexander-Universität Erlangen-Nürnberg · University of California, Berkeley

   Affiliation as printed

   Institute of Transportation Studies, UC Berkeley, Berkeley, CA 94720 USA; Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany

4. Lukas Pflug

   Friedrich-Alexander-Universität Erlangen-Nürnberg

   Affiliation as printed

   Competence Unit for Scientific Computing, Friedrich-Alexander Universität Erlangen-Nürnberg, Martensstr. 5a, 91058 Erlangen, Germany