CONSERVATION LAWS WITH NONLOCAL VELOCITY: THE SINGULAR LIMIT PROBLEM

Friedrich J, Gottlich S, Keimer A, Pflug L (2024)

Publication Type: Journal article

Publication year: 2024

Journal

SIAM Journal on Applied Mathematics Society for Industrial and Applied Mathematics

Book Volume: 84

Pages Range: 497-522

Journal Issue: 2

DOI: 10.1137/22m1530471

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 with CRIS profile

Lukas Pflug FAU Competence Center Scientific Computing (FAU CSC)

Involved external institutions

Universität Mannheim DE Germany (DE) University of California, Berkeley US United States (USA) (US) Rheinisch-Westfälische Technische Hochschule (RWTH) Aachen DE Germany (DE)

How to cite

APA:

Friedrich, J., Gottlich, S., Keimer, A., & Pflug, L. (2024). CONSERVATION LAWS WITH NONLOCAL VELOCITY: THE SINGULAR LIMIT PROBLEM. SIAM Journal on Applied Mathematics, 84(2), 497-522. https://dx.doi.org/10.1137/22m1530471

MLA:

Friedrich, Jan, et al. "CONSERVATION LAWS WITH NONLOCAL VELOCITY: THE SINGULAR LIMIT PROBLEM." SIAM Journal on Applied Mathematics 84.2 (2024): 497-522.

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