Nonlocal Transport Equations-Existence and Uniqueness of Solutions and Relation to the Corresponding Conservation Laws

Coron JM, Keimer A, Pflug L (2020)

Publication Type: Journal article

Publication year: 2020

Journal

SIAM Journal on Mathematical Analysis Society for Industrial and Applied Mathematics

Book Volume: 52

Pages Range: 5500-5532

Journal Issue: 6

DOI: 10.1137/20M1331652

Abstract

In this contribution, we study the existence and uniqueness of nonlocal transport equations. The term "nonlocal" refers to the fact that the flux function's derivative will be integrated over a neighborhood of the corresponding space-time coordinate. We will demonstrate existence and uniqueness of weak solutions for TV ∩ L∞ initial datum and provide stability estimates. Moreover, we investigate the convergence of the nonlocal transport equation to the corresponding local conservation law when the nonlocal reach tends to zero. For quadratic flux functions (including Burgers' equation and the Lighthill-Whitham-Richards traffic flow model), we establish convergence to a weak solution of the local conservation law for "symmetric" nonlocal terms. For specific quasi-convex and quasiconcave initial datum we even obtain convergence to the local entropy solution. We demonstrate that for "nonsymmetric" nonlocal approximations the solution cannot converge to the entropy solution or even a weak solution. We conclude with additional numerical examples showing that convergence appears to hold for more general initial datum.

Authors with CRIS profile

Lukas Pflug Lehrstuhl für Angewandte Mathematik

Involved external institutions

Sorbonne Université FR France (FR) University of California, Berkeley US United States (USA) (US)

How to cite

APA:

Coron, J.M., Keimer, A., & Pflug, L. (2020). Nonlocal Transport Equations-Existence and Uniqueness of Solutions and Relation to the Corresponding Conservation Laws. SIAM Journal on Mathematical Analysis, 52(6), 5500-5532. https://dx.doi.org/10.1137/20M1331652

MLA:

Coron, Jean Michel, Alexander Keimer, and Lukas Pflug. "Nonlocal Transport Equations-Existence and Uniqueness of Solutions and Relation to the Corresponding Conservation Laws." SIAM Journal on Mathematical Analysis 52.6 (2020): 5500-5532.

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