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Nonlocal conservation laws with p-norm, the singular limit problem and applications to traffic flow

Chiarello FA, Keimer A, Pflug L (2027)

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Publication Type: Journal article

Publication year: 2027

Journal

Book Volume: 565

Article Number: 130960

Journal Issue: 1

DOI: 10.1016/j.jmaa.2026.130960

Abstract

In this contribution, we study scalar nonlocal conservation laws with the p -norm. Here, “nonlocal” means that the velocity of the conservation law depends on an integral term in space. Typically, the nonlocal term is an L¹-type weighted average of the solution whereas here we study the case when the solution is integrated in the L^p-norm. We also consider the case of the L^p-quasi-norm when p∈(0,1) and establish, for an initial datum which is uniformly bounded away from zero, the existence and uniqueness of weak solutions. We then demonstrate that weak solutions also exist for initial data that may vanish under more restrictive assumptions. Furthermore, we investigate the singular limit, i.e., what happens when the nonlocal kernel converges to a Dirac distribution. Indeed, for the one-sided exponential kernel, we recover the (entropy) solution of the corresponding local conservation law for all p∈(0,∞) with further restrictions for p∈(0,1). This generalizes the celebrated singular limit result for nonlocal conservation laws for p=1 significantly and showcases the robustness of the approximation of local conservation laws by nonlocal ones. We also investigate the monotonicity of the solution when assuming that the initial datum is monotone. Finally, we prove the convergence of solutions for p→0, resulting in a different kind of nonlocal conservation law. The paper concludes with numerical studies illustrating the effect of p on singular-limit convergence and related phenomena.

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APA:

Chiarello, F.A., Keimer, A., & Pflug, L. (2027). Nonlocal conservation laws with p-norm, the singular limit problem and applications to traffic flow. Journal of Mathematical Analysis and Applications, 565(1). https://doi.org/10.1016/j.jmaa.2026.130960

MLA:

Chiarello, Felisia Angela, Alexander Keimer, and Lukas Pflug. "Nonlocal conservation laws with p-norm, the singular limit problem and applications to traffic flow." Journal of Mathematical Analysis and Applications 565.1 (2027).

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