Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics

Abstract

We study a nonlocal traffic flow model with an Arrhenius type look-ahead interaction. We show a sharp critical threshold condition on the initial data which distinguishes the global smooth solutions and finite time wave break-down.2010 Mathematics Subject Classification. 35L65, 35L67.

Upper-thresholds for shock formation in two-dimensional weakly restricted Euler–Poisson equations

“…The critical threshold in [7] describes the conditional stability of the 1D Euler-Poisson system, where the answer to the question of global vs. local existence depends on whether the initial data crosses a critical threshold. Following [7], critical thresholds have been identified for several one dimensional models, including 2 × 2 quasi-linear hyperbolic relaxation systems [15], Euler equations with non-local interaction and alignment forces [3], and traffic flow models [14].…”

Threshold for shock formation in the hyperbolic Keller–Segel model

“…This concept of critical threshold(CT) and associated methodology is originated and developed in a series of papers by Engelberg, Liu and Tadmor [5,15,16] for a class of Euler-Poisson equations. Following their CT concept, the authors [12] identified sub-thresholds for finite time shock formation in a class of nonlocal conservation laws, which we summarize here. Under the following two assumptions: (H 1 ).…”

“…The improved interaction potential is intended to take into account the fact that a vehicle's speed is affected more by nearby vehicles that distant ones. Sub-thresholds for finite time shock formation in the traffic flow models with constant interaction potential and the improved linear interaction potential are identified in [12].…”

“…For the hyperbolic Keller-Segel problem (1.1), local existence of smooth solutions was established in [4]; see also [2] for the multi-dimensional case. For the general class of nonlocal conservation laws (1.2), C 1 solution regularity was established in [12] by some standard contraction argument, which we summarize here. Theorem 1.4.…”

“…Theorem 1.4. (Local existence [12]) Suppose F ∈ C 3 (R, R) and K ∈ W 1,1 . If u 0 ∈ H 2 , or u 0 ∈ L ∞ and u 0x ∈ H 1 , then there exists T > 0, depending on the initial data, such that (1.2) admits a unique solution u ∈ C 1 ([0, T ) × R).…”