Wave breaking in a class of non-local conservation laws

Abstract

For models describing water waves, Constantin and Escher [4]'s works have long been considered as the cornerstone method for proving wave breaking phenomena. Their rigorous analytic proof shows that if the lowest slope of flows can be controlled by its highest slope initially, then the wave-breaking occur for the Whitham-type equation. Since this breakthrough, there have been numerous refined wave-breaking results established by generalizing the kernel which describes the dispersion relation of water waves. Even though the proofs of these involve a system of coupled Riccati-type differential inequalities, however, little or no attention has been made to a generalization of this Riccati-type system. In this work, from a rich class of non-local conservation laws, a Riccati-type system that governs the flow's gradient is extracted and investigated. The system's leading coefficient functions are allowed to change their values and signs over time as opposed to the ones in many of other previous works are fixed constants. Up to the author's knowledge, the blow-up analysis upon this structural generalization is new and is of theoretical interest in itself as well as its application to various non-local flow models. The theory is illustrated via the Whitham-type equation with nonlinear drift. Our method is applicable to a large class of non-local conservation laws.

On a class of new nonlocal traffic flow models with look-ahead rules

“…3. Indeed, the macroscopic model ( 24) is very close to (25) with a small a = L M = 0.004. We also note that in Figs.…”

“…The wave breakdown phenomenon for the SK model (2) and related nonlocal models has been studied in [25,27]. Recently, it is shown in [28] that the nonlocal slowdown effect can help avoid traffic jams for a family of initial configurations.…”