On the Singular Local Limit for Conservation Laws with Nonlocal Fluxes

Abstract

We give an answer to a question posed in [2], which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one in general does not have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.

Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel

“…On the contrary, we have currently no hint on the limit γ → 0, which was investigated numerically in [2,4,11], since in this case the constants in (1.6) blow up. The counterexamples provided recently in [5] do not cover the problem studied here.…”

A non-local traffic flow model for 1-to-1 junctions

“…In the first part we analyze how our model behaves for a fixed look ahead distance η > 0. For non-local conservation laws, it is still an open question whether the model tends to the corresponding local equation for η tending to zero (see for example [11] for a recent overview). For this reason, we will investigate the limit question as η → 0 from the numerical point of view in Section 6.2.…”

“…As mentioned above, the behaviour of solutions for η tending to zero is of special interest for non-local conservation laws. Concerning non-local LWR traffic flow models as in [8,17], or model (2.1) with v 1 ≡ v 2 , so far the convergence to the classical LWR traffic flow model [27,28] can only be proven for monotone initial data (see [11,24]), since the solution is monotonicity preserving and therefore has a strict maximum principle and a bounded total variation, uniformly in η. Unfortunately, similar results do not hold for model (2.1) with v 1 = v 2 , since the model is, in general, not monotonicity preserving even for constant initial data.…”

“…Since it is still an open question whether the solution of the non-local model tends to the solution of the corresponding local equation when the support of the kernel function tends to zero, see for example [11] for an overview, we investigate this issue only from the numerical point of view.…”

Regularity results for the solutions of a non-local model of traffic flow

“…xxx ρ ε . Notice that a similar approximation was used in [5] to establish a convergence property for the singular limit where the (smooth) convolution kernel is replaced by a Dirac delta, in the viscous case. Here, we will study the properties of smooth solutions ρ ε of this equation corresponding to a fixed initial datum ρ 0 , and then we will recover properties for ρ passing to the limit as ε → 0.…”