Existence, uniqueness and regularity results on nonlocal balance laws

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.

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“…In this section we establish an existence and uniqueness result that slightly extends the well-posedness result in [2] (see also [10,17]…”

Section: Well-posedness Of the Cauchy Problem For A Continuity Equatimentioning

confidence: 87%

On the Singular Local Limit for Conservation Laws with Nonlocal Fluxes

Colombo

Crippa

Spinolo

2019

Arch Rational Mech Anal

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“…Due to the assumptions on λ a , the propagation speed over the entire network can never reach zero and the model does not have shocks; in particular, the weak solution of the model is unique without any additional entropy condition [34].…”

Section: B Non-local Pde Modelmentioning

confidence: 99%

Information Patterns in the Modeling and Design of Mobility Management Services

et al. 2018

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The development of sustainable transportation infrastructure for people and goods, using new technology and business models can prove beneficial or detrimental for mobility, depending on its design and use. The focus of this article is on the increasing impact new mobility services have on traffic patterns and transportation efficiency in general. Over the last decade, the rise of the mobile internet and the usage of mobile devices has enabled ubiquitous traffic information. With the increased adoption of specific smartphone applications, the number of users of routing applications has become large enough to disrupt traffic flow patterns in a significant manner. Similarly, but at a slightly slower pace, novel services for freight transportation and city logistics improve the efficiency of goods transportation and change the use of road infrastructure.The present article provides a general four-layer framework for modeling these new trends. The main motivation behind the development is to provide a unifying formal system description that can at the same time encompass system physics (flow and motion of vehicles) as well as coordination strategies under various information and cooperation structures. To showcase the framework, we apply it to the specific challenge of modeling and analyzing the integration of routing applications in today's transportation systems. In this framework, at the lowest layer (flow dynamics) we distinguish app users from non-app users. A distributed parameter model based on a non-local partial differential equation is introduced and analyzed.The second layer incorporates connected services (e.g., routing) and other applications used to optimize the local performance of the system. As inputs to those applications, we propose a third layer introducing the incentive design and global objectives, which are typically varying over the day depending on road and weather conditions, external events etc. The high-level planning is handled on the fourth layer taking social long-term objectives into account.We illustrate the framework by considering its ability to model at two different levels. Specific to vehicular traffic, numerical examples enable us to demonstrate the links between the traffic network layer and the routing decision layer. With a second example on optimized freight transport, we then discuss the links between the cooperative control layer and the lower layers. The congestion pricing in Stockholm is used to illustrate how also the social planning layer can be incorporated in future mobility services.A. Keimer and A. Bayen are with the Institute of Transportation Studies,

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“…An overview over conservation laws with several other types of nonlocal flux functions can be found in [14,21] and the references therein. See [27] for a recent result on uniqueness and regularity results on nonlocal balance laws and [28] for multi-dimensional nonlocal balance laws with damping. Further relevant references can be found in [17,29].…”

Section: Introductionmentioning

confidence: 99%

Traveling waves for nonlocal models of traffic flow

Ridder

Shen²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We consider several non-local models for traffic flow, including both microscopic ODE models and macroscopic PDE models. The ODE models describe the movement of individual cars, where each driver adjusts the speed according to the road condition over an interval in the front of the car. These models are known as the FtLs (Follow-the-Leaders) models. The corresponding PDE models, describing the evolution for the density of cars, are conservation laws with non-local flux functions. For both types of models, we study stationary traveling wave profiles and stationary discrete traveling wave profiles. (See definitions 1.1 and 1.2, respectively.) We derive delay differential equations satisfied by the profiles for the FtLs models, and delay integro-differential equations for the traveling waves of the nonlocal PDE models. The existence and uniqueness (up to horizontal shifts) of the stationary traveling wave profiles are established. Furthermore, we show that the traveling wave profiles are time asymptotic limits for the corresponding Cauchy problems, under mild assumptions on the smooth initial condition. w(y − z i (t)) dy, k ≥ 0, 1, 2, · · · .(1.8)Notice that the summation in (1.7) actually contains only finitely many non-zero terms. Indeed, if (m + 1) ≥ h, then for every k > m one hasHence, by (1.3), w i,k = 0. With the above definition, from (1.3) it also follows m k=0 w i,k (t) = 1, w i,k (t) ≥ 0 ∀t ≥ 0.(1.9)

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Existence, uniqueness and regularity results on nonlocal balance laws

Cited by 81 publications

References 21 publications

On the Singular Local Limit for Conservation Laws with Nonlocal Fluxes

On the Singular Local Limit for Conservation Laws with Nonlocal Fluxes

Information Patterns in the Modeling and Design of Mobility Management Services

Traveling waves for nonlocal models of traffic flow

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