Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping

Keimer, Alexander; Pflug, Lukas; Spinola, Michele

doi:10.1016/j.jmaa.2018.05.013

Cited by 42 publications

(30 citation statements)

References 27 publications

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“…Notice that similar regularity of solutions is obtained in [11] in the one-dimensional setting and in [12] for non-local equations in multi space-dimension. However, in [11] regularity results are obtained assuming that the convolution kernel has no jumps, unlike the case we are considering, where the kernel ω is discontinuous at x = 0 and possibly at x = η.…”

supporting

confidence: 73%

“…However, in [11] regularity results are obtained assuming that the convolution kernel has no jumps, unlike the case we are considering, where the kernel ω is discontinuous at x = 0 and possibly at x = η. Besides, in [12] the non-local area of integration does not depend on x. Indeed, the above mentioned results are obtained relying on the characteristics method, and they thus need some regularity assumptions to hold along characteristics.…”

mentioning

confidence: 99%

“…This research was conducted while the first author was in secondment at Inria Sophia Antipolis. The authors thank the anonymous referees for relevant remarks and for pointing out reference [12].…”

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confidence: 99%

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

Berthelin¹,

Goatin²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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confidence: 73%

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confidence: 99%

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

Berthelin¹,

Goatin²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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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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“…In [35,36], it has been generally shown that the presented model is mathematically well posed. To obtain an efficient optimization scheme, gradient information will be computed with the help of a so-called adjoint equation [37,38].…”

Section: Gradient-based Optimization Of Particulate Ripening Processesmentioning

confidence: 95%

Model‐Based Optimization of Ripening Processes with Feedback Modules

et al. 2020

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In order to obtain high-quality particulate products with tailored properties, process conditions and their evolution in time must be chosen appropriately. Although the efficiency of these products depends on their dispersity in several dimensions, in established processes the particle size is usually the decisive variable to adjust. As part of the synthesis of these products, feedback modules are often incorporated so that a time-dependent ratio of the obtained product can flow back into the system. Moreover, the synthesis should be an energy-and resource-efficient process. To provide a means of ensuring this requirement, a model-and gradient-based, numerically efficient optimization tool for particle synthesis is presented which was developed to describe population balance equations incorporating feedback terms.

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Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping

Cited by 42 publications

References 27 publications

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

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

Traveling waves for nonlocal models of traffic flow

Model‐Based Optimization of Ripening Processes with Feedback Modules

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