On a nonlocal dispersive equation modeling particle suspensions

Zumbrun, Kevin

doi:10.1090/qam/1704419

Cited by 49 publications

(55 citation statements)

References 21 publications

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“…A similar result was proved in [3] for kernel functions in C 2 (R)∩W 2,∞ (R) instead of C 1 ([0, h]). Similar nonlocal conservation laws with symmetric kernel functions have been studied by [38], and [8] in the context of sedimentation modeling. Multi-dimensional versions and systems were studied as models for crowd dynamics [1,2,[11][12][13]16].…”

Section: Introductionmentioning

confidence: 82%

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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Section: Introductionmentioning

confidence: 82%

Traveling waves for nonlocal models of traffic flow

Ridder

Shen²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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“…However, the method of proof and even the definition of solutions are different from this paper. Another scalar conservation law with nonlocal velocity is to model sedimentation of particles in a dilute fluid suspension, see [32] for the well-posedness of the Cauchy problem. In this model, the nonlocal velocity is due to a convolution of the unknown function with a symmetric smoothing kernel.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

Shang

Wang

2011

Journal of Differential Equations

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International audienceIn this paper, we study a scalar conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. As a generalization of Coron et al. (2010) [14], the velocity function possesses both the local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with initial and boundary data in L∞L∞. We also obtain the stability (continuous dependence) of both the solution and the out-flux with respect to the initial and boundary data. Finally, we prove the existence of an optimal control that minimizes, in the LpLp-sense with 1⩽p⩽∞1⩽p⩽∞, the difference between the actual out-flux and a forecast demand over a fixed time period

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“…Lemma 6 leads to the following energy estimate: Proposition 1. Take Assumptions 1, 2 and 3 as given and define g n as in (19).…”

Section: 4mentioning

confidence: 99%

Dispersion vs. anti-diffusion: Well-posedness in variable coefficient and quasilinear equations of KdV-type

Ambrose¹,

Wright²

2013

Indiana Univ. Math. J.

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Abstract. We study the well-posedness of the initial value problem on periodic intervals for linear and quasilinear evolution equations for which the leading-order terms have three spatial derivatives. In such equations, there is a competition between the dispersive effects which stem from the leadingorder term, and anti-diffusion which stems from the lower-order terms with two spatial derivatives. We show that the dispersive effects can dominate the backwards diffusion: we find a condition which guarantees well-posedness of the initial value problem for linear, variable coefficient equations of this kind, even when such anti-diffusion is present. In fact, we show that even in the presence of localized backwards diffusion, the dispersion will in some cases lead to an overall effect of parabolic smoothing. By contrast, we also show that when our condition is violated, the backwards diffusion can dominate the dispersive effects, leading to an ill-posed initial value problem. We use these results on linear evolution equations as a guide when proving well-posedness of the initial value problem for some quasilinear equations which also exhibit this competition between dispersion and anti-diffusion: a Rosenau-Hyman compacton equation, the Harry Dym equation, and equations which arise in the numerical analysis of finite difference schemes for dispersive equations. For these quasilinear equations, the well-posedness theorem requires that the initial data be uniformly bounded away from zero.

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On a nonlocal dispersive equation modeling particle suspensions

Cited by 49 publications

References 21 publications

Traveling waves for nonlocal models of traffic flow

Traveling waves for nonlocal models of traffic flow

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

Dispersion vs. anti-diffusion: Well-posedness in variable coefficient and quasilinear equations of KdV-type

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