We study a class of traffic flow models with nonlocal look-ahead interactions. The global regularity of solutions depend on the initial data. We obtain sharp critical threshold conditions that distinguish the initial data into a trichotomy: subcritical initial conditions lead to global smooth solutions, while two types of supercritical initial conditions lead to two kinds of finite time shock formations. The existence of non-trivial subcritical initial data indicates that the nonlocal look-ahead interactions can help avoid shock formations, and hence prevent the creation of traffic jams.
We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $ρ$ ahead. The averaging kernel is of exponential type: $w_\varepsilon(s)=\varepsilon^{-1} e^{-s/\varepsilon}$. For any decreasing velocity function $v$, we prove that, as $\varepsilon\to 0$, the limit of solutions to the nonlocal equation coincides with the unique entropy-admissible solution to the scalar conservation law $ρ_t + (ρv(ρ))_x=0$.
This paper focuses on the proof of the stability of entropy weak solutions of a nonlocal balance law modeling vehicular traffic flow on a road with on- and off-ramps. The stability is obtained with respect to a kernel function in the source term. We get an estimate of the L1-dependence of the solution with respect to the initial datum, the on-ramp rate, the off-ramp rate and the mentioned kernel function. We also show numerical experiments in order to perform an optimization problem in traffic flow with on-ramps.
Consider a nonlocal conservation where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (a…
Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counter-examples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular l…
We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $ρ$ ahead. The averaging kernel is of exponential type: $w_\varepsilon(s)=\varepsilon ^{-1} e^{-s/\varepsilon}$. By a transformation of coordinates, the problem can be reformulated as a $2\times 2$ hyperbolic system with relaxation. Uniform BV bounds on the solution are thus obtained, independent of the scaling parameter $\varepsilon $. Letting $\varepsilon\to 0$, the limit yields a weak solution to the corresponding conservation law $ρ_t + ( ρv(ρ))_x=0$. In…
We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model one considers an averaged downstream velocity. The road condition is piecewise constant with a jump at $x=0$. We study stationary traveling wave profiles cross $x=0$, for all possible cases. We show that, depending on the case, there could exit infinitely many profiles, a unique profile, or no profiles at all. Furthermore, some of the profiles are time asym…
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 profil…
We investigate a class of nonlocal conservation laws with the nonlinear advection coupling both local and nonlocal mechanism, which arises in several applications such as the collective motion of cells and traffic flows. It is proved that the $C^1$ solution regularity of this class of conservation laws will persist at least for a short time. This persistency may continue as long as the solution gradient remains bounded. Based on this result, we further identify sub-thresholds for finite time shock formation in traffic flow models with Arrhenius look-ahead dynamics.
We study a nonlocal traffic flow model with an Arrhenius type look-ahead interaction. We show a sharp critical threshold condition on the initial data which distinguishes the global smooth solutions and finite time wave break-down.
In this paper, we proceed to study the nonlocal diffusion problem proposed by Li and Wang [8], where the left boundary is fixed, while the right boundary is a nonlocal free boundary. We first give some accurate estimates on the longtime behavior by constructing lower solutions, and then investigate the limiting profiles of this problem when the expanding coefficient of free boundary converges to $0$ and $\yy$, respectively. At last, we focus on two important kinds of kernel functions, one of which is compactly supported and the other behaves like $|x|^{-γ}$ with $γ\in(1,2]$ near infinity. With…
We present a network formulation for a traffic flow model with nonlocal velocity in the flux function. The modeling framework includes suitable coupling conditions at intersections to either ensure maximum flux or distribution parameters. Based on an upwind type numerical scheme, we prove the maximum principle and the existence of weak solutions on networks. We also investigate the limiting behavior of the proposed models when the nonlocal influence tends to infinity. Numerical examples show the difference between the proposed coupling conditions and a comparison to the Lighthill-Whitham-Richa…
We will consider the nonlocal Hénon equation $$(-Δ)^s u= |x|^α u^{p},\quad \mathbb{R}^{N},$$ where $(-Δ)^s$ is the fractional Laplacian operator with $0<s<1$, $-2s<α$, $p>1$ and $N>2s$. We prove a nonexistence result for positive solutions in the optimal range of the nonlinearity, that is, when $$1<p<p^*_{α, s}:=\frac{N+2α+2s}{N-2s}.$$ Moreover, we prove that a bubble solution, that is a fast decay positive radially symmetric solutions, exists when $p=p_{α, s}^{*}$.
We study a nonlocal particle model describing traffic flow on rough roads. In the model, each driver adjusts the speed of the car according to the condition over an interval in the front, leading to a system of nonlocal ODEs which we refer to as the FtLs (follow-the-leaders) model. Assuming that the road condition is discontinuous at the origin, we seek stationary wave profiles (see Definition 1.1) for the system of ODEs across this discontinuity. We derive a non-local delay differential equation with discontinuous coefficient, satisfied by the profiles, together with conditions on the asympto…
We present a non-local version of a scalar balance law modeling traffic flow with on-ramps and off-ramps. The source term is used to describe the traffic flow over the on-ramp and off-ramps. We approximate the problem using an upwind-type numerical scheme and we provide L^\infty and BV estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar balance laws. Some numerical simulations illustrate the behaviour of solutions in sample cases.
We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its $C^2$-distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate.
A nonlocal Cahn-Hilliard model with a nonsmooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg-Landau energy functional is defined which recovers the classical (local) model for vanishing nonlocal interactions. In contrast to the local Cahn-Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. Here, the lack of smoothness of the potential is essential to guarantee the afo…
Following the seminal paper by Bourgain, Brezis and Mironescu, we focus on the asymptotic behavior of some nonlocal functionals that, for each $u\in L^2(\mathbb{R}^N)$, are defined as the double integrals of weighted, squared difference quotients of $u$. Given a family of weights $\{ρ_ε\}$, $ε\in (0,1)$, we devise sufficient and necessary conditions on $\{ρ_ε\}$ for the associated nonlocal functionals to converge as $ε\to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.
A Nonlocal $p$-Laplacian Interface Model with Sharp Interface
China Jiliang University（中国计量大学）；Tsinghua University（清华大学）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）；The University of Hong Kong（香港大学）
We study the asymptotic behavior for nonlocal diffusion equations $\partial_tu=\mathcal{J}u-χ_0u$ in $\mathbb{R}^n\times(0,\infty)$ and obtain a sufficient condition so that solutions of the Cauchy problem decay in time at the rate of a regular varying function. In the sufficient condition, a sharp bound of certain forms is required for the $k$-fold iterations $\mathcal{J}^ku_0$ or the kernels $J_k$. We prove the desired decay rate by analyzing the asymptotic behavior of a regular varying modified exponential series. Then we verify that the sufficient condition is true for most of the known ra…
We prove a chain rule of local type for a class of fractional hypoelliptic equations of Kolmogorov-Fokker-Planck type. We introduce a semigroup based notion of nonlocal \emph{carré du champ} which works successfully in situations in which the infinitesimal generator of the semigroup itself does not necessarily possess a gradient. Our results extend and sharpen the original 2004 chain rule due to A. Córdoba and D. Córdoba.
We investigate which nonlocal-interaction energies have a ground state (global minimizer). We consider this question over the space of probability measures and establish a sharp condition for the existence of ground states. We show that this condition is closely related to the notion of stability (i.e. $H$-stability) of pairwise interaction potentials. Our approach uses the direct method of the calculus of variations.
We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal{L} u= f(u)$, posed in a bounded domain $Ω\subset {\mathbb R}^N$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $\mathcal{L}$ may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian $(-Δ)^s$ ($0<s<1$) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity $f$ is increasing and looks like a …
Principal eigenvalues of nonlocal operators with advection: sharp scaling limits and spectral phase transitions
Department of Mathematics, International University, Ho Chi Minh City, Vietnam（国际大学数学系）；Vietnam National University Ho Chi Minh City, Ho Chi Minh City, Vietnam（越南国立大学胡志明市）
We present a new class of macroscopic models for pedestrian flows. Each individual is assumed to move towards a fixed target, deviating from the best path according to the instantaneous crowd distribution. The resulting equation is a conservation law with a nonlocal flux. Each equation in this class generates a Lipschitz semigroup of solutions and is stable with respect to the functions and parameters defining it. Moreover, key qualitative properties such as the boundedness of the crowd density are proved. Specific models are presented and their qualitative properties are shown through numeric…
This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form $\partial_t u + {\mathcal L}u^m=0$, $m>1$, where the operator ${\mathcal L}$ belongs to a general class of linear operators, and the equation is posed in a bounded domain $Ω\subset{\mathbb R}^N$. As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, ${\mathcal L}$ can be a power of a uniformly elliptic operator …
Blow-up analysis and extremal functions for nonlocal interaction functionals in dimension $N$
Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro（巴里阿尔多·莫罗大学数学系）；Institut für Mathematik, Goethe-Universität Frankfurt（法兰克福歌德大学数学研究所）
We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data. We show that for small values of the parameter the corresponding solutions decay to 0, while for large values the related solutions converge to 1 uniformly on compacts. Moreover, we prove that the transition from extinction (converging to 0) to propagation (converging to 1) is sharp. Numerical results are provided to verify the theo…
We are interested in the threshold phenomena for propagation in nonlocal diffusion equations with some compactly supported initial data. In the so-called bistable and ignition cases, we provide the first quantitative estimates for such phenomena. The outcomes dramatically depend on the tails of the dispersal kernel and can take a large variety of different forms. The strategy is to combine sharp estimates of the tails of the sum of i.i.d. random variables (coming, in particular, from large deviation theory) and the construction of accurate sub-and super-solutions.
