Traveling Waves for Nonlocal Models of Traffic Flow

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. 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.


Introduction
We consider two conservation laws with nonlocal flux describing traffic flow, ρ(t, x) t + ρ(t, x) · v x+h x ρ(t, y)w(y − x) dy In both models, ρ(t, x) is the density of cars at (t, x), and h > 0 is a real number. We have the following assumptions on the functions w(·) and v(·): (A1) The weight function w(x) is nonnegative and continuous on (0, h), and satisfies The convergence of microscopic models to their macroscopic equivalent is of fundamental interest. In this paper, we aim to study this question in terms of traveling wave profiles. Since the equations (1.1) and (1.2) are rather similar, as well as the systems (1.7) and (1.10), we analyze in detail (1.1) and (1.7). The analysis for (1.2) and (1.10) will only be presented briefly.
To fix ideas and simplify the discussion, we seek stationary traveling wave profiles Q(x) for (1.1). Traveling waves with non-zero speed can be transformed into a stationary profile using a coordinate translation (see Example 1 in Section 5). We derive a delay integro-differential equation, see (3.4), satisfied by the stationary profiles Q(x). The existence and uniqueness (up to horizontal shifts) of these monotone profiles are established. Under mild regularity assumptions on the initial data, we show that the profiles Q(x) are the time asymptotic limits of solutions of the Cauchy problem for (1.1).
On the other hand, we also seek stationary "discrete traveling wave profiles" P (x) for the FtLs model (1.7), defined as follows. Let (z i (t), ρ i (t)) be solutions of (1.7), with certain initial data. We seek a function P (x) such that the position of the point (z i (t), ρ i (t)) traces along the graph of the function P (x) as time t evolves, for every i and t ≥ 0, i.e., P (z i (t)) = ρ i (t), ∀t ≥ 0, ∀i ∈ Z. (1.11) We derive a delay differential equation with a summation term, see (2.2), satisfied by P (x). Similar to the profile Q(x), we establish the existence and uniqueness (up to horizontal shift) of the profiles P (x). Furthermore, we show that these profiles provide attractors to the solutions of the FtLs model (1.7), for initial data with rather mild assumptions.
The profile P (x) depends on the parameter , i.e., the length of the cars. Taking the limit → 0, we prove the convergence of traveling wave solutions P (x) for the particle model to the profile Q(x) for the nonlocal conservation laws.
A completely similar result is proved for the system (1.10) and the conservation law (1.2), with only small modifications in the analysis.
For the local follow-the-leader model, where the speed of each car is determined solely by the leader ahead, the traveling wave profiles have been studied in a recent work by Shen & Shikh-Khalil [28], where existence, uniqueness and stability of traveling waves were established. For the general Cauchy problems, convergence to the corresponding local macroscopic model, where f (ρ) = ρv(ρ), (1.12) has been studied in various literatures [17,22,23]. For the nonlocal model (1.1), well-posedness of the Cauchy problem and convergence of a finite difference scheme has been established by Blandin and Goatin [9]. Amorim et. al [3] proved a similar result for kernel functions in C 2 (IR) ∩ W 2,∞ (IR) instead of C 1 ([0, h]). Both papers also show numerically how the solutions change when the kernel function has support around or behind the car, i.e., modeling backward looking drivers.
An overview over conservation laws with several other types of nonlocal flux functions can be found in, e.g., [14,20] and the references therein. For classical results on delay differential equations, we refer to [18,19].
The paper is organized as follows. In Section 2 we study the non-local FtLs model (1.7), where we prove existence and uniqueness of the traveling wave profiles. We also show that these profiles are time asymptotic limits for initial value problems of the FtLs model. Similar results are proved for the non-local PDE model (1.1) in Section 3. In Section 4 we prove the convergence of the profiles of the FtLs model (1.7) to those of the PDE model (1.1), as the car length tends to 0. In Section 5 we highlight a couple of cases of instability. In Section 6 we treat the alternative system (1.10) and the conservation law (1.2), and prove very similar results. Finally, concluding remarks are given at the end in Section 7.

Non-local Follow-the-Leaders models
We begin with the derivation of the equation satisfied by P (x). Note that (1.7) can be rewritten as a system of ODEs for the discrete density functions ρ i (t), Differentiating both sides of (1.11) in t, and using (1.7) and (2.1), one gets Here P * (z i ) is the weighted average of P around z i , similar to ρ * i , defined as We see that the profile P satisfies a delay differential equation (2.2) with a summation term of P * defined in (2.3). We introduce some notation. Given a profile P (x), we define an operator L P (x) for the position of the leader of the car at x, and for a general index k ≥ 1, We also define an averaging operator A P as Setting x = z i , we can write the delay differential equation in (2.2)-(2.3) as
Lemma 2.1 (Asymptotic limits). Assume that P (x) is a bounded solution of (2.6) whose asymptotic limits satisfy Then, the following holds.
The rate λ − satisfies the estimate Proof. We consider the limit x → +∞, and linearize (2.6) at ρ + . Writing where η(x) is a first order perturbation, we have To simplify the notation, below we use "≈" to denote a first order approximation, such as The weights are now Note that the approximated weightsŵ k are independent of the index i. We havê We compute Plugging all these approximations into (2.6), we obtain Using the positive coefficients a, b defined in (2.8), we can write the linearization of (2.6) as where the linearized weightsŵ k are given in (2.10). Note that (2.11) is a linear delay differential equation, which can be solved explicitly using its characteristic equation. Seeking solutions of the form where M is an arbitrary constant (positive or negative), we have the characteristic equation Thus, the rate λ must satisfy the equation where R(0)= lim λ→0 R(λ) = 1. (2.12) The functions L(λ) and R(λ) have the properties and L (λ) < 0 ∀λ, We conclude that there exists exactly one solution λ for (2.12). Furthermore, we observe: • If b > 1, the solution λ is positive, which we denote by λ + > 0; • if b = 1, the solution is λ = 0, which leads to the trivial solution P (x) ≡ ρ + .
• if b < 1, the solution λ is negative, which leads to an unstable asymptote.
Thus, we obtain a stable asymptote at x → +∞ if and only if b > 1, which is equivalent to To get an estimate on λ + , we define the monotone function therefore H has a unique zeros which is positive. Using the inequality Moreover, sinceŵ k = 0 for ka ≥ h, for λ > 0 we have (2.14) Combining (2.13)-(2.14), we have The function H(λ) has the properties Therefore H has a unique positive root λ , satisfying the rough estimate Using the fact that λ + > λ , we obtain the estimate (2.8).
Definition 2.1. Let the function x → P ∈ (0, 1) be given for x ∈ R. We call a sequence of car positions {z i } a distribution generated by P (x) if Note that for a given P (x), there exist infinitely many car distributions. However, if we fix the position of one car, say z 0 = 0, then the distribution is unique.
The next lemma follows immediately. If a solution {z i (t)} traces along a profile P (x), then it exhibits a periodic behavior, defined as follows. Definition 2.3. Let {z i (t)} be the solution of (1.7) with initial data {z i (0)}. We say that {z i (t)} is periodic if there exists a constant, t p independent of i and t, such that We refer to t p as the period.  Proof.
Step 1. We first show that, if {z i (t)} is periodic, then P (x) satisfies (2.6). Fix a time t ≥ 0. Consider By the periodicity we have Differentiating both sides of the above equation in x, we get which implies (2.6).
Step 2. We now show the reverse implication. Assume that P (x) satisfies (2.6). Fix a time t ≥ 0, and let {z i (t)} be a car distribution generated by P (x) such that z i+1 (t) = L P (z i (t)) for all i. The time it takes for the car at z i to reach z i+1 is dz.
By the argument in Step 1, we conclude that t p (z i ) = 0 for all i, therefore t p (z i ) is constant for any z i . Therefore, {z i (t)} is periodic.
One can connect the period t p to the asymptotic limits ρ ± . The period t p satisfies The next lemma follows immediately.
Lemma 2.4. Let P (x) be a solution of (2.6). Let {z i (0)} be an initial car distribution generated by P (x), and let {z i (t)} be the corresponding solution of (1.7) with period t p . Let ρ − , ρ + be two states such that Then, we have

Initial value problems
We assume that the assumptions (1.3)-(1.4) hold. Fix x 0 , and let the "initial data" Ψ(x) be given for P (x) on [x 0 , +∞), i.e., We assume that Ψ(x) is a smooth monotone function, defined on x ≥ x 0 , with Then, the equation (2.6) can be solved backwards in x, for x < x 0 . We refer to this as the "initial value problem" for (2.6). Note that, since Ψ(x) might not satisfy (2.6), the derivative P might not be continuous at x 0 . Therefore, it is understood that the derivative P (x) in (2.6) denotes the left derivative, i.e., P (x−).
Lemma 2.5 (Monotonicity and positivity). Let P (x) be a solution of the initial value problem with initial data Ψ given on x ≥ x 0 , satisfying (2.19). Then, we have (2.20) Proof. We first prove the monotonicity by contradiction. Assume that P fails to be monotone on x < x 0 . Then there exists a pointx < x 0 such that Since P (x) is monotone on x >x, and A P is an averaging operator, we have By (2.6) we get P (x) > 0, contradicting (2.21). The positivity of P (x) follows from the fact that equation (2.6) is "autonomous" and P = 0 is a critical point.
The next theorem states the existence and uniqueness for the initial value problem.

22)
and where the operator A Ψ is defined as in (2.5), replacing P with Ψ.
Proof. Since (2.6) is a delay differential equation with delay at least , the existence and uniqueness can be established by the method of steps. We define the intervals Consider first I 0 . For x ∈ I 0 , the right hand side of (2.6) is given as initial data. Since Ψ is monotone and positive, by Lemma 2.5 the solution Q(x) is monotone and positive on I 0 . By standard theory for ODE, the solution P (x) exists and is unique on I 0 . Then, an inductive argument leads to the existence and uniqueness of solution on all subsequent intervals I k for k = 1, 2, 3, · · · , and thus for all x ≤ x 0 . By the periodic property from Lemma 2.3, we have Setting x = x 0 , and together with Lemma 2.4, we obtain (2.22).

Existence and uniqueness of the two-point boundary value problem
Then there exist monotone solutions P (x) on x ∈ R. Furthermore, the solutions are unique up to horizontal shifts. Proof.
Step 1. Existence. The existence of solutions is obtained through the convergence of approximate solutions. Let λ + be the exponential rate in Lemma 2.1 for the asymptote at x → +∞, and let Here M is an arbitrary positive constant, which we may simply set as M = 1. Let {x n } be a sequence of points such that and let P (n) (x) be the unique solution for the initial value problem of (2.6) with initial data Ψ(x) given on x ≥ x n . Then, P (n) is positive and monotone on x < x n . As n → ∞, P (n) (x) converges to a limit function P (x), with It remains to check the limit as x → −∞. Denote By Lemma 2.1 we have ρ − n <ρ. We further claim that Let ε > 0. There exists an N , sufficiently large, such that e −λ + xn < ε for all n > N . Using (2.24), we compute, for n > N , Since ε > 0 is arbitrary, we conclude that which proves (2.25). We note that, if Q(x) is a solution to the two-point boundary value problem, then any horizontal shift Q(x + c) is another solution, for any constant c.
Step 2. Uniqueness. Assume that there exist two solutions P 1 , P 2 that are distinct after any horizontal shift. Then, we can consider some shifted versions of P 1 , P 2 such that their graphs cross each other. Letx be the rightmost point where they cross, i.e., assume Denote by A P 1 , A P 2 and L P 1 , L P 2 the averaging operators and the leader operators corresponding to P 1 , P 2 , respectively. By these assumptions, we have, for all x ≥x, .
On both profiles, the position for the leader of the car atx is the same, i.e., Since both profile admits the same period, we have dz, a contradiction to (2.26). Thus, we conclude that the solutions of the two-point boundary value problem are unique, up to horizontal shifts.
Sample profiles. Sample profiles for P (x) with various (ρ − , ρ + ) values and w(x) functions are given in Figure 1. The profiles are generated using the approximate solutions described in Theorem 2.1.
, we observe that the profiles look rather similar qualitatively.

Stability of the traveling waves
It is natural to assume that the road condition right in front of the driver is more important than the condition further ahead. This leads to the additional assumption which gives a convex flux f < 0. With these additional assumptions, the traveling wave profiles turn out to be the time asymptotic limits for the solutions of the FtLs model (1.7), under mild assumptions on the initial data. .
let {z i (t)} be the solution of (1.7) with initial data {z i (0)}, and {ρ i (t)} be the corresponding discrete densities. Assume furthermore that the initial data satisfies where P 1 (x) and P 2 (x) are (horizontally shifted) solutions to (2.6) with the boundary conditions Proof.
Step 1. We first observe that assumption (2.29) implies Fix a time t ≥ 0, and let {z i (t), ρ i (t)} be the solution of the FtLs model. Denote byP (x) the smallest profile bigger than {z i (t), ρ i (t)}, i.e., Let k be an index such that We claim thatρ Indeed, (2.30) and v ≤ 0 imply Equation (2.6) can be written aŝ On the other hand, equations (1.7) and (2.1) lead tȯ By (2.30) and (2.32), we have A 2 < A 1 . Since v ≤ 0, by (2.32) we have B 2 ≤ B 1 . Finally, to compare C 1 and C 2 , let {y i } be the car distribution generated by the profileP (x) with y k = z k , where we also have y k+1 = z k+1 . Define the piecewise constant functionsP (x) and ρ (x, t) asP Since w ≤ 0, we have w(y − z k+1 ) − w(y − z k ) ≥ 0. Using (2.30) and (2.34), we conclude that C 2 ≤ C 1 . This proves (2.31).
On the other hand, letP (x) be the largest profile smaller than {z i (t), ρ i (t)}, i.e., Let k be an index such that Then, by a totally similar argument one concludeṡ Step 2. The global asymptotic stability is a consequence of (2.31) and (2.36). Given ρ − , ρ + , there exists a monotone stationary profile P (x), say, with P (0) =ρ. Since any horizontal shift of P (x) is also a profile, we have a family of non-intersecting profiles generated by horizontal shifts of P (x). Then, in the (x, P )-plane, any point (x, ρ) with ρ − < ρ < ρ + must lie on a unique profile. This motivates the definition of the mapping Φ(x, ρ)= P (0), where P (x) is a profile such that P (x) = ρ. (2.37) Let {z i (t), ρ i (t)} be the solution of (1.7), as in the setting of the theorem. Define the functions Fix a time t ≥ 0. Let k min and k max be the points where {φ i (t)} attains its minimum and maximum values, respectively, i.e., By the results in Step 1, we now have This further implies that lim i.e., {φ i (t)} approaches a constant value, as t → ∞, proving the theorem.
Numerical simulations. We consider an initial data which oscillates between ρ − and ρ + several times around x = 0, see Figure 2. Typical solutions of the FtLs model ρ(t, x) at various time t are given in the same figure, with different weight function w(x). We observe that if w < 0, the oscillations damp out quickly and the solution approaches some profile P (x) as t grows. On the other hand, when w < 0, the solution becomes more oscillatory as t grows, indicating instability of the profiles P (x).
Remark 2.2. We remark that the initial data used in the simulation in Figure 2 does not satisfy the assumptions in Theorem 2.3. Nevertheless, we observe clear stability in the results, indicating that the basin of attraction is probably larger than the assumptions in Theorem 2.3.  In the case where lim This is a delay integro-differential equation. Here the terms w(h−) and w(0+) take into account that w(x) may be discontinuous at x = 0 and x = h. Also, the derivative Q (x) is understood as the left limit, i.e., Q (x−). We remark that, for smooth solutions of Q(x), (3.3) and (3.4) are equivalent.

Asymptotic limits
Lemma 3.1 (Asymptotic limits). Assume that Q(x) is a solution of (3.4) which satisfies Then, the following holds.
i) As x → +∞, Q(x) approaches ρ + with an exponential rate if and only if ρ + >ρ, wherê ρ is the stagnation point satisfying (2.7). The rate λ + satisfies the estimate ii) As x → −∞, Q(x) approaches ρ − with an exponential rate if and only if ρ − <ρ. The rate λ − satisfies the estimate Proof. Consider the limit x → +∞, and assume that Q(x) → ρ + . We linearize (3.3) at ρ + and write where (x) is a small perturbation. We have Putting these in (3.3), one gets Keeping only the first order terms, we obtain a linear delay integro-differential equation for for the perturbation (x): . The function G(·) has the properties Thus, G(λ) has a unique positive solution if and only if G(0) > 0, i.e., We denote this solution by λ + . For any given ρ + , h, and w(x), an estimate for λ + can be obtained by observing which implies (3.6). A completely symmetric argument leads to the result in ii) for the limit x → −∞. (2) On the other hand, if ρ + → 1, we have that β −1 → 0, therefore λ + → ∞. Similarly, if ρ − → 0, then λ − → ∞ as well. Thus, the only traveling wave connecting ρ − = 0, ρ + = 1 is the unit step function, taking the step at an arbitrary position x 0 .

Initial value problem
Assume that w(x) satisfies (1.3) and v(ρ) satisfies (1.4). Fix x 0 . Consider the initial value problem of (3.3) or equivalently (3.4), with initial data Q(x) = Φ(x) given on x ≥ x 0 . Assume that the initial data satisfies We denote the "global flux" at x 0 aŝ Proof. We first prove the monotonicity. Observe that if Q > 0 for x > x 0 , then (3.4) implies Q (x 0 ) > 0. We now proceed with a contradiction argument. Assume that the solution Q(x) is not monotone on x < x 0 , and letŷ < x 0 be the largest local minimum such that Q (ŷ) = 0 and Q (x) > 0 for all x ≥ŷ. But by (3.4), this implies Q (ŷ) > 0, a contradiction. The positivity of the solution follows from the fact that the equation is autonomous where 0 is a critical value. Finally, by (3.3), we easily have (3.13).  (1.4). Consider the initial value problem of (3.3) with initial data Φ(x) satisfying (3.11), on x ≥ x 0 . Then there exists a unique solution Q(x) on x ≤ x 0 .
Proof. Step 1. Existence. For delay differential equations with strictly non-zero delays, the existence and uniqueness of solutions can be proved by the method of steps, cf [18,19]. Unfortunately, for (3.3) this method does not apply since the delay here is arbitrarily small.
The existence of solution Q(x) can be achieved through convergence of approximate solutions. We discretize (3.3) as follows. Fix a step size ∆x, and define For each i, with the function Q(x) given for all x ≥ x i , we generate the value Q i−1 iteratively. The function Q(x) is reconstructed as a piecewise linear interpolation of the grid values Q i−1 , Q i , i.e., We now describe the procedure for computing Q i−1 with Q(x) given for x ≥ x i . The value Q i−1 satisfies the nonlinear equation (3.14) Numerically this can be computed efficiently using Newton iterations with Q i as the initial guess. We also remark that (3.14) can be viewed as a finite difference approximation for (3.4), with mixed forward/backward Euler steps: Since (3.14) is a nonlinear equation, we need to verify that it has a unique solution. By (3.4) we see that Q (x i ) > 0, so we seek a solution Q i−1 < Q i . Furthermore, we have Moreover, Then, for ∆x sufficiently small, we have By (3.15)-(3.16) we conclude that there exists a unique solution Q i−1 of (3.14) in the interval (0, Q i ).
Iterating the above step, one can generate solutions Q i for all i < 0, satisfying By varying the mesh size ∆x, this generates a sequence of continuous solutions Q ∆x (x) that are bounded and monotone. Taking the limit ∆x → 0, the functions Q ∆x converge to a limit function Q(x), preserving the monotonicity and the bound 0 < Q(x) < Q(x 0 ). The property (3.17) implies that Q(x) satisfies (3.3) for every x. This establishes the existence of solutions.
Step 2. Uniqueness. The uniqueness of solutions can be proved by contradiction. Let Q and Q be two distinct solutions of the initial value problem, with the same initial data Φ(x) on x ≥x. Without loss of generality, we assume that Q(x) > Q(x) on some nonempty interval on the left ofx. Since the solutions are monotone, there exist x 1 , x 2 , with x 1 < x 2 <x < x 1 + h and see Figure 3 for an illustration. This implies We now have Since both Q(x), Q(x) are solutions of (3.3), we have a contradiction to (3.18). Thus, we conclude that Q(x) ≡ Q(x) for all x ≤ x 0 .
Combined with the result from Lemma 3.1, we immediately obtain the asymptotic limit ρ − as x → −∞.

The two-point boundary value problem
Then, there exists a solution for the two-point boundary value problem. Furthermore, the solution is unique up to horizontal shifts. Proof.
Step 1: Existence. The proof for existence is similar to that in Theorem 2.2, by constructing a convergent sequence of approximate solutions. Let λ + > 0 be the rate for the stationary traveling wave profile as in Lemma 3.1. Let {x n } for n = 0, 1, 2, · · · be an increasing sequence such that lim n→∞ x n = +∞. Let the initial data be given as and let Q n (x) be the unique solution of the initial value problem of (3.3), with initial data Q(x) = Φ(x) on x ≥ x n . By Theorem 3.1, Q n exists and is unique. Furthermore, the global flux is constant along the solution on x < x n , By Corollary 3.1, we have Using the exact expression of Φ(x), we compute This finally implies that lim n→∞ ρ − n = ρ − , establishing the existence of solutions.
Step 2: Uniqueness. Let Q 1 and Q 2 be two distinct profiles that connect ρ ± at x → ±∞ (respectively) such that after some horizontal shift, the graphs of Q 1 and Q 2 intersect. Let y be the rightmost intersection point, i.e., Then, we have A(Q 1 ; y) > A(Q 2 ; y), so On the other hand, by (3.3) we must have which leads to a contradiction.
Sample profiles. Sample profiles for Q(x) with various (ρ − , ρ + ) values and w(x) functions are given in Figure 4.

Stability of the traveling waves
Under the additional assumptions (2.27) and (2.28), we now show that the stationary profiles are the stable time asymptotic limit for the solutions of the Cauchy problem of the non-local conservation law, under mild assumptions on the initial data. .
Let ρ(t, x) be the solution of the Cauchy problem for (1.1), with initial data ρ(0, x) satisfying the assumptions
Step 1. We observe that assumption (3.21) implies Fix a time t. LetQ(x) be the smallest profile that is larger than ρ(t, x), i.e., and letx be the point wherê We claim that ρ t (t,x) < 0, i.e., [ρ(t,x)v(A(ρ; t,x))] x > 0. (3.23) Indeed, we have the estimate and using w (x) ≤ 0, we get By using (3.3), we compute, at (t,x), Similarly, letQ(x) be a stationary traveling profile such thatQ(x) ≤ ρ(t, x) for all x, and letx be the largest x value whereQ equals ρ. Then, ρ t (t,x) > 0. The proof is completely similar and we omit the details.
Step 2. The time asymptotic stability follows from the properties in Step 1, with very similar arguments as in Step 2 of the proof of Theorem 2.3.
Numerical simulations. Solutions for the nonlocal conservation law (1.1) using oscillatory initial data are computed with a finite difference method, at various time t. See Figure 5, where two cases of the weight function w(x) are tested. In the plots in the top row we have w < 0. Here, we observe that the oscillations are quickly damped and that the solution approaches the profile Q(x) as t grows. This confirms the result in Theorem 3.3. On the other hand, in the plots in the bottom row we have w > 0, and we observe that the solution becomes more oscillatory as t grows, indicating instability of the profiles. This phenomenon is analyzed in more detail in Section 5.
Remark 3.2. Again, we remark that the initial data used in the above simulation does not satisfy the assumptions in Theorem 3.3, but the numerical simulation clearly indicates stability. This suggests that Theorem 3.3 might apply to a much wider class of initial data.  Then, as n → ∞, the sequence of functions P (n) (x) converges to a unique limit Q(x), where Q(x) is the stationary profile for the non-local conservation law, satisfying Proof.
Step 1. Let λ + > 0 be the exponential rate for the profile P (x) as x → ∞, derived in Lemma 2.1, where λ + is the unique solution of (2.12). Let λ + be the corresponding rate for the profile Q(x) as in Lemma 3.1, where λ + is the unique solution of (3.9). Recalling (2.12), we define the function (4.2) Recall that a = /ρ + . Since the first term on the righthand side of (4.2) is an approximate Riemann sum of the integral For λ ≤ λ = max{λ + , λ + }, we have Then, it holds Step 2. Fix a small > 0, and let P (x) be a profile that satisfies (2.6) with boundary conditions P (±∞) = ρ ± . Since P is monotone and bounded, there exists a limit function as → 0. Define the global flux for the profile P (x) as Recall that the global flux for the profile Q(x) is constant in x. However, this does not hold for P (x). Observing that (2.6) can be rewritten as Since the solution P (x) is smooth, we have and therefore lim This further implies (4.1), completing the proof.

Examples
For simplicity of the argument, we established the results for stationary profiles. In the case where a traveling wave has non-zero speed, a coordinate translate can be used.
Example 1. We consider traveling wave profiles with speed σ. Assume that f is convex with We consider the nonlocal conservation law (1.1). Let ξ = x − σt. We seek profile Q(ξ) such that ρ(t, x) = Q(x − σt) is a solution for (1.1). This gives where A is the averaging operator defined in (3.2). This leads to the integro-equation The analysis in Section 3 can be applied to (5.1) in a similar way, achieving similar results.
For the FtLs model, we seek profile P (ξ) such that Differentiating this in t, and after direct computation we arrive at Here L P is the operator in (2.4) and A P is defined in (2.5). Note the similarity between (5.2) and (2.6). Again, similar results are achieved by applying the same approach as in Section 2.
Below we discuss a few cases where the traveling wave profiles either are unstable or converge to entropy-violating shocks.
Example 2. Consider that on the interval [0, h] ahead of a driver, the situation further ahead is more important than the one closer to the driver. In terms of the weight function, this implies that Of course, from a practical point of view, this assumption is rather obscure, and we expect the mathematical models to exhibit erroneous behavior. Indeed, as we have observed in the numerical simulations (see Figure 5), when w (x) > 0 on x ∈ (0, h), the solution of the Cauchy problem for (1.1) does not approach any profile Q(x) as t grows. Here we offer a simple analysis.
To fix ideas, assume that w(0) = 0 and w (h) ≥ c o > 0 on (0, h). We revisit the proof of Theorem 3.3 and observe that (3.27) and (3.24) give Since the profile Q(x) approaches ρ + as x → +∞, therefore, forx sufficiently large, Q (x) and A(Q;x) x become arbitrarily small, and we have This shows that the solution ρ(t, x) can never settle around Q(x) for x large, confirming the instability we observe in the numerical simulation in Figure 5.
Example 3. We now consider the case where also the situation behind a driver influences the behavior of the driver. Although in reality one only adjusts the speed according to the behavior of the leaders, there has been an interest in nonlocal models where the integral kernel has support both in front and behind of the driver. This leads to the non-local conservation law where the weight function w(x) has support on [−h 1 , h 2 ]. One motivation for (5.3) stems from possible extensions of the one-dimensional conservation law into several space dimensions, for applications such as pedestrian flow and flock flow. In this connection, the analysis is much simpler when the kernel w is time-independent, i.e., it does not depend on the direction of the movement. Then, the corresponding one-dimensional kernel w(x) is necessarily an even function, which would result in a radially symmetric kernel in several space dimensions. Therefore, we consider (5. We seek stationary monotone profilesQ(x) that connect ρ ± , i.e., The profileQ(x) satisfies the integral equatioñ Carrying out a similar asymptotic analysis as in Lemma 3.1, for the limit x → ∞, equation (3.9) is modified to Therefore, a positive root λ + exists if and only if β < 1, i.e., ρ + <ρ. Thus, as x → +∞, the profileQ converges to ρ + exponentially if and only if ρ + <ρ. A completely similar argument shows that as x → −∞, the profileQ converges to ρ − exponentially if and only if ρ − >ρ.
The existence of solutions for (5.6) is not obvious. However, if monotone solutions should exist, then the above argument indicates that ρ + <ρ < ρ − (instead of ρ + >ρ > ρ − as in Lemma 3.1). In the limit as → 0+, these profiles converge to a downward jump with ρ − > ρ + , violating the entropy condition.
6 Another non-local model: averaging the velocity In this section we consider the conservation law (1.2) and the corresponding FtLs model (1.10). Note that (1.10) now giveṡ The same results on stationary profiles as in Sections 3 and 4 apply to these models. The proofs are very similar, with only mild modifications. Below we go through the analysis briefly, focusing mainly on the differences.

The FtLs model
We seek stationary "discrete traveling wave profiles" P(x) such that We define the operators L P (x)= x + P(x) , A(v(P(z i )))= m k=0 w i,k v(P((L P ) k (z i ))). (6.3) After a similar derivation, we find that P(x) satisfies a delay differential equation, P (x) = − P 2 (x) · A(v(P(x))) A(v(P(L P (x)))) − A(v(P(x))) . (6.4) The asymptotic limits at x → ±∞, the periodic behavior, the existence and uniqueness of solutions of the initial value problems, and the existence and uniqueness of two-point boundary value problem all follow in almost the same way as those in Section 3.
Stability. The analysis for the stability of the profiles is slightly different. In the same setting as in the proof of Theorem 2.3, we let k be the index such that P(z k (t)) = ρ k (t),P(z i (t)) > ρ i (t) ∀i > k, andP(z i (t)) ≥ ρ i (t) ∀i, and claim thatρ k z k <P (z k ). (6.5) Indeed, using LP (z k ) = z k+1 , we havê A stationary solution for (1.2) satisfies the equation Q(x) · A(v(Q; x)) ≡f = constant = f (ρ ± ). (6.6) This can also be written in the form of a delay integro-differential equation, The asymptotic limits are analyzed in the same way as for Section 3.
Approximate solution for initial value problem. Given Q i for i ≥ k, we compute Q k−1 by solving the nonlinear equation where on x ∈ [x k−1 , x k ] the function Q(x) is reconstructed by linear interpolation. The above nonlinear equation has a unique zero, if it is monotone. We claim that, for ∆x sufficiently small, Indeed, we have proving the claim. The existence and uniqueness of the initial value problem and the two-point boundary value problem follow in a very similar way as those in Section 4.
Remark 6.2. Note again that we do not need the assumption v ≤ 0 in the above proof.
Finally, as → 0, the profiles P (x) converges to Q(x), following the same argument as in the proof of Theorem 4.1. We omit the details.

Concluding remarks
In this paper we analyze existence, uniqueness and stability of several non-local models for traffic flow, for both particle models and PDE models. Furthermore, we prove the convergence of the traveling waves of the FtLs models to those of the corresponding non-local conservation laws. However, the existence of solutions for the initial value problem of the FtLs models remains open, as well as the convergence of solutions of the non-local microscopic model to the macroscopic model. We recall that, for the local models, the micro-macro limits are well treated in the literature, see [15,17,22,23]. Existence of solutions for the Cauchy problem of the non-local conservation laws is also well studied, cf. [9]. In general, the non-local flux has a smoothing effect on the solutions, resulting in higher regularity. We speculate that an adaptation of the approach in [22] combined with the results in [9] could yield the micro-macro limit. Details may come in a future work.
It is also interesting to study stationary profiles for the case where the road condition is discontinuous, for example where the speed limit has a jump at x = 0. Preliminary results in [27] show that the profiles for the models (1.1) and (1.2) are very different. In both cases, some profiles are non-monotone, some are non-unique, and some are also unstable, (similar to the results in [26]), portraying a much more complicated picture.
Codes for the numerical simulations used in this paper can be found: http://www.personal.psu.edu/wxs27/TrafficNL/