Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics

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.


Introduction
In this work we investigate a class of nonlocal conservation laws, where u is the unknown, F is a given smooth function, andū is given by (1.2)ū(t, x) = (K * u)(t, x) = R K(x − y)u(t, y) dy, where K is assumed in W 1,1 (R). The advection couples both local and nonlocal mechanism. This class of conservation laws appears in several applications including traffic flows [8,20], the collective motion of biological cells [3,16,17], dispersive water waves [23,6,2,10], the radiating gas motion [5,19,15] and high-frequency waves in relaxing medium [7,21,22]. We are interested in the persistence of the C 1 solution regularity for (1.1). As is known that the typical well-posedness result asserts that either a solution of a time-dependent PDE exists for all time or else there is a finite time such that some norm of the solution becomes unbounded as the life span is approached. The natural question is whether there is a critical threshold for the initial data such that the persistence of the C 1 solution regularity depends only on crossing such a critical threshold. This concept of critical threshold and associated methodology is originated and developed in a series of papers by Engelberg, Liu and Tadmor [4,12,13] for a class of Euler-Poisson equations.
In this paper we attempt to study such a critical phenomena in (1.1). C 1 solution regularity is shown to persist at least for finite time. Moreover, such persistency may continue as long as the solution gradient remains bounded. We also identify sub-thresholds for finite time shock formation in some special traffic flow models, as well as (1.1) with one sided interaction kernels. These together partially confirm the critical threshold phenomenon in non-local conservation laws (1.1).
The traffic flow model that motivated this study is the one with looking ahead relaxation introduced by Sopasakis and Katsoulakis [20]: where u(t, x) represents a vehicle density normalized in the interval [0, 1] and the relaxation kernel is the constant interaction potential, where γ is a positive constant proportional to the look-ahead distance and K 0 is a positive interaction strength. We set K 0 = 1 since in our study this parameter is not essential.
This linear potential is intended to take into account the fact that a car's speed is affected more by nearby vehicles than distant ones. The authors in [8] carried out some careful numerical study of the traffic flow model (1.3), through three examples: red light traffic, traffic jam on a busy freeway and a numerical breakdown study. In the case of a good visibility (large γ), their numerical studies suggest that (1.3) with the modified potential (1.5) yields solutions that seem to better correspond to reality. The objective of this article is therefore twofold : i) to establish local wellposedness of smooth solutions for (1.1); ii) to identify threshold conditions for the finite time shock formation of the traffic flow model (1.3) subject to two different potentials (1.4) and (1.5), respectively. The finite time shock formation of solutions in traffic flows are understood as congestion formation.
We use X to denote a space X(R) for X = H 2 , W 1,1 and L ∞ + H 2 . The main results are collectively stated as follows.
then u x must blow up at some finite time.
then u x must blow up at some finite time.
Regarding these results several remarks are in order.
i) Our threshold results in Theorems 1.2 and 1.3 are valid for any 0 < γ < ∞. When the look-ahead distance γ → ∞, both threshold conditions are reduced to sup x∈R [u ′ 0 (x)] > 0. On the other hand, when γ → ∞, model (1.3) is reduced to the classical Lightwill-Whitham-Richards(LWR) model [14,18], This local model can be verified to have finite time shock formation if initial data has positive slope u ′ 0 > 0 at some point. Therefore, the threshold conditions identified are consistent with that of the LWR model.
ii) In a recent work [9] D. Li and T. Li presents several finite time shock formation scenarios of solutions to (1.3) with (1.4). Their approach is to analyze the solutions along two characteristic lines defined by 0 = u(t, X 1 (t)) and 1 = u(t, X 2 (t)), with which they justified that if there exist two points α 1 < α 2 , such that u 0 (α 1 ) = 0 and u 0 (α 2 ) = 1, then u x must blow up at some finite time. Compare to their result, our shock formation conditions in Theorems 1.2 and 1.3 may be viewed in the perspective of critical thresholds.
iii) The shock formation conditions in Theorem 1.2 and 1.3 are consistent with the numerical results obtained in [8]. Indeed, a numerical comparison in [8] of solutions to (1.3) with (1.4) for γ = 0.1 and γ = 1 indicates that the solution with γ = 0.1 remains smooth, while the solution with γ = 1 seems to contain a shock discontinuity.
iv) The threshold in (1.7) is bigger than that in (1.6). This observation suggests that under certain initial configuration, the traffic flow model with constant interaction potential may develop a congestion formation, while the model with the linear interaction potential may not. Roughly speaking, it is understood that the drivers with the linear potential are 'smarter' than the drivers with the constant potential. v) For fixed γ > 0, both (1.6) and (1.7) reflect some balance between sup x∈R [u ′ 0 (x)] and inf x∈R [u ′ 0 (x)] for the finite time shock formation: if the non-positive term inf x∈R [u ′ 0 (x)] is relatively small, then sup x∈R [u ′ 0 (x)] needs to be large for the finite time shock formation. It indicates that not only the car density behind the traffic jam but also the car density ahead of the traffic jam contribute to the formation of congestion.
We now summarize the main arguments in our proofs to follow. For the proof of Theorem 1.1, we apply the Banach fixed-point theorem to the transformation S defined We show that there exists T > 0 depending on initial data such that the mapping v = S(u) exists and is a contraction. In so showing, detailed estimates of non-local terms are crucial, and allow us to track the dependence of T on the initial data. For the proofs of Theorem 1.2-1.3, we trace the Lagrangian dynamics of d := u x , which can be obtained from the Eulerian formulation: The right hand side is quadratic in d, the a priori bound 0 ≤ u ≤ 1 ensures the boundedness of both u andū x involved in the coefficients. The key in our approach is to bound the nonlocal termū xx in terms of ]. This way we are able to obtain weakly coupled differential inequalities for both M and N, which yield the desired sub-thresholds.
From the proofs of Theorem 1.2-1.3 we observe that the one-sided interaction property of kernels (1.4) and (1.5) is crucial. Hence our threshold analysis for the traffic flow models is applicable to the class of nonlocal conservation laws (1.1) under the following assumptions: (H 1 ). F ∈ C 3 (R, R), and the kernel K(r) ∈ W 1,1 satisfying The result can be stated as follows.
then u x must blow up at some finite time.
We should point out that it was the threshold analysis for traffic flow models that led us to the thresholds (1.6), (1.7) in the first place, which in turn was then extended to the general class (1.1) as summarized in Theorem 1.4.
We now conclude this section by outlining the rest of the paper. In section 2, we prove local wellposedness for the class of nonlocal conservation laws (1.1). In section 3, we investigate sub-thresholds for nonlocal traffic flow models. We finally sketch the proof of Theorem 1.4 in the end of this paper.

Local wellposedness and regularity
In this section, we study the local well-posedness of (1.1). We consider a solution space x ), which allows u to be non-zero at far field. By transformation U = u − u 0 , we find the following equation for U ∈ B T , This lies in the same class as (1.1). With this in mind, from now on, we shall consider . We prove the local wellposedness result by the fixed point argument. That is, we first define a transformation S as v = S(u), where v is solved from the following equation and then show this mapping has a fixed point. We begin by verifying the existence of v = S(u), which is carried out in a series of Lemmata 2.1-2.3. For simplicity, we take We bound a and b in terms of u in the following lemma.
Proof. We begin with some key inequalities forū: and (2.5) These together lead to (2.2). We also calculate, These estimates give (2.3).
Lemma 2.2 (A priori estimates). Suppose u ∈ B T . A sufficiently smooth solution v of (2.1) must satisfy the energy estimates where c 1 is an embedding constant.
Proof. Apply ∂ l x to the first equation of (2.1) to obtain, Multiplying (2.8) by ∂ l x v and integrating over R, we obtain, This with l = 0 leads to which upon integration gives (2.6). Next, summing (2.9) for l = 0, 1, 2, we obtain (2.10) which upon integration again gives (2.7). Lemma 2.3. Suppose the initial data v(x, 0) = u 0 ∈ H 2 . Then for each u ∈ B T , there exists a unique solution v ∈ B T of (2.1).
Assume that u ∈ B T R , we then have where c 0 and c 1 are the embedding constants. We first show that S maps B T R into B T R for some T small. From (2.7), it follows that , then difference of (2.1) for v 2 and v 1 , respectively, leads to (2.14)ṽ t + a(u 1 )ṽ x =b,ṽ(0, x) = 0 Applying (2.6) we have In order to find a time interval such that the contraction property (2.13) holds, we need to estimate ∂ x a(u 1 ) ∞ and b (·, t) L 2 . First we have (2.17) The first term in (2.15) is bounded as This can be seen from the following calculation: If we assume Fū(0, ·) = 0, then the last term in (2.15) has a similar bound: To obtain this bound, we decompose it the following way Applying the mean value property to the remaining terms gives that Therefore, for 0 < T < T * with x ) norm and thus possesses a unique fixed point u which is the unique solution of (1.1).
We prove the second part of Theorem 1.1 through the following corollary: where c 1 is the embedding constant. This infers that only one of the following occurs i) T = ∞ and u is a global solution; ii) 0 < T < ∞ and lim Proof. We use again the estimate in (2.10), setting v ≡ u, Together with the estimates of a xx L 2 and b H 2 in (2.5) and (2.3), respectively, we obtain Upon integration, we obtain (2.21). The claim in ii) follows from a contradiction argument: If lim t→T − u x ∞ < ∞, it would lead to the boundedness of u H 2 . One may therefore extend the solution for someT > T , which contradicts the assumption that T < ∞ is a maximal existence interval.
3. Sub-thresholds for finite time shock formation 3.1. Proof of Theorem 1.2. In this subsection, we consider the traffic flow model with Arrhenius look-ahead dynamics: whereū(t, x) = 1 γ x+γ x u(t, y) dy. Here γ > 0 denotes look-ahead distance. In the theory of traffic flow, u(t, x) represents a vehicle density normalized in the interval [0, 1].
We want identify some threshold condition for the shock formation of solutions to (3.1). From Corollary 2.4 we know that it suffices to track the dynamics of u x . Our idea is based on tracing M(t) := sup x∈R [u x (x, t)] and N(t) := inf x∈R [u x (x, t)]. The existence and differentiability (in almost everywhere sense) of M(t) and N(t) are proved in [1].
We also state a useful result, which is proved in [11].
Lemma 3.1. (Lemma 3.1. in [11]) Consider the following quadratic equality for A(t) with a(t) > 0, b 1 (t) ≤ b 2 (t) and that a(t), b 1 (t), b 2 (t) are uniformly bounded. i) If A 0 > max b 2 , then A(t) will experience a finite time blow-up. ii) If there exists a constantb such that then (3.2) admits a unique global bounded solution satisfying With this result we obtain the following: Lemma 3.2. Consider the following quadratic inequality, Proof. i) Subtracting (3.2) from (3.3) gives Integration leads to We remark that Lemma 3.2 remains valid even if the quadratic inequality holds almost everywhere. Now, we are ready to prove Theorem 1.2.

3.2.
Proof of Theorem 1.3. We rewrite the traffic flow model (1.3) with the linear potential as Let d := u x and apply ∂ x to (3.15), Here,ũ The existence of ξ(t) and η(t) is justified by Theorem 2.1 in [1]. Then, along (t, ξ(t)), (3.17) can be written as, where the last inequality follows from the fact that And along (t, η(t)), (3.17) can be written as, where the last inequality follows from the fact thatũ xx (t, η) = 2 γ (ū x − N) ≥ 0. (3.21) can be written as We note that N 1 ≤ 0 ≤ N 2 because 0 ≤ u(t) ≤ 1. By using the fact that 0 ≤ u ≤ 1, and −2 ≤ γũ x ≤ 2, it can be shown that N 1 is uniformly bounded from below, The verification of this inequality is similar to the one in the proof (3.10), details are omitted. With the lower bound of N 1 (t), Lemma 3.2 (ii) when applied to (3.22) gives Substituting this lower bound into (3.20), we obtaiṅ In order to apply Lemma 3.2 (i) to (3.24), we proceed to find the upper bound of M 2 (≥ M 1 ). Let v := γ ·ũ x = −2(u −ū), then from the fact that 0 ≤ u,ū ≤ 1, we know that −2 ≤ v ≤ 2. We also let Ω := {(u, v) ∈ R 2 | 0 ≤ u ≤ 1, −2 ≤ v ≤ 2} then M 2 and it's upper bound are given by 3.3. Proof of Theorem 1.4. We only sketch the proof since it is entirely similar to that in the previous sections. Let d := u x and apply ∂ x to the first equation of (1.1) to obtain (3.26) (∂ t + F u · ∂ x )d = −F uu d 2 − 2F uūūx d − Fūūū 2 x − Fūū xx . It can be shown that 0 ≤ u ≤ m, and therefore |ū| ≤ m K W 1,1 , |ū x | ≤ m K W 1,1 .
To find the bound ofū xx , we define for t ∈ [0, T ),