REGULARITY RESULTS FOR THE SOLUTIONS OF A NON-LOCAL MODEL OF TRAFFIC FLOW

. We consider a non-local traﬃc model involving a convolution prod-uct. Unlike other studies, the considered kernel is discontinuous on R . We prove Sobolev estimates and prove the convergence of approximate solutions solving a viscous and regularized non-local equation. It leads to weak, C ([0 ,T ] , L 2 ( R )), and smooth, W 2 , 2 N ([0 ,T ] × R ), solutions for the non-local traﬃc model.

1. Introduction. We consider the non-local traffic model introduced in [4,8] to account for the reaction of drivers to downstream traffic conditions. It consists in the following scalar conservation law, where the traffic velocity depends on a weighted mean of the density: where (ρ * ω)(t, x) = η 0 ρ(t, x + y)ω(y) dy = x+η x ρ(t, y)ω(y − x) dy.
For traffic flow applications, it is reasonable to assume that v is non-increasing, even if monotonicity is not required in this paper. We also recall that a similar model, considering a weighted mean of downstream speeds, has been recently introduced in [7]. More generally, model (1) belongs to the class of conservation laws with non-local flux functions, which appear in several applications, see for example [3,6,9,10,16]. We remark that most of the available well-posedness results concern equations involving smooth convolution kernels [1,2], and are based on the construction of finite-volume approximations and the use of Kružkov's doubling of variable technique [13]. In particular, these results rely on the concept of entropy solutions. Only recently, alternative proofs based on fixed point theorems have been proposed for specific cases [11,15], allowing to get rid of the entropy requirement.
The considered smooth solutions ρ ε are infinitely differentiables and the functions and its derivatives tend towards 0 when x goes to ±∞. Indeed, in the proofs, we only need that it is true for ρ ε , ∂ x ρ ε ∂ 2 xx ρ ε and ∂ 3 xxx ρ ε . Notice that a similar approximation was used in [5] to establish a convergence property for the singular limit where the (smooth) convolution kernel is replaced by a Dirac delta, in the viscous case. Here, we will study the properties of smooth solutions ρ ε of this equation corresponding to a fixed initial datum ρ 0 , and then we will recover properties for ρ passing to the limit as ε → 0.
We have the following result.
. Let ρ ε be smooth solution of (4) with initial datum ρ 0 . We assume ρ 0 ∈ W 1,4 (R)∩H 2 (R). Then, for T > 0 sufficiently small, ρ ε converges in In particular, this provides an alternative proof of existence of weak solutions, locally in time.
To prove this result, in Section 2 we first establish estimates on the non-local term and we derive L p (R), p > 1, estimates for ρ ε , then we get estimates in W 1,2N (R) for ρ ε with respect to x. This allows to prove that there exists T > 0 such that the sequence ρ ε is uniformly bounded with respect to ε in L ∞ (R) on [0, T ]. Then we prove uniform space estimates in W 2,2N (R) for ρ ε , which allows to derive estimates on ∂ t ρ ε . The proof of Theorem 1.2 is deferred to Section 3.
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. 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. In this paper, we are able to overcome these limitations by the kernel regularization ω ε and the viscosity approximation procedure given by (4).
Moreover, notice that we have

2.1.
Estimates of the non-local term. We start by proving the following estimates on the non-local term.

From
Then we get Furthermore, using Fubini's Theroem we have then we get the announced formula.

L p estimates for the viscous case.
We turn now to estimates on solutions solving the viscous and regularized non-local equation. First, we deal with L p estimates.
Proof. The equation (4) can be rewritten as Multiplying (9) by ρ p−1 ε , then integrating with respect to x, we obtain We observe that using the fact that the function ρ ε tends to 0 at ±∞ and that v is bounded, and since ρ ε and ∂ x ρ ε tend to 0 at ±∞. Therefore We use (6) to control the right hand side of (10) and we get which implies with By integration of (12) with respect to t ∈ [0, T ], we get 2.3. W 1,p estimates for p = 2N in the viscous case. We turn now to Sobolev estimates. Let N ∈ N * and set p = 2N .
We estimate now each of these terms.
• By (6) we get • Again by (6) we get where we have used Young's inequality uv ≤ 1 p u p + 1 q v q with q = p/(p − 1).
• Similarly, We now observe that, by convexity of the function x → x p for any u, v, w > 0 and p > 0.
Estimate (7) of Proposition 1 and inequality (16) give These bounds give finally the estimate
Proof. Let ρ ε be a smooth solution of (4) with the same initial datum ρ 0 ∈ H 1 . The relation (18) for N = 1 gives for some constant C that does not depend on ε (since W ε is uniformly bounded). If no uniform L ∞ -bound on ρ ε is available, we can use the Sobolev injection of H 1 (R) in L ∞ (R) and get possibly updating the constant C. We set u ε (t) = ρ ε (t, ·) 2 H 1 , then u ε ≤ C(u ε +u 2 ε ), which leads to We obtain Notice that the initial datum is the same for all the sequence and then u 0 and C 0 do not depend on ε. Setting T <T : Therefore, by Sobolev injection, ρ ε ∈ L ∞ ([0, T ] × R). Using the estimates of Propositions 2 and 3, we get (22) with bounds independents of ε.
2.5. W 2,p estimate for p = 2N . To pass to the limit, we need also estimates in W 2,p , which will provide, in the next section, with the help of the equation, the necessary regularity in time. As in Section 2.3, let N ∈ N * and set p = 2N .
Proof. We differentiate (15) with respect to x, which gives Multiplying this relation by (∂ 2 xx ρ ε ) p−1 , then integrating with respect to x, we obtain 1 p using the fact that the function ∂ 2 xx ρ ε tends to 0 at ±∞ and that v is bounded.
We estimate now each of these terms.