Specified homogenization of a discrete traffic model leading to an effective junction condition

In this paper, we focus on deriving traffic flow macroscopic models from microscopic models containing a local perturbation such as a traffic light. At the microscopic scale, we consider a first order model of the form "follow the leader" i.e. the velocity of each vehicle depends on the distance to the vehicle in front of it. We consider a local perturbation located at the origin that slows down the vehicles. At the macroscopic scale, we obtain an explicit Hamilton-Jacobi equation left and right of the origin and a junction condition at the origin (in the sense of [ 25 ]) which keeps the memory of the local perturbation. As it turns out, the macroscopic model is equivalent to a LWR model, with a flux limiting condition at the junction. Finally, we also present qualitative properties concerning the flux limiter at the junction.

example a traffic light. The schematic representation of the microscopic model is given in Figure   1 1.
2 Perturbation: radius = r We denote by U j (t) the position of the j th vehicle and we assume that the velocity of each 3 vehicle is given by the function V . In order to obtain our homogenization result, we proceed as in 4 [9, 10, 11, 12, 13] and rescale the microscopic model which describes the dynamics of each vehicle, 5 to obtain a macroscopic model that describes the density of vehicles. If the local perturbation 6 is located around zero, at the macroscopic scale it is natural to get an Hamilton-Jacobi equation 7 with a junction condition at the origin (see Figure 2, u 0 x is the primitive of the density of vehicles 8 and the effective Hamiltonian H is defined later in the paper), since the size of the perturbation 9 goes to zero when we do the rescaling. This junction condition keeps the memory of the presence 10 of the local perturbation. Recently, the theory of Hamilton-Jacobi equations with junction or more generally on networks 12 has known important developments in particular since the works of Achdou, Camilli, Cutri, and 13 Tchou [1] and Imbert,Monneau,and Zidani [20]. In this direction, we would like to mention the 14 recent work of Imbert and Monneau [18] in which they give a suitable definition of (viscosity) 15 solutions at the junction which allows to prove comparison principle, stability and so on. 16 In this paper, we will use the ideas developed in [10] in order to pass from microscopic models to 17 macroscopic ones. In particular, we will show that this problem can be seen as an homogenization 18 result. The difficulty here is that, due to the local perturbation, we are not in a periodic setting 19 and so the construction of suitable correctors is more complicated. In particular, we will use the 20 idea developped by Achdou and Tchou in [2], by Galise, Imbert, and Monneau in [14] , and in  In this paper, we are interested in a first order microscopic model of the form 26U j (t) = V (U j+1 (t) − U j (t)) · φ (t, U j (t)) , (2.1) assumption. We have added assumption (A4) to work with V with a bounded support. But by 23 modifying slightly the classical optimal velocity functions, we obtain a function that satisfies all 24 the assumptions. For instance, in the case of the Greenshields based models [15](see also [5]): with n ∈ N\{0}. Another optimal velocity function, based on the Newell model [24](see also [8]), 26 is given by: with n ∈ N\{0} and b ∈ [0, +∞). See Figure 3 for a schematic representation of an optimal 28 velocity function satisfying assumption (A). 29 Remark 2.2. We will give an example of the function φ. We will define φ on the interval [0, 1] 30 since it's a Z-periodic function. For t ∈ [0, 1], where φ 0 is defined in the following form

Injecting the system of ODEs into a single PDE 2
In this paper, we will study the traffic flow when the number of vehicles per unit length tends to 3 infinity by introducing the rescaled "cumulative distribution function" of vehicles, ρ ε , defined by Theorem 8.1 for the proof of this result) the following non-local equation where M ε is a non-local operator defined by vehicles and we have for all j ∈ Z, ρ 1 (t, U j (t)) = −(j + 1). This implies (see Section 8) that we This result helped us inject the system of ODEs into a single PDE. The function ρ ε is simply the 3 rescaling of the function ρ 1 .

4
In the rest of this paper, we couple equation (2.4) with the following initial condition We also assume that the initial condition satisfies the following assumption: 6 (A0) (Gradient bound) The function u 0 is Lipschitz continuous and satisfies (2.8)

Remark 2.4. This condition ensures that initially the vehicles have a security distance between
8 them and since we are working with a first order model, this security distance will be preserved.

9
In fact, h 0 (from assumption (A3)) is called the safety distance. However, since we work with 10 Eulerian coordinates, we use k 0 which is the inverse of the safety distance. We choose u 0 a regular 11 function such that for all ε, with f (ε) → 0 as ε goes to 0. This is explain in (2.17).

13
Remark 2.5 (Lagrangian formulation). Another way to treat this problem is to consider a La-14 grangian formulation, like in [12], considering the function, This function satisfies for all (t, y) The difficulty with this formulation is that the function φ is evaluated at v(t, y) and not at a 17 physical point of the road. The notion of junction in this case is not well defined and this is why 18 we use the formulation (2.4) (where the perturbation function is evaluated at a point of the road) 19 instead of (2.9). This will allow us to use the results of Imbert and Monneau [18] concerning 20 quasi-convex Hamiltonians with a junction condition. 22 We recall that k 0 = 1/h 0 and we define H : R → R, by  We denote by

Convergence result
and we refer to Figure 4 for a schematic representation of H.

2
The main purpose of this article is to prove that the viscosity solution of (2.4)-(2.7) converges 3 uniformly on compact subsets of (0, +∞) × R as ε goes to 0 to the unique viscosity solution of the 4 following problem where A has to be determined and F A is defined by The following theorems are the main results of this paper, and their proof are postponed. The 8 proofs of Theorem 2.6 and Theorem 2.10 are done in Section 5 and the proof of Theorem 2.7 is 9 done in Section 8.

10
Theorem 2.6 (Junction condition by homogenisation We also assume that there exists a constant R > 0 such that, for (2. 18) In fact, in Proposition 2.13, we prove that v ≡ 0 is a suitable corrector which in turn will imply 16 that for a fixed p ∈ [−k 0 , 0], the effective Hamiltonian is given by −V (−1/p)|p| = H(p). 18 We define the non-local operator M p by

Effective Hamiltonian and effective flux-limiter
We then have the following result where we have used assumption (A3) for the second line, the definition of E and J (see (2.6))  To construct the effective flux-limiter A, we consider the following cell problem: find λ ∈ R 6 such that there exists a solution w of the following Hamilton-Jacobi equation More precisely, we have the following result, whose proof is postponed until Section 6. we present in Section 4 whose construction is presented in Section 6.  1 We denote by M the non-local operator M ε (defined in (2.5)) in the case ε = 1. To each operator 2 M , we associate the operatorM which is defined in the same way except that the function E is 3 replaced by the functionẼ, defined by

Notations
(2.22) Remark 2.17. Using the fact that E and V are bounded, we get that for every function U and 5 every x ∈ R, we have We also use the following notations for the upper and lower semi-continuous envelopes of a

Organization of the article 9
Section 3 contains the definition of the viscosity solutions for the problems we consider in the 10 entire article and it also contains some results for those problems. In Section 4 we present some 11 results on the correctors at the junction (Theorem 4.1) that will be used in Section 5 to prove 12 Theorem 2.6. Section 6 contains the proof of Theorem 4.1. In Section 7 we give the proof of the

17
In order to give a general definition for all the non-local problems we consider, we will give the 18 definition for the following equation, with p ∈ R, for all (t, x) ∈ (0, +∞) × R, with ψ : R → [0, 1] a Lipschitz continuous function.
We say that a function u is a viscosity solution of (3.1) if u * and u * are respectively a sub-solution 26 and a super-solution of (3.1).

27
Remark 3.2. We use this definition in order to have a stability result for the non-local term. 28 We refer to [6,26] Definition 3.4 (Viscosity solutions for (2.13)). Let H be given by (2.10) and A ∈ R. An upper ) and for all We say that a function u is a viscosity solution of (2.13) if u * and u * are respectively a sub-solution 10 and a super-solution of (2.13). We refer to this solution as an A−flux limited solution.

1) and v be a super-solution of (3.1). Let us also assume that there exists a constant
(3.2) Proof. The only difficulty in proving the comparison principle comes from the non-local term, but 17 in our case the proof is similar to the proof of [10, Theorem 4.4] and we skip it.

19
We now give a comparison principle on bounded sets, to do this, we define for a given point Proof. The proof of this theorem is similar to the one of Proposition 3.5, so we skip it.

4
Using assumption (A0) and the form of the non-local operator and of H, we have where we used (2.23) and (2.12). The proof for u − is simpler, it uses (2.23) and (2.12), Perron's method for problems with non-local terms), joint to the comparison principle, we obtain 9 the following result.

11
Then, there exists a unique continuous solution u of (3.1) which satisfies (for some constant K 1 )

13
Now we recall an equivalent definition (see [18,Theorem 2.5]) for sub and super solutions at the 14 junction. We will also consider the following problem, Theorem 3.9 (Equivalent definition for sub/super-solutions). Let H given by (2.10) and consider let us fix any time independent test function φ 0 (x) satisfying Given a function u : (0, T ) × R → R, the following properties hold true. for any test function ϕ touching u from above at 26 test function ϕ satisfying (3.5) touching u from above at (t 0 , 0) we have

is a lower semi-continuous super-solution of (3.3) and if for any
then u is an A-flux limited super-solution at (t 0 , 0).

28
Proof. The proof of Theorem 3.9 can be founded in [18, Theorem 2.5].
Proof. In this proof we used the barriers given by Lemma 3.7 (with p = 0 and ψ ≡ 1), which 5 means that the solution u of (2.4)-(2.7) with ε = 1 satisfies for all (t, In the rest of the proof we will use the following notation: Proof of the upper inequality for the control of the space oscillations. We introduce, We want to prove that M ≤ 0. We argue by contradiction and assume that M > 0.

9
Step 1: the test function. For η, α > 0, small parameters, we define Using (3.7), we have that where we used assumption (A0) for the second inequality. Therefore we have 12 lim |x|,|y|→+∞ Since ϕ is upper-semi continuous, it reaches a maximum at a point that we denote by (t,x,ȳ) ∈ Ω.

13
Classically we have for η and α small enough, Step 2:t > 0 andx >ȳ. By contradiction, assume first thatt = 0. Then we have where we used that u 0 is non-increasing, and we get a contradiction. The fact thatx >ȳ, comes 16 directly from the fact that ϕ(t,x,ȳ) > 0.

17
Step 3: utilisation of the equation. By doing a duplication of the time variable and 18 passing to the limit in this duplication parameter, we get that passing to the limit as α goes to 0, we obtain a contradiction.

20
Proof of the lower inequality for the control of the space oscillations Let us introduce, We want to prove that M ≤ 0. We argue by contradiction and assume that M > 0.

10
Step 3: Utilisation of the equation By duplicating the time variable and passing to 11 the limit we have that there exists two real numbers a, b, such that (a, −k 0 + 2αȳ) ∈ D + u(t,ȳ), (3.8) Using that u is a sub-solution of (2.4)-(2.7) (with ε = 1), we get Ifȳ + z <x, using the fact that ϕ(t,x,ȳ + z) ≤ ϕ(t,x,ȳ), for α small enough, we obtain Injecting this in the non-local term, we deduce the claim.
Finally, using (3.8), we obtain which is a contradiction. This ends the proof. The key ingredient to prove the convergence result is to construct correctors for the junction. The then (along a subsequence ε n → 0) we have that w ε converges locally uniformly towards a function In particular, we have (with This section contains the proof of the main homogenization result (Theorem 2.6). This proof relies 2 on the existences of correctors (Proposition 2.13 and Theorem 4.1).

3
We begin with two useful lemmas for the proof of Theorem 2.6. The first result is a direct 4 consequence of Perron's method and Lemma 3.7.
The following lemma is a direct result of Theorem 3.10. 10 satisfies for all t > 0, for all x, y ∈ R, x ≥ y, Before passing to the proof of Theorem 2.6, let us show how it allows us to prove Theorem 13 2.10.
14 Proof of Theorem 2.10. We want to prove that for all t ∈ [0, +∞) and for all x, y ∈ R, x ≥ y, Using Lemma 5.2, we have that the solution u ε of (2.4)-(2.7), satisfies for all (t, x, y) Now using Theorem 2.6, passing to the limit as ε → 0, we obtain the result.

19
We now turn to the proof of Theorem 2.6.

20
Proof of Theorem 2.6. We introduce Thanks to Lemma 5.1, we know that these functions are well defined. We want to prove that u and u are respectively a sub-solution and a super-solution of (2.13). In this case, the comparison 23 principle [18, Theorem 1.4] will imply that u ≤ u. But, by construction, we have u ≤ u, hence we 24 will get u = u = u 0 , the unique solution of (2.13).

25
Let us prove that u is a sub-solution of (2.13) (the proof for u is similar and we skip it). We 26 argue by contradiction and assume that there exists a test function ϕ ∈ C 1 (J ∞ ) (in the sense of 27 Definition 3.3), and a point (t,x) ∈ (0, +∞) × R such that outside Qr ,r (t,x) with η > 0 ϕ t (t,x) + H(x, ϕ x (t,x)) = θ with θ > 0, Given Lemma 5.2 and (5.3), we can assume (up to changing ϕ at infinity) that for ε small enough, Using the previous lemmas we get that the function u satisfies for all t > 0 and x, y ∈ R, x ≥ y, (5.5) First case:x = 0. We only considerx > 0, since the other case (x < 0) is treated in the same 5 way. We define p = ϕ x (t,x) that according to (5.5) satisfies We chooser small enough so thatx − 2r > 0. Let us prove that the test function ϕ satisfies 7 in the viscosity sense, the inequality Let us notice that for ε small enough we have For all (t, x) ∈ Qr ,r (t,x), we have forr small enough where we have used (5.4). We recall that for −k 0 ≤ p ≤ 0, Moreover, for all z ∈ [h 0 , h max ], and for ε andr small enough we have that 12 where we have used the fact that ϕ ∈ C 2 and that z ∈ [h 0 , h max ]. Now using the fact thatẼ is 13 decreasing we have Using this result and replacing the non-local operators in ( (1) Injecting (5.9) in (5.8) and choosing ε andr, we obtain where we have used assumption (A1) for the second line.
2 Getting a contradiction. By definition, we have for ε small enough, Using the comparison principle on bounded subsets for (2.4) (Theorem 3.6), we get 4 u ε ≤ ϕ − η on Qr ,r (t,x).
Passing to the limit as ε → 0, we get u ≤ ϕ − η on Qr ,r (t,x) and this contradicts the fact that Second case:x = 0. Using Theorem 3.9, we may assume that the test function has the following Let us consider the solution w of (2.21) provided by Theorem 4.1, and let us denote by We would like to prove that this function satisfies in the viscosity sense, forr and ε small 11 enough, Let h be a test function touching ϕ ε from below at (t 1 , x 1 ) ∈ Qr ,r (t, 0), so we have and 14 w(s, y) ≥ 1 ε (h(εs, εy) − g(εs)) , for (s, y) in a neighbourhood of This implies that (using (5.11) and takingr small enough) Now for ε small enough such that εh max ≤r, we deduce from the previous inequality and using 3 the fact thatM is a non-local operator with a bounded support, that we have Getting the contradiction. We have that for ε small enough Using the fact that w ε → W , and using (4.4), we have for ε small enough 6 u ε + η 2 ≤ ϕ ε on Q 2r,2r (t, 0)\Qr ,r (t, 0).
Combining this with (5.12), we get that By the comparison principle on bounded subsets (Theorem 3.6) the previous inequality holds in 8 Qr ,r (t, 0). Passing to the limit as ε → 0 and evaluating the inequality in (t, 0), we obtain 9 u(t, 0) + η 2 ≤ ϕ(t, 0) = u(t, 0), which is a contradiction. . The difficulty in our non-local case is that it is non-standard to well define 15 boundary conditions. In order to overcome this difficulty, we will replace the non-local operator 16 by a local one near the boundary. More precisely, for l ∈ (r, +∞), r << l and r ≤ R << l, we 17 want to find λ l,R , such that there exists a solution w l,R of To G R , we associateG R which is defined in the same way but the operator M is replaced byM .

21
Remark 6.1. The operator G R is used to have a local operator near the boundary and then to 1 well define the boundary conditions. 2 6.1 Comparison principle for a truncated problem 3 Proposition 6.2 (Comparison principle on truncated domains). Let us consider the following 4 problem for r < l 1 < l 2 and λ ∈ R, with and l 2 >> R.

Existence of correctors on a truncated domain
with H 0 = min H. 18 Proof. In order to construct a corrector on the truncated domain, we will classically consider the 19 approximated problem Step 1: construction of barriers. Using that 0 and δ −1 C 0 are respectively sub and super-21 solution of (6.7) with C 0 = |H 0 |, the comparison principle and Perron's method for 1-periodic 22 solutions, we deduce that there exists a continuous viscosity solution, v δ of (6.7) which satisfies Lemma 6.5. The function v δ satisfies for all t ∈ R and for all x, y ∈ [−l, l], x ≥ y, Proof of Lemma 6.5. In the rest of the proof we will use the following notation, Step 2.1: proof of the upper inequality. Let ε > 0. We want to prove that We argue by contradiction and assume that M > 0. We then consider Since M > 0, we deduce that M ν > 0. Remark also that we consider the supremum of a continuous, ). Therefore, we can use the viscosity inequalities for (6.7).
14 -Ifx ∈ (−l, l) andȳ = −l, we obtain where we used the fact that H For every value ofx andȳ we obtain a contradiction, therefore we have M ≤ 0.

18
Step 2.2: proof of the lower inequality. We want to prove that We argue by contradiction and assume that M > 0. We then consider Since M > 0, we get M ν > 0. Remark also that we consider the supremum of a continuous, Therefore, we can use the viscosity inequalities for (6.7).
Moreover, whetherx ∈ (−l, l) orx = l, since the non-local operator is negative and H + (−k 0 ) < 0, 15 we have that We deduce that which is a contradiction.

18
Case 2:ȳ = −l. In this situation, the viscosity inequality becomes Using the fact that H − (−k 0 ) = H(−k 0 ) = 0, and as in the previous case, we obtain a contradiction.

20
This ends the proof of the lemma.

22
Step 3: control of the time oscillations of v δ . 1 Lemma 6.6. The function v δ satisfies for all x ∈ [−l, l] and for all t, s ∈ R, Proof.
Since v δ is 1-periodic in t, it is sufficient to show that for all x ∈ [−l, l] and for all t, s ∈ R 4 such that t ≥ s, we have that In order to prove that, we will fix x 0 ∈ (−l, l) and s 0 ∈ R, and we will prove Using the space oscillation of v δ , we have that v δ (s 0 , x) ≤ w δ (s 0 , x). On the other hand, we can 8 check that w δ is a super solution of (6.7) on (s 0 , +∞) × [−l, l] using that Finally, using the comparaison principle on [s 0 , +∞) × [−l, l], we deduce that In particular, for x = x 0 , we obtain (6.11). We deduce that (6.10) is true even if x = ±l because 11 v δ is continuous. The proof is now complete.

12
Step 4: construction of a Lipschitz estimate.

13
Lemma 6.7. There exists a Lipschitz continuous function m δ , such that there exists a constant 14 C, (independent of l, R and δ) such that Proof of Lemma 6.7. Let us define m δ as an affine function in each interval of the form Since m δ , v δ (0, ·) are non-increasing and |v δ (0, and for all x, y ∈ [−l, l], Using the time oscillations of v δ , we deduce that Step 4: passing to the limit as δ goes to 0. Using (6.8) and (6.12), we deduce that there 1 exists δ n → 0 such that 2 δ n v δn (0, 0) → −λ l,R as n → +∞, m δn − m δn (0) → m l,R as n → +∞, the second convergence being locally uniform. Let us consider, v δn (0, 0)). Therefore, we have that λ l,R , m l,R , w l,R and w l,R satisfy (6.13) By stability of the solutions we have that w l,R − 2C and w l,R are respectively a sub-solution and 5 a super-solution of (6.1) and By Perron's method we can construct a solution w l,R of (6.1) and thanks to (6.8) and (6.13), m l,R , 7 λ l,R and w l,R satisfy (6.6).

8
The uniqueness of λ l,R is classical so we skip it. This ends the proof of Proposition 6.4. (6.14) Moreover, we have 14 Proof. This results comes from the fact that we have the following bound on λ l,R which is inde- Similarly, for all x ≤ −r and h ≥ 0, Proof. We only prove (6.15) since the proof for (6.16) is similar. For µ > 0 small enough, we 1 denote by p µ + the real number such that Using that we deduce that p µ + exists, is unique and satisfies −k 0 ≤ p µ + ≤ 0 for µ small enough.

4
Let us now consider the function w + = p µ + x that satisfies We also have For all x ∈ (r, l), using that φ(t, x) = 1, we deduce that and so the restriction of w + to (r, l] satisfies Let us denote by g(t, x) = w l,R (t, x) − w l,R (0, x 0 ) and u(x) = w + (x) − w + (x 0 ) − C, for some 8 x 0 ∈ (r, l) and C defined as in Proposition 6.4. Then we have 9 g(t, x 0 ) ≥ −C = u(x 0 ).
Using that g is a solution of (6.4) and u is a solution of (6.5) (with ε 0 = µ) joint to the comparison 10 principle (Proposition 6.2) we get that This implies that for all h ≥ 0 and for all x ∈ (r, l), Finally, if we choose γ 0 < |p 0 −p + | (with p 0 defined in (2.12)), then and we can choose µ > 0 such that This implies inequality (6.15).

16
Proof of Theorem 4.1. The proof is performed two steps.

17
Step 1: proof of i) and ii). The goal is to pass to the limit as l → +∞ and then as R → +∞. 1 Using Proposition 6.4, there exists l n → +∞, such that 2 m ln,R − m ln,R (0) → m R as n → +∞, the convergence being locally uniform. We also define Thanks to (6.6), we know that w R and w R are finite and satisfy By stability of viscosity solutions, w R − 2C and w R are respectively a sub and a super-solution of Therefore, using Perron's method, we can construct a solution w R of (6.17) with m R , A R and w R (6.18) Using Proposition 6.10, if A > H 0 , we know that there exists a γ 0 and a constant C, such that for 8 all γ ∈ (0, γ 0 ), We now pass to the limit as R → +∞. We consider (up to some subsequence) The last convergence being locally uniform. Thanks to (6.18), we know that w and w are finite 11 and satisfy By stability of viscosity solutions, w − 2C and w are respectively a sub and a super-solution of 13 (2.21) with λ = A. Using Perron's method, we can then construct a solution w of (2.21) with Step 2: proof of iii). We are now interested in the rescaled function w ε (t, x) = εw Using (4.2), we have that Therefore, we can find a sequence ε n → 0, such that 3 w εn → W locally uniformly as n → +∞, with W (0) = 0. Like in [18], arguing as in the proof of convergence away from the junction point, For all γ ∈ (0, γ 0 ), we have that if A > H 0 and x > 0, where we have used (4.2). Therefore we get this result remains valid even if A = H 0 (in this particular case W x = p 0 ). Similarly, we get which implies (4.3) and (4.4). This ends the proof of Theorem 4.1. We argue by contradiction and assume that there exists a λ < A and a function w λ ∈ S solution 15 of (2.21). We assume that w λ (0, 0) = 0 (if we are not in this situation, we do a translation since 16 we have w λ − w λ (0, 0) ∈ S). Arguing as in the proof of Theorem 4.1, we deduce that the function which means that for all x > 0, (6.20) Similarly we have for all x < 0, These inequalities imply that for all γ > 0, there exists a constantC γ > 0 such that for all x ∈ R, we have Let us now introduce, u(t, x) = w(t, x) + CR − At and u λ (t, x) = w λ (t, x) − λt both solutions of 6 (2.4) with ε = 1 and u λ (0, x) ≤ u(0, x). Therefore, the comparison principle implies Dividing by t and passing to the limit as t goes to infinity, we get which is a contradiction. This section is devoted to the proof of Proposition 2.16.

12
Proof of Proposition 2.16. In order to establish the monotonicity, we have to consider the ap-13 proximated truncated cell problem (6.7). Let us consider v δ 1 and v δ 2 viscosity solutions of (6.7), 14 respectively for φ 1 and φ 2 , with 0 ≤ φ 1 ≤ φ 2 . First, using the fact that the non-local operator is 15 negative, we have Therefore, we have meaning that v δ 1 is a sub-solution of (6.7) with φ 2 . The comparison principle and (6.8) imply that Passing to the limit as δ → 0, we obtain 20 0 ≥ λ 1 l,R ≥ λ 2 l,R ≥ H 0 .
Passing to the limit as l, R → +∞, we get the result.

Link between the system of ODEs and the PDE
Conversely, if u is a bounded and continuous viscosity solution of (8.1) satisfying for some time 6 T > 0, and for all t ∈ (0, T ) then the points U j (t), defined by u(t, U j (t)) = −(j + 1) for j ∈ Z, satisfy the system (2.1) on Before giving the proof of Theorem 8.1, let us do the proof of Theorem 2.7.

10
Proof of Theorem 2.7. We recall that in Theorem 2.7, we have u 0 ( we would like to prove that for all ε > 0, we have with f (ε) → 0 as ε goes to 0. To do this, we define a piece-wise affine function v satisfying Given that for all U i+1 (0) − U i (0) ≥ h 0 , we notice that v is k 0 -Lipschitz continuous and by 14 definition of ρ 1 (0, x), we have Let us consider the integer i 0 ∈ N defined by Using the assumption that for all i ∈ Z such that Let us now consider the integer i 1 ∈ N defined by

Now using the assumption that for all
Moreover, we recall that for all ε > 0, we have ρ ε (0, x) = ερ 1 (0, x/ε), this implies that for all where we have used the fact that εu 0 (x/ε) = u 0 (x). Combining (8.3) and (8.4) and choosing we deduce (8.2). Notice also that thanks to (8.2), we have Using the fact that ρ ε is a viscosity solution of (2.4) and the comparison principle (Proposition 3 3.5) we deduce that (with u ε the continuous solution of (2.4) associated to the initial condition where we have used the fact that (2.4) is invariant by addition of constants to the solutions.

6
Passing to the limit as ε → 0 and using Theorem 2.6 we get that ρ ε → u 0 , which ends the proof 7 of Theorem 2.7.  where E and J are defined in (2.6) and u(t, x) is a continuous function such that 13 u(t, x) = ρ * (t, x) = ρ(t, x) for x = U j (t), j ∈ Z, u is decreasing in x, (8.8) with ρ defined in (2.2) (with ε = 1).
14 Proof. We drop the time dependence to simplify the presentation. Let j ∈ Z. Using the fact that 15 u(U j ) = −(j + 1) and (8.8), we have for all z ∈ [0, +∞), Given that u is continuous, this implies that Combining this result with (2.1), we obtain (8.7).
Noticing that because of (8.8), we have for x = U j (t), j ∈ Z,

1M
[ρ * (t, ·)](x) =M [u(t, ·)](x) = M [u(t, ·)](x), and using Lemma 8.2, and Definition 3.1, we can see that ρ * is a discontinuous viscosity super-2 solution of (8.1). We obtain a similar result for ρ * , therefore, ρ is a discontinuous viscosity solution 3 of (8.1). 4 We prove the converse. For the readers convenience we recall Proposition 4.8 from [10] that 5 we will use later. The proof of this proposition remains almost the same in our case the only 6 difference being the definition of the functions E andẼ. Assume that ε = 1 in (2.4). Consider also a sub-solution (resp. a super-solution) u of (2.4). Then 10 θ(u) is also a sub-solution (resp. a super-solution) of (2.4).
This implies that This proves that U i are viscosity super-solutions of (2.1). The proof for sub-solutions is similar 22 and we skip it. Moreover, sincec i is continuous, we deduce that U i ∈ C 1 and it is therefore a 23 classical solution of (2.1).

26
The authors would like to thank R. Monneau for fruitful discussion during the preparation 27 of this paper. This project was co-financed by the European Union with the European regional 28 development fund (ERDF, HN0002137) and by the Normandie Regional Council via the M2NUM 29 project and by ANR HJNet (ANR-12-BS01-0008-01).