REGULARITY CRITERIA FOR STRONG SOLUTIONS TO THE 3D NAVIER-STOKES EQUATIONS

Keywords: 2010 Mathematics Subject Classification. 35Q30, 35D35, 35B44 Navier-Stokes equations. Strong solutions. Blow-up. 1

.

Introduction

Two of the profound open problems in the theory of three dimensional viscous flows are the unique solvability theorem for all time and the regularity of solutions. For the three-dimensional Navier-Stokes system weak solutions are known to exist by a basic result by Leray from 1934 [START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF], but the uniqueness is still open problem [START_REF] Cao | Regularity criteria for the three dimensional Navier-Stokes equations[END_REF]- [START_REF] Doering | The 3D Navier-Stokes problem[END_REF] and [START_REF] Kasyanov | A criterion for the existence of strong solutions for the 3D Navier-Stokes equations[END_REF]. Furthermore, the strong solutions for the 3D Navier-Stokes equations are unique and can be shown to exist on a certain finite time interval for small initial data and small forcing term, but the global regularity for the 3D Navier-Stokes is still open problems (see [START_REF] Deuring | Strong solutions of the Navier-Stokes system in Lipschitz bounded domains[END_REF]- [START_REF] Kasyanov | A criterion for the existence of strong solutions for the 3D Navier-Stokes equations[END_REF], [START_REF] Sell | Dynamics of Evolutionary Equations[END_REF]- [START_REF] Wang | On global regularity of incompressible Navier-Stokes equations in R 3[END_REF] and references therein). In 1933 [START_REF] Leray | Etude de diverses équations intégrales non linéaires et de quelques problemes que pose l'hydrodynamique[END_REF], Leray showed that in the absence of forcing (f = 0), all solutions of Navier-Stokes equations are eventually smooth (i.e. after some T * > 0 depending on the data). Kato and Fujita [START_REF] Kato | On the nonstationary Navier-Stokes system[END_REF] showed that a smooth solution to the three-dimensional Navier-Stokes equations exists for all time if f is small in some sense and u 0 is small in H 1 2 . In this paper, we give a new condition for global existence in time for strong solution for 3D Navier-Stokes equations with external force. We show that no singularity can occur in finite time for a large class of forcing term. We also give an extension of the time interval of regularity to the 3D Navier-Stokes equations with negligible forces [START_REF] Doering | The 3D Navier-Stokes problem[END_REF][START_REF] Foias | Navier-Stokes Equations and Turbulence[END_REF][START_REF] Lions | Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires[END_REF][START_REF] Temam | Navier-Stokes equations[END_REF]. This result means that the solution does not blow up at T * .

Notations and preliminaries

In this section we introduce notations and the definitions of standard functional spaces that will be used throughout the paper. We denote by H m per (Ω), the Sobolev space of periodic functions. These spaces are endowed with the inner product

(u, v) = |β|≤m (D β u, D β v) L 2 (Ω) and the norm u m = |β|≤m ( D β u 2 L 2 (Ω) ) 1 2 .
We define the spaces V m as completions of smooth, divergence-free, periodic, zeroaverage functions with respect to the H m per norms. V ′ m denote the dual space of V m .

Let P be the orthogonal projection in L 2 per R 3 3 with the range V 0 . Let A = -P △ the Stokes operator. It is easy to check that Au = -△u for every u ∈ D (A). We recall that the operator A is a closed positive self-adjoint unbounded operator.

The eigenvalues of A are {λ j } j=∞ j=1 , 0 < λ 1 ≤ λ 2 ≤ ...and the corresponding orthonormal set of eigenfunctions {w j } j=∞ j=1 is complete in V 0 Aw j = λ j w j , w j ∈ D(A), ∀j.

(2.1)

Let us now define the trilinear form b(., ., .) associated with the inertia terms

b (u, v, w) = 3 i,j=1 Ω u i ∂v j ∂x i w j dx. (2.2)
The continuity property of the trilinear form enables us to define (using Riesz representation Theorem) a bilinear continuous operator

B (u, v); V 2 × V 2 → V ′ 2 will be defined by B (u, v) , w = b (u, v, w) , ∀w ∈ V 2 . (2.3)
Recall that for u satisfying ∇.u = 0 we have

b (u, u, u) = 0 and b (u, v, w) = -b (u, w, v) . (2.4)
We recall some well known inequalities that we will be using in what follows.

Young's inequality

ab ≤ σ p a p + 1 qσ q p b q , a, b, σ > 0, p > 1, q = p p -1 . (2.5)
Poincaré's inequality

λ 1 u 2 ≤ A 1 2 u 2 for all u ∈ V 0 , (2.6)
λ 1 is the first eigenvalue of the Stokes operator.

Navier-Stokes equations

The conventional Navier-Stokes system can be written in the evolution form

∂u ∂t -ν△u + u.∇u = f, t > 0, div u = 0, in Ω × (0, ∞) and u (x, 0) = u 0 , in Ω. (3.1)
We recall that a Leray weak solution of the Navier-Stokes equations is a solution which is bounded and weakly continuous in the space of periodic divergence-free L 2 functions, whose gradient is square-integrable in space and time and which satisfies the energy inequality. The proof of the following theorem is given in [START_REF] Temam | Navier-Stokes equations[END_REF].

Theorem 3.1. Assume that f ∈ L 2 (0, T ; V ′ 1 ) and u 0 ∈ V 0 are given. Then there exists at least one solution u of (3.1) such that u ∈ L 2 (0, T ;

V 1 ) ∩ L ∞ (0, T ; V 0 ) , ∀T > 0.
For strong solutions, we have the following result [START_REF] Temam | Navier-Stokes equations[END_REF].

Theorem 3.2. Assume that u 0 ∈ V 1 and f ∈ V 0 are given. then there exists a T > 0 depending on u 0 1 , ν and f , such that there exists a unique strong solution u ∈ L ∞ (0, T ;

V 1 ) ∩ L 2 (0, T ; V 2 ).
This result was obtained for a type of inequality similar to u (., t)

2 1 ≤ 1 + u 0 2 1 1 -Kt (1 + u 0 1 ) 2 , (3.2)
where K = (2

f 2 ν + c 1 ν 3 ).
Hereafter, c i ∈ N ,will denote a dimensionless scale invariant positive constant which might depend on the shape of the domain. The bound in (3.2) is only finite while

Kt (1 + u 0 1 ) 2 < 1; (3.3)
if we choose T satisfying

T < 1 K 1 + u 0 2 1 . (3.4)
The main result of this paper is given in the following theorem.

Theorem 3.3. Assume that u 0 ∈ V 1 and u is the corresponding strong solution to

(3.1) on [0, T ], then i) If f ∈ V 0 and c 8 T f 2 + c 9 u (0) 2 + arctan u (0) 1 < π 2 , (3.5)
then u exists for each finite time T and remains smooth. ii) If f ∈ L 2 (0, T, V 0 ) and

c 10 T 0 f 2 ds + c 11 u (0) 2 + arctan u (0) 2 1 + arctan u (0) 2 1 < π 2 , (3.6)
then u exists globally (T can be ∞) and remains smooth.

Proof. Multiplying (3.1) by △u, we have

1 2 d dt u (., t) 2 1 + ν △u 2 - Ω (u.∇u) .△u dx = (f, △u) . (3.7)
Using schwartz and Young inequality we get

|(f, △u)| ≤ f L 2 △u L 2 (3.8) ≤ c 3 f 2 L 2 + ν 2 △u 2 L 2 .
For the nonlinear term, we use the Hölder's inequality For f ∈ V 0 , the inequality above gives

Ω (u.∇u) .△udx ≤ c 4 u L 6 ∇u L 3 △u L 2 ≤ c 5 ∇u 3 2 △u 3 2 . ( 3
arctan y (t) ≤ c 8 T f 2 + c 9 u (0) 2 + arctan y (0) . (3.19)
The function tan y is increasing and invertible for -π 2 < y < π 2 with inverse function arctan t. Thus, for

c 8 T f 2 + c 9 u (0) 2 + arctan u (0) 2 1 < π 2 (3.20)
we apply the function tan on (3.19) to get

u (t) 2 1 ≤ tan c 8 T f 2 + c 9 u (0) 2 + arctan u (0) 2 1 . (3.21)
The assumption (3.20) on the initial data guarantees that the right hand side of (3.21) is finite. For f ∈ L 2 (0, T, V 0 ) and for each T > 0, integrate (3.

In particular, for

c 10 T 0 f 2 ds + c 11 u (0) 2 + arctan u (0) 2 1 < π 2 , (3.24)
we apply the function tan on (3.23) to get It follows that the assumption (3.24) guarantees that u (t) 1 < ∞ for all t > 0.

u (t)
Recall that the classical regularity result for f = 0 [13, Theorem 3.12.] was obtained for a type of inequality similar to u (., t) (3.26)

It follows that if u 0 1 is finite, then u (., t) 1 is finite, at least for

t < ν 3 /128 u 0 4 1 . (3.27)
Consequently, we get the following result for the negligible forces.

Corollary 3.4. Assume that u 0 ∈ V 1 and u is the corresponding strong solution to (3.1) on [0, T ], then u exists globally and remains smooth for all T > 0 if Thus the solution associate to u 0 satisfies (3.37) has a global regularity. But for the same value of u (0) 1 occurs a blow up in finite time T * by the usual method (3.26). This property follows easily when u (0) approaches zero. This result gives a simple condition for global regularity and extends the known corresponding result (3.26), where a blow-up criterion in finite time T depend on u 0 for negligible forces, see [START_REF] Doering | The 3D Navier-Stokes problem[END_REF][START_REF] Foias | Navier-Stokes Equations and Turbulence[END_REF][START_REF] Lions | Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires[END_REF][START_REF] Temam | Navier-Stokes equations[END_REF].

c 11 u (0) 2 + arctan u (0) 2 1 < π 2 . ( 3

3 T 0 f 2 ds + c 6 T 0 y 0 f 2

 3026002 14) to get arctan y (t) -arctan y (0) ≤ c (s) ds. (3.22) Therefore, (3.18) implies arctan y (t) ≤ c 10 T ds + c 11 u (0) 2 + arctan y (0) .

2 1 ≤ tan c 10 T 0 f 2

 21002 ds + c 11 u (0) 2 + arctan u (0)

  Since the condition (3.36) is independent of time we get global estimates for u (., t) 1 by this method. An important consequence of this result is that for each finite time T * such that T * < ν 3 /128 u 0

	which is finite thanks to the following assumption
		c 11 u (0) 2 + arctan u (0) 2 1 <	π 2	(3.36)
	and this concludes the proof.
						4 1 ,	(3.37)
	there is a u 0 satisfies
	c 11 u (0)	2 + arctan ν 3 /128T * <	π 2	.	(3.38)
						.28)
	Proof. From (3.11) we get for f = 0
			d dt	u 2 1 ≤ c 6 u 6 1 .	(3.29)
	In particular, we have
	d dt	y ≤ c 2 1 .	(3.30)
	Dividing (3.30) by 1 + y 2 yields
					d dt y 1 + y 2 ≤ c 6 y.	(3.31)
	Integrate this over [0, T ] to get
					T
	arctan y (t) ≤ c 6	y (s) ds + arctan y (0) .	(3.32)
					0
	From (3.18), we find
		T	y (s) ds ≤ c 9 u (0)	2	(3.33)
		0
	this implies that (3.32) is equivalent to
	arctan y (t) ≤ c 11 u (0) 2 + arctan y (0) .	(3.34)
	Now, applying the function tan on (3.34) to get
	u (t)				2 1 ,	(3.35)

6 y 1 + y 2 with y (t) = u (., t) 2 1 ≤ tan c 11 u (0)

2 + arctan u (0)

Suppose first f ∈ V 0 . Setting y (t) = u (., t) Dividing (3.12) by 1 + y 2 , we have

≤ 1 and


Guillaume Costeseque

Jean-Patrick Lebacque

J-P Lebacque

R Monneau

A convergent scheme

Keywords: Hamilton-Jacobi equations, junctions, viscosity solutions, numerical scheme, traffic problems. MSC Classification: 65M12, 65M06, 35F21, 90B20

Introduction

The main goal of this paper is to prove properties of a numerical scheme to solve Hamilton-Jacobi (HJ) equations posed on a junction. We also propose a traffic application that can be directly found in Section 4.

Setting of the PDE problem

In this subsection, we first define the junction, then the space of functions on the junction and finally the Hamilton-Jacobi equations. We follow [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF].

The junction. Let us consider N ≥ 1 different unit vectors e α ∈ R 2 for α = 1, . . . , N . We define the branches as the half-lines generated by these unit vectors J α = [0, +∞)e α and J * α = J α \ {0 R 2 }, for all α = 1, . . . , N, and the whole junction (see Figure 1) as J = α=1,...,N J α .

The origin y = 0 R 2 (we just call it "y = 0" in the following) is called the junction point. For a time T > 0, we also consider the time-space domain defined as J T = (0, T ) × J. Space of test functions. For a function u : J T → R, we denote by u α the "restriction" of u to (0, T ) × J α defined as follows for x ≥ 0 u α (t, x) := u(t, xe α ).

Then we define the natural space of functions on the junction:

(1.1) C 1 * (J T ) = u ∈ C(J T ), u α ∈ C 1 ((0, T ) × [0, +∞)) for α = 1, . . . , N . In particular for u ∈ C 1 * (J T ) and y = xe α with x ≥ 0, we define u t (t, y) = u α t (t, x) = ∂u α ∂t (t, x) and u α x (t, x) = ∂u α ∂x (t, x).

HJ equation on the junction. We are interested in continuous functions u : [0, T ) × J → R which are viscosity solutions (see Definition 3.3) on J T of (1.2) for functions H α and H - α that will be defined below in assumption (A1). We consider an initial condition (1.3) u α (0, x) = u α 0 (x), with x ∈ [0, +∞) for α = 1, . . . , N. We make the following assumptions: (A0) Initial data The initial data u 0 := (u α 0 ) α is globally Lipschitz continuous on J, i.e. each associated u α 0 is Lipschitz continuous on [0, +∞) and u α 0 (0) = u β 0 (0) for any α = β.

               u α t + H α (u α x ) =

(A1) Hamiltonians

For each α = 1, ..., N ,

• we consider functions H α ∈ C 1 (R; R) which are coercive, i.e. lim |p|→+∞ H α (p) = +∞;

• we assume that there exists a p α 0 ∈ R such that H α is non-increasing on (-∞, p α 0 ] and non-decreasing on [p α 0 , +∞), and we set:

(1.4)

H - α (p) =      H α (p) for p ≤ p α 0 H α (p α 0 ) for p ≥ p α 0 and H + α (p) =      H α (p α 0 ) for p ≤ p α 0 H α (p) for p ≥ p α 0
where H - α is non-increasing and H + α is non-decreasing. Remark 1.1 Assumption (A1) allows the Hamiltonians H α to have plateaus, in particular at the minimum of H α . In such a case the value p α 0 is not unique.

Presentation of the scheme

We denote by ∆x the space step and by ∆t the time step. We denote by U α,n i an approximation of u α (n∆t, i∆x) for n ∈ N, i ∈ N, where α stands for the index of the considered branch. We define the discrete space derivatives Then we consider the following numerical scheme corresponding to the discretization of the HJ equation (1.2) for n ≥ 0:

(1.7) It is natural to introduce the following Courant-Friedrichs-Lewy (CFL) condition [START_REF] Courant | On the partial difference equations of mathematical physics[END_REF]:

                     U α,n+1 i -U α,n i ∆t + max H + α (p α,n i,-), H - α (p α,n i,+ ) =
(1.9) ∆x ∆t ≥ sup α=1,...,N i≥0, 0≤n≤nT

|H ′ α (p α,n i,+ )|
where the integer n T is assumed to be defined as n T = T ∆t for a given T > 0.

We then have Proposition 1.2 (Monotonicity of the numerical scheme) Let U n := (U α,n i ) α,i and V n := (V α,n i ) α,i two solutions of (1.7). If the CFL condition (1.9) is satisfied and if U 0 ≤ V 0 , then the numerical scheme (1.7) is monotone, that is U n ≤ V n for any n ∈ {0, ..., n T }.

Our scheme (1.7) is related to the Godunov scheme for conservation laws in one space dimension, as it is explained in our application to traffic in Section 4.

Main results

We first notice that even if we can always check a posteriori the CFL condition (1.9), it is not obvious to satisfy it a priori. Indeed the CFL condition (1.9) depends on the discrete gradients p α,n i,± which are themselves functions of ∆t through the scheme (1.7). For this reason, we will consider below a more restrictive CFL condition (see (1.12)) that can be checked from the initial data. To this end, we need to introduce a few notations.

For sake of clarity we first consider σ ∈ {+1, -1} denoted by abuse of notation σ ∈ {+, -} in the remaining, with the convention -σ =if σ = + and -σ = + if σ = -.

Under assumption (A1), we need to use a sort of inverse of (H ± α ) that we define naturally for σ ∈ {+, -} as:

(1.10) (H -σ α ) -1 (a) := σ inf{σp, H -σ α (p) = a} with the additional convention that (H ± α ) -1 (+∞) = ±∞. We set (1.11)

     p α = (H - α ) -1 (-m 0 ) p α = (H + α ) -1 (-m 0 ) with m 0 = inf β=1,...,N, i∈N W β,0 i
where (W β,0 i ) β,i , defined in (1.6), is given by the scheme (1.7) for n = 0 in terms of (U β,0 i ) β,i (itself defined in (1.8)). It is important to notice that with this construction, p α and p α depend on ∆x, but not on ∆t. We now consider another CFL condition which turns out to be more restrictive than CFL condition (1.9) (see Theorem 1.3). This condition is given by

(1.12) ∆x ∆t ≥ sup α=1,...,N pα∈[p α ,p α ] |H ′ α (p α )|
which is then satisfied for ∆t small enough. Note that by construction we have

-∞ < p α ≤ p α < +∞ because -m 0 ≥ max α=1,...,N min R H α (see also Remark 2.4 (i)).
Our first main result is the following:

Theorem 1.3 (Gradient and time derivative estimates) Assume (A1). If (U α,n i
) is the numerical solution of (1.7)-(1.8) and if the CFL condition (1.12) is satisfied with m 0 finite, then the following two properties hold for any n ≥ 0:

(i) For p α and p α defined in (1.11), we have the following gradient estimate:

(1.13) p α ≤ p α,n i,+ ≤ p α , for all i ≥ 0, and α = 1, ..., N.

(ii) Considering M n = sup α,i W α,n i and m n = inf α,i W α,n i
, we have the following time derivative estimate:

(1.14) m 0 ≤ m n ≤ m n+1 ≤ M n+1 ≤ M n ≤ M 0 .
Remark 1.4 Notice that due to (1.13), the more restrictive CFL condition (1.12) implies the natural CFL condition (1.9) for any n T ≥ 0.

Our second main result is the following convergence result which also gives the existence of a solution to equations (1.2)-(1.3).

Theorem 1.5 (Convergence of the numerical solution up to a subsequence) Assume (A0)-(A1). Let T > 0 and ε := (∆t, ∆x)

such that the CFL condition (1.12) is satisfied. Then there exists a subsequence ε ′ of ε such that the numerical solution (U α,n i ) of (1.7)-(1.8) converges as ε ′ goes to zero, locally uniformly on any compact set

K ⊂ [0, T ) × J, towards a solution u := (u α ) α of (1.2)-(1.
3) in the sense of Definition 3.3, i.e.

(1.15) lim sup

ε ′ →0 sup (n∆t,i∆x)∈K |u α (n∆t, i∆x) -U α,n i | = 0, where the index α in (1.15) is chosen such that (n∆t, i∆x) ∈ K ∩ [0, T ) × J α .
In order to give below sharp Lipschitz estimates on the continuous solution u, we first define L α,-and L α,+ as the best Lipschitz constants for the initial data u α 0 , i.e. satisfying for any x ≥ 0 and a ≥ 0

(1.16) aL α,-≤ u α 0 (x + a) -u α 0 (x) ≤ aL α,+ .

Let us consider

(1.17)

       m 0 0 := inf α=1,...,N L α,-≤pα≤L α,+ -H α (p α ), M 0 0 := max max α=1,...,N -max σ∈{+,-} H -σ α (L α,σ ) , -max α=1,...,N H - α (L α,+ ) ,

and

(1.18)

     p 0 α := (H - α ) -1 (-m 0 0 ), p 0 α := (H + α ) -1 (-m 0 0 ).
Corollary 1.6 (Gradient and time derivative estimates for a continuous solution) Assume (A0)-(A1). Let T > 0 and u := (u α ) α be a solution of (1.2)-(1.3) constructed in Theorem 1.5. Then for all a ≥ 0, for all 0 ≤ t ≤ T and x ≥ 0, the function u satisfies the following properties:

(1.19)      am 0 0 ≤ u α (t + a, x) -u α (t, x) ≤ aM 0 0 , ap 0 α ≤ u α (t, x + a) -u α (t, x) ≤ ap 0 α ,
where m 0 0 , M 0 0 , p 0 α and p 0 α are defined in (1.17) and (1.18). Recall that under the general assumptions of Theorem 1.5, i.e. (A0)-(A1), the uniqueness of a solution u of (1.2)-(1.3) is not known. If we replace condition (A1) by a stronger assumption (A1') below, it is possible to recover the uniqueness of the solution (see [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF] and Theorem 1.7 below). This is the following assumption:

(A1') Strong convexity

There exists a constant γ > 0, such that for each α = 1, ..., N , there exists a lagrangian function

L α ∈ C 2 (R; R) satisfying L ′′ α ≥ γ > 0 such that H α is the Legendre-Fenchel transform of L α , i.e. (1.20) H α (p) = L * α (p) = sup q∈R (pq -L α (q))
and

(1.21) H - α (p) = sup q≤0 (pq -L α (q)) and H + α (p) = sup q≥0 (pq -L α (q)).
We can easily check that assumption (A1') implies assumption (A1).

We are now ready to recall the following result extracted from [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF]:

Theorem 1.7 (Existence and uniqueness for a solution of the HJ problem) Assume (A0)-(A1') and let T > 0. Then there exists a unique viscosity solution u of (1.2)-(1.3) on J T in the sense of the Definition 3.3, satisfying for some constant

C T > 0 |u(t, y) -u 0 (y)| ≤ C T for all (t, y) ∈ J T .
Moreover the function u is Lipschitz continuous with respect to (t, y) on J T .

Our last main result is the following:

|u α (n∆t, i∆x) -U α,n i | = 0,
where the index α in (1. [START_REF] Garavello | Conservation laws with discontinuous flux[END_REF]) is chosen such that (n∆t, i∆x) ∈ K ∩ [0, T ) × J α .

Using our scheme (1.7), we will present in Section 5 illustrations by numerical simulations with application to traffic.

Brief review of the literature

Hamilton-Jacobi formulation. We mainly refer here to the comments provided in [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF] and references therein. There is a huge literature dealing with HJ equations and mainly with equations with discontinuous Hamiltonians. However, concerning the study of HJ equation on a network, there exist a few works: the reader is referred to [START_REF] Achdou | Hamilton-Jacobi equations on networks[END_REF][START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF] for a general definition of viscosity solutions on a network, and [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on Networks[END_REF] for Eikonal equations. Notice that in those works, the Lagrangians depend on the position x and are continuous with respect to this variable. Conversely, in [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF] the Lagrangians do not depend on the position but they are allowed to be discontinuous at the junction point. Even for discontinuous Lagrangians, the uniqueness of the viscosity solution has been established in [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF].

Numerical schemes for Hamilton-Jacobi equations. Up to our knowledge, there are no numerical schemes for HJ equations on junctions (except the very recent work [START_REF] Göttlich | Numerical Discretization of Hamilton-Jacobi Equations on Networks[END_REF], see our Section 4 for more details), while there are a lot of schemes for HJ equations for problems without junctions. The majority of numerical schemes which were proposed to solve HJ equations are based on finite difference methods; see for instance [START_REF] Crandall | Two Approximations of Solutions of Hamilton-Jacobi Equations[END_REF] for upwind and centered discretizations, and [START_REF] Falcone | Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Osher | High order essentially non-oscillatory schemes for HamiltonJacobi equations[END_REF] for ENO or WENO schemes. For finite elements methods, the reader could also refer to [START_REF] Hu | A discontinuous Galerkin finite element method for Hamilton-Jacobi equations[END_REF] and [START_REF] Zhang | High order WENO schemes for HamiltonJacobi equations on triangular meshes[END_REF]. Explicit classical monotone schemes have convergence properties but they require to satisfy a CFL condition and they exhibit a viscous behaviour. We can also cite Semi-Lagrangian schemes [START_REF] Dolcetta | On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming[END_REF][START_REF] Falcone | A numerical approach to the infinite horizon problem of deterministic control theory[END_REF][START_REF] Falcone | Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF]. Anti-diffusive methods coming from numerical schemes adapted for conservation laws were thus introduced [START_REF] Bokanowski | Anti-dissipative schemes for advection and application to Hamilton-Jacobi-Bellmann equations[END_REF][START_REF] Xu | Anti-diffusive high order WENO schemes for Hamilton-Jacobi equations[END_REF]. Some other interesting numerical advances are done along the line of discontinuous Galerkin methods [START_REF] Cheng | Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension[END_REF][START_REF] Bokanowski | A discontinuous Galerkin solver for front propagation[END_REF]. Notice that more generally, an important effort deals with Hamilton-Jacobi-Bellman equations and Optimal Control viewpoint. It is out of the scope here.

Organization of the paper

In Section 2, we point out our first main property, namely Theorem 1.3 about the time and space gradient estimates. Then in Section 3, we first recall the notion of viscosity solutions for HJ equations. We then prove the second main property of our numerical scheme, namely Theorem 1.5 and Theorem 1.8 about the convergence of the numerical solution toward a solution of HJ equations when the mesh grid goes to zero. In Section 4, we propose the interpretation of our numerical results to traffic flows problems on a junction. In particular, the numerical scheme for HJ equations (1.7) is derived and the junction condition is interpreted. Indeed, we recover the well-known junction condition of Lebacque (see [START_REF] Lebacque | The Godunov scheme and what it means for first order traffic flow models[END_REF]) or equivalently those for the Riemann solver at the junction as in the book of Garavello and Piccoli [START_REF] Garavello | Traffic flow on networks[END_REF]. Finally, in Section 5 we illustrate the numerical behaviour of our scheme for a junction with two incoming and two outgoing branches.

Gradient estimates for the scheme

This section is devoted to the proofs of the first main result namely the time and space gradient estimates.

Proof of Proposition 1.2

We begin by proving the monotonicity of the numerical scheme.

Proof of Proposition 1.2: We consider the numerical scheme given by (1.7) that we rewrite as follows for n ≥ 0:

(2.23)

       U α,n+1 i = S α U α,n i-1 , U α,n i , U α,n i+1 for i ≥ 1, α = 1, ..., N, U n+1 0 = S 0 U n 0 , (U β,n 1 ) β=1,...,N for i = 0, where (2.24)
             S α U α,n i-1 , U α,n i , U α,n i+1 := U α,n i -∆t max H + α U α,n i -U α,n i-1 ∆x , H - α U α,n i+1 -U α,n i ∆x , S 0 U n 0 , (U β,n 1 ) β=1,...,N := U n 0 -∆t max β=1,...,N H - β U β,n 1 -U n 0 ∆x .
Checking the monotonicity of the scheme means checking that S α and S 0 are non-decreasing in all their variables. Case 1: i ≥ 1 This case is very classical. It is straightforward to check that S α for any α = 1, ..., N is non-decreasing in U α,n i-1 and U α,n i+1 . We compute

∂S α ∂U α,n i =        1 - ∆t ∆x (H + α ) ′ (p α,n i,-) if max H + α (p α,n i,-), H - α (p α,n i,+ = H + α (p α,n i,-), 1 - ∆t ∆x (H - α ) ′ (p α,n i,+ ) if max H + α (p α,n i,-), H - α (p α,n i,+ = H - α (p α,n i,+ )
which is non-negative if the CFL condition (1.9) is satisfied.

Case 2: i = 0 Similarly, it is straightforward to check that S 0 is non-decreasing in each U β,n 1 for β = 1, ..., N . We compute

∂S 0 ∂U n 0 = 1 - ∆t ∆x (H - α ) ′ (p α,n 0,+ ) if H - α (p α,n 0,+ ) > H - β (p β,n 0,+ ) for all β ∈ {1, ..., N } \ {α}
which is also non-negative due to the CFL condition (1.9).

From cases 1 and 2, we deduce that the scheme is monotone.

Proof of Theorem 1.3

In this subsection, we prove the first main result Theorem 1. ) α,i satisfying for some constant

C n > 0: (2.28) |p α,n i,+ | ≤ C n , for i ≥ 0, α = 1, ...N.
We also consider U α,n+1 i α,i and U α,n+2 i α,i computed using the scheme (1.7).

If we have

(2.29) D α,n i,+ ≤ ∆x ∆t for any i ≥ 0 and α = 1, ..., N, Then it comes that m n ≤ m n+1 ≤ M n+1 ≤ M n .

Proof

Step 0: Preliminaries. We introduce for any n ≥ 0, α = 1, ..., N and for any i ≥ 1, σ ∈ {+, -} or for i = 0 and σ = +:

(2.30)

C α,n i,σ := -σ 1 0 dτ (H -σ α ) ′ (p α,n+1 i,σ + τ (p α,n i,σ -p α,n+1 i,σ
)) ≥ 0.

Notice that C α,n i,σ is defined as the integral of (H -σ α ) ′ over a convex combination of p with p ∈ I α,n i,σ . Hence for any n ≥ 0, α = 1, ..., N and for any i ≥ 1, σ ∈ {+, -} or for i = 0 and σ = +, we can check that

(2.31) C α,n i,σ ≤ sup β=1,...,N j≥0 D β,n j,+ .
We also underline that for any n ≥ 0, α = 1, ..., N and for any i ≥ 1, σ ∈ {+, -} or for i = 0 and σ = +, we have the following relationship:

(2.32)

p α,n i,σ -p α,n+1 i,σ ∆t = -σ W α,n i+σ -W α,n i ∆x .
Let n ≥ 0 be fixed and consider (U α,n i ) α,i with ∆x, ∆t > 0 given. We compute U α,n+1 i α,i and U α,n+2 i α,i using the scheme (1.7).

Step 1: Estimate on m n We want to show that W α,n+1 i ≥ m n for any i ≥ 0 and α = 1, ..., N . It is then sufficient to take the infimum over i ≥ 0 and α = 1, ..., N to conclude that

m n+1 ≥ m n .
Let i ≥ 0 be fixed and we distinguish two cases:

Case 1: Proof of W α,n+1 i ≥ m n for all i ≥ 1 Let a branch α fixed. We assume that (2.33) max H + α (p α,n+1 i,- ), H - α (p α,n+1 i,+ ) = H -σ α (p α,n+1 i,σ
) for one σ ∈ {+, -}.

We have

W α,n+1 i -W α,n i ∆t = 1 ∆t max H + α (p α,n i,-), H - α (p α,n i,+ ) -max H + α (p α,n+1 i,- ), H - α (p α,n+1 i,+ ) ≥ 1 ∆t H -σ α (p α,n i,σ ) -H -σ α (p α,n+1 i,σ ) = 1 ∆t 1 0 dτ (H -σ α ) ′ (p α,n+1 i,σ + τ p)p with p = p α,n i,σ -p α,n+1 i,σ = C α,n i,σ W α,n i+σ -W α,n i ∆x
where we use (2.32) and (2.30) in the last line. Using (2.31) and (2.29), we thus get

W α,n+1 i ≥ 1 -C α,n i,σ ∆t ∆x W α,n i + C α,n i,σ ∆t ∆x W α,n i+σ ≥ min(W α,n i , W α,n i+σ ) ≥ m n . Case 2: Proof of W n+1 i ≥ m n for i = 0 We recall that in this case, we have U β,n 0 =: U n 0 for any β = 1, ..., N . Let us denote W β,n 0 =: W n 0 = U n+1 0 -U n 0 ∆t
for any β = 1, ..., N . Then we define α 0 such that

H - α0 (p α0,n+1 0,+ ) = max α=1,...,N H - α (p α,n+1 0,+
).

We argue like in Case 1 above and we get

W n+1 0 -W n 0 ∆t ≥ C α0,n 0,+ W α0,n 1 -W n 0 ∆x .
Then using (2.31) and (2.29) we conclude that:

W n+1 0 ≥ m n .
Step 2: : Estimate on M n We recall that n ≥ 0 is fixed. The proof for M n is directly adapted from Part 1. We want to show that W α,n+1 i ≤ M n for any i ≥ 0 and α = 1, ..., N . We distinguish the same two cases:

• If i ≥ 1, instead of (2.33) we simply choose σ such that max H + α (p α,n i,-), H - α (p α,n i,+ ) = H -σ α (p α,n i,σ ) for one σ ∈ {+, -}.
• If i = 0, we define α 0 such that

H - α0 (p α0,n 0,+ ) = max α=1,...,N H - α (p α,n 0,+ ).
Then taking the supremum, we can easily prove that

M n+1 ≤ M n , for any n ≥ 0.
By definition of m n and M n for a given n ≥ 0, we recover the result

m n ≤ m n+1 ≤ M n+1 ≤ M n .
The second important result needed for the proof of Theorem 1.3 is the following one:

Lemma 2.2 (Gradient estimate) Assume (A1). Let n ≥ 0 fixed and ∆x, ∆t > 0. We consider that (U α,n i ) α,i is given and we compute U α,n+1 i α,i using the scheme (1.7).

If there exists a constant K ∈ R such that for any i ≥ 0 and α = 1, ..., N , we have

K ≤ W α,n i := U α,n+1 i -U α,n i ∆t
Then it follows for any i ≥ 0 and α = 1, ..., N

(H - α ) -1 (-K) ≤ p α,n i,+ ≤ (H + α ) -1 (-K) with p α,n i,+ defined in (1.5) and (H - α ) -1 , (H + α ) -1 defined in (1.10).

Proof

Let n ≥ 0 be fixed and consider (U α,n i ) α,i with ∆x, ∆t > 0 given. We compute U α,n+1 i α,i using the scheme (1.7).

Let us consider any i ≥ 0 and α = 1, ..., N . We distinguish two cases according to the value of i.

Case 1: i ≥ 1 Assume that we have K ≤ W α,n i = -max σ∈{+,-} H -σ α (p α,n i,σ ).
It is then obvious that we get

-K ≥ H -σ α (p α,n i,σ ), for any σ ∈ {+, -}.
According to (A1) on the monotonicity of the Hamiltonians H α , we obtain (2.34)

     (H + α ) -1 (-K) ≥ p α,n i,-= p α,n i-1,+ (H - α ) -1 (-K) ≤ p α,n i,+
for any i ≥ 1, n ≥ 0 and α = 1, . . . , N.

Case 2: i = 0

The proof is similar to Case 1 because on the one hand we have

K ≤ W α,n 0 =: W n 0 = -max β=1,...,N H - β (p β,n 0,+ )
which obviously leads to (H - α ) -1 (-K) ≤ p α,n 0,+ , where we use the monotonicity of H - α from assumption (A1). On the other hand, from (2.34) we get

(H + α ) -1 (-K) ≥ p α,n 1,-= p α,n 0,+ .
We conclude

(H - α ) -1 (-K) ≤ p α,n i,+ ≤ (H + α ) -1 (-K), for any i, n ≥ 0 and α = 1, ..., N
which ends the proof.

Proof of Theorem 1.3: The idea of the proof is to introduce new continuous Hamiltonians Hα that satisfy the following properties:

(i) the new Hamiltonians Hα are equal to the old ones H α on the segment [p α , p α ],

(ii) the derivative of the new Hamiltonians | H′ α | taken at any point is less or equal to sup

p∈[p α ,p α ] |H ′ α (p)|.
This modification of the Hamiltonians is done in order to show that the gradient stays in the interval [p α , p α ].

Step 1: Modification of the Hamiltonians Let the new Hamiltonians Hα for all α = 1, ..., N be defined as

(2.35) Hα (p) =      g l α (p) for p ≤ p α H α (p) for p ∈ [p α , p α ] g r α (p)
for p ≥ p α with p α , p α defined in (1.11) and g l α , g r α two functions such that

                   g l α ∈ C 1 ((-∞, p α ]), g l α (p α ) = -m 0 , (g l α ) ′ (p α ) = H ′ α (p α ), (g l α ) ′ < 0 on (-∞, p α ), |(g l α ) ′ (p)| < |H ′ α (p α )| for p < p α , g l α → +∞ for p → -∞, and
                   g r α ∈ C 1 ([p α , +∞)), g r α (p α ) = -m 0 , (g r α ) ′ (p α ) = H ′ α (p α ), (g r α ) ′ > 0 on (p α , +∞), |(g r α ) ′ (p)| < |H ′ α (p α )| for p > p α , g r α → +∞ for p → +∞.
We can easily check that

(2.36) 0 < H′ α < sup pα∈[p α ,p α ] |H ′ α (p α )|, on R \ [p α , p α ],
and

(2.37) Hα > -m 0 on R \ [p α , p α ].
We can also check that Hα satisfies (A1). Then Proposition 2.1 and Lemma 2.2 hold true for the new Hamiltonians Hα (especially we can adapt (1.10) to the Hα for defining a sort of inverse).

Let H+ α (resp. Hα ) denotes the non-decreasing (resp. non-increasing) part of Hα . We consider the new numerical scheme for any n ≥ 0 that reads as:

(2.38)                      Ũ α,n+1 i -Ũ α,n i ∆t + max H+ α (p α,n i,-), H- α (p α,n i,+ ) = 0, for i ≥ 1, α = 1, . . . , N, Ũ β,n 0 =: Ũ n 0 , for all β = 1, . . . , N Ũ n+1 0 -Ũ n 0 ∆t + max β=1,...,N H- β (p β,n 0,+ ) = 0 for i = 0, subject to the initial condition (2.39) Ũ α,0 i = U α,0 i = u α 0 (i∆x), i ≥ 0, α = 1, . . . , N.
The discrete time and space gradients are defined such as:

(2.40) pα,n i,+ := Ũ α,n i+1 -Ũ α,n i ∆x and W α,n i := Ũ α,n+1 i -Ũ α,n i ∆t .
Let

|H ′ α (p α )| ≤ ∆x ∆t .
Step 2: First gradient bounds Let n ≥ 0 be fixed. By definition (2.41) and if mn is finite, we have mn ≤ W α,n i , for any i ≥ 0, α = 1, ..., N.

Using Lemma 2.2, it follows that

(2.44) ( H- α ) -1 (-mn ) ≤ pα,n i,+ ≤ ( H+ α ) -1 (-mn )
, for any i ≥ 0 and α = 1, ..., N.

We define C n = max ( Hα ) -1 (-mn ) , ( H+ α ) -1 (-mn ) > 0, and we recover that |p α,n i,+ | ≤ C n , for any i ≥ 0, α = 1, ..., N.

Step 3: Time derivative and gradient estimates For any n ≥ 0, (2.43) holds true. Moreover, if mn is finite, then there exists C n > 0 such that

|p α,n i,+ | ≤ C n , for any i ≥ 0, α = 1, ..., N.
Then using Proposition 2.1 we get

(2.45) mn ≤ mn+1 ≤ M n+1 ≤ M n for any n ≥ 0.
In particular, mn+1 is also finite.

Using the assumption that m 0 is finite and according to (1.11), Lemma 2.2 and the scheme (1.7), we can check that (2.46) p α ≤ p α,0 i,+ ≤ p α for any i ≥ 0 and α = 1, ..., N.

From (2.39), we have p α,0 i,+ = pα,0 i,+ . Therefore, from (2.35), (1.7) and (2.46), we deduce that W α,0 i = W α,0 i and we obtain that m0 = m 0 .

According to (2.45), we deduce that m 0 ≤ W α,n i for any i ≥ 0, n ≥ 0 and α = 1, ..., N . Then using Lemma 2.2 and (2.37), we conclude that for all i ≥ 0, n ≥ 0 and α = 1, ..., N (2.47)

p α ≤ pα,n i,+ ≤ p α .
Step 4: Conclusion If (2.47) holds true, then Hα (p α,n i,+ ) = H α (p α,n i,+ ) for all i ≥ 0, n ≥ 0 and α = 1, ..., N . Thus the modified scheme (2.38) is strictly equivalent to the original scheme (1.7) and U α,n i = Ũ α,n i . We finally recover the results for all i ≥ 0, n ≥ 0 and α = 1, ..., N :

(i) (Time derivative estimate) m n ≤ m n+1 ≤ M n+1 ≤ M n , (ii) (Gradient estimate) p α ≤ p α,n i,+ ≤ p α .
Remark 2.3 (Do the bounds (1.14) always give informations on the gradient?)

We assume that the assumptions of Theorem 1.3 hold true.

(i) (Bounds on m n ) From the scheme (1.7), we can rewrite

m n = inf α,i min σ∈{+,-} -H -σ α (p α,n i,σ ) .
It is then obvious that -m 0 ≥ min pα∈R H α (p α ) for α = 1, ..., N, which ensures that the bound from below in (1.14) always gives an information on the gradient (p α,n i,+ ). (ii) (Bounds on M n ) For the bounds from above in (1.14), we get

(2.48) H α p α,n i,+ ≥ -M 0 for all α = 1, ..., N, i ≥ 0 and n ≤ n T . Then (2.48) is always true if -M 0 ≤ min R H α . Therefore for each α = 1, ..., N , (2.48) gives an information on the (p α,n i,+ ) only if -M 0 > min pα∈R H α (p α ).

Remark 2.4 (Extension to weaker assumptions on H α than (A1))

All the results of this paper can be extended if we consider weaker conditions than (A1) on the Hamiltonians H α . Indeed, we can assume that the H α for any α = 1, ..., N are locally Lipschitz instead of being C 1 . This assumption is more adapted for our traffic application (see Section 4).

We now focus on what should be modified if we do so. How to modify CFL condition (1.9)?

The main new idea is then to consider the closed convex hull for the discrete gradient defined by

I α,n := Conv(p α,n i,+ ) i≥0 .
Then the Lipschitz constant L α,n of the considered H α is a natural upper bound

|H α (p + q) -H α (p)| ≤ L α,n |q| with p, p + q ∈ I α,n .
Then the natural CFL condition which replaces (1.9) is the following one:

(2.49) ∆x ∆t ≥ sup α=1,...,N n≤nT L α,n .
With such a condition, we can easily prove the monotonicity of the numerical scheme.

How to modify CFL condition (1.12)? Assume that CFL condition (1.12) is replaced by the following one

(2.50) ∆x ∆t ≥ ess sup α=1,...,N pα∈[p α ,p α ] |H ′ α (p α )|,
where ess sup denotes the essential supremum.

In the proof of Theorem 1.3, the time derivative estimate uses the integral of H ′ α which is defined almost everywhere if H α is at least Lipschitz. The remaining of the main results of Section 1.3 do not use a definition of H ′ α , except in the CFL condition. We just need to satisfy the new CFL condition (2.50).

3 Convergence result for the scheme

Viscosity solutions

We introduce the main definitions related to viscosity solutions for HJ equations that are used in the remaining. For a more general introduction to viscosity solutions, the reader could refer to Barles [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF] and to Crandall, Ishii, Lions [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF].

Let T > 0. We set u := (u α ) α=1,...,N ∈ C 1 * (J T ) where C 1 * (J T ) is defined in (1.1) and we consider the additional condition u α (t, 0) = u β (t, 0) =: u(t, 0) for any α, β.

Remark 3.1 Following [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF], we recall that (1.2) can be rigorously rewritten as

(3.51) u t + H(y, u y ) = 0, for (t, y) ∈ [0, T ) × J, with
H(y, p) :=      H α (p), for p ∈ R, if y ∈ J * α , max α=1,...,N H - α (p α ), for p = (p 1 , ..., p N ) ∈ R N , if y = 0,
subject to the initial condition (3.52) u(0, y) = u 0 (y) := (u α 0 (x)) α=1,...,N , for y = xe α ∈ J with x ∈ [0, +∞).

Definition 3.2 (Upper and lower semi-continuous envelopes)

For any function u : [0, T ) × J → R, upper and lower semi-continuous envelopes are respectively defined as:

u * (t, y) = lim sup (t ′ ,y ′ )→(t,y) u(t ′ , y ′ ) and u * (t, y) = lim inf (t ′ ,y ′ )→(t,y) u(t ′ , y ′ ).
Moreover, we recall A function u : J T → R is a viscosity subsolution (resp. supersolution) of (1.2) on J T if it is an upper semicontinuous (resp. lower semi-continuous) function, and if for any P = (t, y) ∈ J T and any test function ϕ := (ϕ α ) α ∈ C 1 * (J T ) such that uϕ ≤ 0 (resp. uϕ ≥ 0) at the point P , we have

           u is
(3.53) ϕ α t (t, x) + H α (ϕ α x (t, x)) ≤ 0 if y = xe α ∈ J * α , (3.54) resp. ϕ α t (t, x) + H α (ϕ α x (t, x)) ≥ 0 if y = xe α ∈ J * α , (3.55) ϕ t (t, 0) + max α=1,...,N H - α (ϕ α x (t, 0)) ≤ 0 if y = 0, (3.56) resp. ϕ t (t, 0) + max α=1,...,N H - α (ϕ α x (t, 0)) ≥ 0 if y = 0 .
ii) Viscosity sub and super-solution on [0, T ) × J A function u : [0, T ) × J → R is said to be a viscosity subsolution (resp. supersolution) of (1.2)-(1.3) on [0, T ) × J, if u is upper semi-continuous (resp. lower semi-continuous), if it is a viscosity subsolution (resp. supersolution) of (1.2) on J T and if moreover it satisfies:

       u(0, y) ≤ u 0 (y)
for all y ∈ J, resp. u(0, y) ≥ u 0 (y) for all y ∈ J , when the initial data u 0 is assumed to be continuous.

iii) Viscosity solution on J T and on [0, T ) × J A function u : [0, T ) × J → R is said to be a viscosity solution of (1.2) on J T (resp. of (1.2)-(1.3) on [0, T ) × J) if u * is a viscosity subsolution and u * is a viscosity supersolution of (1.2) on J T (resp. of (1.2)-(1.3) on [0, T ) × J).

Hereafter, we recall two properties of viscosity solutions on a junction that are extracted from [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF]:

Proposition 3.4 (Comparison principle) Assume (A0)-(A1'
) and let T > 0. Assume that u and u are respectively a viscosity subsolution and a viscosity supersolution of (1.2)-(1.3) on [0, T ) × J in the sense of Definition 3.3. We also assume that there exists a constant C T > 0 such that for all (t, y)

∈ [0, T ) × J u(t, y) ≤ C T (1 + |y|) (resp. u(t, y) ≥ -C T (1 + |y|)) .
Then we have u ≤ u on J T .

Proposition 3.5 (Equivalence with relaxed junction conditions) Assume (A1') and let T > 0. A function u : [0, T ) × J → R is a viscosity subsolution (resp. a viscosity supersolution) of (1.2) on J T if and only if for any function ϕ := (ϕ α ) α ∈ C 1 * (J T ) and for any P = (t, y) ∈ J T such that uϕ ≤ 0 (resp. uϕ ≥ 0) at the point P , we have the following properties

• if y = xe α ∈ J * α , then ϕ α t (t, x) + H α (ϕ α x (t, x)) ≤ 0 (resp. ≥ 0)
• if y = 0, then either there exists one index α ∈ {1, ..., N } such that

ϕ α t (t, 0) + H α (ϕ α x (t, 0)) ≤ 0 (resp. ≥ 0)
or (3.55) (resp. (3.56)) holds true for y = 0.

We skip the proof of Proposition 3.4 and Proposition 3.5 which are directly available in [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF].

Convergence of the numerical solution

We assume (A0), (A1') and we set ε := (∆t, ∆x) satisfying the CFL condition (1.12). This section first deals with a technical result (see Lemma 3.6) that is very useful for the proof of Theorem 1.8 that is the convergence of the numerical solution of (1.7)-(1.8) towards a solution of (1.2)-(1.3) when ε goes to zero. According to Theorem 1.7, we know that the equation (1.2)-(1.3) admits a unique solution in the sense of Definition 3.3. For Theorem 1.5, we extend the convergence proof, assuming the weakest assumption (A1) instead of (A1'). We close this subsection with the proof of gradient estimates for the continuous solution (see Corollary 1.6).

For α = 1, ..., N , i ≥ 0 and n ≥ 0, recall that (U α,n i ) solves the numerical scheme (1.7)-(1.8). This numerical solution U α,n i is expected to be a certain approximation of u α (n∆t, i∆x) where u α is the unique viscosity solution of (1.2)-(1.3) given by Theorem 1.7. For sake of clarity, we then denote our numerical solution as follows u α ε (n∆t, i∆x) := U α,n i , for any α = 1, ..., N and any i ≥ 0, n ≥ 0, and we recall u α ε (n∆t, 0) =: u ε (n∆t, 0), for any α = 1, ..., N. We also denote by G α ε the set of all grid points (n∆t, i∆x) on [0, T ) × J α for any branch α = 1, ..., N , and we set

(3.57) G ε = α=1,...,N G α ε
the whole set of grid points on [0, T ) × J, with identification of the junction points (n∆t, 0) of each grid G α ε . We call u ε the function defined by its restrictions to the grid points of the branches

u ε = u α ε on G α ε .
For any point (t, y) ∈ [0, T ) × J, we define the half relaxed limits

(3.58) u(t, y) = lim sup ε→0 Gε∋(t ′ ,y ′ )→(t,y) u ε (t ′ , y ′ ), (3.59)   resp. u(t, y) = lim inf ε→0 Gε∋(t ′ ,y ′ )→(t,y) u ε (t ′ , y ′ )   .
Thus we have that u := (u α ) α (resp. u := (u α ) α ) is upper semi-continuous (resp. lower semi-continuous). Lemma 3.6 (ε-uniform space and time gradient bounds) Assume (A0)-(A1). Let T > 0 and ε = (∆t, ∆x) such that the CFL condition (1.12) is satisfied. If (U α,n i ) is the numerical solution of (1.7)-(1.8), then for any α = 1, ..., N , i ≥ 0 and n ≥ 0, we have

(3.60) p 0 α ≤ U α,n i+1 -U α,n i ∆x ≤ p 0 α and m 0 0 ≤ U α,n+1 i -U α,n i ∆t ≤ M 0 0 ,
where the quantities (independent of ε) m 0 0 , M 0 0 , p 0 α and p 0 α are respectively defined in (1.17) and (1.18).

Proof of Lemma 3.6:

Let ε = (∆t, ∆x) be fixed such that the CFL condition (1.12) is satisfied.

Step 1: Proof of m 0 ≥ m 0 0 , p 0 α ≤ p α and p α ≤ p 0 α

We first show that (3.61) m 0 ≥ m 0 0 .

Indeed using (A1) and the fact that H α (p) = max {H - α (p), H + α (p)} for any p ∈ R, we get

m 0 = inf α,i -H α (p α,0 i ) ≥ inf α pα∈[L α,-,L α,+ ] {-H α (p α )} =: m 0 0 ,
where we recall that L α,-and L α,+ are the best Lipschitz constants defined in (1.16) that implies (3.62) L α,-≤ p α,0 i,+ ≤ L α,+ , for any i ≥ 0.

From (1.18) and the monotonicity of H ± α , we deduce (3.63) p 0 α ≤ p α and p α ≤ p 0 α , for any α = 1, ..., N.

Step 2: Proof of M 0 ≤ M 0 0

Recall the definitions

M 0 := sup α,i W α,0 i = max{A, B}, with      A := min α=1,...,N -H - α (p α,0 0,+ ) , B := sup α,i≥1 min σ∈{+,-} -H -σ α (p α,0 i,σ
) .

and

M 0 0 := max max α=1,...,N min σ∈{+,-} -H -σ α (L α,σ ) , min α=1,...,N -H - α (L α,+ ) .
Let us show that

(3.64) M 0 ≤ M 0 0 .
We distinguish two cases according to the value of M 0 :

• If M 0 = A, then M 0 0 ≥ min α=1,...,N (-H - α (L α,+ )) ≥ min α=1,...,N (-H - α (p α,0 0,+ )) = A = M 0 ,
where we use (3.62) and the monotonicity of

H - α . • If M 0 = B, then M 0 0 ≥ max α=1,...,N min σ∈{+,-} (-H -σ α (L α,σ )) ≥ sup α,i≥1 min σ∈{+,-} (-H -σ α (p α,0 i,σ )) = B = M 0 .
which comes from (3.62) and the following inequality (due to (3.62))

min σ∈{+,-} (-H -σ α (L α,σ )) ≥ min σ∈{+,-}
(-H -σ α (p α,0 i,σ )), for any i ≥ 1.

Step

3: Conclusion

The estimates (3.60) directly follow from (3.61), (3.64) and (3.63) and Theorem 1.3.

Proof of Theorem 1.8:

Step 0: Preliminaries Let T > 0 be fixed and let ε := (∆t, ∆x) satisfy the CFL condition (1.12). Assume that u ε is the numerical solution of (1.7)-(1.8). We consider u and u respectively defined in (3.58) and (3.59). By construction, we have u ≤ u.

We will show in the following steps that u (resp. u) is a viscosity supersolution (resp. viscosity subsolution) of equation (1.2)-(1.3), such that there exists a constant C T > 0 such that for all (t, y)

∈ [0, T ) × J u(t, y) ≥ -C T (1 + |y|) (resp. u(t, y) ≤ C T (1 + |y|)) ,
and such that u(0, y) ≥ u 0 (y) (resp. u(0, y) ≤ u 0 (y)) for all y ∈ J.

Using the comparison principle (Proposition 3.4), we obtain u ≤ u ≤ u.

Thus from Definition 3.3, we can conclude that u = u = u. This implies the statement of Theorem 1.8.

Step 1: First bounds on the half relaxed limits From Lemma 3.6, we deduce that for any α = 1, ..., N , any i ≥ 0 and any n ≥ 0, we have

m 0 0 n∆t ≤ U α,n i -U α,0
i ≤ M 0 0 n∆t. Passing to the limit with ε → 0 (always satisfying CFL condition (1.12)), we get u 0 (y) + m 0 0 t ≤ u(t, y) ≤ u(t, y) ≤ u 0 (y) + M 0 0 t, for (t, y) ∈ [0, T ) × J. This implies that In next step, we show that u is a supersolution of (1.2)-(1.3) in the viscosity sense. We skip the proof that u is a viscosity subsolution because it is similar.

Step 2: Proof of u being a viscosity supersolution Let us consider u = (u α ) α=1,...,N as defined in (3.59) and a test function

ϕ := (ϕ α ) α ∈ C 1 * ([0, T ) × J) satisfying u ≥ ϕ on [0, T ) × J, u = ϕ at P 0 = (t 0 , y 0 ) ∈ [0, T ) × J, with t 0 > 0.
Thus up to replacing ϕ(P ) by φ(P ) = ϕ(P ) + |P -P 0 | 2 , we can assume that u > ϕ on B r (P 0 ) \ {P 0 }, u = ϕ at P 0 .

We set B r (P 0 ) the open ball in [0, T ) × J centred at P 0 with fixed radius r > 0 i.e.

B r (P 0 ) := (t, y) ∈ [0, T ) × J, (tt 0 ) 2 + d(y, y 0 ) 2 < r where d(•, •) denotes the natural distance on J. We also set Ω ε defined as the intersection between the closed ball centred on P 0 and the grid points G ε (defined in (3.57)), i.e.

Ω ε := B r (P 0 ) ∩ G ε .
Note that for ε small enough, we have Ω ε = ∅. Up to decreasing r, we can assume that B r (P 0 ) ⊂ [0, T -r)×J.

Define also

M ε = inf Ωε (u ε -ϕ) = (u ε -ϕ)(P ε ),
where

P ε = (t ε , y ε ) ∈ [0, T ) × J αε with y ε = x ε e αε and t ε := n ε ∆t x ε := i ε ∆x .
By the definition of u in (3.59), it is classical to show that if ε → 0 we get the following (at least for a subsequence)

(3.66)    M ε = (u ε -ϕ)(P ε ) → M 0 = inf Br (P0) (u -ϕ) = 0, P ε → P 0 .
Let us now check that u is a viscosity supersolution of (1.2). To this end, using Proposition 3.5 we want to show that

• if y 0 = x 0 e α0 ∈ J * α0 for a given α 0 , then

ϕ α0 t + H α0 (ϕ α0 x ) ≥ 0 at (t 0 , x 0 ),
• if y 0 = 0, then either there exists one index α 0 such that

ϕ α0 t + H α0 (ϕ α0
x ) ≥ 0 at (t 0 , 0), or we have

ϕ t + sup α=1,...,N H - α (ϕ α x ) ≥ 0 at (t 0 , 0).
Because t 0 > 0 and P ε → P 0 , this implies in particular that n ε ≥ 1 for ε small enough. We have to distinguish two cases according to the value of y 0 . Case 1: P 0 = (t 0 , y 0 ) with y 0 = 0 We distinguish two subcases, up to subsequences. Using the definitions (2.23), (2.24) and the numerical scheme (1.7), we recall that for all n ≥ 0 and for any α = 1, ..., N U α,n+1

0 =: U n+1 0 = U n 0 -∆t max α=1,...,N H - α U α,n 1 -U n 0 ∆x = S 0 U n 0 , (U α,n 1 ) α=1,...,N
where S 0 is monotone under the CFL condition (1.12) (see Proposition 1.2).

Let

ϕ ε := M ε + ϕ such that ϕ ε (P ε ) = u ε (P ε ) = U nε 0 = S 0 U nε-1 0 , U α,nε-1 1 α=1,...,N ≥ S 0 ϕ ε ((n ε -1)∆t, 0), (ϕ α ε ((n ε -1)∆t, ∆x)) α=1,...,N
where we use the monotonicity of the scheme in the last line and the fact that u ε ≥ ϕ ε on Ω ε . Thus we have

ϕ ε (n ε ∆t, 0) -ϕ ε ((n ε -1)∆t, 0) ∆t + max α=1,...,N H - α ϕ α ε ((n ε -1)∆t, ∆x) -ϕ α ε ((n ε -1)∆t, 0) ∆x ≥ 0. This implies (ϕ ε ) t + max α=1,...,N H - α ((ϕ α ε ) x ) + o ε (1) ≥ 0 at (t ε , 0)
and passing to the limit with ε → 0, we get the supersolution condition at the junction point

(3.67) ϕ t + max α=1,...,N H - α (ϕ α x ) ≥ 0 at (t 0 , 0). Subcase 1.2: Pε = (tε, yε) with yε ∈ J * αε
In this case, the infimum M ε is reached for a point on the branch α ε which is different from the junction point. Thus the definitions (2.23), (2.24) and the numerical scheme (1.7) give us that for all n ≥ 0 and i ≥ 1

U αε,n+1 i = U αε,n i + ∆t min {-H - αε (p αε,n i,+ ), -H + αε (p αε,n i,-)} = S αε [U αε,n i-1 , U αε,n i , U αε,n i+1 ]. Let ϕ αε ε := M ε + ϕ αε such that ϕ αε ε (P ε ) = u αε ε (P ε ) = U αε,nε iε = S αε [U αε,nε-1 iε-1 , U αε,nε-1 iε , U αε,nε-1 iε+1 ] ≥ S αε [ϕ αε ε ((n ε -1)∆t, (i ε -1)∆x), ϕ αε ε ((n ε -1)∆t, i ε ∆x), ϕ αε ε ((n ε -1)∆t, (i ε + 1)∆x)]
where we use the monotonicity of the scheme and the fact that u αε ε ≥ ϕ αε ε in the neighbourhood of (t ε , x ε ).

Thus we have that for any ε = (∆t, ∆x)

0 ≤ ϕ αε ε (n ε ∆t, i ε ∆x) -ϕ αε ε ((n ε -1)∆t, i ε ∆x) ∆t + max H + αε ϕ αε ε (n ε ∆t, i ε ∆x) -ϕ αε ε (n ε ∆t, (i ε -1)∆x) ∆x , H - αε ϕ αε ε (n ε ∆t, (i ε + 1)∆x) -ϕ αε ε (n ε ∆t, i ε ∆x) ∆x . Since H α (p) = max {H + α (p), H - α (p)}, this implies (ϕ αε ) t + H αε ((ϕ αε ) x ) + o ε (1) ≥ 0 at (t ε , x ε ).
Up to a subsequence, we can assume that α ε is independent of ε and equal to α 0 . Thus passing to the limit with ε → 0, we obtain (3.68)

ϕ α0 t + H α0 (ϕ α0 x ) ≥ 0 at (t 0 , 0).
Case 2: P 0 = (t 0 , y 0 ) with y 0 = x 0 e α0 ∈ J * α0

As P ε → P 0 from (3.66), we can always consider that for ε small enough, we can write P ε = (t ε , y ε ) with y ε ∈ J * αε . Thus the proof for this case is similar to the one in Subcase 1.2. We then conclude (3.69)

ϕ α0 t + H α0 (ϕ α0 x ) ≥ 0 at (t 0 , x 0 ).
Step

3: Conclusion

The results (3.65), (3.67), (3.68) and (3.69) imply that u is a viscosity supersolution of (1.2)- (1.3). This ends the proof of Theorem 1.8.

Proof of Theorem 1.5: The proof is quite similar to the proof of Theorem 1.8. However it differs on some points mainly because we do not know if the comparison principle from Proposition 3.4 holds for (1.2).

• We recall from Lemma 3.6 that u α ε (n∆t, i∆x) := U α,n i with ε = (∆t, ∆x) enjoys some discrete Lipschitz bounds in time and space, independent of ε.

• It is then possible to extend the discrete function u ε , defined only on the grid points, into a continuous function ũε , with the Q 1 quadrilateral finite elements approximation for which we have the same Lipschitz bounds. We recall that the approximation is the following: consider a map (t, x) → u(t, x) that takes values only on the vertex of a rectangle ABCD with A = (0, 0), B = (0, 1), C = (1, 1) and D = (1, 0) (for sake of simplicity we take ∆t = 1 = ∆x). Then we extend the map u to any point (t, x) of the rectangle such that

u(t, x) = [u A + x(u B -u A )](1 -t) + [u D + x(u C -u D )]t.
• In this way we can apply the Ascoli theorem which shows that there exists a subsequence ũε ′ which converges towards a function u, uniformly on every compact set (in time and space).

• We can then conclude that u is a viscosity super and subsolution of (1.2)-(1.3) repeating the proof of Theorem 1.8.

This ends the proof.

Proof of Corollary 1.6: The proof combines the gradient and time derivative estimates from Lemma 3.6 and the results of convergence from Theorem 1.5. Indeed, passing to the limit in (3.60) for a subsequence ε ′ , using the convergence result of Theorem 1.5, we finally get (1.19).

Application to traffic flow

As our motivation comes from traffic flow modelling, this section is devoted to the traffic interpretation of the model and the scheme. Notice that [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF] has already focused on the meaning of the junction condition in this framework.

Settings

We first recall the main variables adapted for road traffic modelling as they are already defined in [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF]. We consider a junction with N I ≥ 1 incoming roads and N O ≥ 1 outgoing ones. We also set that N I + N O =: N .

Densities and scalar conservation law. We assume that the vehicles densities denoted by (ρ α ) α solve the following scalar conservation laws (also called LWR model for Lighthill, Whitham [START_REF] Lighthill | On kinetic waves. II. Theory of Traffic Flows on Long Crowded Roads[END_REF] and Richards [START_REF] Richards | Shock Waves on the Highway[END_REF]):

(4.70)

ρ α t + (f α (ρ α )) X = 0, for (t, X) ∈ [0, +∞) × (-∞, 0), α = 1, ..., N I , ρ α t + (f α (ρ α )) X = 0, for (t, X) ∈ [0, +∞) × (0, +∞), α = N I + 1, ..., N I + N O ,
where we assume that the junction point is located at the origin X = 0. We assume that for any α the flux function f α : R → R reaches its unique maximum value for a critical density ρ = ρ α c > 0 and it is non decreasing on (-∞, ρ α c ) and non-increasing on (ρ α c , +∞). In traffic modelling, ρ α → f α (ρ α ) is usually called the fundamental diagram. Let us define for any α = 1, ..., N the Demand function f α D (resp. the Supply function f α S ) such that (4.71)

f α D (p) = f α (p) for p ≤ ρ α c f α (ρ α c ) for p ≥ ρ α c resp. f α S (p) = f α (ρ α c ) for p ≤ ρ α c f α (p) for p ≥ ρ α c .
We assume that we have a set of fixed coefficients 0 ≤ (γ α ) α ≤ 1 that denote:

• either the proportion of the flow from the branch α = 1, ..., N I which enters in the junction,

• or the proportion of the flow on the branch α = N I + 1, ..., N exiting from the junction.

We also assume the natural relations

NI α=1 γ α = 1 and NI +NO β=NI +1 γ β = 1.
Remark 4.1 We consider that the coefficients (γ α ) α=1,...,N are fixed and known at the beginning of the simulations. Such framework is particularly relevant for "quasi stationary" traffic flows.

Vehicles labels and Hamilton-Jacobi equations. Extending for any N I ≥ 1 the interpretation and the notations given in [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF] for a single incoming road, let us consider the continuous analogue u α of the discrete vehicles labels (in the present paper with labels increasing in the backward direction with respect to the flow)

(4.72)            u α (t, x) = u(t, 0) - 1 γ α -x 0 ρ α (t, y)dy, for x := -X > 0, if α = 1, ..., N I , u β (t, x) = u(t, 0) - 1 γ β x 0 ρ β (t, y)dy, for x := X > 0, if β = N I + 1, ..., N,
with equality of the functions at the junction point (x = 0), i.e.

(4.73) u α (t, 0) = u β (t, 0) =: u(t, 0) for any α, β.

where the common value u(t, 0) is nothing else than the (continuous) label of the vehicle at the junction point.

Following [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF], for a suitable choice of the function u(t, 0), it is possible to check that the vehicles labels u α satisfy the following Hamilton-Jacobi equation:

(4.74) u α t + H α (u α x ) = 0, for (t, x) ∈ [0, +∞) × (0, +∞), α = 1, ..., N where (4.75) H α (p) :=          - 1 γ α f α (γ α p) for α = 1, ..., N I , - 1 γ α f α (-γ α p) for α = N I + 1, ..., N I + N O .
Following definitions of H - α and H + α in (1.4) we get (4.76)

H - α (p) =          - 1 γ α f α D (γ α p) for α ≤ N I , - 1 γ α f α S (-γ α p) for α ≥ N I + 1,
and

H + α (p) =          - 1 γ α f α S (γ α p) for α ≤ N I , - 1 γ α f α D (-γ α p) for α ≥ N I + 1.
The junction condition in (1.2) that reads

(4.77) u t (t, 0) + max α=1,...,N H - α (u x (t, 0 + )) = 0.
is a natural condition from the traffic point of view. Indeed condition (4.77) can be rewritten as

(4.78) u t (t, 0) = min min α=1,...,NI 1 γ α f α D (ρ α (t, 0 -)), min β=NI +1,...,N 1 γ β f β S (ρ β (t, 0 + )) .
The condition (4.78) claims that the passing flux is equal to the minimum between the upstream demand and the downstream supply functions as they were presented by Lebacque in [START_REF] Lebacque | Les modèles macroscopiques du trafic[END_REF] and [START_REF] Lebacque | The Godunov scheme and what it means for first order traffic flow models[END_REF] (also for the case of junctions). This condition maximises the flow through the junction. This is also related to the Riemann solver RS2 in [START_REF] Garavello | Conservation laws on complex networks[END_REF] for junctions.

Review of the literature with application to traffic

Junction modelling. There is an important and fast growing literature about junction modeling from a traffic engineering viewpoint: see [START_REF] Khoshyaran | Internal state models for intersections in macroscopic traffic flow models[END_REF][START_REF] Tampere | A generic class of first order node models for dynamic macroscopic simulations of traffic flows[END_REF][START_REF] Flötteröd | Operational macroscopic modeling of complex urban intersections[END_REF] for a critical review of junction models. The literature mainly refers to pointwise junction models [START_REF] Khoshyaran | Internal state models for intersections in macroscopic traffic flow models[END_REF][START_REF] Lebacque | Macroscopic flow models ( First order macroscopic traffic flow models for networks in the context of dynamic assignment)[END_REF][START_REF] Lebacque | First-order macroscopic traffic flow models: intersection modeling, network modeling[END_REF]. Pointwise models are commonly restated in many instances as optimization problems.

Scalar one dimensional conservation laws and networks. Classically, macroscopic traffic flow models are based on a scalar one dimensional conservation law, e.g. the so-called LWR model (Lighthill, Whitham [START_REF] Lighthill | On kinetic waves. II. Theory of Traffic Flows on Long Crowded Roads[END_REF] and Richards [START_REF] Richards | Shock Waves on the Highway[END_REF]). The literature is also quite important concerning hyperbolic systems of conversation laws (see for example [START_REF] Bressan | Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem[END_REF][START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF][START_REF] Lax | Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves[END_REF][START_REF] Serre | Systems of Conservation Laws I: Hyperbolicity, entropies, shock waves[END_REF] and references therein) but these books also propose a large description of the scalar case. It is well-known that under suitable assumptions there exists a unique weak entropy solution for scalar conservation laws without junction.

Until now, existence of weak entropy solutions for a Cauchy problem on a network has been proved for general junctions in [START_REF] Garavello | Conservation laws on complex networks[END_REF]. See also Garavello and Piccoli's book [START_REF] Garavello | Traffic flow on networks[END_REF]. Uniqueness for scalar conservation laws for a junction with two branches has been proved first in [START_REF] Garavello | Conservation laws with discontinuous flux[END_REF] and then in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] under suitable assumptions. Indeed [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] introduces a general framework with the notion of L 1 -dissipative admissibility germ that is a selected family of elementary solutions. To the best authors' knowledge, there is no uniqueness result for general junctions with N ≥ 3 branches and a differential characterization of the solution.

The conservation law counterpart of model (4.74),(4.73),(4.77) has been studied in [START_REF] Garavello | Conservation laws on complex networks[END_REF] as a Riemann solver called RS2. In [START_REF] Garavello | Conservation laws on complex networks[END_REF] an existence result is presented for concave flux functions, using the Wave Front Tracking (WFT) method. Moreover the Lipschitz continuous dependence of the solution with respect to the initial data is proven. This shows that the process of construction of a solution (here the WFT method) creates a single solution. Nevertheless, up to our knowledge, there is no differential characterization of this solution.

Therefore the uniqueness of this solution is still an open problem.

Numerical schemes for conservation laws. According to [START_REF] Godlewski | Hyperbolic systems of conservation laws[END_REF] and [START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF], the numerical treatment of scalar conservation laws mainly deals with first order numerical schemes based on upwind finite difference method, such as the Godunov scheme [START_REF] Godunov | A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics[END_REF] which is well-adapted for the LWR model [START_REF] Lebacque | The Godunov scheme and what it means for first order traffic flow models[END_REF].

As finite difference methods introduce numerical viscosity, other techniques were developed such as kinetic schemes that derive from the kinetic formulation of hyperbolic equations [START_REF] Perthame | Kinetic formulation of conservation laws, Oxford lecture series in mathematics and its applications[END_REF]. Such kinetic schemes are presented in [START_REF] Aregba-Driollet | Discrete kinetic schemes for multidimensional systems of conservation laws[END_REF] and they are applied to the traffic case in [START_REF] Bretti | Fast algorithms for the approximation of a fluid-dynamic model on networks[END_REF][START_REF] Bretti | Numerical algorithms for simulations of a traffic model on road networks[END_REF][START_REF] Bretti | A Fluid-Dynamic Traffic Model on Road Networks[END_REF].

In [START_REF] Göttlich | Numerical Discretization of Hamilton-Jacobi Equations on Networks[END_REF] the authors apply a semidiscrete central numerical scheme to the Hamilton-Jacobi formulation of the LWR model. The equivalent scheme for densities recovers the classical Lax-Friedrichs scheme. Notice that the authors need to introduce at least two ghost-cells on each branch near the junction point to counterstrike the dispersion effects when computing the densities at the junction.

Derived scheme for the densities

The aim of this subsection is to properly express the numerical scheme satisfied by the densities in the traffic modelling framework. Let us recall that the density denoted by ρ α is a solution of (4.70).

Let us consider a discretization with time step ∆t and space step ∆x. Then we define the discrete car density ρ α,n i ≥ 0 for n ≥ 0 and i ∈ Z (see Figure 2) by

(4.79) ρ α,n i :=    γ α p α,n |i|-1,+ , for i ≤ -1, α = 1, ..., N I , -γ α p α,n i,+ , for i ≥ 0, α = N I + 1, ..., N I + N O ,
where we recall and the segments for (ρ α,n i ) i∈Z .

p α,n j,+ := U α,n j+1 -U α,n j ∆x , for j ∈ N, α = 1, ..., N. J λ x > 0 x < 0 -2 -1 2 1 J NI+NO J NI+1 J 1 J β J NI ρ β,n
We have the following result:

Lemma 4.2 (Derived numerical scheme for the density)

If (U α,n i
) stands for the solution of (1.7)-(1.8), then the density (ρ α,n i ) defined in (4.79) is a solution of the following numerical scheme for α = 1, ..., N (4.80)

∆x ∆t {ρ α,n+1 i -ρ α,n i } =                      F α (ρ α,n i-1 , ρ α,n i ) -F α (ρ α,n i , ρ α,n i+1 ) for i ≤ -1 if α ≤ N I , i ≥ 1 if α ≥ N I + 1, F α 0 (ρ β,n -1 ) β≤NI , (ρ λ,n 0 ) λ≥NI +1 -F α (ρ α,n i , ρ α,n i+1 ) for i = 0, α ≥ N I + 1, F α (ρ α,n i-1 , ρ α,n i ) -F α 0 (ρ β,n -1 ) β≤NI , (ρ λ,n 0 ) λ≥NI +1 for i = -1, α ≤ N I ,
where we define the fluxes by

(4.81)                        F α (ρ α,n i-1 , ρ α,n i ) := min f α D (ρ α,n i-1 ), f α S (ρ α,n i ) for i ≤ -1 if α ≤ N I , i ≥ 1 if α ≥ N I + 1, F α 0 (ρ β,n -1
) β≤NI , (ρ λ,n 0 ) λ≥NI +1 := γ α F 0 for α = 1, ..., N,

F 0 := min min β≤NI 1 γ β f β D (ρ β,n -1 ), min λ≥NI +1 1 γ λ f λ S (ρ λ,n 0 ) .
and f α S , f α D are defined in (4.71). The initial condition is given by

(4.82) ρ α,0 i :=          γ α u α 0 (|i|∆x) -u α 0 ((|i| -1)∆x) ∆x , for i ≤ -1, α = 1, ..., N I , γ α u α 0 (i∆x) -u α 0 ((i + 1)∆x) ∆x , for i ≥ 0, α = N I + 1, ..., N I + N O .
Remark 4.3 Notice that (4.80) recovers the classical Godunov scheme [START_REF] Godunov | A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics[END_REF] for i = 0, -1 while it is not standard for the two other cases i = 0, -1. Moreover we can check that independently of the chosen CFL condition, the scheme (4.80) is not monotone (at the junction, i = 0 or i = -1) if the total number of branches N ≥ 3 and is monotone if N = 2 for a suitable CFL condition.

m 0 = min min α≤NI i≤-1 min 1 γ α f α D (ρ α,0 i-1 ), 1 γ α f α S (ρ α,0 i ) , min α≥NI +1 i≥1 min 1 γ α f α D (ρ α,0 i-1 ), 1 γ α f α S (ρ α,0 i ) , min min α≤NI 1 γ α f α D (ρ α,0 -1 ), min α≥NI +1 1 γ α f α S (ρ α,0 0 ) ,
with the first part dealing with incoming branches, the second with outgoing branches and the third with the junction point. As f α (p) = min {f α S (p), f α D (p)} for any p, the latter can be rewritten as the minimal initial flux

m 0 = min    min α≤NI i≤-1 1 γ α f α (ρ α,0 i ) , min α≥NI +1 i≥0 1 γ α f α (ρ α,0 i )    .
We set for any α = 1, ..., N ρ

- α := (f α D ) -1 (γ α m 0 ), ρ + α := (f α S ) -1 (γ α m 0 ),
From Theorem 1.3 and Remark 1.4, if (1.12) is satisfied then it is easy to check that

ρ - α ≤ ρ α,n i ≤ ρ + α , for all n ≥ 0.
Then the CFL condition (1.12) can be rewritten for the densities as

(4.83) ∆x ∆t ≥ sup α=1,...,N ρ α ∈[ρ - α ,ρ + α ] |(f α ) ′ (ρ α )| .
Proof of Lemma 4.2: We distinguish two cases according to if we are either on an incoming or an outgoing branch. We investigate the incoming case. The outgoing case can be done similarly. Let us consider any α = 1, ..., N I , n ≥ 0 and i ≤ -1.

According to (4.79), for i ≤ -2 we have that:

ρ α,n+1 i -ρ α,n i ∆t = γ α ∆x∆t U α,n+1 |i| -U α,n+1 |i|-1 -U α,n |i| -U α,n |i|-1 ) = γ α ∆x min -H - α (p α,n |i|,+ ), -H + α (p α,n |i|,-) -min -H - α (p α,n |i|-1,+ ), -H + α (p α,n |i|-1,-) = 1 ∆x min f α D (ρ α,n i-1 ), f α S (ρ α,n i ) -min f α D (ρ α,n i ), f α S (ρ α,n i+1 )
where we use the numerical scheme (1.7) in the second line and (4.76) in the last line.

We then recover the result if we set the fluxes functions F α as defined in (4.81).

For the special case of i = -1, we have

ρ α,n+1 -1 -ρ α,n -1 ∆t = γ α ∆x U α,n+1 1 -U α,n 1 ∆t - U α,n+1 0 -U α,n 0 ∆t = γ α ∆x min -H - α (p α,n 1,+ ), -H + α (p α,n 1,-) -min β=1,...,N -H - β (p β,n 0,+ ) = 1 ∆x min f α D (ρ α,n -2 ), f α S (ρ α,n -1 ) -γ α min min β=1,...,NI 1 γ β f β D (ρ β,n -1 ), min λ=NI +1,...,N 1 γ λ f λ S (ρ λ,n 0 )
where in the last line we have used (4.76). Setting the flux function F α 0 for i = 0 as defined in (4.81), we also recover the result in that case.

Numerical extension for non-fixed coefficients (γ α )

Up to now, we were considering fixed coefficients γ := (γ α ) α and the flux of the scheme at the junction point at time step n ≥ 0 was

F 0 (γ) := min min β≤NI 1 γ β f β D (ρ β,n -1 ), min λ≥NI +1 1 γ λ f λ S (ρ λ,n 0 ) .
In certain situations, we want to maximize the flux F 0 (γ) for γ belonging to an admissible set Γ. Indeed we can consider the set A := argmax γ∈Γ F 0 (γ).

In the case where this set is not a singleton, we can also use a priority rule to select a single element γ * ,n of A. This defines a map (ρ β,n -1 ) β≤NI , (ρ λ,n 0 ) λ≥NI +1 → γ * ,n . At each time step n ≥ 0 we can then choose this value γ = γ * ,n in the numerical scheme (4.80), (4.81).

Simulation

In this section, we present a numerical experiment. The main goal is to check if the numerical scheme (1.7),(1.8) (or equivalently the scheme (4.80),(4.82)) is able to illustrate the propagation of shock or rarefaction waves for densities on a junction.

Settings

We consider the case of a junction with N I = 2 = N O , that is two incoming roads denoted α = 1 and 2 and two outgoing roads denoted α = 3 and 4. For the simulation, we consider that the flow functions are equal on each branch f α =: f for any α = 1, ..., 4. Moreover the function f is bi-parabolic (and only Lipschitz) as depicted on Figure 3. It is defined as follows (5.84)

f (ρ) =            (1 -k)f max ρ 2 c ρ 2 + kf max ρ c ρ, for ρ ≤ ρ c , (1 -k)f max (ρ max -ρ c ) 2 ρ 2 + (kρ c + (k -2)ρ max )f max (ρ max -ρ c ) 2 ρ - ρ max (kρ c -ρ max )f max (ρ max -ρ c ) 2 , for ρ > ρ c ,
with the jam density ρ c = 20 veh/km, the maximal ρ max = 160 veh/km, the maximal flow f max = 1000 veh/h and k = 1.5. The Hamiltonians H α for α = 1, ..., 4 are defined in (4.75) according to the flow function f given in (5.84). Because f is not C 1 at ρ c , the Hamiltonians H α do not satisfy assumption (A1). Nevertheless, we can use Remark 2.4 and the fact that f is Lipschitz to extend our results for those Hamiltonians. We also assume that the coefficients (γ α ) are all identical γ α = 1 2 for any α = 1, ..., 4.

Notice that the computations are carried out for different ∆x. In each case the time step ∆t is set to the maximal possible value satisfying the CFL condition (1.12). We consider branches of length L = 200 m and we have N b := L ∆x points on each branch such that i ∈ {0, ..., N b }.

Initial and boundary conditions

Initial conditions. In traffic flow simulations it is classical to consider Riemann problems for the vehicles densities at the junction point. We not only consider a Riemann problem at the junction but we also choose the initial data to be discontinuous (with two values of the densities (left and right)) on the outgoing branch number 3 (see Table 1 where left (resp. right ) stands for the left (resp. right) section of branch 3). We then consider initial conditions (u α 0 (x)) α=1,...,N corresponding to the primitive of the densities depicted on Figure 6 (a). We also take the initial label at the junction point such that u α 0 (0) =: u 0 (0) = 0, for any α.

We can check that the initial data (u α 0 (x)) α=1,...,N satisfy (A0). We are interested in the evolution of the densities. We stop to compute once we get a stationary final state as shown on Figure 6 (f). The values of densities and flows are summarized in Table 1. Boundary conditions. For any i ≤ N b we use the numerical scheme (1.7) for computing (U α,n i ). Nevertheless at the last grid point i = N b , we have ), the boundary conditions are easily translatable to the scheme (4.80) for the densities. For incoming roads, the flow that can enter the branch is given by the minimum between the supply of the first cell and the demand of the virtual previous cell which correspond to the value of f evaluated for the initial density on the branch ρ α 0 (see Table 1). For outgoing roads, the flow that can exit the branch is given by the minimum between the demand of the last cell H + α (p α,n N b ,-) and the supply of the virtual next cell H - α (p α,n N b ,+ ) which is the same than the supply of the last cell H - α (p α,n N b -1,+ ).

Simulation results

Vehicles labels and trajectories. Notice that here the computations are carried out for the discrete variables (U α,n i ) while the densities (ρ α,n i ) are computed in a post-treatment using (4.79). It is also possible to compute directly the densities (ρ α,n i ) according to the numerical scheme (4.80). Hereafter we consider ∆x = 5m (that corresponds to the average size of a vehicle) and ∆t = 0.16s. The numerical solution (U α,n i ) is depicted on Figure 4 (a). The vehicles trajectories are deduced by considering the iso-values of the labels surface (U α,n i ) (see Figure 4 (b)). In this case, one can observe that the congestion (described in the next part) induces a break in the velocities of the vehicles when going through the shock waves. The same is true when passing through the junction. We can also recover the gradient properties of Theorem 1.3. On Figure 5, the gradients p α,n i,+ are plotted as a function of time. We numerically check that the gradients stay between the bounds p α and p α . Propagation of waves. We describe hereafter the shock and rarefaction waves that appear from the considered initial data (see Figure 6). At the initial state (Figure 6 Figure 6 numerically illustrates the convergence of the numerical solution (ρ α,n i ) when the grid size (∆x, ∆t) goes to zero. The rate of convergence is let to further research.

2 J 3 Figure 1 :

 231 Figure 1: Junction model

  0 on (0, T ) × (0, +∞), for α = 1, . . . , N, u β =: u, for all β = 1, . . . , N u t + max β=1,...,N H - β (u β x ) = 0 on (0, T ) × {0},

  0, for i ≥ 1, α = 1, . . . , N, U β,n 0 =: U n0 , for all β = 1, . . . ) for i ≥ 0, α = 1, . . . , N.

Theorem 1 . 8 (

 18 Convergence of the numerical solution under uniqueness assumption) Assume (A0)-(A1'). Let T > 0 and ε = (∆t, ∆x) such that the CFL condition (1.12) is satisfied. If u := (u α ) α is the unique solution of (1.2)-(1.3) in the sense of Definition 3.3, then the numerical solution (U α,n i ) of (1.7)-(1.8) converges locally uniformly to u when ε goes to zero, on any compact set K ⊂ [0, T )× J, i.e.

  upper semi-continuous if and only if u = u * , u is lower semi-continuous if and only if u = u * , u is continuous if and only if u * = u * . Definition 3.3 (Viscosity solutions) i) Viscosity sub and super-solution on J T = (0, T ) × J

( 3 .

 3 65) u(0, y) ≤ u 0 (y) ≤ u(0, y), for all y ∈ J, and u(t, y) ≤ C T (1 + |y|) and u(t, y) ≥ -C T (1 + |y|), with C T = max |m 0 0 |T, |M 0 0 |T + |u 0 (0)| + L and L = max α,σ∈{±} |L α,σ |.

Subcase 1 . 1 :

 11 Pε = (tε, yε) with yε = y0 = 0

Figure 2 :

 2 Figure 2: Discretization of the branches with the nodes for U α,n |i| i∈Z

Remark 4 . 4

 44 From (1.11), (1.7) and (4.76), we can easily show that

Figure 3 :

 3 Figure 3: Graph of the function f

  + α (p α,n N b ,-), H - α (p α,n N b ,+ ) = 0, for α = 1, . . . , N,where p α,n N b ,-is defined in (1.5) and we set the boundary gradient as followsp α,n N b ,+ =    ρ α 0 γ α , if α ≤ N I , p α,n N b ,-= p α,n N b -

  Discrete labels (U α,n i ) i,n(b) Trajectories of some vehicles

Figure 4 :

 4 Figure 4: Numerical solution and vehicles trajectories

  (a)), the traffic situation on roads 1, 2 and 4 is fluid (ρ {1,2,4} 0 ≤ ρ c ) while the road 3 is congested (ρ 3 0 ≥ ρ c ). Nevertheless the demands at the

Figure 5 :

 5 Figure 5: Bounds p α and p α on the gradient

Table 1 :

 1 Values of densities and flows for initial and final states on each branch These boundary conditions are motivated by our traffic application. Indeed while they are presented for the scheme (1.7) on (U α,n i

		Initial state	Final state
	Branch	Density	Flow	Density	Flow
		(veh/km) (veh/h) (veh/km) (veh/h)
	1	15	844	90	625
	2	15	844	90	625
	3 (left)	30	962	90	625
	3 (right)	90	625	90	625
	4	5	344	10	625

1,+ , if α ≥ N I + 1.

Aknowledgements

The authors are grateful to C. Imbert for indications about the literature, M. Hustić for his suggestions to simplify certain parts of the proofs and L. Paszkowski for valuable comments on the presentation. The authors also thank the two anonymous referees for their comments and corrections that improved the manuscript. This work was partially supported by the ANR (Agence Nationale de la Recherche) through HJnet project ANR-12-BS01-0008-01.