Paola Goatin
email: paola.goatin@inria.fr

Alessandra Rizzo
email: alessandra.rizzo1@unime.it

Instabilities in generic second order traffic models with relaxation

Keywords: 2020 Mathematics Subject Classification: 35L65, 90B20 Macroscopic traffic flow models, hyperbolic systems of conservation laws, relaxation, sub-characteristic condition

We prove the existence of weak solutions for a class of second order traffic models with relaxation, without requiring the sub-characteristic stability condition to hold. Therefore, large oscillations may arise from small perturbations of equilibria, capturing the formation of stop-and-go waves observed in reality. An analysis of the corresponding travelling waves completes the study.

Introduction

Macroscopic traffic flow models have been used since several decades [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF] to describe the spatio-temporal evolution of aggregated quantities, like vehicle density and average speed. In particular, second order models [START_REF] Aw | Resurrection of "second order" models of traffic flow[END_REF][START_REF] Payne | Models of freeway traffic and control[END_REF][START_REF] Whitham | Linear and nonlinear waves[END_REF][START_REF] Zhang | A non-equilibrium traffic model devoid of gas-like behavior[END_REF], which consist in the mass conservation equation coupled with a momentum balance equation accounting for the speed dynamics, have been developed to describe specific traffic characteristics that cannot be captured by scalar models, such as scattered fundamental diagrams and the appearance and persistence of stop-and-go waves. Indeed, traffic equilibria are generally unstable [START_REF] Bando | Dynamical model of traffic congestion and numerical simulation[END_REF][START_REF] Sugiyama | Traffic jams without bottlenecks-experimental evidence for the physical mechanism of the formation of a jam[END_REF], and small perturbances in drivers' behaviour can generate so called "phantom" jams with no apparent cause. The same behaviour can be observed at a macroscopic scale considering relaxation source terms that violate the usual stability conditions [START_REF] Flynn | Self-sustained nonlinear waves in traffic flow[END_REF][START_REF] Seibold | Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models[END_REF].

In this paper, we study a Generic Second Order Model (GSOM) [START_REF] Lebacque | Generic second order traffic flow modelling[END_REF] with relaxation, which extends the well-known Aw-Rascle-Zhang system [START_REF] Aw | Resurrection of "second order" models of traffic flow[END_REF][START_REF] Zhang | A non-equilibrium traffic model devoid of gas-like behavior[END_REF], allowing for more general speed formulation. In the literature, hyperbolic systems of balance laws with relaxation source terms are usually studied under a stability condition ensuring well-posedness and convergence results as the relaxation parameter tends to zero [START_REF] Chen | Hyperbolic conservation laws with stiff relaxation terms and entropy[END_REF][START_REF] Chen | Zero relaxation and dissipation limits for hyperbolic conservation laws[END_REF]. We recall that the stability condition is related to the dissipativity of the evolution equation obtained by a first order expansion around equilibria, as observed in [START_REF] Chen | Hyperbolic conservation laws with stiff relaxation terms and entropy[END_REF]. Under this hypothesis, one can recover uniform compactness estimates, which ensure convergence to the equilibrium equation, see e.g. [START_REF] Lattanzio | The zero relaxation limit for the hydrodynamic Whitham traffic flow model[END_REF][START_REF] Li | Global solutions and zero relaxation limit for a traffic flow model[END_REF][START_REF] Goatin | The zero relaxation limit for the Aw-Rascle-Zhang traffic flow model[END_REF][START_REF] Schochet | The instant-response limit in Whitham's nonlinear traffic-flow model: uniform wellposedness and global existence[END_REF] for traffic flow applications.

Aiming at capturing the traffic flow instabilities observed in reality, here we drop the stability assumption, and we study the model properties for a fixed positive relaxation parameter. In particular, we provide a global existence proof for weak solutions based on wave-front tracking approximations and an analysis of travelling waves for the system and its Chapman-Enskog expansion. Our results are general enough to hold for a class of traffic flow models, without too specific assumptions, and they show that these models are able to capture the formation and persistence of stop-and-go waves. This paves the way to the design of control strategies to dampen traffic oscillations in a macroscopic framework, in the spirit of [START_REF] Garavello | A multiscale model for traffic regulation via autonomous vehicles[END_REF].

The presentation is organized as follows. We introduce the GSOM system with relaxation in Section 2 and we describe the corresponding Riemann problem in Section 3. Section 4 provides the compactness estimates on the sequence of approximate solutions, which are used to pass to the limit providing the existence of weak solutions in Section 5. Travelling waves are studied in Sections 6 and 7.

Model description

We consider the system

     ∂ t ρ + ∂ x (ρv) = 0, ∂ t (ρw) + ∂ x (ρwv) = ρ V (ρ) -v τ , x ∈ R, t > 0, (2.1)
defined on an invariant domain of the form

Ω := U = (ρ, w) ∈ R 2 : ρ ∈ [0, R(w max )], w ∈ [w min , w max ] , (2.2)
for some 0 < w min ≤ w max < +∞. Above, the average speed of vehicles is a function of the density ρ = ρ(t, x) and of a Lagrangian vehicle property w = w(t, x), namely v = V(ρ, w) for some speed function V : Ω → R ≥0 satisfying [START_REF] Fan | Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model[END_REF]:

V(ρ, w) ≥ 0, V(0, w) = w, (2.3a)
2V ρ (ρ, w) + ρV ρρ (ρ, w) < 0 for w > 0, (2.3b)

V w (ρ, w) > 0, (2.3c) ∀w > 0 ∃ R(w) > 0 : V(R(w), w) = 0. (2.3d)
As in [START_REF] Fan | Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model[END_REF], we observe that (2.3b) implies that Q(ρ, w) := ρV(ρ, w) is strictly concave and

V ρ (ρ, w) < 0 for w > 0, if V is a C 2 function in ρ.
We also remark that in (2.3d) we can have R(w) = R for all w > 0. Moreover, we assume the equilibrium speed

V : R + → R + is a non-increasing function (V ′ (ρ) ≤ 0) such that V (ρ) = 0 for ρ ≥ ρ max > 0.
Notice that, setting V(ρ, w) = w -p(ρ) for a suitable "pressure" function p, system (2.1) corresponds to the Aw-Rascle-Zhang (ARZ) model [START_REF] Aw | Resurrection of "second order" models of traffic flow[END_REF][START_REF] Zhang | A non-equilibrium traffic model devoid of gas-like behavior[END_REF]. We also remark that, taking w = w constant, we recover the classical Lighthill-Whitham-Richards (LWR) model [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF].

Under the above hypotheses, system (2.1) is strictly hyperbolic for ρ > 0, with eigenvalues

λ 1 (ρ, w) = V(ρ, w) + ρV ρ (ρ, w), λ 2 (ρ, w) = V(ρ, w), (2.4)
and corresponding eigenvectors

r 1 (ρ, w) = -1 0 , r 2 (ρ, w) = V w (ρ, w) -V ρ (ρ, w) , (2.5)
with the first characteristic field being genuinely non-linear and the second linearly degenerate.

The associated Riemann invariants [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF]Chapter 7.3] are

z 1 (ρ, w) = V(ρ, w), z 2 (ρ, w) = w.
Since V ρ (ρ, w) < 0 and V(0, w) = w, the range of v = V(ρ, w) is given by v ∈ [0, w] for any w ∈ [w min , w max ]. Therefore, the inverse function ρ = R(v, w) is uniquely defined in the invariant domain

W := W = (v, w) ∈ R 2 : 0 ≤ v ≤ w, w ∈ [w min , w max ] .
(2.6)

We remark that, in the (v, w) coordinates, the curves ρ = R(v, w) = const are parallel to the first bisector [START_REF] Rascle | An improved macroscopic model of traffic flow: derivation and links with the Lighthill-Whitham model[END_REF]Section 5]. Indeed, we get:

R(v, w) = const =⇒ R v + R w = 0 V(R(v, w), w) = v =⇒ V ρ R v = 1 and V ρ R w + V w = 0 Replacing R v = 1/V ρ and R w = -V w /V ρ in the first relation, we get V w = 1 on R(v, w) level curves.
We recall that weak solutions of (2.1) corresponding to an initial datum U (0, x) = U 0 (x) = (ρ 0 (x), w 0 (x)) ∈ L ∞ (R, E) are defined as

Definition 1. Let u 0 = (ρ 0 , ρ 0 w 0 ) T ∈ L 1 (R; R 2 + ) and T > 0 be given. Then u = (ρ, ρw) T ∈ C 0 [0, T ], L 1 (R; R 2 + ) is a weak solution to the Cauchy problem for (2.1) if, for all φ ∈ C 1 c ] -∞, T [ ×R; R , it holds R u 0 (x)φ(0, x) dx+ T 0 R u∂ t φ + F (u)∂ x φ (t, x) dxdt+ T 0 R G(u)φ(t, x) dxdt = 0, (2.7)
where

F (u) = ρv ρwv , G(u) =   0 ρ V (ρ) -v τ   .
For smooth solutions, system (2.1) is equivalent to:

     ∂ t ρ + ∂ x (ρv) = 0, ∂ t v + λ 1 (ρ, w)∂ x v = V (ρ) -v τ , x ∈ R, t > 0, (2.8)
and

     ∂ t ρ + ∂ x (ρv) = 0, ∂ t w + λ 2 (ρ, w)∂ x w = V (ρ) -v τ , x ∈ R, t > 0.
(2.9)

We also recall that the sub-characteristic condition [START_REF] Chen | Hyperbolic conservation laws with stiff relaxation terms and entropy[END_REF][START_REF] Chen | Zero relaxation and dissipation limits for hyperbolic conservation laws[END_REF] writes

V ρ (ρ, w) ≤ V ′ (ρ) ≤ 0 for w s.t. V(ρ, w) = V (ρ), (2.10)
see e.g. Figure 1. If (2.10) is satisfied, it is known [START_REF] Goatin | The zero relaxation limit for the Aw-Rascle-Zhang traffic flow model[END_REF] that (weak entropic) solutions to (2.1) converge to the (weak entropic) solutions of

∂ t ρ + ∂ x (ρV (ρ)) = 0, x ∈ R, t > 0, (2.11)
for τ ↘ 0 or t → ∞ (using the change of variables t = t/τ and x = x/τ ).

ρ v 0 R V max V (ρ) V min ρ ρv 0 R ρV (ρ) ρ v 0 R V max V (ρ) V min ρ ρv 0 R ρV (ρ)
Figure 1: Speed-density (left) and flow-density (right) fundamental diagrams for the GSOM model (2.1) corresponding to the choices (satisfying the sub-characteristic condition (2.10)):

Top: V(ρ, w) = w -αρ, with w ∈ [V min , V max ] and V (ρ) = β(R -ρ) with α ≥ β > 0; Bottom: V(ρ, w) = w(R -αρ), with w ∈ [V min , V max ] and V (ρ) = β(R -ρ) with α ≥ β > 0. The equilibrium curve V (ρ) = V (R(v, w)) is implicitly defined by V (R(v, ϕ(v))) = v.
Differentiating w.r.to v we obtain

V ′ (ρ) R v + R w ϕ ′ (v) = 1, which gives ϕ ′ (v) = 1 R w 1 V ′ (ρ) -R v = 1 V w 1 - V ρ V ′ (ρ) . Since V w > 0, we get ϕ ′ (v) ≤ 0 if and only if V ρ ≤ V ′ (ρ), see (2.10).
We are interested in investigating the behaviour of (2.1) when (2.10) is violated. Therefore, we assume that there exists a point v cr , resp.

w cr = ϕ(v cr ), ρ cr = R(v cr , w cr ), such that ϕ ′ (v cr ) = 0, i.e. V ρ (ρ cr , w cr ) = V ρ (ρ cr , ϕ(v cr )) = V ′ (ρ cr ).
(2.12)

We will distinguish the two cases:

ϕ ′ (v) < 0 if v < v cr , ϕ ′ (v) > 0 if v > v cr , (2.13a
)

and ϕ ′ (v) > 0 if v < v cr , ϕ ′ (v) < 0 if v > v cr . (2.13b)
Moreover, we observe that, for v > v cr for case (2.13a), resp. v < v cr for case (2.13b),

ϕ ′ (v) < 1 if V ′ (R(v, ϕ(v))) > V ρ (R(v, ϕ(v)), ϕ(v)) + V ′ (R(v, ϕ(v)))V w (R(v, ϕ(v)), ϕ(v)).
In case (2.13a), the equilibrium curve intersects the vacuum line v = w at v = ŵ implicitly defined by V(0, ŵ) = V (0), see Figure 2, left. We remark that in this case all invariant domains must include vacuum states.

In case (2.13b), the equilibrium curve intersects the congestion line v = 0 at ŵ implicitly defined by V(R( ŵ), ŵ) = 0 = V (ρ max ), see Figure 2, right. We remark that in this case all invariant domains must include congestion states. We also define ρ, v such that

v = V(ρ, ŵ) = V (ρ). (2.14) v w v ŵ v cr w = ϕ(v) 0 w max v w max w min w cr E(0, ρ) v w v ŵ v cr w = ϕ(v) 0 w max w max w min w cr E(ρ, R( ŵ)

The Riemann problem

Let us consider (2.1) with initial data

(ρ, w)(0, x) = U L = (ρ L , w L ) if x < 0, U R = (ρ R , w R ) if x > 0, (3.1)
and set v L = V(ρ L , w L ), v R = V(ρ R , w R ).
We observe that, if the subcharacteristic condition (2.10) is satisfied, weak solutions of (2.1), (3.1) tend to weak solutions of the Riemann problem

           ∂ t ρ + ∂ x (ρV (ρ)) = 0, x ∈ R, t > 0, ρ(0, x) = ρL = if x < 0, ρR if x > 0, as τ ↘ 0 (or t → ∞), where ρL,R = R(v L,R , wL,R ) are the projection of U L , U R on the equilibrium curve implicitly defined by vL,R = V R(v L,R , vL,R + w L,R -v L,R ) , wL,R = vL,R + w L,R -v L,R , see Figure 3. v w U L U R V (ρ) 0 w max w min w max w min ρ ρv 0 U L U R R(w max ) R(w min )
Figure 3: Illustration of the limit behaviour of the solutions to the Riemann problem (2.1), (3.1) under the subcharacteristic condition (2.10).

The homogeneous Riemann problem

We recall here the detailed construction of weak (entropy) solutions of problem

∂ t ρ + ∂ x (ρv) = 0, ∂ t (ρw) + ∂ x (ρwv) = 0, x ∈ R, t > 0, (3.2)
with initial conditions (3.1).

In general, the left state U L is connected to U M = (ρ M , w M ) by a first family wave (rarefaction or shock), i.e.

z 2 (ρ L , w L ) = w L = w M = z 2 (ρ M , w M ), and U M is connected to U R by a contact-discontinuity with z 1 (ρ M , w M ) = v M = V(ρ M , w M ) = v R = z 1 (ρ R , w R ). Thus, the intermediate state U M is identified by the system of equations        w M = w L , v M = v R , ρ M = R(v R , w L ). If w L ≤ v R , we set ρ M = 0, meaning that U M corresponds to the vacuum.
The propagation speed σ of a shock wave between two states U -and U + is given by the Rankine-Hugoniot condition

σ(U -, U + ) = ρ + v + -ρ -v - ρ + -ρ - . (3.3)
In this work, we will rely on the following definition of solutions of (2.1), (3.1), see also [START_REF] Fan | Data-fitted generic second order macroscopic traffic flow models[END_REF]. Since the presence of vacuum states prevents uniqueness even if an entropy condition is enforced, we refer to [START_REF] Andreianov | A second-order model for vehicular traffics with local point constraints on the flow[END_REF] for an alternative construction. Definition 2. For any U L , U R ∈ Ω, the Riemann solver

RS : Ω × Ω → C 0 ]0, +∞[ ; L 1 loc (R; Ω) , (U L , U R ) → RS(U L , U R )
is defined as follows:

1. If (v R , w R ) ∈ W and v L ≥ v R , then RS(U L , U R )(t, x) =        U L if x < σ(U L , U M )t, U M if σ(U L , U M )t < x < v R t, U R if x > v R t,
with σ defined in (3.3).

If

(v L , w L ) ∈ W \ {v = w}, and v L ≤ v R < w L , then RS(U L , U R )(t, x) =            U L if x < λ 1 (ρ L , w L )t, Û if λ 1 (ρ L , w L )t < x < λ 1 (ρ M , w M )t, U M if λ 1 (ρ M , w M )t < x < v R t, U R if x > v R t, with Û = (ρ, w L ) solving λ 1 (ρ, w L ) = x t . 3. If (v L , w L ) ∈ W \ {v = w} and w L ≤ v R , then RS(U L , U R )(t, x) =            U L if x < λ 1 (ρ L , w L )t, Û if λ 1 (ρ L , w L )t < x < λ 1 (0, w L )t, U M if λ 1 (0, w L )t < x < v R t, U R if x > v R t, with Û = (ρ, w L ) solving λ 1 (ρ, w L ) = x t and U M = (0, w L ). 4. If v L = w L and v R = w R , then RS(U L , U R )(t, x) = U L if x < 0, U R if x > 0. (3.4)

The relaxation Riemann problem

We now turn to the solution of the relaxation step, consisting of the system of ODEs

     ∂ t ρ = 0, ∂ t v = V (ρ) -v τ = ∂ t w, x ∈ R, t > 0, (3.5)
with Riemann-like initial datum (3.1), whose solutions are given by

ρ(t, x) = ρ(0, x), v(t, x) = V (ρ(0, x)) + v(0, x) -V (ρ(0, x)) e -t/τ , x ∈ R, t > 0. (3.6)
Invariant domains for this step must include segments parallel to the line v = w joining any point of the domain with the equilibrium curve.

From the above analysis, we easily see that the invariant domains for the relaxed Riemann problem (2.1), (3.1) must have edges parallel to the Riemann invariant level curves and contain properly a portion of the equilibrium curve, see Figure 2 and definitions (4.1), (4.2) below.

Wave-front tracking approximations

The construction of wave-front tracking (WFT) approximate solutions is based on a two step process, that successively solves the homogeneous system (3.2) for a piece-wise constant initial datum, and then integrates the source term contained in the ODE (3.5), see [START_REF] Goatin | The zero relaxation limit for the Aw-Rascle-Zhang traffic flow model[END_REF] and references therein.

To define the invariant domains, we need to distinguish the two cases illustrated in Figure 2, namely the case in which the equilibrium curve w = ϕ(v) is a convex function and the case in which it is concave. Under the hypothesis (2.13a), invariant domains take the form

E = E[0, ρ] = U = (ρ, w) : ρ ∈ [0, ρ], V(ρ, w) ∈ [v, v M ], w ∈ [w m , w] , (4.1)
with ρ ≥ ρ, v M ≥ v, w m ≤ w cr and v = V(ρ, w) = V (ρ), where ρ, v and w cr are chosen as in (2.12), (2.14). Under the hypothesis (2.13b), invariant domains take the form

E = E[ρ, R( ŵ)] = U = (ρ, w) : ρ ∈ [ρ, R(w M )], V(ρ, w) ∈ [0, v], w ∈ [ w, w M ] , (4.2)
with ρ ≤ ρ, w M ≥ w cr and v = V(ρ, w) = V (ρ), where ρ and w cr are chosen as in (2.12), (2.14).

Let us consider initial data U 0 = (ρ 0 , w 0 ) : R → E such that TV(w 0 ) + TV(V(ρ 0 , w 0 )) < +∞. For any T > 0, we consider a sequence of time-steps ∆t ν > 0, ν ∈ N such that ∆t ν → 0 and we partition the interval [0, T [ in intervals of the form [n∆t ν , (n + 1)∆t ν [, n ∈ N. We denote with U ν (t, x) = (ρ ν , w ν )(t, x), t ∈ [0, T ], x ∈ R, the sequence of WFT approximate solutions of (2.1) constructed as detailed below:

1. Define a sequence of piece-wise constant functions

U ν 0 = (ρ ν 0 , w ν 0 ) ∈ E satisfying TV(V(ρ ν 0 , w ν 0 )) ≤ TV(V(ρ 0 , w 0 )) , V(ρ ν 0 , w ν 0 ) -V(ρ 0 , w 0 ) L ∞ ≤ 1 ν , ρ ν 0 -ρ 0 L 1 ≤ 1 ν , TV(w ν 0 ) ≤ TV(w 0 ) , w ν 0 -w 0 L ∞ ≤ 1 ν , w ν 0 -w 0 L 1 ≤ 1 ν ,
and, for each ν ∈ N, the piece-wise constant function U ν 0 has a finite number of discontinuities.

2. Solve the homogeneous system (3.2) corresponding to the Riemann problems arising at discontinuities for t ∈ [0, ∆t ν [ using the WFT method and name U ν (t, .), t ∈ [0, ∆t ν [, the corresponding piece-wise constant function [START_REF] Baiti | The semigroup generated by a Temple class system with large data[END_REF].

3. At t = ∆t ν , we define

ρ ν (∆t ν , •) = ρ ν (∆t ν -, •) , w ν (∆t ν , •) = w ν (∆t ν -, •) + ∆t ν τ V (ρ ν (∆t ν , •)) -V(U ν (∆t ν -, •)) .
Note that ρ is conserved during this second step, while w (and v = V(ρ, w)) is updated according to (3.5).

Observe that we also have

V(U ν (∆t ν , •)) = V(U ν (∆t ν -, •)) + ∆t ν τ V (ρ ν (∆t ν , •)) -V(U ν (∆t ν -, •)) , (4.3)
see (3.5).

4. Treat U ν (∆t ν , .) as a new piece-wise constant initial condition and repeat the previous steps 2-3 to define the solution U ν (t, .) for each t ∈ [0, T ], for any T > 0 fixed.

L ∞ estimates

Proposition 1. For ∆t ν ≤ τ / max 1, ∥V w ∥ ∞ , the set E is an invariant domain for the proposed WFT scheme.

Proof. Let us assume U 0 (x) ∈ E for all x ∈ R. System (3.2) being of Temple class, Step 2 above clearly preserves the inequalities on the Riemann invariants w, V(ρ, w).

Concerning the relaxation Step 3., if v -≥ V (ρ) we have that

V (ρ) = 1 - ∆t τ V (ρ) + ∆t τ V (ρ) ≤ 1 - ∆t τ v -+ ∆t τ V (ρ) = v + ≤ v -,
where we used (4.3) and the hypothesis ∆t ≤ τ . Hence,

v + ∈ [v, v M ] (resp. v + ∈ [0, v]).
On the other hand

w + = w -+ ∆t τ V (ρ) -v -≤ w - and, developing V(ρ, w -) = V(ρ, ϕ(V (ρ))) + V w (ρ, w) w --ϕ(V (ρ)) = V (ρ) + V w (ρ, w) w --ϕ(V (ρ))
for some w,

w + = w -- ∆t τ V(ρ, w -) + ∆t τ V (ρ) = w -1 - ∆t τ V w (ρ, w) + ∆t τ V w (ρ, w) ϕ(V (ρ)) ≥ ϕ(V (ρ)),
where we used ∆t ≤ τ /∥V w ∥ ∞ . Therefore,

w + ∈ [w m , w] (resp. w + ∈ [ w, w M ]).
The case v -< V (ρ) is treated analogously.

BV estimates

Proposition 2. Let us assume there exists c > 0 such that V ρ (ρ, w) ≤ -c for all (ρ, w) ∈ E.

For any t > 0, the total variation of the Riemann invariants

TV(W (U ν (t, •))) := TV(w ν (t, •)) + TV(V(U ν (t, •)))
of the WFT approximate solution satisfies the uniform bound

TV(W (U ν (t, •))) ≤ TV(W (U 0 )) e Kt/τ , (4.4)
where

K := 2 c V ′ ∞ max 1, ∥V w ∥ ∞ .
In particular, when τ ↘ 0, the estimate (4.4) blows up.

Proof. Notice that, when we solve the homogeneous system (3.2), the total variation of both Riemann invariants is non-increasing in time since we are dealing with a Temple-class system. We thus focus on the evolution of the total variation at step 3, corresponding to (3.5). We recall that, at t n = n∆t ν and dropping the index ν for simplicity,

ρ(t n +, x) = ρ(t n -, x), v(t n +, x) = v(t n -, x) + ∆t τ (V (ρ) -v)(t n -, x), w(t n +, x) = w(t n -, x) + ∆t τ (V (ρ) -v)(t n -, x).
Therefore, at each jump in the approximate solution, we have

|v + r -v + l | = v - r -v - l + ∆t τ V (ρ r ) -V (ρ l ) - ∆t τ (v - r -v - l ) = 1 - ∆t τ (v - r -v - l ) + ∆t τ V (ρ r ) -V (ρ l ) ≤ 1 - ∆t τ v - r -v - l + ∆t τ V ′ ∞ |ρ r -ρ l | .
From the relation ρ = R(v, w), we get

|ρ r -ρ l | ≤ ∥R v ∥ ∞ v - r -v - l + ∥R w ∥ ∞ w - r -w - l ≤ 1 c v - r -v - l + ∥V w ∥ ∞ c w - r -w - l by the relations R v = 1/V ρ and R w = -V w /V ρ . Hence |v + r -v + l | ≤   1 + ∆t τ V ′ ∞ c -1   v - r -v - l + ∆t τ V ′ ∞ ∥V w ∥ ∞ c w - r -w - l . (4.5)
Similar estimates lead to

|w + r -w + l | ≤ ∆t τ V ′ ∞ c + 1 v - r -v - l + 1 + ∆t τ V ′ ∞ ∥V w ∥ ∞ c w - r -w - l . (4.6)
Summing (4.5) and (4.6), we obtain

TV(W (U (t n +, •))) ≤ 1 + K ∆t τ TV(W (U (t n -, •)) with K := 2 c max V ′ ∞ , V ′ ∞ ∥V w ∥ ∞ ,
which gives (4.4).

L 1 Lipschitz continuity in time

To get the BV estimates in space and time, following [START_REF] Bressan | Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem[END_REF], see also [START_REF] Goatin | The zero relaxation limit for the Aw-Rascle-Zhang traffic flow model[END_REF]Lemma 2.3], we need the next result.

Proposition 3. Under the same assumption as in Proposition 2, let us assume TV(W (U 0 )) = M < +∞. Then there exist a constant C M > 0 independent of τ and a constant L τ > 0, such that, ∀ a < b and ∀ 0 ≤ s < t, the WFT approximate solutions satisfy

a |ρ ν (t, x) -ρ ν (s, x)| ≤ C M e Kt/τ (t -s), (4.7)
b a |w ν (t, x) -w ν (s, x)| ≤ C M e Kt/τ + L τ (t -s + ∆t). (4.8)
In particular, both estimates above blow up as τ ↘ 0.

Proof. Let s and t ∈ R such that 0 ≤ s < t. If there are no time-steps between s and t, (4.7) and (4.8) are true for any L τ ≥ 0, as a direct application of Temple-class system properties, see [START_REF] Goatin | The zero relaxation limit for the Aw-Rascle-Zhang traffic flow model[END_REF]. We suppose now that there are N + 1 time-steps between s and t:

s ≤ k∆t ≤ (k + 1)∆t ≤ • • • ≤ (N + k)∆t ≤ t, so that N ∆t ≤ t -s.
Let a < b given and x ∈ ]a, b[. We can then estimate

ρ ν (t, x) -ρ ν (s, x) = ρ ν (t, x) -ρ ν ((N + k)∆t, x) + k+N -1 i=k [ρ ν ((i + 1)∆t, x) -ρ ν (i∆t, x)] + ρ ν (k∆t, x) -ρ ν (s, x) ≤ ρ ν (t, x) -ρ ν ((N + k)∆t, x) + k+N -1 i=k ρ ν ((i + 1)∆t, x) -ρ ν (i∆t, x) + ρ ν (k∆t, x) -ρ ν (s, x) .
Since ρ ν does not change through the splitting process, we can apply the previous property between two consecutive time-steps to obtain

a ρ ν (t, x) -ρ ν (s, x) dx ≤ C M e Kt/τ   t -(N + k)∆t + N +k-1 i=k (i + 1)∆t -i∆t + k∆t -s   = C M e Kt/τ t -(N + k)∆t + (N + k)∆t -k∆t + k∆t -s = C M e Kt/τ (t -s).
Concerning w ν , which is modified at each splitting step, we have to consider an additional term:

N +k i=k b a w ν (i∆t+, x) -w ν (i∆t-, x) dx = ∆t τ N +k i=k b a (V (ρ ν ) -v ν )(i∆t-, x) dx ≤ ∆t τ (N + 1)(b -a) sup U ν ∈E V (ρ ν ) -v ν ≤ L τ (t -s + ∆t), with L τ = b -a τ sup U ν ∈E V (ρ ν ) -v ν .
Summing to the other terms, we get (4.8).

Existence of weak solutions

The uniform bounds derived in Section 4 allow to apply Helly's Theorem to state the existence of a subsequence of WFT approximate solutions, still denoted by {W ν } ν , converging in L 1 loc to a function W . It now remains to prove that u = u(W ) is a weak solution of (2.1).

Theorem 1. Let U 0 : R → E with TV(W (U 0 )) < +∞ and let W be the limit function of the sequence {W ν } ν of WFT approximate solutions as ν → ∞. Then u = u(W ) is a weak solution of (2.1) with initial data u 0 = u(W (U 0 )) in the sense of Definition 1.

Proof. Let T > 0 be a given finite time horizon and consider φ ∈ [START_REF] Crasta | Viscosity solutions and uniqueness for systems of inhomogeneous balance laws[END_REF], we observe that

C 1 c ] -∞, T [ ×R; R . We define N ν so that T = N ν ∆t ν + β ν , β ν ∈ [0, ∆t ν [. Following
(k+1)∆t ν k∆t ν R u(W ν )∂ t φ + F (u(W ν ))∂ x φ (t, x) dxdt = R φ((k + 1)∆t ν , x)u(W ν ((k + 1)∆t ν -, x)) dx - R φ(k∆t ν , x)u(W ν (k∆t ν +, x)) dx,
since u ν := u(W ν ) are weak solutions of the homogeneous system (3.2) in each interval ]k∆t, (k + 1)∆t[ ×R by construction. Therefore, remembering that ρ ν (k∆t+) = ρ ν (k∆t-), we get

T 0 R u(W ν )∂ t φ + F (u(W ν ))∂ x φ (t, x) dxdt =   N ν -1 k=0 (k+1)∆t ν k∆t ν + T N ν ∆t ν   R u(W ν )∂ t φ + F (u(W ν ))∂ x φ (t, x) dxdt = N ν -1 k=0 R φ((k + 1)∆t ν , x)u(W ν ((k + 1)∆t ν -, x)) dx - N ν k=0 R φ(k∆t ν , x) u(W ν (k∆t ν -, x)) + ∆tG(u(W ν (k∆t ν -, x))) dx = - N ν k=0 ∆t R φ(k∆t ν , x)G(u(W ν (k∆t ν -, x))) dx - R φ(0, x)u(W ν 0 (x)) dx
Passing to the limit as ν → +∞ in the above equality, by Lebesgue dominated convergence theorem we obtain

T 0 R u(W )∂ t φ + F (u(W ))∂ x φ (t, x) dxdt = - T 0 R φ(t, x)G(u(W (t, x))) dx - R φ(0, x)u(W 0 (x)) dx ,
which concludes the proof.

Remark 1. In the present setting, the violation of the subcharacteristic condition (2.10) contradicts the existence of an entropy, entropy-flux pairs for system (2.1), i.e. functions η, q : R 2 + → R such that ∇η T (z)DF (z) = ∇ T q(z), ∇η T (z)G(z) ≤ 0, for z ∈ R 2 + . We refer to [7, Theorem 2.1] for details.

Traveling wave analysis

In this section, we aim at investigating the behaviour of travelling waves for system (2.1) violating the stability condition (2.10). We refer to [START_REF] Flynn | Self-sustained nonlinear waves in traffic flow[END_REF][START_REF] Seibold | Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models[END_REF][START_REF] Mascia | L 1 nonlinear stability of traveling waves for a hyperbolic system with relaxation[END_REF] for a similar analysis conducted on other models. We look for traveling wave solutions of (2.8) of the form (ρ(ξ), v(ξ)), where ξ = (x-σt)/τ :

-σρ ′ + (ρv) ′ = 0, (6.1a)

-σv ′ + λ 1 (ρ, w)v ′ = V (ρ) -v. (6.1b)
Integrating (6.1a), we get

ρv = σρ + m =⇒ ρ = m v -σ = m V(ρ, w) -σ ⇐⇒ V(ρ, w) = σ + m ρ .
for some m ∈ R. From (6.1b), we then get

v ′ = V (ρ) -v λ 1 (ρ, w) -σ = (v -σ)(V -v) (v -σ) 2 + mV ρ or ρ ′ = ρ σ -v V (ρ) -v λ 1 (ρ, w) -σ . (6.2)
Note that, if (2.10) holds, then the denominator λ 1 (ρ, w) -σ ̸ = 0, otherwise there exists a sonic point v σ such that

v σ -σ = ± -mV ρ =⇒ v σ = σ ± -mV ρ .
Since V ρ ≤ 0, we need to take m > 0. We regularize (6.2) requiring v σ = V (m/(v σ -σ)).

In a periodic setting, two states (ρ ± , v ± ) will be connected alternatively by a jump discontinuity satisfying the Rankine-Hugoniot condition

σ = ρ + v + -ρ -v - ρ + -ρ - , (6.3)
and a solution of (6.2). We recall that entropy admissible jump discontinuities must satisfy v -≥ v + . In particular, if v -> v + (shock) we must have w -= w + . Therefore, for the solution of (6.2) going from v + to v -, we must have v ′ ≥ 0. Besides, we have m = ρ ± (v ± -σ) ≥ 0 since v ± ≥ σ. This confirms the existence of a sonic point v σ . (Note that in the case of a contact discontinuity, it holds v -= v + = σ and therefore v ′ = 0).

Let us analyze (6.2) better. We can rewrite the denominator as

(v -σ) 2 -( -mV ρ ) 2 = (v -σ) --mV ρ (v -σ) + -mV ρ .
Since v > σ, the denominator has a unique zero of multiplicity one at

v σ = σ + -mV ρ .
Solution of (6.2) being increasing along ξ, we have

v + < σ + -mV ρ =⇒ (v + -sσ) 2 -( -mV ρ ) 2 < 0, v -> σ + -mV ρ =⇒ (v --σ) 2 -( -mV ρ ) 2 > 0.
Therefore, to have v ′ ≥ 0 in (6.2), the numerator must satisfy

V (ρ) ≤ v = V(ρ, w ± ) for ρ σ ≤ ρ ≤ ρ + , V (ρ) ≥ v = V(ρ, w ± ) for ρ -≤ ρ ≤ ρ σ . In particular, V (ρ σ ) = v σ = σ + m ρ σ . See Figure 4.
To summarize, the recipe to construct a periodic wave oscillating between states (ρ ± , v ± ) (refereed to as "jamiton" in [START_REF] Flynn | Self-sustained nonlinear waves in traffic flow[END_REF][START_REF] Seibold | Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models[END_REF]) is the following: 1. For a prescribed downstream state (ρ

+ , v + ) such that v + = V(ρ + , w + ) > V (ρ + ) compute ρ σ implicitly defined by V (ρ σ ) = V(ρ σ , w + ) = v σ . Then, from the identities        v + = σ + m ρ + , v σ = σ + m ρ σ , we recover        m = ρ σ ρ + ρ + -ρ σ (v σ -v + ), σ = v + ρ + ρ + -ρ σ + v σ ρ σ ρ σ -ρ + . ρ v ρ σ ρ + ρ - 0 V(ρ, w ± ) V (ρ) ρ ρv ρ σ ρ + ρ - 0 ρV (ρ)
ρV(ρ, w ± ) Figure 4: Speed-density (left) and flow-density (right) representations of the curves involved in the construction of the traveling wave profiles, where we set

V(ρ, w) = w -ρ, V (ρ) = V max 1 -exp(C(1 -R/ρ)) .
2. Determine the upstream state (ρ -, v -), with v -= V(ρ -, w + ) using the Rankine-Hugoniot condition (6.3).

Integrate

(6.2) from v(0) = v + to v(π) = v -.
The density is then given by

ρ(ξ) = m v(ξ) -σ .
The period of the travelling wave is given by π.

Chapman-Enskog expansion

Another classical way of investigating the stability of equilibria of relaxation systems is to perform a formal expansion with respect to the relaxation parameter, see e.g. [START_REF] Bedjaoui | On the validity of Chapman-Enskog expansions for shock waves with small strength[END_REF][START_REF] Chen | Hyperbolic conservation laws with stiff relaxation terms and entropy[END_REF][START_REF] Corli | Viscous profiles in models of collective movement with negative diffusivity[END_REF].

We consider the equations in (ρ, v) coordinates (2.8), which we recall here ∂ t ρ + ∂ x (ρv) = 0, (7.1a)

∂ t v + λ 1 (ρ, w)∂ x v = V (ρ) -v τ . (7.1b)
Assuming smooth solutions, from (7.1b) we recover: Replacing this last expression in (7.1a) and truncating to the first order we get

v = V (ρ) -τ ∂ t v + λ 1 (ρ, w)∂ x v = V (ρ) -τ V ′ (ρ)∂ t ρ + λ 1 (ρ, w)∂ x v .
∂ t ρ + ∂ x (ρV (ρ)) = τ ∂ x ρV ′ (ρ) λ 1 (ρ, w) -(ρV (ρ)) ′ ∂ x ρ . (7.2)
Since V ′ (ρ) ≤ 0, the diffusive equation (7.2) is stable if and only if λ 1 (ρ, w) ≤ (ρV (ρ)) ′ , i.e. (2.10) holds. Alternatively, the formal Chapman-Enskog expansion in v = ∞ k=0 τ k v k leads to:

∂ t ρ + ∞ k=0 τ k ∂ x (ρv k ) = 0, ∞ k=0 τ k ∂ t v k + λ 1 (ρ, w) ∞ k=0 τ k ∂ x v k = V (ρ) τ - ∞ k=0 τ k-1 v k .
The second identity yields v 0 = V (ρ),

∂ t v 0 + λ 1 (ρ, w)∂ x v 0 = -v 1 , ∂ t v k + λ 1 (ρ, w)∂ x v k = -v k+1 .
Replacing in the first identity we get

∂ t ρ + ∂ x (ρV (ρ)) = ∂ t ρ + ∂ x (ρv 0 ) = - ∞ k=1 τ k ∂ x (ρv k ) = ∞ k=0 τ k+1 ∂ x ρ ∂ t v k + λ 1 (ρ, w)∂ x v k ,
which, to the first order k = 0, is

∂ t ρ + ∂ x (ρV (ρ)) = τ ∂ x ρ ∂ t v 0 + λ 1 (ρ, w)∂ x v 0 = τ ∂ x ρV ′ (ρ) ∂ t ρ + λ 1 (ρ, w)∂ x ρ = τ ∂ x ρV ′ (ρ) λ 1 (ρ, w) -(ρV (ρ)) ′ ∂ x ρ (7.3)
as in (7.2).

Considering travelling wave solutions of (7.2) of the form ρ(t, x) = ρ(ξ), where ξ = (x -σt)/τ , we get -σρ ′ + (ρV (ρ)) ′ = ρV ′ (ρ) λ 1 (ρ, w) -(ρV (ρ)) ′ ρ ′ ′ . (7.4)

By integration, we obtain

ρ ′ = ρ(V (ρ) -σ) -m ρV ′ (ρ) λ 1 (ρ, w) -(ρV (ρ)) ′ , (7.5)
which is coherent with (6.2) setting m = ρ(σ -v) and assuming σ ∼ (ρV (ρ)) ′ , which gives ρV ′ (ρ) = σ -V (ρ) ∼ σ -v.

Figure 2 :

 2 Figure 2: Examples of non-monotone equilibrium curves w = ϕ(v) and minimal invariant domains E (continuous red line) and general invariant domains (dashed red line): left, convex case (2.13a); right, concave case (2.13b).

  b

  b

4 .

 4 The total number of vehicles involved is then given by

Figure 5 :

 5 Figure 5: Density and speed profiles of the jamiton solution of (6.1).

Substituting ( 7 .

 7 1a), i.e. ∂ t ρ = -∂ x (ρv) = -∂ x (ρV (ρ)) + O(τ ) and ∂ x v = ∂ x V (ρ) + O(τ ), we get v = V (ρ) -τ λ 1 (ρ, w)∂ x V (ρ) -V ′ (ρ)∂ x (ρV (ρ)) + O(τ 2 ).

Acknowledgements

This research was partially conducted during a three months visit of AR at the Inria Centre of Université Côte d'Azur. We also acknowledge the financial support of GNFM of the Istituto Nazionale di Alta Matematica. AR's research was partially supported by INdAM-GNFM through "Progetto Giovani GNFM 2023" entitled "p-systems in elastodinamica non lineare".