Oana Hotescu
email: oana.hotescu@enseeiht.fr

Katia Jaffrès-Runser

Jean-Luc Scharbarg
email: jean-luc.scharbarg@enseeiht.fr

Christian Fraboul
email: christian.fraboul@enseeiht.fr

Multiplexing Avionics and additional flows on a QoS-aware AFDX network

AFDX is the standard switched Ethernet solution for the transmission of avionics flows. Today's AFDX deployments in commercial aircrafts are lightly loaded to ensure the determinism of control and command operations. Manufacturers envision to take advantage of the remaining AFDX bandwidth to transmit additional non avionics flows (video, audio, service). These flows must not compromise the in-time transmission of avionics ones: constraints on jitter at source end system and end-to-end latency have to be insured for each avionics flow. In this paper, we investigate the scheduling of avionics and additional flows, mainly at the end system level. We show that an event-triggered strategy is better than a time-triggered one for additional flows at source level, but it might compromise the jitter constraint of avionics flows and increase the end-to-end latency of additional ones. We consider two time-triggered scheduling strategies, i.e. an optimal one and a simpler one based on a heuristic. We show that the later one performs nearly as well as the former one and that, for both of them, the difference with an event-triggered strategy at source level is limited and can be statically bounded.

I. INTRODUCTION

The drastic increase of embedded systems and avionics functions in civil avionics has led, at the turn of the century, to the design of the now de-facto networking standard named AFDX [START_REF]ARINC specification 664 P7-1, Aircraft Data Network, Part 7: Avionics Full Duplex Switched Ethernet (AFDX) Network[END_REF] (Avionics Full-DupleX switched Ethernet). This standard offers a deterministic networking service to a large amount of control and command flows. These flows are carried using virtual links (VL) whose bandwidth is preserved using traffic policing at the network entrance. This policing is possible since all flows are generated with a maximum jitter by the source end system.

Several approaches have been proposed to upper bound end-to-end delays, including network calculus and trajectory approach [START_REF] Charara | Methods for bounding end-to-end delays on an AFDX network[END_REF], [START_REF] Bauer | Applying and optimizing trajectory approach for performance evaluation of AFDX avionics network[END_REF]. The bound on the end-to-end delay provided by these methods is shown to be relatively pessimistic in [START_REF] Bauer | Worst-case end-to-end delay analysis of an avionics AFDX network[END_REF], resulting into a lightly loaded network [START_REF] Charara | Methods for bounding end-to-end delays on an AFDX network[END_REF].

Therefore manufacturers envision the network to be shared by regular control command avionics flows and less critical video, audio or service flows related to aircraft monitoring and maintenance operations. Therefore, it is of course mandatory to guarantee that delay as well as jitter constraints are still satisfied for avionics flows. Moreover, if video or audio flows are considered, their timely streaming at the destination is only possible if appropriate guarantees on their end-to-end delay are offered. Thus, advanced scheduling mechanisms have to be implemented both at the switch and end system levels.

At the switch egress ports, Static Priority Queuing (SPQ) scheduling policy with 2 priority levels is currently deployed. It is suitable for the case where a single type of additional flows is carried by the network. Adding more types of flows necessitates the use of advanced scheduling policies such as Deficit Round Robin (DRR) or Weighted Round Robin (WRR). In this paper, we focus on a single type of additional flows.

At source end system, avionics flows have to be regulated as to ensure the fact that they don't use more bandwidth than planned. This regulation requires the jitter at end system output to be upper bounded to a value specified by the standard (typically 500 µs). The scheduling policy implemented in end systems has to guarantee that this constraint is not violated. This paper investigates different candidate scheduling policies for end systems.

To the best of our knowledge, no previous work has been dedicated to the optimization of source scheduling for different levels of QoS in an AFDX network. Work has been dedicated in the last years to the problem of finding an optimal scheduling schema in synchronous time-triggered networks, such as TTEthernet [START_REF] Tȃmaş-Selicean | Design optimization of TTEthernet-based distributed real-time systems[END_REF], [START_REF] Craciunas | Optimal static scheduling of real-time tasks on distributed time-triggered networked systems[END_REF], [START_REF] Craciunas | Combined task-and network-level scheduling for distributed time-triggered systems[END_REF] and TSN [START_REF] Craciunas | Scheduling real-time communication in IEEE 802.1 Qbv time sensitive networks[END_REF]. The main originality of our problem is related to the impossible synchronization of AFDX end systems and switches. Thus, an event-triggered strategy has to be used at switch level.

The main contribution of the paper is to propose and compare event-triggered as well as time-triggered strategies for the scheduling of mixed flows at source end system.

Section II presents the AFDX network with avionics and additional flows. Section III details the proposed scheduling strategies for end systems. These strategies are compared on a realistic case study in Sections IV and V. Section VI gives conclusions and directions for future works.

II. PROBLEM STATEMENT

A. The AFDX network

Avionics Full DupleX switched Ethernet(AFDX) [START_REF]ARINC specification 664 P7-1, Aircraft Data Network, Part 7: Avionics Full Duplex Switched Ethernet (AFDX) Network[END_REF] is a switched Ethernet-based network that has been designed for safety-critical avionics applications. An AFDX network is composed of a set of end systems interconnected by switches and physical links. Each end system is connected to a single switch port by a unique physical link. Data exchange between end systems can occur only through virtual communication channels called Virtual Links (VLs). A VL connects one source end system to one or more destination ones. VLs share the network bandwidth. Thus, each virtual link is assigned an upper bounded bandwidth specified by a Bandwidth Allocation Gap (BAG) and a maximum frame size S max . The BAG represents the minimum time interval between two consecutive frames transmitted through the VL. The BAG value is given in the set of powers of 2: {1, 2, 4, 8, 16, 32, 64, 128} milliseconds. For a given VL, at most one frame of maximum allowed size can be transmitted within a BAG duration.

A subset of an AFDX configuration is depicted in Figure 1. Six VLs v1 . . . v6 are shown. They are transmitted between five end systems through seven switches. Black dashed lines represent sets of links to/from end systems and switches which are not represented in Figure 1. Additional flow a1 will be explained in Section II-B. At end system, the BAG constraint has to be guaranteed for each VL. As depicted in Figure 2 a traffic regulator is associated to each VL. It delays any frame which does not respect the BAG constraint. Then, frames coming from the regulators are multiplexed by the scheduler into a single flow to be sent on the output physical link. If multiple frames from different VLs arrive at scheduler input at the same time, some of them may experience queuing delay or jitter. Figure 3 illustrates this jitter on VL v3 in figure 1. The transmissions of two consecutive v3 frames are shown. The first one is delayed by competing frames at the output port of es1, while the second one is transmitted immediately. Therefore, these two frames arrive at switch sw1 within an interval which is smaller than the BAG, i.e. BAGjitter where jitter is the waiting time of the first frame in es1. This jitter can be problematic.

Indeed the network doesn't trust the end system. Thus, the BAG constraint of each VL is checked at the entrance of the network. The distance between two consecutive frames of a given VL should never be less than the BAG minus the maximum possible jitter for this VL. The ARINC 664 P7-1 standard specifies that the jitter for a given VL must not exceed 500 µs. At the switch level, there are two FIFO queues (high priority, low priority) at each output port. Each incoming frame is buffered in the queue corresponding to its priority. Classically, frames are served in each output port, following a static priority queuing (SPQ) policy (high priority first). Up to now, all VLs are assigned the high priority level.

B. Additional flows on the AFDX network

The AFDX network has to be certified. Typically, a proof of determinism has to be provided: safe upper bounds have to be provided on frame end-to-end delays. Several methods have been proposed to compute such upper bounds, e.g. network calculus [START_REF] Le Boudec | Network calculus: a theory of deterministic queuing systems for the internet[END_REF], [START_REF] Li | Improving end-to-end delay upper bounds on an AFDX network by integrating offsets in worst-case analysis[END_REF], trajectory approach [START_REF] Bauer | Applying Trajectory approach with static priority queuing for improving the use of available AFDX resources[END_REF], forward end-toend delay analysis (FA) [START_REF] Benammar | Forward Endto-End Delay for AFDX Networks[END_REF]. These methods have to take into account very rare worst-case scenarios and they are based on pessimistic assumptions, resulting in a lightly loaded network. Typically, link load in a 100 Mbps AFDX configuration exceeds 10 % for a very small subset of the links and it never exceeds 25 % [START_REF] Charara | Methods for bounding end-to-end delays on an AFDX network[END_REF]. A similar situation occurs for the A350, even if the load is slightly higher since fewer switches are in use. Thus the avionics industry envisions to use the available AFDX bandwidth for the transmission of additional flows of lower criticality such as video, audio or service data flows. Currently, these flows are carried by dedicated communication networks, introducing additional weight and maintenance costs. The goal is to remove these dedicated networks and integrate all flows on the core AFDX backbone.

It is of course mandatory to guarantee that delay as well as jitter constraints are still satisfied for avionics flows. Moreover, if video or audio flows are considered, their timely streaming at the destination is only possible if appropriate guarantees on their end-to-end delay are offered. Thus, advanced scheduling mechanisms have to be implemented both at switch and end system levels.

As previously mentioned, a single type of additional flow is assumed in this paper. Therefore, the already implemented SPQ 2-priority levels is used at switch egress port. In the next section, we describe candidate scheduling policies at the end system level.

III. SCHEDULING AT THE END SYSTEM LEVEL

Figure 2 gives an overview of the architecture of existing end systems. As previously mentioned, regulators ensure that each VL respects its bandwidth limit. A scheduler multiplexes frames from the different VLs on the output link.

Additional video and audio flows are regulated, while service flows might not be regulated. Usually, video flows have higher demand of transmission than audio flows or avionics flows, but they are not authorized to exceed the available bandwidth. Nevertheless, these additional flows have to be scheduled with the avionics VLs. For avionics flows, the main constraint is the maximum allowed jitter at source end system, i.e. 500 µs as specified in the ARINC 664 standard. In the next paragraphs, we present the two classes of solutions for the scheduling of frames at end system level:

• the event-triggered solutions where pending frames are transmitted, based on their arrival time and priority order, • the time-triggered solutions where pending frames are transmitted in dedicated slots.

A. Event-triggered scheduling

In an event-triggered scheduling, pending frames are selected for transmission, based on their priority and the corresponding scheduling policy. Many scheduling policies have been considered, such as FIFO, SPQ, WRR or DRR. In the context of this paper, we assume two types of flows: critical avionics flows and less/not critical additional ones. The impact of additional flows on avionics ones has to be strictly mastered. Thus FIFO scheduling policy cannot be considered, since avionics flows would be delayed by a potentially large number of additional frames (all pending frames at the arrival of the avionics one).

The other scheduling policies are able to control the impact of additional flows on avionics ones. With WRR or DRR, a guaranteed portion of the bandwidth is allocated to each class of flows, leading to an isolation between classes. The portion of bandwidth is the number of frames for WRR and the number of bytes for DRR. Considering SPQ scheduling policy, the impact of additional flows on avionics ones can be limited to one frame with maximum size provided that avionics flows have a higher priority than additional ones. Since existing AFDX hardware implement SPQ with two priority levels and we address one single type of additional flows, we assume such a scheduling policy in the context of this paper.

Figure 4 considers the flows emitted by end systems es1 and es2 in Figure 1. The SPQ part in Figure 4 shows a possible scenario when SPQ scheduling policy is implemented at the end system level. At time 0, one frame from each VL v1, v2, v3, v4, v5 and from additional flow a1 is ready at end system es1. Based on SPQ, VLs have a higher priority than additional flow. Thus, frames from the VLs are transmitted first, in an arbitrary order (v1 1 , v2 1 , v3 1 , v4 1 , v5 1 in Figure 4). Then frames from additional flow a1 are transmitted (a1 1 arrived at 0 and the following ones arrived later). At time 1000 µs, v5 second frame is ready for transmission (v5 BAG is 1 ms) and it is immediately transmitted, since there are no other pending frames in es1.

As previously mentioned, SPQ scheduling policy is used in switch output ports. Thus, VL frames are transmitted in their arrival order, while additional frames are transmitted in their arrival order, provided there are no pending VL frames.

In the scenario in Figure 4, additional frames have no impact on VL ones. The situation might be different if an additional frame is under transmission at the arrival time of a VL one. However, this impact is bounded by the transmission duration of one additional frame. Thus it can be easily taken into account.

Let's now consider the jitter of VLs at the end system level. In Figure 4, v5 is the only VL with two transmitted frames, since it is the only one with a BAG smaller than 2 ms. The first frame of v5 waits till time 160 µs, since there are four other pending VL frames which are arbitrarily transmitted first (they are ready at the same time). Conversely, the second one is transmitted immediately (at time 1000 µs). Thus, the jitter for v5 at its source end system is 160 µs in this scenario. Using such an SPS scheduling, this jitter might increase dramatically when the number of VLs generated by the end system increases. As previously mentioned, this jitter must never exceed 500 µs.

One goal of the time-triggered scheduling presented in the next paragraph is to mitigate this jitter.

B. Time-triggered scheduling

In an AFDX network without additional flows, Timetriggered scheduling can be implemented in end systems by statically reserving slots for each VL [START_REF] Hotescu | Towards Quality of Service Provision with Avionics Full Duplex Switching[END_REF]. It comes to build a table as the one depicted in Figure 5 for es1 in Figure 1. The duration of each slot should be at least the transmission time of any frame in the configuration. In Figure 1, the transmission time of any frame is 40 µs. Thus the slot length in Figure 5 is 40 µs. The scheduling in the table is repeated forever. Therefore the duration of the table is the least common multiple of the BAGs, i.e. 128 ms. In Figure 5, one line duration is 1 ms, leading to a table with 128 lines. The four first lines are represented. v5 is allocated one slot every line, since its BAG is 1 ms. Following the same principle, v3 is allocated one slot every two lines. Considering the table in Figure 5 and the scenario in the upper part in Figure 4, we obtain frame transmission as depicted in the TT part in Figure 4 (we assume that the table starts at time 0).

Since the duration between two consecutive slots allocated to a given VL is exactly a BAG, the duration between two consecutive transmissions of a given VL is never less than its BAG (although it can be more since the BAG is a minimum inter-frame duration). Thus, the jitter constraint is guaranteed by construction for each VL: the upper bound of the jitter is 0.

The main drawback of this time-triggered solution is the waiting time till the next slot allocated to the VL, which can be close to the BAG. In Figure 4, this waiting time is more than 1 ms for VL v1. Depending on the scenario, it could be close to 128 ms for v4.

A classical solution to mitigate this problem is to overreserve slots: for instance, if one slot is allocate to each VL every ms, a frame will never wait more than 1 ms. Thanks to the regulator unit associated to each VL in its source end system (Figure 2), a VL will never use more than one slot within a BAG. In the rest of the paper, we assume such an over-reservation (one slot per ms per VL). It comes to assign one column of the table to each VL. This over-reservation limits the available bandwidth for additional flows. Therefore, a compromise between the over-reservation and the waiting delay till the next slot might be an interesting solution for VLs. This point is left for future work.

When additional flows are considered, their impact on avionics one has to be mastered. The simplest solution is to schedule these additional flows in the free slots of the table. This solution is illustrated in the TT part in Figure 4. Pending additional frames have to wait till the next free slot(s). Obviously the positions of the VL slots in the table will impact this waiting time. In the next paragraphs, we propose two policies for the assignment of slots to VLs in the table.

1) Uniform table scheduling:

The idea of the first policy is to uniformly distribute free slots in the table. It is based on a very simple heuristic. Columns of slots assigned to VLs are spread in the table so that the distance between any two columns is the same. Figure 6 shows such a table for es1 in Figure 1. Since we have 5 VLs and 25 slots per ms, one column is allocated to a VL every 5 slots (columns 1, 6, 11, 16 and 21).

2) Optimal table scheduling:

The second policy aims at choosing the slot assignment of VLs that minimizes the emission lag of additional flows, i.e. the maximum waiting time of any additional frame in the end system. The problem can be formulated as an integer constrained optimization problem. The two types of flows are defined by the following sets:

• A set V of VLs with for which a bounded jitter has to be ensured. These flows need an entire column reservation in table scheduling in order to keep their jitter to zero. • A set A of additional flows whose maximal emission lag has to be minimized. The following definitions are given:

• D s represents the slot duration.

• N represents the number of slots composing the table.

• P Ai is the actual period of frame generation for an additional flow A i ∈ A. • the integer decision variables {x i } i∈[1,N A ] that take their values in the integer set {1, .., N }. x i gives the identifier of the slot where an additional frame i is scheduled. Thus, x = [x 1 , .., x i , .., x N A ] is a sorted set of slot identifiers. • the integer decision variables {y j } j∈[1,|V |] representing the slot at which the first avionics frame of VL V j is scheduled. Following frames of V j are scheduled periodically every line of the table. These variables take their values in the integer set {1, .., C}. There is one variable y j for each VL and one variable x i for each additional frame.

The additional frames generation dates are listed in the ordered set {d i } i∈[1,..,N A ] . The value of d i represents the slot number in which the frame i is generated and thus d i ∈ {1, .., N }.

The optimization criterion minimizes the maximum emission lag over all additional frames. The emission lag is given by δ i = x i -d i representing the number of entire slots between the generation and the actual scheduling date of an additional frame i.

The following mathematical program is defined:

M inimize max i∈[1,..,N A ] (x i -d i )
Subject to:

x i > x j , i > j, ∀i, j ∈ [1, N A ] (1)
x i ≥ d i , ∀i ∈ [1, N A ] (2)
x i = y j + (l -1)×C, ∀i ∈ [1, N A ], ∀j ∈ [1, |V |], ∀l ∈ [1, L] y j = y k , j = k, ∀j, k ∈ [1, |V |] (3)
The set of constraints in equation ( 1) ensures that a slot in the scheduling table can be assigned to exactly one additional frame while the set of constraints in equation ( 2) ensures that a slot can be assigned to an additional frame only after its frame generation date. Equation (3) ensures that a VL can be assigned slots in a single column of the table. The slots on this column can neither be assigned to another VL nor to an additional flow. The problem dimension is given by the number of VLs and the number of frames generated by the additional flows.

IV. CASE STUDY

A. Network configuration

Currently deployed AFDX networks work at 100 Mbps. To be able to carry additional flows such as video or audio data, network data rate needs to be increased. For this reason, aircraft manufacturers envision to increase AFDX data rate to 1 Gbps in future deployments.

In our case study hereafter, we consider 1 Gbps AFDX network carrying additional uncompressed video flows coming from surveillance cameras situated on various locations of the plane.

The video transmission in avionics systems follows the ADVB standard specifications [START_REF]ARINC specification 818-2 Avionics Digital Video Bus (ADVB) High Data Rate[END_REF]. It specifies that only uncompressed digital video streams can be exchanged in the avionics systems to mitigate stream reconstruction delays at the receiver. This protocol is based on Fiber Channel Audio Video (FC-AV) protocol which is particularly suited for high bandwidth communications of high-resolution video streams. The delivery of video data without compression is needed by the avionics systems supporting time-critical functions such as assisted takeoff and landing, navigation, etc. Thus, neither frame losses nor additional latency introduced by compression/decompression operations are tolerated.

A video is composed of a sequence of images referred to as video frames. A video frame is transmitted as a sequence of data packets called ADVB frames. The sequence of ADVB frames form an ADVB container that is specific to a single video frame. ADVB frames are labeled with an object field that represents the type of data carried in the frame:

• Object 0 frame contains video frame header data (image information and auxiliary data), • Object 1 frame contains audio data, • Object 2 or 3 frames contain video data. For the transmission of an image, the first frame is of object 0 type and lists information related to the image (sequence number, type of color coding, dimensions, etc.). ADVB standard specifies that, to encapsulate video data into frames, images are scanned from left to right, line by line. If the last frame's payload for a line is not complete, padding bytes are added to keep the frame size constant.

The ADVB frame is very similar to the frame used for Ethernet with a maximum frame length of 1518 bytes including 1500 bytes of payload. Therefore, AFDX network is a good candidate to deliver video streams in accordance with ADVB specifications.

ADVB specifies several video formats. These formats are listed in Table II. In this paper, we consider VGA and XGA formats. XGA video format with 3-byte color requires a resolution of 1024 pixels × 768 lines per image, progressively scanned from left to right and from top to bottom as 3-byte RGB pixels. This format needs 3072 bytes per line that can be encapsulated into 3 Ethernet frames. To encapsulate the entire image, a total of 2305 frames will be used (2304 frames for the video data and 1 additional frame for the auxiliary information). A video that displays 24 frames per second has to receive 55320 frames per second with a bandwidth requirement of 640.685 Mbps. Since video is uncompressed here, the period of video frame generation equals 18.076µs for XGA format. Similarly, the VGA format with 640 pixels × 480 lines per image leads to a bandwidth requirement of 280 Mbps and a period of Ethernet frame generation of 43.35µs. The network configuration is similar to the one deployed on the Airbus A350 on which a couple of ADVB video flows are added. The basic configuration is composed of 126 ES, 7 switches on each redundant network and 1106 VLs. A unique ES of this configuration is subject to video transmission. This ES is source of 8 VLs emitting frames periodically at their BAG. BAG values of these VLs are in the set of {4, 8, 16, 32} as indicated in Table III. The maximum frame length of avionics frames carried by these VLs is between 115 and 835 bytes. Paths of these VLs cross from 1 to 3 switches. Video flows can be added on any of these paths. The most loaded switch egress port uses around 40 Mbps of the 1Gbps available on the network. The rest of the bandwidth can thus be dedicated to video flows. Thus, video flows originating from the same ES cannot exceed 960 Mbps.

We illustrate here two possible scenarios for the evaluation.

• First, one camera is deployed. A single XGA video flow (768 × 1024) with a requirement of 672 Mbps can be transported into the network. • Second, two cameras are connected to the given ES for redundancy. The screen resolution needs to be decreased making possible the transmission of 2 SVGA flows with a requirement of 350 Mbps each or of 2 VGA flows requiring each 280 Mbps.

Next we discuss the scheduling of these additional video flows at the end system level.

B. Scheduling at end system

We now focus on the source ES and we analyze how the scheduling policy impacts the transmission of video flows.

1) Event-triggered: If SPQ policy is applied at ES level, no bandwidth over provision is required. The total amount of available network (approximately 994 Mbps) can thus be used by the video flows. Therefore, the three scenarios of video transmission established previously are possible(i.e. 1 XGA, 2 SVGA or 2 VGA flows).

2) Time-triggered: The table scheduling is configured with 1Gbps data rate. The table is composed of 128 lines of 1ms. Each line is divided into 64 slots of 15.625µs. This slot duration has been chosen as a function of the maximum frame length of the flows and the data link rate and it is long enough to transmit one frame of any flow. According to this time division, the table can be now represented as a 2D matrix of 128 lines × 64 columns.

When the order of frames is established at ES by the table scheduling, the number of time slots are over provisioned for avionics frames. Also, the slot duration of 15.625 µs is over sized so that a frame with the maximum length can be transmitted in a single slot. The total amount of bandwidth required for slot reservation of the 8 VLs is thus of 315 Mbps.The rest of the available bandwidth of 685 Mbps can be used by the video flows. In this case, only two scenarios of video transmission are still possible without exceeding the network bandwidth: 1 XGA video flow or 2 VGA video flows together with the VLs.

Uniform and optimal table schedules can be applied. Uniform heuristic gives an emission lag of 1 slot to the XGA video flow and of 2 slots for the 2 VGA flows. The VLs distribution in the table corresponding to the columns allocated to them is the following one: {0, 8, 16, 24, 32, 40, 48, 56}. It is the same for the two scenarios of video transmission.

The mathematical optimization program is solved using the CPLEX commercial constraint solver [START_REF]IBM ILOG CPLEX Optimization Studio CP Optimizer User's Manual[END_REF]. The optimal table schedule produces the same maximum emission lag as the uniform one: 1 slot for the XGA flow and 2 slots for the 2 VGA flows. The solving time is around 1 hour and 20 minutes for both scenarios. However, the solutions are different as the VLs distributions in table are different. The VLs are allocated at columns {2, 10, 18, 26, 34, 42, 50, 58} for the transmission of the XGA flow. When 2 VGA are added, the VLs are assigned to columns in the set of {4, 11, 22, 29, 41, 46, 54, 62}.

These table allocations will be compared in Section V.

V. END-TO-END DELAY EVALUATION

In this section, we evaluate the impact of scheduling strategies on VL and video flow delays. First, we give simulation results to evaluate the distribution of video flow delays for both event-triggered and time-triggered scheduling. Second we compute worst-case end-to-end latencies for VLs in order to determine the impact of video flows on VL worst-case delay. Such a worst-case analysis is mandatory for certification.

A. Simulation study

This section investigates the distribution of source, network and end-to-end delay of video and avionics frames for the event-triggered and time-triggered scheduling policies introduced previously in this paper. Next, our settings are introduced. In a second stage, results for two representative scenarios are given and analyzed.

1) Simulation settings: a) Simulator: Following results were produced by an in-house AFDX network simulator whose core relies on the OMNeT++ discrete event framework [START_REF] Varga | An overview of the OMNeT++ simulation environment[END_REF]. Our simulator models the main parameters of AFDX (routing of virtual links, queuing policies, switching latency, etc.). The simulation scenario considers the network architecture described previously. On these architecture, the video flows follow the path of the additional flow in Fig. 1.

b) Scheduling policies: At the switch level, a 2-level SPQ policy is applied. The higher priority is allocated to VLs and the lower one to video flows. At source ES, the delay statistics are compared for the SPQ policy and the uniform and optimal table schedules.

c) Network settings: The network data rate is set to 1 Gbps and the switching delay to 2µs. The results presented herein are made available for the case where the video flows and avionics VLs follow the same path and cross 3 switches. Thus, for a maximum frame size, the minimum expected endto-end delay is 54.576 µs. d) Offsets: There is no global synchronization of ADFX network pieces of equipment. Thus any possible shift is possible between any two end system, depending on their relative starting time and clock drift. We introduce random relative offsets in our simulations to capture this unpredictable shift. Indeed we have to take into consideration that not all end systems start emitting frames simultaneously and we define therefore an end system relative offset of end system X as the delay between the transmission date of the first frame in the network and the date of the first frame transmission by end system X. The sole assumption we make on these offsets is that they cannot exceed the maximum BAG value.

To capture the delay distribution of a wide range of possible scenarios, Monte-Carlo simulations are performed here. The relative offsets of end systems take a new randomly chosen value in the [0..128ms] interval at each new simulation. Source, network and end-to-end delays of video frames are measured for each new simulation. All values are merged to obtain the global distribution of these delays. Repeated simulations have shown that around one hundred simulation runs are enough to retrieve a complete distribution.

e) Simulation duration: The simulation duration has been set to 384ms, corresponding to 3 × 128 ms. It is equivalent to 2 times the duration of the hyper period (128 ms) in addition to the maximum offset value of 128ms. This duration captures the complete delay distribution as after at most two cycles, the system behavior becomes cyclic. Since the end system offset is a random value, it will vary for each simulation run, leading to a variation of the number of generated frames during the same simulation duration. For that reason, we consider the distribution delay for the first 5906 frames arrived at their destination for the VGA display format and the first 14162 frames arrived at their destination for the XGA format.

2) Results: Results in Table IV shows the 99th centile(d 99 ) and the average(d avg ) delay of video flows measured at source ES output, at network, and at destination ES with SPQ, Uniform and Optimal table schedules. The investigated scenarios consider the emission of i) 1 XGA video flow and ii) 2 VGA video flows together with 8 VLs.

The source delay experienced by the XGA flow is around 20.41µs with the SPQ policy and around 31µs with both uniform and optimal tables. As expected, delays with Uniform and Optimal table schedules are similar. The SPQ low delay value is due to the small number of VLs and the reduced length of avionics frames requiring a transmission delay between 0.92 to 6.6µs. Even though, the table scheduling produces a greater source delay because of the fixed time slot duration of 15.625µs, it is beneficial for avionics frames whose jitter is 0, while SPQ generates some delay variation due to the simultaneous emission of the VLs. Table schedule has a positive impact on the network crossing delay and on the endto-end delay that are significantly reduced in comparison with SPQ.

Similar results can be observed on the 2VGA flows. These simulation results show that event-triggered policy can have a better performance for video flows at ES level, but it introduces a delay variation for VLs. Time-triggered policy may be disadvantageous for video flows at source ES, but null jitter is ensured for avionics frames as well as a reduction of the end-to-end delay of video flows.

B. Worst-case analysis for avionics frames

Table V shows the worst-case end-to-end delay for a VL crossing the same path as a video flow. Results have been computed first for the basic A350 network configuration where no video transmission is enabled. The case of a video flow added on this architecture is considered in a second stage. FIFO, SPQ and table scheduling applied at ES are compared. Worst-case analysis is realized with Network Calculus. Following results were produced with the C++ tool developed by A. Soni et al. in [17] implementing the latest version of Network Calculus. For the basic configuration, the worst-case end-to-end delay of the VL is higher with FIFO policy applied at ES level than with table scheduling. The upper bound on the end-to-end delay is reduced with table scheduling as offsets are introduced between VLs at source end system permitting to spread their arrival at the switch level. When a video flow is added, we can see that FIFO produces a large upper bound delay on both VL and video flow as no priority is considered between flows. SPQ allows the use of two priorities reducing the bound on the VL's end-to-end delay. This bound is clearly reduced with table scheduling. Still, the impact on the end-to-end delay of the VL is limited in comparison with the basic video-free configuration.

VI. CONCLUSION

In this paper, we consider an AFDX network shared between avionics and additional (less critical) flows. We investigated event-triggered and time-triggered scheduling policies for the transmission of avionics and additional flows at the end system level. We show that an event-triggered strategy is better for additional flows at source level, but it might dramatically increase the jitter for avionics flows at source level and increase the end-to-end latency of additional flows. Therefore, a time-triggered strategy is the most promising candidate. We also show that a very simple heuristic can efficiently build such a time-triggered scheduling.

In this paper we assume a single type of additional flows. However, different types of additional flows are envisioned by manufacturers. Thus our work has to be extended, first at the switch level where more complex scheduling policies should be considered (such as WRR or DRR), second at the end system level where types of additional flows should be treated differently.

Fig. 1 .

 1 Fig. 1. Illustrative AFDX configuration VLs in Figure 1 have identical frame size (500 bytes), leading to a transmission time of 40 µs at 100 Mb/s. In this paper, we

Fig. 3 .

 3 Fig. 2. End system architecture

Fig. 4 .

 4 Fig. 4. Scheduling at the end system level

  Fig. 6.Table with one slot per ms per VL

  Table I summarizes VL features.

  Table with one slot per ms per VL

  A is the number of frames generated by all additional flows in A during the table duration. • |V | is the number of avionics VLs in V • L is the number of lines in the table • C is the number of columns in the table Two different types of decision variables are defined:

• N

TABLE II ETHERNET

 II FRAME ENCAPSULATION FOR DIFFERENT VIDEO DISPLAY FORMATS IN 3-BYTES RGB AND FRAME SIZE OF 1518 BYTES.

	Display format	Data rate	Period	Number of
	(pixels×pixels)	(Mbps)	(µs)	video frames/s
	QVGA(320 × 240)	93	129.8	7 704
	VGA(640 × 480)	280	43.35	23 064
	SVGA(800 × 600)	350	34.69	28 824
	XGA(1024 × 768)	672	18.076	55 320
	SXGA(1280 × 1024)	896	13.55	73 752

TABLE IV AVERAGE

 IV (davg ) AND 99TH CENTILE(d 99 ) DELAYS FOR VIDEO FLOWS WITH THE SPQ, UNIFORM AND OPTIMAL TABLE SCHEDULES. SCENARIOS CONSIDER 8 VLS SHARING THE BANDWIDTH WITH i) 1 XGA FLOW AND ii) 2 VGA FLOWS. END-TO-END(E2E) DELAY IS COMPOSED OF SOURCE, SWITCH QUEUING AND LINK TRANSMISSION DELAY.

	Scenarios			Src	SPQ delay (µs) Switch	E2E	Uniform delay (µs) Src Switch E2E	Optimal delay (µs) Src Switch E2E
	1 XGA		d 99 davg	20.41 0.023	141.174 54.92	141.174 54.95	31.11 8.62	69.063 54.62	95.491 63.22	31.13 8.62	69.063 54.62	90.34 63.23
			d 99	20.416	130.61	130.61	31.006	69.063	87.48	30.88	60.93	85.45
	2 VGA	1	davg d 99	0.007 32.56	54.74 148.086	54.74 160.23	8.13 46.68	54.61 69.063	62.73 105.86	8.09 46.81	54.6 62.25	62.69 101.38
		2	davg	12.17	54.99		67.17	23.94	54.62	78.55	23.94	54.62	78.57

TABLE V

 V WORST-CASE BOUND ON THE END-TO-END DELAY FOR A VL AND VIDEO FLOW CROSSING THE SAME PATH.

	A350 basic configuration (VLs only)
	Scheduling	E2E delay (µs)
	FIFO	1279.79
	Table	519.284
	A350 video configuration
	Scheduling	E2E delay (µs) VL Video
	FIFO	2204.21 2204.21
	SPQ	1328.43 2928.65
	Table	598.961 603.409


Stability and instability of traveling wave solutions

to scalar balance laws

V. Duchêne *
June 18, 2020

1 Presentation of the results

In the present contribution we discuss the dynamic stability of traveling wave solutions -including constant equilibria-to first order scalar hyperbolic balance laws

∂ t u + ∂ x f (u) = g(u), u : R + × R → R , (1.1)
where f and g are regular real functions, accounting respectively for advection and reaction processes. Our discussion is based on the recent works of the author and L.M. Rodrigues, [START_REF] Duchêne | Large-Time Asymptotic Stability of Riemann Shocks of Scalar Balance Laws[END_REF][START_REF] Duchêne | Stability in scalar balance laws: fronts and periodic waves[END_REF]. 1 Our aim here is to provide an overview of the results therein in a homogenized framework as well as key ingredients of the proofs, while the reader is referred to the original works for comprehensive results and detailed proofs.

We will prove the large-time asymptotic orbital stability under regular perturbations of piecewise regular entropic traveling wave solutions under non-degeneracy hypotheses and sign criteria at key locations of the wave, namely infinities, discontinuities and characteristic points. We also show the spectral and nonlinear instability of bounded piecewise regular traveling waves satisfying the reverse sign criteria.

An important feature of our results is that we measure stability in strong topologies, 2 obstructing in particular the emergence of additional discontinuities from piecewise regular initial data. That this is conceivable is of course due to the presence of the source term, g, since it is well-known that the solution to scalar conservative laws -that is setting g ≡ 0 in (1.1)-emerging from any smooth and decaying at infinity perturbation of a constant state will inevitably lead to the occurrence of a gradient catastrophe in finite time. This is to be compared with the following example:

∂ t u + ∂ x α u 2 2 = -β u with α ∈ R, β > 0.
Using the method of characteristics, one easily obtain the global-in-time existence of a classical solution as well as exponential decay for any C 1 (R) initial data satisfying α ∂ x (u(0, •)) ≥ -β.

Hence asymptotic large-time stability of the trivial equilibrium for initial perturbations with sufficiently small derivative holds when β > 0, while instability is easily seen to hold -as in the purely reactive case, α = 0when β < 0. We aim at providing a comprehensive theory encompassing general advection and reaction terms as well as general traveling wave solutions.

The outline of the present document is as follows. In Section 1.1, we classify, under non-degeneracy assumptions of the functions f and g, the (strictly) entropic piecewise regular traveling wave solutions to (1.1). We then describe precisely the notions of stability and instability which are used in our results, in Section 1.2. In Section 2 we present three mechanisms of instability, which allow to narrow down the possibly stable waves to a handful of candidates, classified in constants, fronts, shocks and composite waves. In Section 3, we establish the stability of these candidates. Section 4 contains a conclusion and additional comments.

Let us introduce a few notations. For n ∈ N , Ω ⊂ R n , and 1 ≤ p ≤ ∞, L p (Ω) is the usual Lebesgue set of p-integrable (or essentially bounded if p = ∞) real functions on Ω, and L p loc (Ω) is the set of locally p-integrable (or locally essentially bounded if p = ∞) real functions on Ω. If Ω ⊂ R n is connected and k ∈ N, W k,p (Ω) (resp. C k (Ω), resp. BU C k (Ω)) is the set of functions such that derivatives up to the order k belong to L p (Ω) (resp. are continuous, resp. are bounded and uniformly continuous). If Ω ⊂ R n is the disjoint union of connected sets, then W k,p (Ω) (resp. C k (Ω), resp. BU C k (Ω)) denotes the set of functions such that the restriction to each connected component, Ω, belongs to W k,p ( Ω) (resp. C k ( Ω), resp. BU C k ( Ω)). For D ⊂ R a closed discrete set, ∞ (D) is the set of bounded real functions on D. We denote

C k+ (R) = f ∈ C k (R) : r → max |x-y|≤r |g (k) (x)-g (k) (y)| r ∈ L 1 loc (R) .
These spaces are endowed with natural norms and topological structure associated with their definition.

Classification of traveling waves

In this section we classify bounded piecewise regular traveling wave solutions to (1.1), under non-degeneracy assumptions on the real functions f ∈ C 2 (R) and g ∈ C 1 (R), and entropy conditions introduced thereafter.

Definition 1.1. A piecewise regular traveling wave solution to (1.1) is a weak solution in the form

u : (t, x) → U (x -σ t) with (U , σ) ∈ C 1 (R \ D) × R
where D is a (possibly empty) closed discrete set.

For (U , σ, D) defining a piecewise regular traveling wave, R \ D is a union of disjoint open intervals, and

∀x ∈ R \ D, f (U (x)) -σ U (x) = g(U (x)). (1.2)
Solving this scalar differential equation motivates the following non-degeneracy assumptions.

Definition 1.2. We say that f ∈ C 2 (R) and g ∈ C 1 (R) are non-degenerate if zeroes of g are isolated, and i. for any u such that g(u ) = 0, one has g (u ) = 0 and f (u ) = 0;

ii. for any u = u † such that g(u

) = g(u † ) = 0, one has f (u ) = f (u † ). Proposition 1.3. Let f ∈ C 2 (R), g ∈ C 1 (R)
be non-degenerate and (U , σ, D) define a piecewise regular traveling wave solution to (1.1). Then

• If u ∈ U (R \ D) is a characteristic value, that is f (u ) = σ, then g(u ) = 0;
• U is either constant or strictly monotonic on connected components of R \ D;

• For any d ∈ D, U possesses left and right limits,

U (d -) ∈ R∪{-∞, +∞} and U (d + ) ∈ R∪{-∞, +∞}; • if a connected component is not lower (resp. upper) bounded then U has a limit u -∞ ∈ R ∪ {-∞, +∞} at -∞ (resp. u +∞ ∈ R ∪ {-∞, +∞} at +∞). If u -∞ ∈ R (resp. u +∞ ∈ R), then g(u -∞ ) = 0, (resp. g(u +∞ ) = 0) and either U is constant on the component, or f (u -∞ ) = σ (resp. f (u +∞ ) = σ).
Proof. The first item is a consequence of (1.2), and one has f (u ) = 0 and g (u ) = 0 by Definition 1.2.

Hence ∀x ∈ R \ D, U (x) = 0 or U (x) = F (U (x))
where for any J connected component of R \ D, F : U (J) → R is the map defined by

∀u ∈ U (J), F (u) = g(u) f (u)-σ if f (u) -σ = 0, g (u) f (u)
otherwise.

The remaining items follow from monotonicity and properties of solutions to scalar differential equations.

A bounded piecewise regular traveling wave defined by (U , σ, D) possesses left and right limits at any discontinuous point d ∈ D, and hence should satisfy the Rankine-Hugoniot condition, that is

σ[U ] d = [f (U )] d . (1.3)
Above, and thereafter, we denote for h ∈ C 1 (V \ {x}) where V is a neighborhood of x:

h(x ± ) = lim δ 0 h(x ± δ) and [h] x = h(x + ) -h(x -) .
We will assume henceforth that the solution is entropic through (strict) Oleinik's conditions.

Definition 1.4. We say that a bounded piecewise regular traveling wave solution to (1.1) defined by

(U , σ, D) is entropic if for any d ∈ D, f (U (d -)) > σ > f (U (d + )) (1.4)
and for any v strictly between U (d + ) and U (d -),

f (v) -f (U (d -)) v -U (d -) > f (v) -f (U (d + )) v -U (d + ) . (1.5)
Remark 1.5. Let us recall that by the theory due to Kružkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF], there exists a unique bounded local-intime entropy-admissible weak solution to (1.1) emerging from any bounded initial data. Oleinik's conditions ensure that the piecewise regular traveling wave solution is indeed entropy-admissible, but are also essential to its stability. Indeed, a spectral analysis of the operator L defined below reveals the role of Lax's entropy condition (1.4) to the spectral stability of a discontinuous traveling wave, and to the well-posedness of the corresponding linearized dynamics. Condition (1.5) would also be crucial to the stability properties of discontinuous traveling waves if one allowed -which is not the case in this work-perturbations breaking the large shock into a "sum" of smaller subshocks; see [START_REF] Bressan | Hyperbolic systems of conservation laws[END_REF]Remark 4.7] for a more detailed discussion.

The strict entropy conditions provides useful information on admissible piecewise regular traveling waves.

Proposition 1.6. Let f ∈ C 2 (R), g ∈ C 1 (R) be non-degenerate and (U , σ, D) define a bounded, piecewise regular entropic traveling wave solution to (1.1). Then the following holds.

• On any bounded connected component of R \ D, U is strictly monotonic, passes through a characteristic value u and g (u ) > 0.

• On a connected component of R \ D bounded from above but not from below, either U is constant with value u = u -∞ , and f (u -∞ ) -σ > 0;

or U is strictly monotonic, possesses a limit u -∞ ∈ R at -∞ with g(u -∞ ) = 0, passes through a characteristic value if and only if g (u -∞ ) < 0, and sgn(f (u -∞ ) -σ) = sgn(g (u -∞ )).

• On a connected component of R \ D bounded from below but not from above, either U is constant with value u = u +∞ , and f (u +∞ ) -σ < 0; or U is strictly monotonic, possesses a limit u +∞ ∈ R at +∞ with g(u +∞ ) = 0, passes through a characteristic value if and only if g (u +∞ ) < 0, and sgn(f (u +∞ ) -σ) = -sgn(g (u +∞ )).

• If D = ∅ and U is not constant, then U is strictly monotonic, possesses limits u ±∞ ∈ R at ±∞ satisfying g(u ±∞ ) = 0, passes through a characteristic value if and only if g (u -∞ )g (u +∞ ) > 0 and f (U ) -σ has the sign of ∓g (u ±∞ ) near ±∞.

Proof. Since we know the sign of f (U ) -σ near discontinuities of U by (1.4), we only need to connect its sign to the sign of g (U ) near ±∞ or near a characteristic point, that is

x ∈ R \ D such that f (U (x )) = σ. At a characteristic point x , we have f (U (x )) U (x ) = g (U (x )) thus f (U (x)) -σ x→x ∼ g (U (x )) (x -x ) .
Near ±∞, if U is defined but not constant, the existence of finite limits stems from monotonicity and boundedness, U does not vanish near ±∞ and

f (U (x)) -σ = g(U (x)) U (x) x→±∞ ∼ g (u ±∞ ) U (x) -u ±∞ U (x) .
The claim easily follows.

Using Proposition 1.6, we may classify bounded piecewise regular entropic traveling wave solution to (1.1).

Corollary 1.7. Let f ∈ C 2 (R), g ∈ C 1 (R) be non-degenerate and (U , σ, D) define u : (t, x) → U (x -σt) a bounded, piecewise regular entropic traveling wave solution to (1.1). Then either 1. D = ∅ and U ≡ u ∈ R with g(u) = 0. The value of σ ∈ R is irrelevant. We say that u is a constant equilibrium.

2. D = ∅ and U is strictly monotonic, with finite limits u ±∞ at ±∞. One has g(u -∞ ) = g(u +∞ ) = 0. We say that u is a front.

= f (u + )-f (u -) u + -u -
. We say that u is a shock.

4.

In any other case, D = ∅ and U is strictly monotonic on at least one connected components of R \ D, and is constant on at most two (unbounded) connected component of R \ D. We say that u is a composite traveling wave. This contains in particular periodic traveling waves.

We may now describe our results. Of course, notions of stability and instability are detailed in precise statements thereafter; they are introduced and discussed in the following section.

Theorem (rough statement). Let f and g be sufficiently regular and non-degenerate, and (U , σ, D) define a bounded piecewise regular entropic traveling wave solution to (1.1). Then u : (t, x) → U (x -σt) is stable if and only if

• U has finite limits u ±∞ at ±∞ and g (u -∞ ) < 0, g (u +∞ ) < 0;

• on connected components such that U is strictly monotonous, U passes through a characteristic value u ∈ R, and g (u ) > 0;

• for any d ∈ D, [g(U )] d [U ] d ≤ 0.
In other words, and following the terminology of Corollary 1.7, stable piecewise regular entropic traveling wave solutions to (1.1) consist in 1. constant equilibria u ∈ R such that g(u) = 0 and g (u) < 0 (we say the equilibria are dissipative);

2. fronts taking a characteristic value u ∈ R, and such that g (u ) > 0;

3. (strictly) entropic shocks between two dissipative equilibria;

4. composite waves satisfying 1. (resp. 2.) on connected components where U is constant (resp. strictly monotonous), and [g(U )] d

[U ] d < 0 and (1.4)-(1.5) at any discontinuity, d. U is constant on one or two (necessarily unbounded) connected components and strictly monotonous on exactly one component.

Remark 1.8. That the two assertions of the Theorem are equivalent is a direct consequence of Proposition 1.6. The only non-trivial task consists in ruling out the possibility of U being strictly monotonous on two consecutive connected components and defining a stable piecewise regular entropic traveling wave solutions. This stems from the fact that if U is strictly monotonic on two consecutive connected components separated by d 0 ∈ D, then it passes through a single characteristic point u in both components, by Proposition 1.6 and the non-degeneracy condition. We can then check that the inequalities g (u ) > 0 and

[ g(U ) ] d 0 [ U ] d 0 ≤ 0 are incompatible.

Notions of stability

Let us now clarify what is meant by "stability" in this work. As aforementioned, closedness will be described with topologies controlling piecewise smoothness. Moreover, while the stability results will encode control on deformations of shape, some re-synchronization of positions is allowed. This is essential in the presence of discontinuities, but will turn out to be useful as well in the presence of characteristic points. We will also distinguish between the spectral stability described through the spectrum of linearized problems, and dynamic nonlinear stability describing the large-time behavior of solutions to (1.1) with close initial data. The main lesson of this work is that, in our framework, the two notions coincide and that the nonlinear stability under regular perturbations may be decided from a handful of sign conditions encoding spectral stability. While such a statement is familiar in the study of dynamic stability, one should mention that this result is far from obvious for convection/reaction equations, due to the fact that the operators at stake do not offer any regularization effects. As a consequence, it is not sufficient to consider the linearized dynamics about the traveling wave, and our proofs rely instead on decay estimates for all nearby linear dynamics.

With the above discussion in mind, for (U , σ, D) defining a bounded piecewise regular traveling wave solution to (1.1), we look for entropic solutions u to (1.1) in the form

u(t, x + σt + ψ(t, x)) = U (x) + u(t, x) (1.6)
with ( u, ψ) small provided they are sufficiently small initially. We always assume that

ψ(t, •) is admissible, that is x ∈ R → x + ψ(t, x
) is increasing and bijective.

Definition 1.9 (Nonlinear stability). Given functional spaces X , Y of locally integrable real functions, a bounded piecewise regular traveling wave solution to (1.1) defined by

(U , σ, d) is nonlinearly stable in X × Y if for any U × V neighborhood of U × {0}
for the X × Y topology, there exists U 0 × V 0 another neighborhood of U ×{0} for the X ×Y topology such that for any (u 0 , ψ 0 ) ∈ U 0 ×V 0 such that ψ 0 is admissible, the unique globalin-time entropy solution emerging from the initial data defined by u(0, x+ψ 0 (x)) = u 0 (x) satisfies that for any t ∈ R + , there exists ψ(t, •) ∈ V admissible such that u(t, •+σt+ψ(t, •)) ∈ U. The traveling wave is nonlinearly unstable in X × Y otherwise. A traveling wave nonlinearly stable in X × Y is orbitally stable with asymptotic phase if, additionally, there exists ψ ∞ ∈ Y admissible such that u(t,

• + σt + ψ(t, •)), ψ(t, •) → U , ψ ∞ in X × Y as t → ∞.
Note that we aim at a space shift ψ(t, •) regular on R and a shape deformation u(t, •) regular on R \ D with limits from both sides at each d ∈ D. This leads us to the following restriction on entropy solutions. Definition 1.10. We say that u ∈ L 1 loc (I × R) is a piecewise regular entropy solution to (1.1) on the time interval I ⊂ R if there exist a closed discrete set D and ψ ∈ C 1 (I × R) admissible for any t ∈ I such that (t, x) → u(t, x + σt + ψ(t, x)) ∈ C 1 (I × (R \ D)), and u satisfies (1.1) in its regular domain, as well as the Rankine-Hugoniot condition

u l (t, d) -u r (t, d) σ + ∂ t ψ(t, d) = f (u l (t, d)) -f (u r (t, d))
and the strict Oleinik entropy conditions

         σ + ∂ t ψ(t, d) > f (u r (t, d)) , f (τ u l (t,d)+(1-τ ) ur(t,d))-f (u l (t,d)) τ u l (t,d)+(1-τ ) ur(t,d)-u l (t,d) > f (τ u l (t,d)+(1-τ ) ur(t,d))-f (ur(t,d)) τ u l (t,d)+(1-τ ) ur(t,d)-ur(t,d)
for any τ ∈ (0, 1) ,

f (u l (t, d)) > σ + ∂ t ψ(t, d) .
(1.7)

where u l (t, d) = lim δ 0 u(t, d + σt + ψ(t, d) -δ) and u r (t) = lim δ 0 u(t, d + σt + ψ(t, d) + δ).
Notice that we assume in Definition 1.10 that discontinuities do not vanish or emerge, and discontinuity paths do not touch on the time interval I. Using identity (1.6) and chain rules, one can check that u being a piecewise regular entropy solution reduces to interior equations

∂ t ( u -ψU ) + ∂ x (f (U ) -σ)( u -ψU ) -g (U )( u -ψU ) = -∂ x (f (U + u) -f (U ) -f (U ) u) + g(U + u) -g(U ) -g (U ) u + ∂ x ψ (g(U + u) -g(U )) -∂ t (∂ x ψ u) + ∂ x (∂ t ψ u)
on R \ D, and at any d ∈ D the Rankine-Hugoniot condition

∂ t ψ [ U ] d -[ (f (U ) -σ) u ] d = [ f (U + u) -f (U ) -f (U ) u) ] d -∂ t ψ [ u ] d
and the Oleinik entropy conditions which we omit to write down. Indeed, since we only consider waves satisfying strict entropy condition, they do not show up at the linearized level. The above suggests to consider the spectral problem associated with the linearized equations, that is

λ (ǔ -ψU ) + ∂ x ((f (U ) -σ)(ǔ -ψU )) -g (U )(ǔ -ψU ) = A + ∂ x (B) on R \ D , λ ψ [ U ] d -[ (f (U ) -σ)(ǔ -ψU ) ] d -ψ [ (f (U ) -σ)U ] d = [ -B ] d at any d ∈ D ,
with ( ψ, ǔ) playing the role of the value at λ of the Laplace transform in time of (ψ, u).

For the sake of tractability we relax the foregoing problem into the problem of the determination of the spectrum of a given operator. To do so we choose X a functional space of locally integrable functions on R \ D and y a space of functions on D. We assume that any w ∈ X such that ∂ x ((f (U ) -σ)w) -g (U )w ∈ X possesses limits from the left and from the right at any point d ∈ D. Then one may define on X × y, the operator with maximal domain

L(w, (y d ) d∈D ) := -∂ x ((f (U ) -σ)w) + g (U )w, y d [ (f (U ) -σ)U ] d [ U ] d + [ (f (U ) -σ)w ] d [ U ] d d∈D .
Definition 1.11 (Spectral stability). We call X × y-spectrum of the linearization about a piecewise regular traveling wave defined by (U , σ, D) the spectrum of L. We say that the wave defined by (U , σ, D) is spectrally stable in X × y if the X × y-spectrum is contained in the set of complex values with negative real parts, and {0}. 3 We say that the wave defined by (U , σ, D) is spectrally unstable if there exists an element of the X × y-spectrum with positive real part.

Instability results

In this section, we exhibit instability mechanisms for piecewise regular traveling wave solutions to (1.1), both at the spectral and nonlinear level.

Instabilities at infinity

Proposition 2.1. Let k ∈ N, f ∈ C k+2 (R), g ∈ C 2 (R) ∩ C k+1 (
R) be non-degenerate and (U , σ, D) define a piecewise regular traveling wave solution to (1.1), u. If D is unbounded from above (resp. from below) and U admits a limit

u +∞ ∈ R at +∞ (resp. u -∞ ∈ R at -∞), then g (u +∞ ) + i(f (u +∞ ) -σ)R (resp. g (u -∞ ) + i(f (u -∞ ) -σ)R) is included in the X × y-spectrum of the linearization about u if for some I ⊂ R neighborhood of +∞ (resp.
-∞) the norm of X restricted to smooth functions compactly supported in I is controlled by the W k,p (I)-norm and controls the L q (I)-norm, for some

1 ≤ p, q ≤ ∞ such that (p, q) = (1, ∞). In particular, if g (u +∞ ) > 0 (resp. g (u -∞ ) > 0) then u is spectrally unstable in BU C k (R \ D) × ∞ (D).
Proof. Since the difference is purely notational we only treat the case where the limit is at +∞. We pick χ : R → R non zero, smooth and compactly supported. For ξ ∈ R and ε > 0, we let

w (ε) : R \ D → C, x → e -i ξx χ ε x - 1 ε .
For ε sufficiently small, w (ε) is supported in I and

(g (u +∞ ) + i(f (u +∞ ) -σ)ξ)w (ε) + ∂ x ((f (U ) -σ)w (ε) ) -g (U )w (ε) W k,p (I) ε 1-1 p w (ε) L q (I) ε -1 q ,
where we used that U -u +∞ and derivatives up to the order k + 2 converge exponentially fast to zero at infinity. Hence (w (εn) , (0) d∈D ) n∈N with (ε n ) n∈N positive and converging to zero defines a Weyl sequence, and the proof is complete.

Proposition 2.2. Let f ∈ C 2 (R), g ∈ C 1 (R) be non-degenerate and (U , σ, D) define a bounded piecewise regular entropy-admissible traveling wave solution to (1.1), u. If D is unbounded from above (resp. from below) and U admits a limit u +∞ at +∞ (resp. u -∞ at -∞) and g (u +∞ ) > 0 (resp. g (u -∞ ) > 0), then the following holds. There exists δ > 0, and a sequence (u n ) n∈N of piecewise regular weak solutions to (1.1) defined for t ∈ [0, T n ] such that for any I ⊂ R neighborhood of +∞ (resp. -∞), one has for n sufficiently large i. u n (0, •) -U is smooth and compactly supported in I, and for any k ∈ N and 1 ≤ q ≤ ∞,

u n (0, •) -U W k,q (I) → 0 as n → ∞, ii. (t, x) → u n (t, •+σt)-U (t, •) ∈ C 1 ([0, T n ]×R), has compact support in [0,
T n ]×I, and for any admissible ψ ∈ C 0 (R) and any 1 ≤ p ≤ ∞,

u n (T n , • + ψ(•)) -U L p (R) ≥ δ.
In particular, u is nonlinearly unstable in

BU C k (R \ D) × BU C k (R) for any (k, k ) ∈ N 2 .
Proof. Since the difference is purely notational we only treat the case where the limit is at +∞. We set u n (0, •) = U + w εn where w ε = ε 2 χ(εx -1 ε ) with χ : R → R non zero, smooth and compactly supported, and (ε n ) n∈N is positive and converging to zero. We may construct locally in time the solution u n via characteristics. For ε > 0 and x ∈ (x ε , +∞) where x ε := inf(supp(w ε )), we define v ε (•, x) and X ε (•, x) by the initial data v ε (0, x) = (U + w ε )(x) and X ε (0, x) = x, and the differential equations

∂ t v ε (t, x) = g v ε (t, x) and ∂ t X ε (t, x) = f v ε (t, x) .
For any α > 0, there exist δ 0 > 0 and ε 0 > 0 such that for any ε ∈ (0, ε 0 ] and for any t > 0 such that

r(t) = sup({|v ε (s, x) -u +∞ | : s ∈ [0, t], x ∈ (x ε , +∞)}) ≤ δ 0 , one has |v ε -u +∞ |(t, x) ≥ |v ε (0, x) -u +∞ |e (g (u+∞)-α)t and |∂ x v ε (t, x)| ≤ |∂ x v ε (0, x)|e (g (u+∞)+α)t .
Choosing α sufficiently small and lowering ε 0 further if necessary, we deduce that there exists T ε ∈ (0, +∞) such that r(T ε ) = δ 0 , and

|∂ x v ε | ≤ 1 and ∂ x X ε ≥ 1/2 on [0, T ε ] × (x ε , +∞).
We then uniquely define

u ε through the relation u ε (t, X ε (t, x)) = v ε (t, x) on {(s, y), 0 ≤ s ≤ T ε , X ε (t, x ε ) < y < ∞}, and set u ε (t, x) = U (x -σt) for x ∈ R \ (X ε (t, x ε ), +∞).
We easily verify that (u εn ) n∈N , for (ε n ) n∈N positive, sufficiently small and converging towards zero satisfies the desired properties.

Instabilities at characteristic points

If x ∈ R \ D is a characteristic point, that is U (x ) = u with f (u ) = σ
and U (x ) = 0, then for any j ∈ {1, . . . , k}, -g (u ) j belongs to the X × y-spectrum of the linearization about u provided that δ

x , • • • , δ (j)
x act continuously on X . In particular, if

g (u ) < 0, then u is spectrally unstable in BU C k (R \ D) × ∞ (D).
Proof. It is sufficient to prove that there exists w a non trivial combination of δ

x , • • • , δ (j) x such that -g (u )j w -(f (U ) -σ)∂ x w -g (U ) w = 0 .
This follows from the fact that f (U (x)) -σ = (x -x )g (u ) + o(x -x )(x → x ) and hence for any j ∈ N

-(f (U ) -σ)δ (j+1) x -g (U )δ (j)
x ∈ j g (u ) δ (j) x + span({δ x , • • • , δ (j-1)

x }) .

Proposition 2.4. Let f ∈ C 2 (R), g ∈ C 1 (R) and (U , σ, D) define a bounded piecewise regular entropyadmissible traveling wave solution to (1.1), u. If x ∈ R \ D is a characteristic point, that is U (x ) = u with f (u ) = σ and U (x ) = 0, and if g (u ) < 0, then the following holds. There exists a sequence (u n ) n∈N of piecewise regular solutions to (1.1) defined for t ∈ [0, T n ) such that for any I ⊂ R neighborhood of x , one has for n sufficiently large i. u n (0, •) -U is smooth and compactly supported in I, and for any k ∈ N and 1 ≤ q ≤ ∞,

u n (0, •) -U W k,q (I) → 0 as n → ∞, ii. (t, x) → u n (t, • + σt) -U (t, •) ∈ C 1 ([0, T n ) × R) ∩ L ∞ ([0, T n ) × R), has support in [0, T n ) × I, and
u n (t, • + σt) W 1,∞ (I) → ∞ as t → T n .
In particular, u is nonlinearly unstable in

BU C k (R \ D) × BU C k (R) for any (k, k ) ∈ (N ) 2 .
Proof. We pick χ : R → R a real smooth compactly supported function such that χ(x ) = 0 and χ (x ) > 0.

For ε ∈ (0, 1), we denote

δ ε := -1 ln ε , χ ε := χ(x + •-x δε ), x - ε := inf(supp(χ ε )), x + ε := sup(supp(χ ε ))
, and for any x ∈ (x - ε , x + ε ) we define v ε (•, x) and X ε (•, x) by the initial data v ε (0, x) = U (x) + εU (x )χ ε (x) and X ε (0, x) = x, and the differential equations

∂ t v ε (t, x) = g v ε (t, x) , ∂ t X ε (t, x) = f v ε (t, x) .
There exists ε 0 > 0 such that for any 0 < ε < ε 0 and any

x ∈ (x - ε , x + ε ), sgn(g(v ε (0, x))) = -sgn(v ε (0, x)-u ), and hence v ε (•, x), X ε (•, x) ∈ C 2 (R + ) and sup({|v ε (t, x) -u | : t ≥ 0}) = |v ε (0, x) -u |. Moreover, v ε (t, x ) = U (x ) = u and X ε (t, x ) = x -σt, and hence the identities ∂ x v ε (t, x ) = ∂ x v ε (0, x )e g (u )t and ∂ x X ε (t, x ) = 1 + f (u ) g (u ) ∂ x v ε (0, x ) e g (u )t -1 hold for any t ≥ 0. Since ∂ x v ε (0, x ) = U (x )(1 + ε δε χ (x )), U (x ) = g (u )
f (u ) and χ (x ) > 0 and g (u ) < 0, we infer that

T ε := sup t > 0 : inf s∈[0,t],x∈(x - ε ,x + ε ) (∂ x X ε (s, x)) > 0 < ∞.
We then uniquely define

u ε by u ε (t, X ε (t, x)) = v ε (t, x) on {(s, y), 0 ≤ s < T ε , X ε (t, x - ε ) ≤ y ≤ X ε (t, x + ε )}, and set u ε (t, x) = U (x -σt) for x ∈ R \ (X ε (t, x - ε ), X ε (t, x + ε )
). We easily verify that (u εn ) n∈N , for (ε n ) n∈N positive, sufficiently small and converging towards zero satisfies the desired properties.

Instabilities of shock positions

Proposition 2.5. Let f ∈ C 2 (R), g ∈ C 1 (R) and (U , σ, D) define a bounded piecewise regular traveling wave solution to (1.1), u. For any

d 0 ∈ D, [ g(U ) ] d 0 [ U ] d 0
belongs to the X × y-spectrum of the linearization about u. In particular, if

[ g(U ) ] d 0 [ U ] d 0 > 0 for some d 0 ∈ D, u is spectrally unstable in BU C k (R \ D) × ∞ (D) for any k ∈ N.
Proof. One readily checks that (w, (y d ) d∈D ) = (0, (δ d,d0 ) d∈D ) provides an eigenvector for

[ g(U ) ] d 0 [ U ] d 0 . Proposition 2.6. Let k ∈ N, f ∈ C k+1 (R)∩C 2 (R), g ∈ C k (R)∩C 1 (R)
and (U , σ, D) define a bounded piecewise regular entropy-admissible traveling wave solution to (1.1), u, such that for some d 0 ∈ D,

[ g(U ) ] d 0 [ U ] d 0 > 0.
Then the following holds. For any I ⊂ R neighborhood of d 0 , there exists δ > 0, a sequence (u n ) n∈N of piecewise regular solutions to (1.1) defined for t ∈ [0, T n ] and a sequence (ψ n ) n∈N of smooth admissible functions compactly supported on [0,

T n ] × I such that (t, x) → u n (t, x + σt + ψ n (t, x)) -U (x) ∈ C 1 ([0, T n ] × (R \ D)) has support in [0, T n ] × (I \ {d 0 }) and i. for any 1 ≤ q ≤ ∞ ψ n (0, •) W k,q (I) → 0 and u n (0, • + ψ n (0, •)) -U W k,q (I\{d0}) → 0 as n → ∞,
ii. for any admissible ψ ∈ C 0 (R) and any 1 ≤ p ≤ ∞,

u n (T n , • + ψ(•)) -U L p (R) ≥ δ.
Moreover, if the strict entropy conditions holds at d ∈ d 0 , then it continues to hold at the shock position x(t) such that x(t)

+ σt + ψ n (t, x(t)) = d 0 . In particular, u is nonlinearly unstable in BU C k (R \ D) × BU C k (R).
Proof. There exists η > 0 and

U -, U + ∈ C 1 ([d 0 -η, d 0 + η]) such that for any x ∈ [d 0 -η, d 0 ), U -(x) = U (x), for any x ∈ (d 0 , d 0 + η], U + (x) = U (x), and f (U ± ) -σ U ± = g(U ± ) on [d 0 -η, d 0 + η]. We notice that f (U + )-f (U -) U + -U - (d 0 ) = [ g(U ) ] d 0 [ U ] d 0
. By continuity, lowering η > 0 if necessary, we may ensure that

α := inf x∈(d0-η,d0+η) f (U + ) -f (U -) U + -U - (x) > 0,
and, if (1.4) and (1.5) hold at d = d 0 , for any x ∈ [d 0 -η, d 0 + η] and v strictly between U -(x) and U + (x),

f (U -(x)) > σ > f (U + (x)) and f (v) -f (U -(x)) v -U -(x) > f (v) -f (U + (x)) v -U + (x) .
For any ε ∈ (-η, η), we define u ε (t, •) for t ∈ [0, T ε ] as

u ε (t, x) =      U -(x -σt) if x -σt ∈ [d 0 -η, d 0 + ψ ε (t)), U + (x -σt) if x -σt ∈ (d 0 + ψ ε (t), d 0 + η], U (x -σt) if x -σt ∈ R \ (D ∪ (d 0 -η, d 0 + η)),
where

ψ ε ∈ C 2 ([0, T ε ]
) is the solution with initial data ψ ε (0) = ε to the Rankine-Hugoniot condition

σ + ψ ε (t) = f (U + ) -f (U -) U + -U - (d 0 + ψ ε (t))
and

T ε = sup({t ≥ 0 : sup s∈[0,t] |ψ ε (s)| < η}) > 0.
From the Rankine-Hugoniot condition we infer |ψ ε |(t) ≥ |ε|e αt . Hence (u εn ) n∈N for (ε n ) n∈N nonzero, sufficiently small and converging towards zero satisfies the desired properties.

Stability results

We now turn to the stability of constants, fronts, shocks and composite waves, under the spectral assumptions stated in the Theorem.

Stable equilibria

In this section, we discuss the spectral and asymptotic stability of constant states u ∈ R with respect to regular perturbations under the condition g(u) = 0 and g (u) < 0 . Besides, it provides exponential decay estimates of regular solutions to

∂ t v + a∂ x v -bv = r (3.2)
for any functions a close to f (u) -σ and b close to g (u) in a suitable sense.

Lemma 3.1. Assume a, b ∈ BU C 0 (R) with a bounded away from zero. Then the following holds.

• L a, b := -a∂ x + b is a closed, densely-defined operator on BU C 0 (R) with domain BU C 1 (R).

• For any λ ∈ C such that (λ) > sup R b(•) ,
for any ř ∈ BU C 0 (R), there exists a unique v(

• ; λ) ∈ BU C 1 (R) such that (λ -L a, b ) v( • ; λ) = ř and moreover v( • ; λ) L ∞ (R) ≤ 1 λ -sup R b(•) ř L ∞ (R) . Moreover if λ ∈ R, λ ∈ (sup R b(•), ∞) and ř ≥ 0, then v( • ; λ) ≥ 0. • Assume moreover that a ∈ BU C 1 (R), b is constant, (λ) > b -inf R a (•) , and ř ∈ W 1,∞ (R). Then ∂ x v( • ; λ) L ∞ (R) ≤ 1 λ -b + inf R a (•) ∂ x ř L ∞ (R) .
Proof. Without loss of generality, we assume that a is positive. Let ř ∈ BU C 0 (R). It is easy to check that when (λ)

> sup R b(•), v( • ; λ) ∈ BU C 1 (R) is uniquely defined by v(x; λ) := x -∞ e x y b(z)-λ a(z) d z ř(y) a(y) d y .
The second item is immediately deduced, in particular thanks to the chain inequalities

|v(x; λ)| ≤ ř L ∞ (R) λ -sup R b(•) x -∞ e x y b(z)-λ a(z) d z λ -b(y) a(y) d y = ř L ∞ (R) λ -sup R b(•) .
The third item is obtained in the same way after differentiation and integration by parts:

∂ x v(x; λ) = ř(x) a(x) + x -∞ b -λ a(x) e x y b-λ a(z) d z ř(y) a(y) d y = x -∞ e x y b-λ a(z) d z ∂ y ř(y) a(x) d y = x -∞ e x y b-λ-a (z) a(z) d z ∂ y ř(y) a(y) d y .
This concludes the proof.

We now turn to nonlinear stability results.

Proposition 3.2. Let f ∈ C 2 (R), g ∈ C 1+ (R) and u ∈ R satisfying (3.1). Then for any C 0 > 1, there exists ε > 0 such that for any v 0 ∈ BU C 1 (R) satisfying

v 0 W 1,∞ (R) ≤ ε ,
the initial data u(0, •) = u + v 0 generates a global unique classical solution to (1.1), u ∈ BU C 1 (R + × R), and it satisfies for any t ≥ 0

u(t, •) -u L ∞ (R) ≤ v 0 L ∞ (R) C 0 e g (u) t , ∂ x u(t, •) L ∞ (R) ≤ ∂ x v 0 L ∞ (R) C 0 e g (u) t .
Using an additional concavity/convexity assumption, we may refine the above result by assuming only asymmetric initial smallness on the derivative of the initial data, consistently with the example of the introduction.

Proposition 3.3. Let f ∈ C 2 (R), g ∈ C 1+ (R) and u ∈ R satisfying (3.1) and f (u) = 0.
Then for any C 0 > 1, there exists ε > 0 such that for any

v 0 ∈ BU C 1 (R) satisfying v 0 L ∞ (R) ≤ ε and (sgn(f (u)) ∂ x v 0 ) -L ∞ (R) ≤ ε ,
the initial data u(0, •) = u + v 0 generates a global unique classical solution to (1.1), u ∈ BU C 1 (R + × R), and it satisfies for any t ≥ 0

u(t, •) -u L ∞ (R) ≤ v 0 L ∞ (R) C 0 e g (u) t , (sgn(f (u)) ∂ x u(t, •)) -L ∞ (R) ≤ (sgn(f (u)) ∂ x v 0 ) -L ∞ (R) C 0 e g (u) t , ∂ x u(t, •) L ∞ (R) ≤ ∂ x v 0 L ∞ (R) C 0 e g (u) t .
We sketch the proof of Proposition 3.3, the proof of Proposition 3.2 being similar.

Proof of Proposition 3.3. From Lemma 3.1, we may apply apply general theorems on evolution systems 4 and deduce from any a ∈ C 0 ([0, T ), BU C 1 (R)) and b ∈ C 0 ([0, T ), BU C 0 (R)) (with T ∈ (0, ∞]) an evolution system S a,b on BU C 0 (R) generated by the family of operators L a(t,•), b(t,•) , and such that for any

v 0 ∈ BU C 0 (R), any 0 ≤ s ≤ t < T S a,b (s, t) v 0 L ∞ (R) ≤ e t s sup R b(τ,•) d τ v 0 L ∞ (R) ,
and

S a,b (s, t) v 0 ≥ 0 if v 0 ≥ 0. If moreover b is constant, then v 0 ∈ BU C 1 (R) yields for any 0 ≤ s ≤ t < T ∂ x S a,b (s, t) v 0 L ∞ (R) ≤ e (t-s) b-t s inf R ∂xa(τ,•) d τ ∂ x v 0 L ∞ (R) . Let ε ∈ (0, 1]. Pick a classical solution u = u + v starting from u + v 0 such that v 0 L ∞ (R) ≤ ε.
Then if u exists (as a classical solution) on [0, t 0 ), for any 0 ≤ t < t 0 , we have the Duhamel formula

v(t, •) = S f (u+v), g (u) v 0 + t 0 S f (u+v), g (u) (s, t) g(u + v) -g(u) -g (u)v (s, •) d s .
Therefore if moreover for any t ∈ [0, t 0 ), v(t, •) L ∞ (R) ≤ 2ε e g (u) t , then for any t ∈ [0, t 0 )

e -g (u) t v(t, •) L ∞ (R) ≤ v 0 L ∞ (R) + t 0 ω g (2εe g (u) s )2εe g (u) s e -g (u) s v(s, •) L ∞ (R) d s where ω g (r) = max |x-y|≤r |g (x)-g (y)| r ∈ L 1 loc (R), so that for any t ∈ [0, t 0 ), v(t, •) L ∞ (R) ≤ v 0 L ∞ (R) e g (u) t e 1 |g (u)| 2ε 0 ωg(r) d r .
Choosing ε sufficiently small, we may ensure exp( 1 |g (u)| 2ε 0 ω g (r) d r) < min({2, C 0 }) and a continuity argument yields that the L ∞ estimate of the Proposition holds as long as u persists as a classical solution. From the identity

∂ x v(t, •) = S f (u+v), g (u+v)-f (u+v)∂xv (0, t) ∂ x v 0 ,
by linearity and preservation of non negativity, we deduce

(sgn(f (u))∂ x v(t, •)) -≤ S f (u+v), g (u+v)-f (u+v)∂xv (0, t) (sgn(f (u))∂ x v 0 ) -.
Proceeding as above and lowering ε is necessary, we deduce the second estimate of the Proposition -again as long as u persists as a classical solution-and the third is obtained in the same way. This in particular rules out finite-time blow-up, and the proof is complete.

Stable fronts

In this section we study the stability of bounded continuous fronts, that is u : (t, x) → U (x -σt) solution to (1.1) with f ∈ C 2 (R) and g ∈ C 1 (R), where σ ∈ R and U ∈ C 1 (R) ∩ L ∞ (R) is strictly monotonous. We assume the existence and uniqueness of a characteristic value:

∃!u ∈ U (R), f (u ) = σ and g(u ) = 0 , (3.3)
and assume the non-degeneracy and (strict) spectral stability condition at the characteristic value:

f (u ) = 0 and g (u ) > 0 . (3.4)
Denoting u ±∞ = lim x→±∞ U (x) ∈ R, we assume the non-degeneracy condition at infinity:

g(u ±∞ ) = 0 , g (u ±∞ ) < 0 and f (u ±∞ ) = f (u ) . (3.5)
We denote

F : U (R) → R the C 0 map defined by ∀u ∈ U (R), F (u) = g(u) f (u)-σ if f (u) -σ = 0, g (u) f (u) otherwise, so that ∀x ∈ R, U (x) = F (U (x)) . (3.6)
Linearizing (1.1) about the traveling solution u yields the linear equation

∂ t v + f (u)∂ x v = g (u) -(f (u)) v, v : R + × R → R .
Studying the spectral stability of U as a solution to (1.1) hence amounts to studying the spectrum of the time-independent operator defined by

L := -(f (U ) -σ) ∂ x - U U .
However, as in Section 3.1, the nonlinear stability result will stem from resolvent estimates on a wider class of linear operators. Based on properties of (f (U ) -σ) -recall Proposition 1.6-we denote5

X 1 (R) := a ∈ BU C 1 (R) : a(0) = 0} and consider a ∈ X 1 (R) such that a(x) > 0 if x > 0 , a(x) < 0 if x < 0 . (3.7)
We then denote

L a := -a ∂ x + a U U the closed, densely-defined operator on BU C 1 (R) with domain BU C 2 (R). Lemma 3.4. Let f ∈ C 2 (R), g ∈ C 1 (R) and (U , σ) satisfying (3.3)-(3.4)-(3.5)-(3.6) such that F ∈ C 1 (U (R))
and U (0) = u , and denote

θ := min({g (u ), -g (u +∞ ), -g (u -∞ )}) > 0.
There

exists χ ∈ L 1 (R) ∩ L ∞ (R) such that inf R (f (U )) -(f (U ) -σ) U U + (f (U ) -σ) χ ≥ θ.
For any a ∈ X 1 (R) satisfying (3.7) and

θ a := inf R a -a U U + a χ > 0 , (3.8)
and for any λ ∈ C such that (λ) > -θ a , for any ř ∈ X 1 (R), there exists a unique v(

• ; λ) ∈ BU C 2 (R) such that (λ -L a ) v( • ; λ) = ř
and moreover v(0) = 0 and

v X 1 (R) ≤ 1 (λ) + θ a ř X 1 (R)
where

• X 1 (R) is a norm equivalent to • W 1,∞ (R) on X 1 (R)
and is defined by

∀v ∈ X 1 (R), v X 1 (R) := e -• 0 χ v - U U v L ∞ (R) . (3.9)
Proof. Since F ∈ C 1 , U ∈ BU C 2 (R), and one readily checks that

χ(x) =      max θ-(f (U )) (x)+(f (U (x))-σ) U (x) U (x) (f (U (x))-σ) , 0 when x > 0, min θ-(f (U )) (x)+(f (U (x))-σ) U (x) U (x) (f (U (x))-σ) , 0 when x < 0, (3.10)
satisfies χ ∈ L 1 (R) ∩ L ∞ (R) as well as the desired inequality.

Dividing by U the resolvent problem (λ -L a ) v = ř, differentiating and then multiplying by U yields

λ + a -a U U v U U + a v U U = ř U U . Local solvability in W 1,∞ (R) of the above yields, when (λ) > -θ a , v U U (x) = x 0 e -x y 1 a (λ+a -a U U ) a(y) ř U U (y) d y , (3.11)
and hence, using (3.7) and (3.8),

e -• 0 χ v - U U v L ∞ (R) ≤ 1 (λ) + θ a e -• 0 χ ř - U U ř L ∞ (R) .
When λ = 0, local solvability of the resolvent problem in W 1,∞ (R) also enforces v(0) = ř(0)/λ and hence

v(x) = 1 λ ř(0) U (0) U (x) + U (x) x 0 1 U (y) v - U U v (y) d y.
Hence the eigenvalue 0 of L a is of multiplicity 1, with spectral projector defined by (Πř)(x) := ř(0) U (0) U (x). When ř(0) = 0 the resolvent problem is uniquely solved in W 2,∞ (R) for any (λ) > -θ a by v(x) = U (x)

x 0 1 U (y) v - U U v (y) d y
where we recall that v -U U v has been uniquely determined in (3.11). There remains to prove that

• X 1 (R) is equivalent to • W 1,∞ (R) on X 1 (R). Since χ ∈ L 1 (R), the key argument consists in proving that, when v ∈ X 1 (R), v L ∞ (R) is controlled up to a multiplicative constant by ∂ x v -U U v L ∞ (R)
. This follows from the identity For any C 0 > 1, there exists ε > 0 such that for any v 0 ∈ BU C 1 (R) satisfying

v = x 0 U (x) U (y) ∂ x v - U U v (
v 0 W 1,∞ (R) ≤ ε, (3.12)
and, denoting x ∈ R the characteristic point such that U (x ) = u ,

v 0 (x ) = 0 , (3.13)
the initial data u(0, •) = U + v 0 generates a global unique classical solution to (1.1), u ∈ BU C 1 (R + × R), and it satisfies for any t ≥ 0

u(t, • + x + σt) -U (• + x ) X 1 (R) ≤ v 0 X 1 (R) C 0 e -θt , (3.14)
where • X 1 (R) is defined in (3.9)-(3.10), and

u(t, x + σt) = U (x ) . (3.15)
Proof. Changing the reference frame, we may assume that x = 0. We shall seek u under the form

u(t, x) = U (x -σt) + v(t, x -σt)
where v satisfies

∂ t v -L a( v) v = N ( v), (3.16
)

denoting a : X 1 (R) → X 1 (R) such that a( v) := f (U + v) -σ, and N : X 1 (R) → X 1 (R) such that N ( v) := (f (U ) -f (U + v)) U U v + g(U + v) -g(U ) -g (U ) v -(f (U + v) -f (U ) -f (U ) v) U .
One readily checks that there exists ε 0 > 0, depending only on f , U and χ -defined as in (3.10)-such that if u, the solution to (1.1) emerging from the initial data u(0, •) = U + v 0 , persists as a classical solution to (1.1) on the time interval [0, t 0 ) and satisfies for any t ∈ [0, t 0 ), v(t, •) ∈ X 1 (R) and

v(t, •) W 1,∞ (R) ≤ ε 0 , (3.17)
then a( v) ∈ C([0, t 0 ); X 1 (R)) satisfies (3.7) and (3.8). Hence by the resolvent estimates obtained in Lemma 3.4 and the aforementioned general theorems on evolution systems, the family of operators L a(t,•) generates an evolution system S a on X 1 (R) such that for any 0 ≤ s ≤ t < T and any v 0 ∈ X 1 (R),

S a (s, t) v 0 X 1 (R) ≤ e -t s θ a(τ,•) d τ v 0 X 1 (R) . (3.18)
Then, v ∈ C([0, t 0 ); X 1 (R)) the classical solution to (3.16) satisfies Duhamel's formula

v(t) = S a( v) (0, t)( v 0 ) + t 0 S a( v) (s, t) N ( v(s)) d s.
We may then proceed as in the proof of Proposition 3.3. Let ε ∈ (0, ε 0 ). Assuming that for any t ∈ [0, t 0 ), v X 1 (R) ≤ 2εe -θt , we deduce from Duhamel's formula and the equivalence of

• X 1 (R) with • W 1,∞ (R) , a
quantitative decay estimate on θ -θ a (t, •) and sharp bounds on N :

X 1 (R) → X 1 (R) which eventually yield v(t, •) X 1 (R) ≤ C ε v 0 X 1 (R) e -θt ,
where C ε > 1 depends only on f , g, U , χ and ε, and can be brought arbitrarily close to 1 provided ε is sufficiently small. Choosing ε > 0 such that C ε < min({2, C 0 }) and such that from v X 1 (R) ≤ 2ε stems (3.17) -using again the equivalence of • X 1 (R) with • W 1,∞ (R) -we infer from a continuity argument that (3.14) holds on [0, t 0 ), and this in particular rules out finite-time blow up. The proof is complete.

The assumption (3.13) is not a restriction to the orbital stability with asymptotic phase of stable fronts since, as stated below, any sufficiently small perturbation is an admissible perturbation of a shifted front. It is interesting to notice that the asymptotic phase is determined at initial time from the location of the characteristic point, contrarily to the more standard situation where the phase is only implicitly defined, as in the case of shocks described in the following section. Lemma 3.6. Let f ∈ C 2 (R), g ∈ C 1 (R) and (U , σ) satisfying (3.3)-(3.4)-(3.6) and such that F ∈ C 1 (U (R)), and denote x ∈ R such that U (x ) = u . For any C 0 > 1, there exists ε > 0 such that for any v

0 ∈ W 1,∞ (R) satisfying v 0 W 1,∞ (R) ≤ ε, there exists a unique x ∈ R such that U (x ) + v 0 (x ) = u , and it satisfies |x -x | ≤ C 0 |U (0)| v 0 L ∞ (R) .
Moreover, denoting Ũ := U (• + x -x ) and ṽ0 := U + v 0 -Ũ , we have Ũ (x ) = u , ṽ0 (x ) = 0 and

ṽ0 W 1,∞ (R) ≤ v 0 W 1,∞ (R) + U W 1,∞ (R) |x -x |.
Proof. We use that U is strictly monotonic, U (x ) = 0 and U ∈ W 2,∞ (R). The existence of x is easily deduced from the intermediate value theorem, while uniqueness and the estimate on |x -x | follows from the mean value theorem. The last estimate proceeds from the triangular and mean value inequalities.

Stable shocks

In this section we show the orbital stability with asymptotic phase under regular perturbations of spectrally stable strictly entropy-admissible Riemann shocks of (1.1), that is

u(t, x) = U (x -(d 0 + σt)) , with initial shock position d 0 ∈ R, speed σ ∈ R and wave profile U U (x) = u -if x < 0 , u + if x > 0 , (3.19)
where (u -, u + ) ∈ R 2 , u + = u -satisfy the equilibrium condition g(u + ) = 0 and g(u -) = 0 ; (3.20) the speed σ ∈ R satisfies the Rankine-Hugoniot condition

f (u + ) -f (u -) = σ(u + -u -) , (3.21)
and stability is ensured by the (strict) entropy admissibility

         σ > f (u + ) , f (τ u -+(1-τ ) u + )-f (u -) τ u -+(1-τ ) u + -u - > f (τ u -+(1-τ ) u + )-f (u + ) τ u -+(1-τ ) u + -u + for any τ ∈ (0, 1) , f (u -) > σ , (3.22)
and the spectral assumptions g (u + ) < 0 and g (u -) < 0 .

One could prove the nonlinear stability following the strategy of Section 3.1, after an analysis of the corresponding spectral problem, taking into account the position of the shock; recall Section 1.2. However it is more effective to rely directly on the result obtained for constant states, Proposition 3. For any C 0 > 1, there exists ε > 0 and C > 0 such that for any ψ 0 ∈ R and v 0 ∈ BU C 1 (R ) satisfying

v 0 W 1,∞ (R ) ≤ ε , (3.24)
there exists a unique global-in-time piecewise regular entropy solution to (1.1), u, emerging from the initial data u(0,

•) = (U + v 0 )(• + ψ 0 ) with U as in (3.19). Moreover, there exists ψ ∈ C 2 (R + ) satisfying ψ(0) = 0 and u ± ∈ BU C 1 (R + × R) such that for any t ≥ 0 u(t, x) = u -(t, x) if x < ψ 0 + σt + ψ(t), u + (t, x) if x > ψ 0 + σt + ψ(t), (3.25)
and one has for any t ≥ 0

u ± (t, •) -u ± L ∞ (R) ≤ v 0 L ∞ (R ± ) C 0 e g (u ± ) t , ∂ x u ± (t, •) L ∞ (R) ≤ ∂ x v 0 L ∞ (R ± ) C 0 e g (u ± ) t , |ψ(t) -ψ ∞ | + |ψ (t)| ≤ v 0 L ∞ (R ) C e max({g (u + ),g (u -)}) t , where ψ ∞ = lim t→∞ ψ(t) and satisfies |ψ ∞ | ≤ v 0 L ∞ (R ) C.
Proof. We first extend the left-and right-components of the initial data so as to introduce u

0,± ∈ BU C 1 (R) such that ∀x ∈ R ± , u 0,± (x) = (U + v 0 )(x + ψ 0 ) and u 0,± -u ± L ∞ (R) ≤ C 1/2 0 v 0 L ∞ (R ± ) and ∂ x u 0,± L ∞ (R) ≤ ∂ x v 0 L ∞ (R ± ) .
By Proposition 3.2 with the amplification factor C 1/2 0 and provided the corresponding constraint on ε holds, we may define u ± ∈ BU C 1 (R + × R) as the global unique classical solutions to (1.1) emerging from the initial data u ± (0, •) = u 0,± , and the desired estimates hold. The solution u is then obtained by patching together u + and u -as in (3.25) where the discontinuity curve is defined through the Rankine-Hugoniot condition Then u is an entropy solution to (1.1), and uniqueness is guaranteed by the theory due to Kružkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF].

(u + -u -)(t, ψ 0 + σt + ψ(t)) × σ + ψ (t) = f (u + ) -f (u -) (t, ψ 0 + σt + ψ(t)) .
Remark 3.8. It should be noted that u ± are not defined uniquely, because we have freedom in the choice of the initial data u ± (0, •) = u 0,± . However, the solution u obtained from (3.25) is of course unique in the class of entropy-admissible solutions.

Stable composite waves

In this section we show the orbital stability with asymptotic phase under regular perturbations of bounded piecewise regular traveling wave solutions to (1.1) defined by (U , σ, D), under the following assumptions. For any connected component of R \ D, J, one has either i. U ≡ u is constant on J with g(u) = 0, and g (u) < 0; or ii. U is strictly monotonous on J, bounded and satisfies, for any x ∈ J, U (x) = F (U (x)) where

F (u) = g(u) f (u)-σ if f (u) -σ = 0 , g (u) f (u)
otherwise, and F ∈ C 1 (U (J)). Moreover, there exists a unique u ∈ U (J) such that f (u ) = σ, and one has g(u ) = 0, f (u ) = 0 and g (u ) > 0. We denote x ∈ R the characteristic point, that is U (x ) = u .

We denote u ±∞ = lim x→±∞ U (x), and one has g(u ±∞ ) = 0 and g (u ±∞ ) < 0.

Recall that under non-degeneracy conditions on f and g, by Proposition 1.6, U may be constant only on unbounded connected components of R \ D and (see Remark 1.8) there exists at most -and hence exactlyone connected component on which U is strictly monotonous. For the sake of exposition, we provide the result only for the case where the number of elements in D is |D| = 2; yet the equivalent statement with |D| = 1 can be easily inferred.

As in Section 3.3, we shall infer our stability result by piecing together regular solutions emerging from extensions of the different components of the initial data. To this aim, it is convenient to introduce for the connected components J ⊂ R \ D such that U is strictly monotonous, Ǔ the maximal solution to Ǔ (x) = F ( Ǔ (x)) such that Ǔ = U on J .

± (t, •) -u ±∞ L ∞ (R ± ) ≤ v 0 (• + d 0,± ) L ∞ (R ± ) C 0 e g (u±∞)t , (∂ x u)(t, •) L ∞ (R ± ) ≤ (∂ x v 0 )(• + d 0,± ) L ∞ (R ± )
C 0 e g (u±∞)t , u ( ) (t, • + x + ψ 0 + σt) -Ǔ (• + x ) X 1 ( Jt ) ≤ v 0 X 1 ( J0 ) C 0 e -g (u )t , where • X 1 ( Jt) is defined by (3.9)-(3.10), replacing θ therein with g (u ) and restricting the L ∞ norms to the domain Jt := (d 0,--x -ψ 0 + ψ -(t), d 0,+ -x -ψ 0 + ψ + (t)).

3 .

 3 D = {d} and U is constant on (-∞, d) and on (d, +∞), and takes two different values u -= u + . One has g(u -) = g(u + ) = 0 and σ

Proposition 2 . 3 .

 23 Let k ∈ N and f ∈ C k+2 (R), g ∈ C k+1 (R)and (U , σ, D) define a piecewise regular traveling wave solution to (1.1), u.

(3. 1 )

 1 Lemma 3.1 below shows in particular the spectral stability in BU C 1 (R) of constant states satisfying (3.1).

Proposition 3 . 5 .

 35 y) d y and the boundedness of U /U . We now turn to the consequence of Lemma 3.4 to the nonlinear stability of continuous fronts. Let f ∈ C 3+ (R), g ∈ C 2+ (R) and (U , σ) satisfying (3.3)-(3.4)-(3.5)-(3.6) and such that F ∈ C 1 (U (R)), and denote θ := min({g (u ), -g (u +∞ ), -g (u -∞ )}) > 0.

2 . 6 Proposition 3 . 7 .

 2637 Let f ∈ C 2 (R), g ∈ C 1+ (R) and (σ, u -, u + ) ∈ R 3 satisfying (3.20)-(3.21)-(3.22)-(3.23).

  Existence and uniqueness of ψ ∈ C 2 (R + ) satisfying the above with initial datum ψ(0) = 0 follows from the standard theory on differential equations. The desired bounds on ψ are easily deduced from the corresponding bounds on |u ± -u ± |, and ψ ∞ = ∞ 0 ψ (t) d t . That (1.7) holds by lessening ε further if necessary follows from the continuity ofS f : R × R × [0, 1] → R, (a, b, τ ) → f (τ a + (1 -τ ) b) -f (a) τ a + (1 -τ ) b -a -f (τ a + (1 -τ ) b) -f (b) τ a + (1 -τ ) b -b .

Hypothesis 3 . 9 .

 39 The set D ⊂ R is finite and non-empty. For any d ∈ D, (1.3)-(1.4)-(1.5) hold and [ g(U ) ] d [ U ] d < 0 .

Proposition 3 . 10 .

 310 Let f ∈ C 3+ (R), g ∈ C 2+ (R) be non-degenerate and (U , σ, D) satisfying Hypothesis 3.9 with D = {d -, d + }. For any C 0 > 1 there exists ε > 0 and C > 0 such that for anyD 0 = {d 0,-, d 0,+ } ⊂ R 2 and v 0 ∈ BU C 1 (R \ D 0 ) satisfying v 0 W 1,∞ (R\D0) + |d 0,--d -| + |d 0,+ -d + | ≤ ε,there exists a unique global piecewise regular entropy solution to (1.1), u, emerging from the initial datau(0, •) =      u -∞ + v 0 (x) if x < d 0,-, Ǔ (x) + v 0 (x) if d 0,-< x < d 0,+ , u +∞ + v 0 (x) if x > d 0,+ . Moreover, there exist ψ ± ∈ C 2 (R + ) with ψ ± (0) = 0 and u -, u + , u ( ) ∈ BU C 1 (R + × R) such that ∀t ≥ 0, u(t, x) t, x) if x < d 0,-+ ψ -(t) + σt , u ( ) (t, x) if d 0,-+ ψ -(t) < x -σt < d 0,+ + ψ + (t) , u + (t, x) if x > d 0,+ + ψ + (t) + σt ,(3.26)and for any t ≥ 0,|ψ ± (t) -ψ 0 | + |ψ ± (t)| ≤ v 0 W 1,∞ (R\D0) + |d 0,--d -| + |d 0,+ -d + | C(1 + t)e -θ±t ,where θ ± = min({-g (u ±∞ ), g (u ), -[ g(U ) ] d ± [ U ] d ± }) > 0 and ψ 0 ∈ R is uniquely determined by (U + v 0 )(x + ψ 0 ) = u = U (x ) ,and satisfies|ψ 0 | ≤ C0 |U (0)| v 0 L ∞ ((d0,-,d0,+)) , and one has for any t ≥ 0, u(t, x + ψ 0 + σt) = u ( ) (t, x + ψ 0 + σt) = u and u

This is in fact the key distinction with earlier works on the subject[START_REF] Fan | Large time behavior in inhomogeneous conservation laws[END_REF][START_REF] Lyberopoulos | Large-time structure of solutions of scalar conservation laws without convexity in the presence of a linear source field[END_REF][START_REF] Lyberopoulos | A Poincaré-Bendixson theorem for scalar balance laws[END_REF][START_REF] Mascia | Continuity in finite time of entropy solutions for nonconvex conservation laws with reaction term[END_REF][START_REF] Mascia | Qualitative behavior of conservation laws with reaction term and nonconvex flux[END_REF][START_REF] Mascia | The perturbed Riemann problem for a balance law[END_REF][START_REF] Sinestrari | Large time behaviour of solutions of balance laws with periodic initial data[END_REF][START_REF] Sinestrari | Asymptotic profile of solutions of conservation laws with source[END_REF][START_REF] Sinestrari | Instability of discontinuous traveling waves for hyperbolic balance laws[END_REF][START_REF] Sinestrari | The Riemann problem for an inhomogeneous conservation law without convexity[END_REF], which describe large-time dynamics in L ∞ (R) topology. Our proofs are also radically different: while the previous references rely on generalized characteristics of Dafermos[START_REF] Dafermos | Generalized characteristics and the structure of solutions of hyperbolic conservation laws[END_REF], we employ tools of spectral analysis which are less devoted to the specific case of scalar hyperbolic balance laws. We let the reader refer to[START_REF] Duchêne | Large-Time Asymptotic Stability of Riemann Shocks of Scalar Balance Laws[END_REF] for a more detailed discussion.

Unless U is constant and D = ∅, (U , (1) d∈D ) ∈ X × y is an eigenfunction associated with eigenvalue 0, pertaining to the translation invariance of the problem.

See for instance[START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] Chapter

5, Theorem 3.1] with X = BU C 0 (R) and Y = BU C 1 (R), and apply [12, Chapter 5, Theorem 2.3] to reduce the verification of assumption (H 2 ) there to the case where b is constant.

Here we implicitly set the characteristic point to x = 0, that is U (0) = u . This can be done without harm thanks to the translation invariance of the problem.

One could obtain in the same way a result based on Proposition 3.2, assuming only asymmetric smallness on the derivative of the perturbation, if one -or both-of the two end-states satisfies f (u) = 0.

Proof. We proceed as in the proof of Proposition 3.7, and the first step is to introduce regular extensions of the initial data for each connected component of R \ D 0 . On the connected components J ± such that u(0, •) = u ±∞ + v 0 , we introduce v 0,± ∈ BU C 1 (R) such that v 0,± (x) = v 0 (x) for any x ∈ J ± , and

By Proposition 3.2 with the amplification factor C 1/2 0 and provided the corresponding constraint on ε holds, we obtain u ± ∈ BU C 1 (R + × R) global unique classical solutions to (1.1) emerging from the initial data u ± (0, •) = u ±∞ + v 0,± , and the desired estimates hold.

On the connected components J such that u(0, •) = Ǔ + v 0 , we first extend Ǔ to Ǔ ∈ BU C 1 (R) by solving the differential equation Ǔ = F ( Ǔ ) after modifying (if necessary) f and g on Ǔ (R \ J ) in order to ensure that f ( Ǔ (x)) = f (u ) if and only if x = x , and -g (lim x→±∞ Ǔ (x)) > g (u ). We then uniquely determine ψ 0 -by Lemma 3.6-as the solution to ( Ǔ + v 0 )(x + ψ 0 ) = u , and set

Finally, we may define ṽ0,( ) so that ṽ0,(

By Proposition 3.5 with the amplification factor C 1/2 0 and provided the corresponding constraint on ε holds, we obtain u ( ) ∈ BU C 1 (R + × R) global unique classical solution to (1.1) emerging from the initial data u ( ) (0, •) = Ǔ (• -ψ 0 ) + ṽ0,( ) , and satisfying the desired inequality.

We now construct u through (3.26), defining ψ ± from the Rankine-Hugoniot condition at discontinuities:

u r -u l where u r (resp. u l ) is the limit from the right (resp. from the left) at the discontinuity, consistently with (3.26). Uniqueness and global existence of ψ ± ∈ C 2 (R + ) satisfying the desired estimates follows from standard results on differential equations, using the previously obtained estimates and the fact that

By construction, u is a global-in-time piecewise regular entropy solution to (1.1) provided Oleinik's (strict) entropy conditions hold on discontinuity curves, but this follows as in the proof of Proposition 3.7.

Conclusion

In Section 3, we proved the nonlinear orbital stability with asymptotic phase of constant equilibria, fronts, shocks and composite wave solutions to (1.1) in BU C 1 (R\D)×BU C 1 (R) (where D is the set of discontinuities of the traveling wave) provided that the entropy and spectral stability assumptions stated in the Theorem hold. We have also proved in Section 2 the spectral and nonlinear instability of all other -under the nondegeneracy assumptions on f and g-bounded piecewise regular entropic traveling wave solutions to (1.1). Further results using different functional spaces may be obtained. Firstly, the nonlinear asymptotic stability BU C k (R \ D) × BU C k (R) for k ≥ 2, without assuming smallness on higher derivatives, is easily deduced from the corresponding one when k = 1, after differentiating the equation. Furthermore, using the extension/patching strategy employed when dealing with discontinuities in Section 3.3 and 3.4, we may prove stability under perturbations admitting small strictly entropic discontinuities. Moreover, Proposition 3.3 allows to prove, by a classical approximation/compactness argument, the asymptotic stability of dissipative equilibria u ∈ R such that f (u) = 0 (or strictly entropic shocks between such equilibria) in BV (R) -the space of functions of bounded variation-allowing discontinuous initial data generating small rarefaction waves as well. We let the reader refer to [START_REF] Duchêne | Large-Time Asymptotic Stability of Riemann Shocks of Scalar Balance Laws[END_REF][START_REF] Duchêne | Stability in scalar balance laws: fronts and periodic waves[END_REF] for precise statements and comprehensive discussions.