Romain Serizel

Nicolas Turpault

Ankit Shah

Justin Salamon

SOUND EVENT DETECTION IN SYNTHETIC DOMESTIC ENVIRONMENTS

Keywords: Sound event detection, synthetic data, weakly labeled data, semi-supervised learning

We present a comparative analysis of the performance of state-ofthe-art sound event detection systems. In particular, we study the robustness of the systems to noise and signal degradation, which is known to impact model generalization. Our analysis is based on the results of task 4 of the DCASE 2019 challenge, where submitted systems were evaluated on, in addition to real-world recordings, a series of synthetic soundscapes that allow us to carefully control for different soundscape characteristics. Our results show that while overall systems exhibit significant improvements compared to previous work, they still suffer from biases that could prevent them from generalizing to real-world scenarios.

INTRODUCTION

We are constantly surrounded by sounds and we rely heavily on these sounds to obtain important information about what is happening around us [1]. Ambient sound analysis aims at automatically extracting information from these sounds. It encompasses disciplines such as sound scene classification (in which context does this happen?) or sound event detection and classification (SED) (what happens during this recording?). This area of research has been attracting a continuously growing attention during the past years as it can have a great impact in many applications in noise monitoring in smart cities [START_REF] Bello | SONYC: A system for the monitoring, analysis and mitigation of urban noise pollution[END_REF][START_REF] Juan | Sound analysis in smart cities[END_REF], surveillance [4], urban planning [START_REF] Bello | SONYC: A system for the monitoring, analysis and mitigation of urban noise pollution[END_REF], multimedia information retrieval [START_REF] Wold | Content-based classification, search, and retrieval of audio[END_REF][START_REF] Jin | Event-based video retrieval using audio[END_REF]; and domestic applications such as smart homes, health monitoring systems and home security solutions [START_REF] Serizel | Large-Scale Weakly Labeled Semi-Supervised Sound Event Detection in Domestic Environments[END_REF][START_REF] Debes | Monitoring activities of daily living in smart homes: Understanding human behavior[END_REF][START_REF] Zigel | A method for automatic fall detection of elderly people using floor vibrations and sound-proof of concept on human mimicking doll falls[END_REF].

In Task 4 of the Detection and Classification of Acoustic Scenes and Events (DCASE) 2019 challenge [START_REF] Turpault | Sound event detection in domestic environments with weakly labeled data and soundscape synthesis[END_REF], an extension of the same task from the previous year [START_REF] Serizel | Large-Scale Weakly Labeled Semi-Supervised Sound Event Detection in Domestic Environments[END_REF], we proposed to investigate the added value of synthetic soundscapes with strong labels when training a system to perform SED (with time boundaries) in domestic environments. That is, a system had to detect the presence of a sound event as well as predict the onset and offset times of each occurrence of the event. We generated strongly annotated synthetic soundscapes using the Scaper library [START_REF] Salamon | Scaper: A library for soundscape synthesis and augmentation[END_REF].

The ranking of the task was performed on real audio clips extracted from YouTube and Vimeo. However, the performance analy-This work was made with the support of the French National Research Agency, in the framework of the project LEAUDS "Learning to understand audio scenes" (ANR-18-CE23-0020) and the French region Grand-Est. Experiments presented in this paper were carried out using the Grid5000 testbed, supported by a scientific interest group hosted by Inria and including CNRS, RENATER and several Universities as well as other organizations (see https://www.grid5000).

sis and the study of the system's behavior under scenarios when using real-world data collected from the internet is limited, because we have no control over the characteristics of the soundscapes. Therefore, we provided an additional evaluation set composed of synthetic soundscapes. This dataset was inspired by our analysis of the performance of systems submitted to DCASE 2018 task 4 [START_REF] Serizel | Sound Event Detection from Partially Annotated Data: Trends and Challenges[END_REF]. Using synthetic soundscapes allowed us to design specific scenarios to test the robustness of the submitted systems to corrupt signals, in terms of discrimination between long sound events and short sound events, or in terms on sound event segmentation (localizing a sound event in time regardless of its class). This latter point, in particular, is crucial for SED from weakly labeled data. However, it remains under-investigated.

The synthetic soundscapes evaluation dataset included multiple subsets comprised of the same set of synthetic soundscapes, each with a different type of data degradation applied via the audio degradation toolbox [START_REF] Mauch | The audio degradation toolbox and its application to robustness evaluation[END_REF] or with for different foreground-to-background signal-to-noise ratio (FBSNR), that is the ratio between the loudness of the sound event (foreground) and the loudness of the background noise. We also considered other subsets where the same sound event would be localized at different time instants within a sound clip and subsets using the long sound event classes as background noise. All these scenarios are realistic and represent conditions where the submitted systems are likely to fail. However, gathering enough real data that would cover each of these aspects is hardly feasible. Not to mention that in some cases (e.g., varying FBSNR) it would require recording new data to ensure that only the tested parameter is changing between experiments. Synthetic soundscapes then offer a flexible and realistic alternative to performance preliminary tests before considering further investigation on real data if need be.

In this paper, we present the results obtained by the participants of DCASE 2019 task 4 on the evaluation composed of synthetic soundscapes. This performance was neither presented nor analysed within the challenge. We propose an analysis of the robustness of the ten top-performing approaches (on the evaluation set composed of synthetic soundscapes) to degradation of the recording quality and to varying FBSNR and a study of the robustness of the segmentation process implemented in the submitted systems.

DESED DATASET AND TASK SETUP

The DESED1 dataset is composed of 10-sec audio clips recorded in a domestic environment [START_REF] Serizel | Large-Scale Weakly Labeled Semi-Supervised Sound Event Detection in Domestic Environments[END_REF][START_REF] Turpault | Sound event detection in domestic environments with weakly labeled data and soundscape synthesis[END_REF] or synthesized to simulate a domestic environment. In this paper, we focus on the synthetic subset of the DESED dataset.

DESED synthetic soundscapes evaluation set

The DESED synthetic soundscapes evaluation set is comprised of 10-second audio clips generated with Scaper [START_REF] Salamon | Scaper: A library for soundscape synthesis and augmentation[END_REF], a Python library for soundscape synthesis and augmentation. Scaper operates by taking a set of foreground sounds and a set of background sounds and automatically sequencing them into random soundscapes sampled from a user-specified distribution controlling the number and type of sound events, their duration, signal-to-noise ratio, and several other key characteristics. This set is used for analysis purposes and its design is motivated by the analysis of DCASE 2018 task 4 results [START_REF] Serizel | Sound Event Detection from Partially Annotated Data: Trends and Challenges[END_REF]. In particular, most submissions from DCASE 2018 task 4 performed poorly in terms of event segmentation, that is they were not able to localize sound events properly in time (regardless of the sound event classes).

The foreground events are obtained from the Freesound Dataset (FSD) [START_REF] Font | Freesound technical demo[END_REF][START_REF] Fonseca | Freesound datasets: a platform for the creation of open audio datasets[END_REF]. Each sound event clip was verified by a human to ensure that the sound quality and the event-to-background ratio were sufficient to be used as an isolated sound event. We also controlled for whether the sound event onset and offset were present in the clip. Each selected clip was then segmented when needed to remove silences before and after the sound event and between sound events when the file contained multiple occurrences of the sound event class. The number of unique isolated sound events per class used to generate the subset of synthetic soundscapes is presented in Turpault et al. [START_REF] Turpault | Sound event detection in domestic environments with weakly labeled data and soundscape synthesis[END_REF].

Background sounds are extracted from YouTube videos under a Creative Commons license and from the Freesound subset of the MUSAN dataset [START_REF] Snyder | MU-SAN: A Music, Speech, and Noise Corpus[END_REF]. These recordings were selected because they contain a low amount of sound events from our 10 target foreground sound event classes. However, there is no guarantee that these sound event classes are completely absent from the background clips. DESED synthetic soundscapes evaluation set is further divided into several subsets (described below) for a total of 12,139 audio clips synthesized from 314 isolated events. The synthetic soundscapes are annotated with strong labels that are automatically generated by Scaper [START_REF] Salamon | Scaper: A library for soundscape synthesis and augmentation[END_REF].

Varying foreground-to-background SNR

A subset of 754 soundscapes is generated with Scaper scripts are designed such that the distribution of sound events per class, the number of sound events per clip (depending on the class) and the sound event class co-occurrence are similar to that of the validation set which is composed of real recordings. The foreground event signal-to-noise ratio (SNR) parameter was uniformly drawn between 6 dB and 30 dB. Four versions of this subset are generated varying the value of the background SNR parameter:

• 0 dB (the FBSNR is between 6 dB and 30 dB);

• 6 dB (the FBSNR is between 0 dB and 24 dB);

• 15 dB (the FBSNR is between -9 dB and 15 dB);

• 30 dB (the FBSNR is between -24 dB and 0 dB).

In the remainder of the paper, these subsets will be referred to as fbsnr 30dB, fbsnr 24dB, fbsnr 15dB and fbsnr 0dB, respectively. This subset is designed to study the impact of the SNR on the SED systems performance. Related results are discussed in Section 3.2.

Audio degradation

Six alternative versions of the subset fbsnr 30dB are generated introducing artificial degradation with the Audio Degradation Toolbox [START_REF] Mauch | The audio degradation toolbox and its application to robustness evaluation[END_REF]. The signal degradations are generated to simulate degradation faced in real environments. The following degradations are used (with default parameters) : "smartPhonePlayback", "smartPhoneRecording", "unit applyClippingAlternative", "unit applyDynamicRangeCompression", "unit applyLowpassFilter" and "unit applyHighpassFilter". In the remainder of the paper, these subsets will be referred to as phone play, phone record, clipping, compression, lowpass and highpass, respectively. This subset is designed to study the robustness of the SED to audio degradation. Related results are discussed in Section 3.1.

Varying onset time

A subset of 750 soundscapes is generated with uniform sound event onset distribution and only one event per soundscape. The parameters are set such the FBSNR is between 6 dB and 24 dB. Three variants of this subset are generated with the same isolated events, only shifted in time. In the first version, all sound events have an onset located between 250 ms and 750 ms, in the second version the sound event onsets are located between 4.75 s and 5.25 s and in the last version the sound event onsets are located between 9.25 s and 9.75 s. In the remainder of the paper, these subsets will be referred to as 500ms, 5500ms and 9500ms, respectively. This subset is designed to study of the sensibility of the SED segmentation to the sound event location in time. In particular, we wanted to control if SED systems were learning a bias in terms of time localization depending on the event length (e.g., long sound events would most often start at the beginning of the sound clip). Related results are discussed in Section 4.

Long sound events vs. short sound events

A subset with 522 soundscapes is generated where the background is selected from one of the five long sound event classes (Blender, Electric shaver/toothbrush, Frying, Running water and Vacuum cleaner). The foreground sound events are selected from the five short sound event classes (Alarm/bell/ringing, Cat, Dishes, Dog and Speech). Three variants of this subset are generated with the same sound event scripts and varying values of the background SNR parameter. In a first subset, the resulting FBSNR is 0 dB, the FBSNR is 15 dB in the second and 30 dB in the last subset. In the remainder of the paper, these subsets will be referred to as ls 0dB, ls 15dB and ls 30 dB, respectively. This subset is designed to study the impact of a sound event being in the background or the foreground on SED performance [START_REF] Salamon | Feature learning with deep scattering for urban sound analysis[END_REF]. Related results are discussed in Section 3.2.

Evaluation metrics

Submissions were evaluated with event-based measures for which the system output is compared to the reference labels event by event [START_REF] Mesaros | Metrics for polyphonic sound event detection[END_REF]. The correspondence between sound event boundaries are estimated with a 200 ms tolerance collar on onsets and a tolerance collar on offsets that is a maximum of 200 ms and 20 % of the duration of the sound event. When sound event classes are taken into account, the overall F1-score is the unweighted average of the class-wise F1-scores (macro-average). The metrics are computed using the sed eval library [START_REF] Mesaros | Metrics for polyphonic sound event detection[END_REF].

ROBUSTNESS TO NOISE AND DEGRADATIONS

In this section, we focus on the impact of signal degradation on the SED and the FBSNR performance. Each participant was allowed to submit up to four different systems. Only the F1-score for the topperforming system (on fbsnr 24dB) for each participant is presented here. We limit the analysis to the 10 top-performing systems. Of the top 10 performing systems, 9 were based on CNN and 7 included recurrent layers, the most common input features were log-mel energies and the most common post-processing approach was median filtering. For further details about each system see the submission reports [START_REF] Agnone | Virtual adversarial training system for dcase 2019 task 4[END_REF][START_REF] Cances | Multi task learning and post processing optimization for sound event detection[END_REF][START_REF] Teck | Non-negative matrix factorization-convolution neural network (nmf-cnn) for sound event detection[END_REF][START_REF] Delphin | Mean teacher with data augmentation for dcase 2019 task 4[END_REF][START_REF] Kiyokawa | Sound event detection with resnet and self-mask module for dcase 2019 task 4[END_REF][START_REF] Lim | Sound event detection in domestic environments using ensemble of convolutional recurrent neural networks[END_REF][START_REF] Lin | Guided learning convolution system for dcase 2019 task 4[END_REF][START_REF] Shi | Hodgepodge: Sound event detection based on ensemble of semi-supervised learning methods[END_REF][START_REF] Yan | Weakly labeled sound event detection with resdual crnn using semi-supervised method[END_REF][START_REF] Zhang Zhang | An improved system for dcase 2019 challenge task 4[END_REF].

Simulated degradations

The F-1 score obtained on the degraded subsets is presented in Table 1. The performance on fbsnr 24dB subset are presented here for comparison purpose. The system are ordered alphabetically.

Some systems seem to have over-fitted the synthetic soundscapes subset of the training set and as a result, their performance decreased for most of the degradations. Otherwise, the trend is similar for most of the systems. The submitted systems seem to be rather robust to smartphone-related degradations and compression which can be related to the fact that they have been trained on audio data extracted for YouTube and that has most probably been recorded with smartphones. On the other hand, all systems seem to be very sensitive to low-pass and high-pass filtering which tends to indicate that systems are not robust to changes in the frequency range of the input representation between training and test.

Foreground-to-background Signal-to-noise ratio

In Figure 1, we present the F1-score performance for the 10 topperforming systems mentioned above under varying FBSNR (see Section 2.1.1). The trend for all systems is similar, so no submission stands out in terms of robustness to noise. Interestingly, on fbsnr 15dB where FBSNR should be distributed almost evenly around 0 dB, F1-score performance are still acceptable for most systems and remain in the range of what was obtained on real recording clips [START_REF] Turpault | Sound event detection in domestic environments with weakly labeled data and soundscape synthesis[END_REF]. Unsurprisingly, on fbsnr 0dB, the FBSNR is always negative and the performance for all systems collapses.

We then propose to analyze the systems' performance when the background is actually one event from the long sound event classes and the foreground sound events are selected in the short sound event classes (see Section 2.1.4). In Figure 2, we present the F1-score performance for the 3 top performing systems (on this particular task) together with the performance averaged over all systems. Fig. 2: SED performance depending on the FBSNR when the soundscape is composed of a long event and multiple short events. In all cases, when the FBSNR is low, all systems consistently obtain better performance on long sound event classes. Whereas when the FBSNR is high, all systems obtain better performance on short sound event classes. When the FBSNR is 0 dB most of the systems perform similarly on short sound event classes and long sound event classes. This tends to show that the bias toward long event classes observed in DCASE 2018 [START_REF] Serizel | Large-Scale Weakly Labeled Semi-Supervised Sound Event Detection in Domestic Environments[END_REF][START_REF] Serizel | Sound Event Detection from Partially Annotated Data: Trends and Challenges[END_REF] is less important this year. The trend is confirmed by the performance on short or long sound event classes that are within the same range in the most favorable cases (0 dB FBSNR for the long sound event classes, 30 dB for the short sound event classes). However, the system submitted by Lin et al. [START_REF] Lin | Guided learning convolution system for dcase 2019 task 4[END_REF] does not follow this trend and mostly performs better on long events. This could be due to the guided learning methods that biases the mean teacher model. The teacher labels are converted to 0/1 predictions instead of probabilities which may increase the number of with a positive labels and introduce a bias towards long events.

SEGMENTATION

In this section, we focus on the analysis of the systems' performance in terms of segmentation, that is, the ability of the submitted system to localize a sound event in time (regardless of the sound event class). Sound event segmentation then relies on finding the time instant for Table 1: F1-score performance on the degraded synthetic soundscapes both the sound event onset and offset. In particular, we consider the scenario described in Section 2.1.3 in which three versions of a sound clip are generated with the same background and the same sound event starting either at the beginning, in the middle or at the end of the sound clip. The F1-score performance is presented for the 3 top-performing systems (on this particular task) together with the performance of the baseline system [START_REF] Turpault | Sound event detection in domestic environments with weakly labeled data and soundscape synthesis[END_REF].

For short sound event classes, the F1-score performance is similar wherever the sound event is located within the segment. we will focus on long sound event classes. In Figure 3, we present the F1score in terms of segmentation, onset and offset detection for long sound event classes (but regardless of the sound event class).

For the long sound event classes, the sound event position within the clip seems to have a large impact as performance dramatically decreases when the sound event is located towards the end of the audio clip. One possible explanation could be that in the training set, long sound events have onsets and offsets are mostly located at the beginning of the audio clips. However, as shown in Figure 4, the onset distribution over time is similar for long and short sound event classes. However, the offsets of long sound event classes are often located toward the end of the sound clip. Therefore, if any bias was introduced by the training it should probably have led to a better offset detection for long sound event classes toward the end of the sound clips. This is confirmed by the fact that all systems are able to detect quite accurately the offsets of long sound event classes when the sound event onset is located toward the middle of the sound clip. However, in this case the sound event offsets are located toward the end of the sound clip in most of the time (see Figure 5). One alternative explanation is that the submitted systems are simply not able to detect a long sound event class toward the end of the sound clip. For example, median filtering with variable length that are used in most of the submissions (more than 0.5 s for long sound event classes in some cases [START_REF] Lin | Guided learning convolution system for dcase 2019 task 4[END_REF][START_REF] Delphin | Mean teacher with data augmentation for dcase 2019 task 4[END_REF]) would make it unlikely to detect a long sound event class at the end of the sound clip. This hypothesis tends to be confirmed by the fact that the baseline that is using fixed length median filtering as post-processing performs similarly wherever the sound event is located.

CONCLUSION

In this paper, we present an analysis of the performance of the stateof-the-art SED systems. All submissions to DCASE 2019 task 4 were evaluated on a subset composed of synthetic soundscapes. The analysis shows that training SED on sound clips extracted from internet video makes the systems robust towards degradation related to recording and playing sound on a smartphone. Additionally, we emphasize that even though performance has drastically improved since DCASE 2018 task 4, SED systems still rely on biases (in particular for segmentation) that would probably prevent from generalizing to real case conditions. A first step towards solving this problem was taken in DCASE 2019 where the evaluation set included a real recording from an unseen source (Vimeo). A solution regarding the segmentation problem would be to design the evaluation set such that it includes the limit cases exhibited in the paper together with providing the isolated sound events to the participants (instead of the soundscapes) such that they could design more diverse training examples that would allow for removing so of the bias introduced during the design and learning phases of their systems.

Fig. 1 :

 1 Fig. 1: SED performance depending on the FBSNR.

Fig. 3 :

 3 Fig. 3: Segmentation performance for the long sound event classes depending on the event localization in time.

  (a) Long sound event classes.

Fig. 4 :

 4 Fig. 4: Time distribution of the onsets and the offsets in the synthetic soundscapes subset of DESED training set.

  (a) 500ms subset distribution. (b) 5500ms subset distribution.

Fig. 5 :

 5 Fig. 5: Time distribution of the offsets for long event classes in the subsets 500ms and 5500ms of DESED Evaluation set.

https://project.inria.fr/desed/


Stéphane Vento
email: stephane.vento@univ-paris-est.fr

Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations

Keywords: dissipative dispersive equations, well-posedness, ill-posedness AMS Classification : 35Q55, 35A05, 35M10

We study the Cauchy problem for the dissipative Benjamin-Ono equations u t + Hu xx + |D| α u + uu x = 0 with 0 ≤ α ≤ 2. When 0 ≤ α < 1, we show the ill-posedness in H s (R), s ∈ R, in the sense that the flow map u 0 → u (if it exists) fails to be C 2 at the origin. For 1 < α ≤ 2, we prove the global well-posedness in H s (R), s > -α/4. It turns out that this index is optimal.

1 Introduction, main results and notations

Introduction

In this work we consider the Cauchy problem for the following dissipative Benjamin-Ono equations

u t + Hu xx + |D| α u + uu x = 0, t > 0, x ∈ R, u(0, •) = u 0 ∈ H s (R), (dBO)
with 0 ≤ α ≤ 2, and where H is the Hilbert transform defined by

Hf (x) = 1 π pv 1 x * f (x) = F -1 -i sgn(ξ) f (ξ) (x)
and |D| α is the Fourier multiplier with symbol |ξ| α .

When α = 0, (dBO) is the ordinary Benjamin-Ono equation derived by Benjamin [START_REF] Benjamin | Internal waves of permanent form in fluids of great depth[END_REF] and later by Ono [START_REF] Ono | Algebraic solitary waves in stratified fluids[END_REF] as a model for one-dimensional waves in deep water. The Cauchy problem for the Benjamin-Ono equation has been extensively studied these last years. It has been proved in [START_REF] Saut | Sur quelques généralisations de l'équation de Korteweg-de Vries[END_REF] that (BO) is globally well-posed in H s (R) for s ≥ 3, and then for s ≥ 3/2 in [START_REF] Ponce | On the global well-posedness of the Benjamin-Ono equation[END_REF] and [START_REF] Iório | On the Cauchy problem for the Benjamin-Ono equation[END_REF]. In [START_REF] Tao | Global well-posedness of the Benjamin-Ono equation in H 1 (R)[END_REF], Tao get the well-posedness of this equation for s ≥ 1 by using a gauge transformation (which is a modified version of the Cole-Hopf transformation). Recently, combining a gauge transformation together with a Bourgain's method, Ionescu and Kenig [START_REF] Ionescu | Global well-posedness of the Benjamin-Ono equation in low-regularity spaces[END_REF] shown that one could go down to L 2 (R), and this seems to be, in some sense, optimal. It is worth noticing that all these results have been obtained by compactness methods. On the other hand, Molinet, Saut and Tzvetkov [START_REF] Molinet | Ill-posedness issues for the Benjamin-Ono and related equations[END_REF] proved that, for all s ∈ R, the flow map u 0 → u is not of class C 2 from H s (R) to H s (R). Furthermore, building suitable families of approximate solutions, Koch and Tzvetkov proved in [START_REF] Koch | Nonlinear wave interactions for the Benjamin-Ono equation[END_REF] that the flow map is actually not even uniformly continuous on bounded sets of H s (R), s > 0. As an important consequence of this, since a Picard iteration scheme would imply smooth dependance upon the initial data, we see that such a scheme cannot be used to get solutions in any space continuously embedded in C([0, T ]; H s (R)).

When α = 2, (dBO) is the so-called Benjamin-Ono-Burgers equation u t + (H -1)u xx + uu x = 0.

(BOB)

Edwin and Robert [START_REF] Edwin | The Benjamin-Ono-Burgers equation: an application in solar physics[END_REF] have derived (BOB) by means of formal asymptotic expansions in order to describe wave motions by intense magnetic flux tube in the solar atmosphere. The dissipative effects in that context are due to heat conduction. (BOB) has been studied in many papers, see [START_REF] Dix | Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation[END_REF][START_REF] Fokas | Global solutions and their asymptotic behavior for Benjamin-Ono-Burgers type equations[END_REF][START_REF] Zhang | Local Lipschitz continuity of a nonlinear bounded operator induced by a generalized Benjamin-Ono-Burgers equation[END_REF].

Working in Bourgain's spaces containing both dispersive and dissipative effects1 , Otani showed in [START_REF] Otani | Bilinear estimates with applications to the generalized Benjamin-Ono-Burgers equations[END_REF] that (BOB) is globally well-posed in H s (R), s > -1/2. In this paper, we prove that this index is in fact critical since the flow map u 0 → u is not of class C 3 from H s (R) to H s (R), s < -1/2. Intriguingly, this index coincides with the critical Sobolev space for the Burgers equation u t -u xx + uu x = 0, see [START_REF] Dix | Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation[END_REF][START_REF] Bekiranov | The initial-value problem for the generalized Burgers' equation[END_REF]. This result is in a marked contrast with what occurs for the KdV-Burgers equation which is well-posed above H -1 (R), see [START_REF] Molinet | On the low regularity of the Korteweg-de Vries-Burgers equation[END_REF]. Now consider the general case 0 ≤ α ≤ 2. By running the approach of [START_REF] Molinet | On the low regularity of the Korteweg-de Vries-Burgers equation[END_REF] combined with the smoothing relation obtained in [START_REF] Otani | Bilinear estimates with applications to the generalized Benjamin-Ono-Burgers equations[END_REF], we can only get that the problem (dBO) is well-posed in H s (R) for 3/2 < α ≤ 2 and s > 1/2 -α/2. This was done by Otani in [START_REF] Otani | Well-posedness of the generalized Benjamin-Ono-Burgers equations in Sobolev spaces of negative order[END_REF]. Here we improve this result by showing that (dBO) is globally well-posed in H s (R), for 1 < α ≤ 2 and s > -α/4. It is worth comparing (dBO) with the pure dissipative equation

u t + |D| α u + uu x = 0. (1.1)
In the Appendix, we show that (1.1) with 1 < α ≤ 2 is well-posed in H s (R) as soon as s > 3/2 -α. The techniques we use are very common in the context of semilinear parabolic problems and can be easily adapted to (dBO). In particular when α = 2, this provides an alternative (and simpler) proof of our main result. When α < 2, clearly we see that the dispersive part in (dBO) plays a key role in the low regularity of the solution.

We are going to perform a fixed point argument on the integral formulation of (dBO) in the weighted Sobolev space

u X b,s α = i(τ -ξ|ξ|) + |ξ| α b ξ s Fu(τ, ξ) L 2 (R 2 ) . (1.2)
This will be achieved by deriving a bilinear estimate in these spaces. By Plancherel's theorem and duality, it reduces to estimating a weighted convolution of L 2 functions. In some regions where the dispersive effect is too weak to recover the lost derivative in the nonlinear term at low regularity (s > -α/4), in particular when considering the high-high interactions, we are led to use a dyadic approach. In [START_REF] Tao | Multilinear weighted convolution of L 2 -functions, and applications to nonlinear dispersive equations[END_REF], Tao systematically studied some nonlinear dispersive equations like KdV, Schrödinger or wave equation by using such dyadic decomposition and orthogonality. Following the spirit of Tao's works, we shall prove some estimates on dyadic blocks, which may be of independent interest. Indeed, we believe that they could certainly be used for other equations based on a Benjamin-Ono-type dispersion.

Next, we show that our well-posedness results turn out to be sharp. Adapting the arguments used in [START_REF] Molinet | Ill-posedness issues for the Benjamin-Ono and related equations[END_REF] to prove the ill-posedness of (BO), we find that the solution map u 0 → u (if it exists) cannot be C 3 at the origin from H s (R) to H s (R) as soon as s < -α/4. See also [START_REF] Bourgain | Periodic Korteweg de Vries equation with measures as initial data[END_REF][START_REF] Molinet | On the low regularity of the Korteweg-de Vries-Burgers equation[END_REF][START_REF] Molinet | Well-posedness results for the generalized Benjamin-Ono equation with small initial data[END_REF][START_REF] Vento | Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity[END_REF] for situations where this method applies. Note that we need to prove the discontinuity of the third iterative term to obtain the condition s < -α/4, whereas the second iterate is usually sufficient to get an optimal result. On the other hand, we prove using similar arguments, that in the case 0 ≤ α < 1, the solution map fails to be C 2 in any H s (R), s ∈ R. This is mainly due to the fact that the operator |D| α is too weak to counterbalance the lost derivative which appears in the nonlinear term ∂ x u 2 .

Main results

Let us now formally state our results.

Theorem 1.1. Let 1 < α ≤ 2 and u 0 ∈ H s (R) with s > -α/4. Then for any T > 0, there exists a unique solution u of (dBO) in

Z T = C([0, T ]; H s (R)) ∩ X 1/2,s α,T . Moreover, the map u 0 → u is smooth from H s (R) to Z T and u belongs to C((0, T ], H ∞ (R)).
Remark 1.1. The spaces X b,s α,T are restricted versions of X b,s α defined by the norm (1.2). See Section 1.3 for a precise definition.

Remark 1.2. In [START_REF] Otani | Well-posedness of the generalized Benjamin-Ono-Burgers equations in Sobolev spaces of negative order[END_REF], Otani studied a larger family of dispersive-dissipative equations taking the form

u t -|D| 1+a u x + |D| α u + uu x = 0 (1.3)
with a ≥ 0 and α > 0. He showed that (1.3) is globally well-posed in H s (R) provided a + α ≤ 3, α > (3 -a)/2 and s > -(a + α -1)/2. If a = 0, it is clear that we get a better result, at least when α < 2. It will be an interesting challenge to adapt our method of proofs to (1.3) in the case a > 0.

Remark 1.3. Another interesting problem should be to consider the periodic dissipative BO equations

u t + Hu xx + |D| α u + uu x = 0, t > 0, x ∈ T, u(0, •) = u 0 ∈ H s (T), (1.4)
Recall that in [START_REF] Molinet | Global well-posedness in L 2 for the periodic Benjamin-Ono equation[END_REF], Molinet proved the global well-posedness of the periodic BO equation in L 2 (T). To our knowledge, equation (1.4) in the case α > 0 has never been investigated.

Theorem 1.1 is sharp in the following sense.

Theorem 1.2. Let 1 ≤ α ≤ 2 and s < -α/4. There does not exist T > 0 such that the Cauchy problem (dBO) admits a unique local solution defined on the interval [0, T ] and such that the flow map u 0 → u is of class C 3 in a neighborhood of the origin from H s (R) to H s (R).

In the case 0 ≤ α < 1, we have the following ill-posedness result.

Theorem 1.3. Let 0 ≤ α < 1 and s ∈ R. There does not exist T > 0 such that the Cauchy problem (dBO) admits a unique local solution defined on the interval [0, T ] and such that the flow map u 0 → u is of class C 2 in a neighborhood of the origin from H s (R) to H s (R). The structure of our paper is as follows. We introduce a few notation in the rest of this section. In Section 2, we recall some estimates related to the linear (dBO) equations. Next, we prove the crucial bilinear estimate in Section 3, which leads to the proof of Theorem 1.1 in Section 4. Section 5 is devoted to the ill-posedness results (Theorems 1.2 and 1.3). Finally, we briefly study the dissipative equation (1.1) in the Appendix.

Notations

When writing A B (for A and B nonnegative), we mean that there exists C > 0 independent of A and B such that A ≤ CB. Similarly define A B and A ∼ B. If A ⊂ R N , |A| denotes its Lebesgue measure and χ A its characteristic function. For f ∈ S ′ (R N ), we define its Fourier transform F(f ) (or f ) by

Ff (ξ) = R N e -i x,ξ f (x)dx.
The Lebesgue spaces are endowed with the norm

f L p (R N ) = R N |f (x)| p dx 1/p , 1 ≤ p < ∞
with the usual modification for p = ∞. We also consider the space-time Lebesgue spaces L p x L q t defined by

f L p x L q t = f L q t (R) L p x (R)
.

For b, s ∈ R, we define the Sobolev spaces H s (R) and their space-time versions H b,s (R 2 ) by the norms

f H s = R ξ 2s | f (ξ)| 2 dξ 1/2 , u H b,s = R 2 τ 2b ξ 2s | u(τ, ξ)| 2 dτ dξ 1/2 , with • = (1 + | • | 2 ) 1/2
. Let V (•) be the free linear group associated to the linear Benjamin-Ono equation, i.e.

∀t ∈ R, F x (V (t)ϕ)(ξ) = exp(itξ|ξ|) ϕ(ξ), ϕ ∈ S ′ .
We will mainly work in the X b,s α space defined in (1.2), and in its restricted version X b,s α,T , T ≥ 0, equipped with the norm

u X b,s α,T = inf w∈X b,s α { w X b,s α , w(t) = u(t) on [0, T ]}. Note that since F(V (-t)u)(τ, ξ) = u(τ + ξ|ξ|, ξ), we can re-express the norm of X b,s α as u X b,s α = iτ + |ξ| α b ξ s u(τ + ξ|ξ|, ξ) L 2 (R 2 ) = iτ + |ξ| α b ξ s F(V (-t)u)(τ, ξ) L 2 (R 2 ) ∼ V (-t)u H b,s + u L 2 t H s+αb x .
Finally, we denote by S α the semigroup associated with the free evolution of (dBO),

∀t ≥ 0, F x (S α (t)ϕ)(ξ) = exp[itξ|ξ| -|ξ| α t] ϕ(ξ), ϕ ∈ S ′ ,
and we extend S α to a linear operator defined on the whole real axis by setting

∀t ∈ R, F x (S α (t)ϕ)(ξ) = exp[itξ|ξ| -|ξ| α |t|] ϕ(ξ), ϕ ∈ S ′ . (1.5)

Linear estimates

In this section, we collect together several linear estimates on the operators S α introduced in (1.5) and L α defined by

L α : f → χ R + (t)ψ(t) t 0 S α (t -t ′ )f (t ′ )dt ′ .
Recall that (dBO) is equivalent to its integral formulation

u(t) = S α (t)u 0 - 1 2 t 0 S α (t -t ′ )∂ x (u 2 (t ′ ))dt ′ . (2.1)
It will be convenient to replace the local-in-time integral equation (2.1) with a global-in-time truncated integral equation. Let

ψ be a cutoff function such that ψ ∈ C ∞ 0 (R), supp ψ ⊂ [-2, 2], ψ ≡ 1 on [-1, 1]
, and define ψ T (•) = ψ(•/T ) for all T > 0. We can replace (2.1) on the time interval [0, T ], T < 1 by the equation

u(t) = ψ(t) S α (t)u 0 - χ R + (t) 2 t 0 S α (t -t ′ )∂ x (ψ 2 T (t ′ )u 2 (t ′ ))dt ′ . (2.2)
Proofs of the results stated here can be obtained by a slight modification of the linear estimates derived in [START_REF] Molinet | On the low regularity of the Korteweg-de Vries-Burgers equation[END_REF].

Lemma 2.1. For all s ∈ R and all ϕ ∈ H s (R),

ψ(t)S α (t)ϕ X 1/2,s α ϕ H s . (2.3) Lemma 2.2. Let s ∈ R. For all 0 < δ < 1/2 and all v ∈ X -1/2+δ,s α , χ R + (t)ψ(t) t 0 S α (t -t ′ )v(t ′ )dt ′ X 1/2,s α v X -1/2+δ,s α . (2.4)
To globalize our solution, we will need the next lemma.

Lemma 2.3. Let s ∈ R and δ > 0. Then for any f ∈ X

-1/2+δ,s α , t -→ t 0 S α (t -t ′ )f (t ′ )dt ′ ∈ C(R + ; H s+αδ ). Moreover, if (f n ) is a sequence satisfying f n → 0 in X -1/2+δ,s α , then t 0 S α (t -t ′ )f n (t ′ )dt ′ L ∞ (R + ;H s+αδ ) -→ 0.
3 Bilinear estimates

Dyadic blocks estimates

We introduce Tao's [k; Z]-multipliers theory [START_REF] Tao | Multilinear weighted convolution of L 2 -functions, and applications to nonlinear dispersive equations[END_REF] and derive the dyadic blocks estimates for the Benjamin-Ono equation.

Let Z be any abelian additive group with an invariant measure dη. For any integer k ≥ 2 we define the hyperplane

Γ k (Z) = {(η 1 , ..., η k ) ∈ Z k : η 1 + ... + η k = 0}
which is endowed with the measure

Γ k (Z) f = Z k-1 f (η 1 , ..., η k-1 , -(η 1 + ... + η k-1 ))dη 1 ...dη k-1 . A [k; Z]-multiplier is defined to be any function m : Γ k (Z) → C. The multiplier norm m [k;Z]
is defined to be the best constant such that the inequality

Γ k (Z) m(η) k j=1 f j (η j ) ≤ m [k;Z] k j=1 f j L 2 (Z) (3.1)
holds for all test functions f 1 , ..., f k on Z. In other words,

m [k;Z] = sup f j ∈S(Z) f j L 2 (Z) ≤1 Γ k (Z) m(η) k j=1 f j (η j ) .
In his paper [START_REF] Tao | Multilinear weighted convolution of L 2 -functions, and applications to nonlinear dispersive equations[END_REF], Tao used the following notations. Capitalized variables N j , L j (j = 1, ..., k) are presumed to be dyadic, i.e. range over numbers of the form 2 ℓ , ℓ ∈ Z. In this paper, we only consider the case k = 3, which corresponds to the quadratic nonlinearity in the equation. It will be convenient to define the quantities N max ≥ N med ≥ N min to be the maximum, median and minimum of N 1 , N 2 , N 3 respectively. Similarly, define L max ≥ L med ≥ L min whenever L 1 , L 2 , L 3 > 0. The quantities N j will measure the magnitude of frequencies of our waves, while L j measures how closely our waves approximate a free solution.

Here we consider [3; R × R]-multipliers and we parameterize R × R by η = (τ, ξ) endowed with the Lebesgue measure dτ dξ. Define

h j (ξ j ) = ξ j |ξ j |, λ j = τ j -h j (ξ j ), j = 1, 2, 3,
and the resonance function

h(ξ) = h 1 (ξ 1 ) + h 2 (ξ 2 ) + h 3 (ξ 3 ).
By a dyadic decomposition of the variables ξ j , λ j , h(ξ), we will be led to estimate

X N 1 ,N 2 ,N 3 ,H,L 1 ,L 2 ,L 3 [3;R×R] (3.2)
where

X N 1 ,N 2 ,N 3 ,H,L 1 ,L 2 ,L 3 = χ |h(ξ)|∼H 3 j=1 χ |ξ j |∼N j χ |λ j |∼L j . (3.3)
From the identities

ξ 1 + ξ 2 + ξ 3 = 0 (3.4)
and

λ 1 + λ 2 + λ 3 + h(ξ) = 0
on the support of the multiplier, we see that (3.3) vanishes unless

N max ∼ N med (3.5)
and

L max ∼ max(H, L med ). (3.6) Lemma 3.1. On the support of X N 1 ,N 2 ,N 3 ,H,L 1 ,L 2 ,L 3 , one has H ∼ N max N min . (3.7) Proof. Recall that h(ξ) = ξ 1 |ξ 1 | + ξ 2 |ξ 2 | + ξ 3 |ξ 3 |.
By symmetry, we can assume |ξ 3 | ∼ N min . This forces by (3.4) ξ 1 ξ 2 < 0. Suppose for example ξ 1 > 0 and ξ 2 < 0 (the other case being similar). Then if

ξ 3 > 0, h(ξ) = ξ 2 1 -ξ 2 2 + ξ 2 3 = ξ 2 1 -(ξ 1 + ξ 3 ) 2 + ξ 2 3 = -2ξ 1 ξ 3 and in this case |h(ξ)| ∼ N max N min . Now if ξ 3 < 0, then h(ξ) = ξ 2 1 -ξ 2 2 -ξ 2 3 = (ξ 2 + ξ 3 ) 2 -ξ 2 2 -ξ 2 3 = 2ξ 2 ξ 3
and it follows again that |h(ξ)| ∼ N max N min .

We are now ready to state the fundamental dyadic blocks estimates for the Benjamin-Ono equation.

Proposition 3.1. Let N 1 , N 2 , N 3 , H, L 1 , L 2 , L 3 > 0 satisfying (3.5), (3.6), (3.7).
1. In the high modulation case L max ∼ L med ≫ H, we have

(3.2) L 1/2 min N 1/2 min .
(3.8)

In the low modulation case

L max ∼ H, (a) ((++) coherence) if N max ∼ N min , then (3.2) L 1/2 min L 1/4 med , (3.9)
(b) ((+-) coherence) if N 2 ∼ N 3 ≫ N 1 and H ∼ L 1 L 2 , L 3 , we have for any γ > 0 (3.2) L 1/2 min min(N 1/2 min , N 1/2-1/2γ max N -1/2γ min L 1/2γ med ).
(3.10)

Similarly for permutations of the indexes {1, 2, 3}.

(c) In all other cases, the multiplier (3.3) vanishes.

Proof. First we consider the high modulation case L max ∼ L med ≫ H. Suppose for the moment that L 1 ≥ L 2 ≥ L 3 and N 1 ≥ N 2 ≥ N 3 . By using the comparison principle (Lemma 3.1 in [START_REF] Tao | Multilinear weighted convolution of L 2 -functions, and applications to nonlinear dispersive equations[END_REF]), we have

(3.2) χ |ξ 3 |∼N 3 χ |λ 3 |∼L 3 [3;R×R] .
By Lemma 3.14 and Lemma 3.6 in [START_REF] Tao | Multilinear weighted convolution of L 2 -functions, and applications to nonlinear dispersive equations[END_REF],

(3.2)

χ |λ 3 |∼L 3 [3;R] χ |ξ 3 |∼N 3 [3;R] L 1/2 3 N 1/2 3 .
It is clear from symmetry that (3.8) holds for any choice of L j and N j , j = 1, 2, 3. Now we turn to the low modulation case H ∼ L max . Suppose for the moment that N 1 ≥ N 2 ≥ N 3 . The ξ 3 variable is currently localized to the annulus {|ξ 3 | ∼ N 3 }. By a finite partition of unity we can restrict it further to a ball {|ξ 3 -ξ 0 3 | ≪ N 3 } for some |ξ 0 3 | ∼ N 3 . Then by box localisation (Lemma 3.13 in [START_REF] Tao | Multilinear weighted convolution of L 2 -functions, and applications to nonlinear dispersive equations[END_REF]) we may localize ξ 1 , ξ 2 similarly to regions

{|ξ 1 -ξ 0 1 | ≪ N 3 } and {|ξ 2 -ξ 0 2 | ≪ N 3 } where |ξ 0 j | ∼ N j .
We may assume that |ξ 0 1 + ξ 0 2 + ξ 0 3 | ≪ N 3 since we have ξ 1 + ξ 2 + ξ 3 = 0. We summarize this symmetrically as

(3.2) χ |h(ξ)|∼H 3 j=1 χ |ξ j -ξ 0 j |≪N min χ |λ j |∼L j [3;R×R]
for some ξ 0 j satisfying

|ξ 0 j | ∼ N j for j = 1, 2, 3; |ξ 0 1 + ξ 0 2 + ξ 0 3 | ≪ N min .
Without loss of generality, we assume L 1 ≥ L 2 ≥ L 3 . By Lemma 3.6, Lemma 3.1 and Corollary 3.10 in [START_REF] Tao | Multilinear weighted convolution of L 2 -functions, and applications to nonlinear dispersive equations[END_REF], we get

(3.2) χ |h(ξ)|∼H 3 j=2 χ |ξ j -ξ 0 j |≪N min χ |λ j |∼L j [3;R×R] |{(τ 2 , ξ 2 ) : |ξ 2 -ξ 0 2 | ≪ N min , |τ 2 -h 2 (ξ 2 )| ∼ L 2 , |ξ -ξ 2 -ξ 0 3 | ≪ N min , |τ -τ 2 -h 3 (ξ -ξ 2 )| ∼ L 3 }| 1/2
for some (τ, ξ) ∈ R × R. For fixed ξ 2 , the set of possible τ 2 ranges in an interval of length O(L 3 ) and vanishes unless

h 2 (ξ 2 ) + h 3 (ξ -ξ 2 ) = τ + O(L 2 ).
On the other hand, inequality |ξ -

ξ 2 -ξ 0 3 | ≪ N min implies |ξ + ξ 0 1 | ≪ N min , hence (3.2) L 1/2 3 |Ω ξ | 1/2 for some ξ such that |ξ + ξ 0 1 | ≪ N min (in particular |ξ| ∼ N 1
) and with

Ω ξ = {ξ 2 : |ξ 2 -ξ 0 2 | ≪ N min , h 2 (ξ 2 ) + h 3 (ξ -ξ 2 ) = τ + O(L 2 )}. Let us write Ω ξ = Ω 1 ξ ∪ Ω 2 ξ with Ω 1 ξ = {ξ 2 ∈ Ω ξ : ξ 2 (ξ -ξ 2 ) > 0} Ω 2 ξ = {ξ 2 ∈ Ω ξ : ξ 2 (ξ -ξ 2 ) < 0}.
We need only to consider the three cases

N 1 ∼ N 2 ∼ N 3 , N 2 ∼ N 3 ≫ N 1 and N 1 ∼ N 2 ≫ N 3 (the case N 1 ∼ N 3 ≫ N 2 follows by symmetry). Estimate of |Ω 1 ξ | :
In Ω 1 ξ we can assume ξ 2 > 0 and ξ -ξ 2 > 0 (the other case being similar). Then we have

h 2 (ξ 2 ) + h 3 (ξ -ξ 2 ) = ξ 2 2 + (ξ -ξ 2 ) 2 = 2 ξ 2 - ξ 2 2 + ξ 2 2
and thus [START_REF] Molinet | Global well-posedness in L 2 for the periodic Benjamin-Ono equation[END_REF], we see that we must have

2 ξ 2 - ξ 2 2 + ξ 2 2 = τ + O(L 2 ). (3.11) If N 1 ∼ N 2 ∼ N 3 , we see from (3.11) that ξ 2 variable is contained in the union of two intervals of length O(L 1/2 2 ) at worst. Therefore |Ω 1 ξ | L 1/2 2 in this case. If N 1 ∼ N 2 ≫ N 3 , then ξ 2 - ξ 2 + ξ 0 1 2 ≤ ξ 2 -ξ 0 2 - ξ + ξ 0 1 2 -ξ 0 3 + |ξ 0 1 + ξ 0 2 + ξ 0 3 | ≤ |ξ 2 -ξ 0 2 | + 1 2 |ξ + ξ 0 1 | + |ξ 0 3 | + |ξ 0 1 + ξ 0 2 + ξ 0 3 | N 3 and we get |ξ 2 -ξ 2 | ∼ N 1 . From (3.
N 2 1 = O(L 2 ), which is in contradiction with L 2 L 1 ∼ N max N min .
We deduce that the multiplier vanishes in this region. If N 2 ∼ N 3 ≫ N 1 , then we have obviously |ξ 2 -ξ 2 | ∼ N 2 and, in the same way, the multiplier vanishes.

Estimate of |Ω 2

ξ | : We can assume ξ 2 > 0 and ξ -ξ 2 < 0. It follows that

h 2 (ξ 2 ) + h 3 (ξ -ξ 2 ) = ξ 2 2 -(ξ -ξ 2 ) 2 = 2ξ ξ 2 - ξ 2 = τ + O(L 2 ). (3.12) If N 1 ∼ N 2 ∼ N 3 ,

Bilinear estimate

In this section we prove the following crucial bilinear estimate.

Theorem 3.1. Let 1 < α ≤ 2 and s > -α/4. For all T > 0, there exist δ, ν > 0 such that for all u, v ∈ X

1/2,s α with compact support (in time) in [-T, +T ], ∂ x (uv) X -1/2+δ,s α T ν u X 1/2,s α v X 1/2,s α . (3.13)
To get the required contraction factor T ν in our estimates, the next lemma is very useful (see [START_REF] Otani | Well-posedness of the generalized Benjamin-Ono-Burgers equations in Sobolev spaces of negative order[END_REF]).

Lemma 3.2. Let f ∈ L 2 (R 2 ) with compact support (in time) in [-T, +T ].
For any θ > 0, there exists ν = ν(θ) > 0 such that

F -1 f (τ, ξ) τ -ξ|ξ| θ L 2 xt T ν f L 2 xt .
Proof of Theorem 3.1. By duality, Plancherel and Lemma 3.2, it suffices to show that

ξ 3 ξ 3 s ξ 1 -s ξ 2 -s |λ 1 | + |ξ 1 | α 1/2 |λ 2 | + |ξ 2 | α 1/2 |λ 3 | + |ξ 3 | α 1/2-δ [3;R×R] 1.
By dyadic decomposition of the variables ξ j , λ j , h(ξ), we may assume |ξ j | ∼ N j , |λ j | ∼ L j and |h(ξ)| ∼ H. By the translation invariance of the [k, Z]multiplier norm, we can always restrict our estimate on L j 1 and N max 1. The comparison principle and orthogonality reduce our estimate to show that

Nmax∼N med ∼N L 1 ,L 2 ,L 3 1 N 3 N 3 s N 1 -s N 2 -s (L 1 + N 1 α ) 1/2 (L 2 + N 2 α ) 1/2 (L 3 + N 3 α ) 1/2-δ × X N 1 ,N 2 ,N 3 ,Lmax,L 1 ,L 2 ,L 3 [3;R×R] (3.14)
and

Nmax∼N med ∼N Lmax∼L med H≪Lmax N 3 N 3 s N 1 -s N 2 -s (L 1 + N 1 α ) 1/2 (L 2 + N 2 α ) 1/2 (L 3 + N 3 α ) 1/2-δ × X N 1 ,N 2 ,N 3 ,H,L 1 ,L 2 ,L 3 [3;R×R] (3.15)
are bounded, for all N 1. We first show that (3.15) 1. For s > -1/2, one has

N 3 N 3 s N 1 -s N 2 -s N min -s N max
and we get from (3.8), (3.15)

Nmax∼N Lmax≫N N min N min -s N L 1/2 min N 1/2 min L 1/2 min (L max + N α ) 1/2-δ (L max + N min α ) 1/2-δ L δ max N min >0 N 1/2 min N min -s N (N N min + N α ) 1/2-δ (N N min + N min α ) 1/2-δ .
When N min 1, we get (3.15)

N min 1 N 1/2 min N N α/2-αδ (N N min ) 1/2-δ N min 1 N δ min N (1-α)/2+δ(α+1)
1 for δ ≪ 1 and α > 1. When N min 1, then (3.15)

N min 1 N 1/2-s min N (N N min ) 1/2-δ-ε N αε (N N min ) 1/2-δ N min 1 N -1/2-s+2δ+ε min N 2δ-ε(α-1) 1 for ε = 2δ/(α -1) > 0, δ ≪ 1 and s > -1/2.
Now we show that (3.14) 1. We first deal with the contribution where (3.9) holds. In this case N min ∼ N max and we get (3.14)

Lmax∼N 2 N 1-s L 1/2 min L 1/4 med L 1/2 min (L med + N α ) 1/2 (L max + N α ) 1/2-2δ L δ max N 1-s N α/4 N 1-4δ N -s-α/4+4δ 1 for s > -α/4 and δ ≪ 1.
Now we consider the contribution where (3.10) applies. By symmetry it suffices to treat the two cases

N 1 ∼ N 2 ≫ N 3 , H ∼ L 3 L 1 , L 2 , N 2 ∼ N 3 ≫ N 1 , H ∼ L 1 L 2 , L 3 .
In the first case, estimate (3.10) applied with γ = 1 yields

(3.2) L 1/2 min min(N 1/2 3 , N -1/2 3 L 1/2 med ) L 1/2 min N 1/4 3 N -1/4 3 L 1/4 med ∼ L 1/2 min L 1/4 med and thus (3.14) N 3 >0 Lmax∼N N 3 N 3 N 3 s N -2s L 1/2 min L 1/4 med L 1/2 min (L med + N α ) 1/2 (L max + N min α ) 1/2-2δ L δ max N 3 >0 N 3 N 3 s N -2s N α/4 (N N 3 ) 1/2-2δ N 3 >0 N 1/2+2δ 3 N 3 s N -2s-α/4-1/2+2δ .
Since -2s -α/4 -1/2 + 2δ < 0, we may write (3.14)

N 3 >0 N 1/2+2δ 3 N 3 -s-α/4-1/2+2δ N 3 1 N 1/2+2δ 3 + N 3 1 N -s-α/4+4δ 3 1 for δ ≪ 1 and s > -α/4. Finally consider the case N 2 ∼ N 3 ≫ N 1 , H ∼ L 1 L 2 , L 3 . Let 0 < γ ≪ 1. If we assume N 1/2 min N 1/2-1/2γ max N -1/2γ min L 1/2γ med , i.e. L med N 1-γ max N 1+γ
min , then we get from (3.10) that (3.14)

N 1 >0 Lmax∼N N 1 N 1 -s N L 1/2 min N 1/2 1 L 1/2 min (L med + N α ) 1/2-δ L 1/2-δ max L δ max N 1 >0 N 1/2 1 N 1 -s N (N 1-γ N 1+γ 1 + N α ) 1/2-δ (N N 1 ) 1/2-δ N 1 >0 N δ 1 N 1 -s N 1/2+δ (N 1-γ N 1+γ 1 + N α ) 1/2-δ . If N 1 1, then (3.14) N 1 1 N δ 1 N (1-α)/2+δ(1+α) 1 for δ ≪ 1 and α > 1. If N 1 1, then (3.14) N 1 1 N -s+δ 1 N 1/2+δ (N 1-γ N 1+γ 1 ) 1/2-δ-ε N αε N 1 1 N -s-1/2+(1+γ)(δ+ε)+δ-γ/2 1 N γ(1/2-δ)+2δ-ε(α-1+γ) 1 for δ, γ ≪ 1, s > -1/2 and ε = [2δ + γ(1/2 -δ)]/(α -1 + γ) > 0. If we assume N 1/2 min N 1/2-1/2γ max N -1/2γ min L 1/2γ med , i.e. L med N 1-γ max N 1+γ min , we get (3.14) N 1 >0 Lmax∼N N 1 N 1 -s N L 1/2 min N 1/2-1/2γ N -1/2γ 1 L 1/2γ med L 1/2 min (L med + N α ) 1/2-δ L 1/2-δ max L δ max N 1 >0 L med N 1-γ N 1+γ 1 N -1/2γ-1/2+δ 1 N 1 -s N 1-1/2γ+δ L 1/2γ med (L med + N α ) 1/2-δ .
When N 1 1, we have (3.14)

N 1 1 N -1/2γ-1/2+δ 1 N 1-1/2γ+δ N -α/2+αδ (N 1-γ N 1+γ 1 ) 1/2γ N 1 1 N δ 1 N (1-α)/2+δ(1+α) 1
for δ ≪ 1 and α > 1. When N 1 1, then (3.14)

N 1 1 N -s-1/2-1/2γ+δ 1 N 1-1/2γ+δ (N 1-γ N 1+γ 1 ) 1/2γ-1/2+δ+ε N -αε N 1 1 N -s-1/2+(1+γ)(δ+ε)+δ-γ/2 1 N γ(1/2-δ)+2δ-ε(α-1+γ) 1
as previously. This completes the proof of Theorem 3.1.

Proof of Theorem 1.1

In this section, we sketch the proof of Theorem 1.1 (see for instance [START_REF] Molinet | On the low regularity of the Korteweg-de Vries-Burgers equation[END_REF] for the details).

Actually, local existence of a solution is a consequence of the following modified version of Theorem 3.1. Proposition 4.1. Given s + c > -α/4, there exist ν, δ > 0 such that for any s ≥ s + c and any u, v ∈ X

1/2,s α with compact support in [-T, +T ], ∂ x (uv) X -1/2+δ,s α T ν ( u X 1/2,s + c α v X 1/2,s α + u X 1/2,s α v X 1/2,s + c α
). (4.1) Estimate (4.1) is obtained thanks to (3.13) and the triangle inequality

∀s ≥ s + c , ξ s ≤ ξ s + c ξ 1 s-s + c + ξ s + c ξ -ξ 1 s-s + c .
Let u 0 ∈ H s (R) with s > -α/4. Define F (u) as

F (u) = ψ(t) S α (t)u 0 - χ R + (t) 2 t 0 S α (t -t ′ )∂ x (ψ 2 T (t ′ )u 2 (t ′ ))dt ′ .
We shall prove that for T ≪ 1, F is contraction in a ball of the Banach space

Z = {u ∈ X 1/2,s α : u Z = u X 1/2,s + c α + γ u X 1/2,s α < +∞},
where γ is defined for all nontrivial ϕ by

γ = ϕ H s + c ϕ H s .
Combining (2.3), (2.4) as well as (4.1), it is easy to derive that

F (u) Z ≤ C( u 0 H s + c + γ u 0 H s ) + CT ν u 2 Z and F (u) -F (v) Z ≤ CT ν u -v Z u + v Z
for some C, ν > 0. Thus, taking T = T ( u 0 H s + c ) small enough, we deduce that F is contractive on the ball of radius 4C u 0 H s + c in Z. This proves the existence of a solution u to u = F (u) in X 1/2,s α,T . Following similar arguments of [START_REF] Molinet | On the low regularity of the Korteweg-de Vries-Burgers equation[END_REF], it is not too difficult to see that if

u 1 , u 2 ∈ X 1/2,s α,T
are solutions of (2.2) and 0 < δ < T /2, then there exists ν > 0 such that

u 1 -u 2 X 1/2,s α,δ T ν u 1 X 1/2,s α,T + u 2 X 1/2,s α,T u 1 -u 2 X 1/2,s α,δ
, which leads to u 1 ≡ u 2 on [0, δ], and then on [0, T ] by iteration. This proves the uniqueness of the solution.

It is straightforward to check that

S α (•)u 0 ∈ C(R + ; H s (R))∩C(R * + ; H ∞ (R)
). Then it follows from Theorem 3.1, Lemma 2.3 and the local existence of the solution that

u ∈ C([0, T ]; H s (R)) ∩ C((0, T ]; H s+αδ (R))
for some T = T ( u 0 H s + c ). By induction, we have u ∈ C((0, T ]; H ∞ (R)). Taking the L 2 -scalar product of (dBO) with u, we obtain that t → u(t) H s + c is nonincreasing on (0, T ]. Since the existence time of the solution depends only on the norm u 0 H s + c , this implies that the solution can be extended globally in time.

Ill-posedness results

This section is devoted to the proof of Theorems 1.2 and 1.3. We adopt the notation p(ξ) = ξ|ξ|.

Assume that u is a solution of (dBO) such that the solution map

u 0 → u is of class C k (k = 2 or k = 3) at the origin from H s (R) to H s (R). The relation F (u, ϕ) := u(t, ϕ) -S α (t)ϕ + 1 2 t 0 S α (t -t ′ )∂ x (u 2 (t ′ , ϕ))dt ′ ≡ 0 combined with implicit function theorem gives u 1 (t, x) := ∂u ∂ϕ (t, x, 0)[h] = S α (t)h u 2 (t, x) := ∂ 2 u ∂ϕ 2 (t, x, 0)[h, h] = t 0 S α (t -t ′ )∂ x (u 1 (t ′ )) 2 dt ′ u 3 (t, x) := ∂ 3 u ∂ϕ 3 (t, x, 0)[h, h, h] = t 0 S α (t -t ′ )∂ x (u 1 (t ′ )u 2 (t ′ ))dt ′ etc
Since the solution map is C k , we must have

u k (t) H s h k H s , ∀h ∈ H s (R).
(5.1)

In the sequel, we will show that (5.1) fails in the case 0 ≤ α < 1 and k = 2, and in the case 1 ≤ α ≤ 2, k = 3 and s > -α/4.

The case 0 ≤ α < 1

It suffices to show the following lemma.

Lemma 5.1. Let 0 ≤ α < 1 and s ∈ R. There exists a sequence of functions {h N } ⊂ H s (R) such that for all T > 0,

h N H s 1,
and

lim N →∞ sup [0,T ] t 0 S α (t -t ′ )∂ x (S α (t ′ )h N ) 2 dt ′ H s = +∞.
Proof. We define h N by its Fourier transform

1 h N (ξ) = γ -1/2 χ I 1 (ξ) + γ -1/2 N -s χ I 2 (ξ)
with

I 1 = [γ/2, γ], I 2 = [N, N + γ] and N ≫ 1, γ ≪ N to be chosen later.
Then it is clear that h N H s ∼ 1. Computing the Fourier transform of u 2 (t) leads to

F x (u 2 (t))(ξ) = cξ t 0 e i(t-t ′ )p(ξ) e -(t-t ′ )|ξ| α (e it ′ p(ξ) e -t ′ |ξ| α h N ) * 2 (ξ)dt ′ = cξe itp(ξ) e -t|ξ| α R h N (ξ 1 ) h N (ξ -ξ 1 )
× t 0 e it ′ (p(ξ1 )+p(ξ-ξ 1 )-p(ξ)) e -t ′ (|ξ

1 | α +|ξ-ξ 1 | α -|ξ| α ) dt ′ dξ 1 = cξe itp(ξ) e -t|ξ| α R h N (ξ 1 ) h N (ξ -ξ 1 )
× e it(p(ξ 1 )+p(ξ-ξ 1 )-p(ξ)) e -t(|ξ

1 | α +|ξ-ξ 1 | α -|ξ| α ) -1 i(p(ξ 1 ) + p(ξ -ξ 1 ) -p(ξ)) -(|ξ 1 | α + |ξ -ξ 1 | α -|ξ| α ) dξ 1 . Set χ(ξ, ξ 1 ) = i(p(ξ 1 ) + p(ξ -ξ 1 ) -p(ξ)) -(|ξ 1 | α + |ξ -ξ 1 | α -|ξ| α ).
By support considerations, we have u 2 (t)

H s ≥ v 2 (t) H s with F x (v 2 (t))(ξ) = cN -s γ -1 ξe itp(ξ) e -t|ξ| α K ξ e tχ(ξ,ξ 1 ) -1 χ(ξ, ξ 1 ) dξ 1 (5.2)
and

K ξ = {ξ 1 : ξ 1 ∈ I 1 , ξ -ξ 1 ∈ I 2 } ∪ {ξ 1 : ξ 1 ∈ I 2 , ξ -ξ 1 ∈ I 1 }.
We easily see that if ξ 1 ∈ K ξ , then ξ ∈ [N + γ/2, N + 2γ] and

p(ξ 1 ) + p(ξ -ξ 1 ) -p(ξ) = 2ξ 1 (ξ 1 -ξ) ∼ γN, |ξ 1 | α + |ξ -ξ 1 | α -|ξ| α N α .
We deduce that for γ = N α-1 ≪ N , we have |χ(ξ, ξ 1 )| ∼ N α . Now define

t N = (N + 2γ) -α-ε ∼ N -α-ε
so that e -t N |ξ| α 1. By a Taylor expansion of the exponential function,

e t N χ(ξ,ξ 1 ) -1 χ(ξ, ξ 1 ) = t N + R(t N , ξ, ξ 1 ) (5.3) with |R(t N , ξ, ξ 1 )| k≥2 t k N |χ(ξ, ξ 1 )| k-1 k! N -α-2ε .
Therefore the main contribution of (5.3) in (5.2) is given by t N , and since

|K ξ | ∼ γ, it follows that |F x (v 2 (t N ))(ξ)| N -s+1 γ -1 e -(N +2γ) -ε γN -α-ε χ [N +γ/2,N +2γ] (ξ) N -s+1-α-ε χ [N +γ/2,N +2γ] (ξ).
We get the lower bound for the H s -norm of u 2 (t N )

u 2 (t N ) H s N -s+1-α-ε N +2γ N +γ/2 (1+|ξ| 2 ) s dξ 1/2 ∼ N 1-α-ε γ 1/2 ∼ N (1-α)/2-ε , which leads to lim N →∞ sup [0,T ] u 2 (t) H s = +∞
for ε ≪ 1 and α < 1, as we claim.

The case 1 ≤ α ≤ 2

Let 1 ≤ α ≤ 2 and s < -α/4. As previously, it suffices to find a suitable sequence {h N } such that h N H s 1 and lim

N →∞ sup [0,T ] u 3 (t) H s = +∞.
For this purpose, we define the real-valued function h N by

h N (ξ) = N -s γ -1/2 (χ I N (ξ) + χ I N (-ξ)) (5.4)
with I N = [N, N + 2γ], N ≫ 1 and γ ≪ N to be chosen later. We have

F x (u 3 (t))(ξ) = cξ t 0 e i(t-t ′ )p(ξ) e -(t-t ′ )|ξ| α F x (S α (t ′ )h N ) * F x (u 2 (t ′ ))(ξ)dt ′ and F x (S α (t ′ )h N ) * F x (u 2 (t ′ ))(ξ) = c R 2 h N (ξ 1 ) h N (ξ 2 -ξ 1 ) h N (ξ -ξ 2 )ξ 2 × e it ′ (p(ξ-ξ 2 )+p(ξ 2 )) e -t ′ (|ξ-ξ 2 | α +|ξ 2 | α ) e tχ(ξ 2 ,ξ 1 ) -1 χ(ξ 2 , ξ 1 ) dξ 1 dξ 2 .
Hence, we can write u 3 = v 3 -w 3 with

F x (v 3 (t))(ξ) = cξe itp(ξ) e -t|ξ| α R 2 h N (ξ 1 ) h N (ξ 2 -ξ 1 ) h N (ξ -ξ 2 ) ξ 2 χ(ξ 2 , ξ 1 ) × t 0 e it ′ (p(ξ 1 )+p(ξ 2 -ξ 1 )+p(ξ-ξ 2 )-p(ξ)) e -t(|ξ 1 | α +|ξ 2 -ξ 1 | α +|ξ-ξ 2 | α -|ξ| α ) dt ′ dξ 1 dξ 2 = cξe itp(ξ) e -t|ξ| α R 2 h N (ξ 1 ) h N (ξ 2 -ξ 1 ) h N (ξ -ξ 2 ) ξ 2 χ(ξ 2 , ξ 1 ) e tλ(ξ,ξ 1 ,ξ 2 ) -1 λ(ξ, ξ 1 , ξ 2 ) dξ 1 dξ 2 and F x (w 3 (t))(ξ) = cξe itp(ξ) e -t|ξ| α R 2 h N (ξ 1 ) h N (ξ 2 -ξ 1 ) h N (ξ -ξ 2 ) ξ 2 χ(ξ 2 , ξ 1 ) × t 0 e t ′ χ(ξ,ξ 2 ) dt ′ dξ 1 dξ 2 = cξe itp(ξ) e -t|ξ| α R 2 h N (ξ 1 ) h N (ξ 2 -ξ 1 ) h N (ξ -ξ 2 ) ξ 2 χ(ξ 2 , ξ 1 ) e tχ(ξ,ξ 2 ) -1 χ(ξ, ξ 2 ) dξ 1 dξ 2
where we set

λ(ξ, ξ 1 , ξ 2 ) = i(p(ξ 1 )+p(ξ 2 -ξ 1 )+p(ξ-ξ 2 )-p(ξ))-(|ξ 1 | α +|ξ 2 -ξ 1 | α +|ξ-ξ 2 | α -|ξ| α ).
Let t N = (N + 4γ) -α-ε for some 0 < ε ≪ 1. We get

|F x (v 3 (t N ))(ξ)|χ [N +3γ,N +4γ] (ξ) N -3s+1 γ -3/2 K ξ ξ 2 χ(ξ 2 , ξ 1 ) e t N λ(ξ,ξ 1 ,ξ 2 ) -1 λ(ξ, ξ 1 , ξ 2 ) dξ 1 , dξ 2 where K ξ = K 1 ξ ∪ K 2 ξ ∪ K 3 ξ and K 1 ξ = {(ξ 1 , ξ 2 ) : ξ 1 ∈ I N , ξ 2 -ξ 1 ∈ I N , ξ -ξ 2 ∈ -I N }, K 2 ξ = {(ξ 1 , ξ 2 ) : ξ 1 ∈ I N , ξ 2 -ξ 1 ∈ -I N , ξ -ξ 2 ∈ I N }, K 3 ξ = {(ξ 1 , ξ 2 ) : ξ 1 ∈ -I N , ξ 2 -ξ 1 ∈ I N , ξ -ξ 2 ∈ I N }. If ξ ∈ [N + 3γ, N + 4γ] and (ξ 1 , ξ 2 ) ∈ K ξ , we easily see that ξ 2 χ(ξ 2 , ξ 1 ) ∼ N -1 and p(ξ 1 ) + p(ξ 2 -ξ 1 ) + p(ξ -ξ 2 ) -p(ξ) ∼ γ 2 , |ξ 1 | α + |ξ 2 -ξ 1 | α + |ξ -ξ 2 | α -|ξ| α ∼ N α .
Thus we are led to choose γ

= N α/2 ≪ N for N ≫ 1 so that |λ(ξ, ξ 1 , ξ 2 )| ∼ N α . Then it follows that e t N λ(ξ,ξ 1 ,ξ 2 ) -1 λ(ξ, ξ 1 , ξ 2 ) = |t N | + O(N -α-2ε ).
Consequently,

|F x (v 3 (t N ))(ξ)|χ [N +3γ,N +4γ] (ξ) N -3s+1 γ -3/2 N -1 γ 2 N -α-ε χ [N +3γ,N +4γ] (ξ) ∼ N -3s-α-ε γ 1/2 χ [N +3γ,N +4γ] (ξ) ∼ N -3s-3α/4-ε χ [N +3γ,N +4γ] (ξ), since |K ξ | ∼ γ 2 .
Concerning w 3 , we verify that for (ξ 1 , ξ 2 ) ∈ K ξ , we have |χ(ξ, ξ 2 )| γN and then

|F x (w 3 (t N ))(ξ)|χ [N +3γ,N +4γ] (ξ) N -3s+1 γ -3/2 γ 2 N -1 (γN ) -1 χ [N +3γ,N +4γ] (ξ) ∼ N -3s-1 γ -1/2 χ [N +3γ,N +4γ] (ξ) ∼ N -3s-1-α/4 χ [N +3γ,N +4γ] (ξ)
Since -3s -1 -α/4 < -3s -3α/4 -ε for α < 2, we deduce that the main contribution in the H s -norm of u 3 is given by v 3 H s , that is,

u 3 (t N ) H s N -3s-3α/4-ε γ 1/2 N s ∼ N -2s-α/2-ε ,
and we find the condition -2s -α/2 > 0, i.e. s < -α/4.

When α = 2, the contributions of v 3 and w 3 are equivalent, and we must proceed with a bit more care, by considering directly the difference u 3 = v 3 -w 3 . More precisely, for γ = εN ≪ N , we have

|λ(ξ, ξ 1 , ξ 2 )| ∼ |χ(ξ, ξ 2 )| ∼ N 2 . Noticing that λ(ξ, ξ 1 , ξ 2 ) -χ(ξ, ξ 2 ) = χ(ξ 2 , ξ 1 ),
we deduce

e t N λ(ξ,ξ 1 ,ξ 2 ) -1 λ(ξ, ξ 1 , ξ 2 ) - e t N χ(ξ,ξ 2 ) -1 χ(ξ, ξ 2 ) = t 2 N |χ(ξ 2 , ξ 1 )| + O(t 3 N N 2 |χ(ξ 2 , ξ 1 )|)
Setting again t N = N -2-ε , and since |ξ 2 | ∼ N , it follows that

|F x (u 3 (t N ))(ξ)|χ [N +3γ,N +4γ] N -3s+1 γ -3/2 γ 2 N N -4-2ε χ [N +3γ,N +4γ] (ξ)
and thus u 3 (t N ) H s N -2s-2-2ε γ ∼ N -2s-1-2ε , which tends to infinity as soon as -2s -1 > 0, i.e. s < -1/2.

A Appendix

We prove here that the pure dissipative equation u t + |D| α u + uu x = 0 (A.1) for 1 < α ≤ 2 is well-posed in H s (R), s > s α where

s α = 3 2 -α,
and that the solution map fails to be smooth when s < s α . The method of proof is classical and is based on the smoothing properties of the generalized heat kernel G α (t, x) = 1 2π R e ixξ e -t|ξ| α dξ, t > 0.

and in the same way,

F (u) -F (v) X T T ν ( u X T + v X T ) u -v X T .
This proves that for T ≪ 1, F is contractive in a ball of X T . Now we solve (A. D -s G α (t) L 2/α u 0 H s , and it follows from (A.4) that t β D -s G α (t) L 2/α t β (t -(2-α)/2α + t -(2-α)/2α+s/α ) T -s/α . Now we deal with the nonlinear term. Using the Sobolev embedding L ( 1 2 -s) -1 (R) ֒→ H s (R) valid for any -1/2 < s < 0, we obtain Proof. The proof is similar to that of Theorems 1.2 and 1.3. Define h N as in (5.4) and consider the high-high interactions in the convolution product (e -t|ξ| α h N ) * (e -t|ξ| α h N ). We get that for ξ ∈ [2N, 2N + 4γ], γ = N 1-ε and t N ∼ N -α-ε ,

t 0 G α (t -t ′ ) * ∂ x u 2 (t ′ ) H s dt ′ t 0 ∂ x G α (t -t ′ ) L ( 5 2 -s-α) -1 u 2 (t ′ ) L 1/(α-
|F x (u 2 (t N ))(ξ)| N -2s-α+1-ε χ [2N,2N +4γ] (ξ)
where u 2 is defined by

u 2 (t) = t 0 G α (t -t ′ ) * ∂ x (G α (t ′ ) * h N ) 2 dt ′ .
We conclude that u 2 (t N ) H s N -s-α+1-ε γ 1/2 N -s+3/2-α-3ε/2 → +∞ as soon as s < 3/2 -α.

Remark 1 . 4 .

 14 At the end-point α = 1, our proof of Theorem 1.3 fails. However, Theorem 1.2 provides the ill-posedness in H s (R), for s < -1/4. So, it is still not clear of what happens to (dBO) when α = 1 and s ≥ -1/4.

1 )

 1 in the case 3/2 < α ≤ 2 and s α < s < 0. DefineY T = C([0, T ]; H s (R)) ∩ C β ([0, T ]; L 2/(α-1) (R)) equipped with the norm u Y T = sup t∈[0,T ] u(t) H s + sup t∈[0,T ] t β u(t) L 2/(α-1) .By Young inequality, we getG α (t) * u 0 L 2/(α-1) = D -s G α (t) * D s u 0 L 2/(α-1)

T ν u 2 Y T with ν = 1 -

 21 t ′ ) -(α+1)/2α t ′-2β dt ′ u 2 Y T (α + 1)/2α -β > 0. Finally, one has F (u) Y T T ν u 0 H s + T ν u 2 Y Tand the claim follows.Remark A.1. Let U α (t) = F -1 ξ (e itξ|ξ| e -t|ξ| α ) be the fundamental solution of the linear (dBO) equation. Using that |F x U α (t)| = |F x G α (t)| as well as the well-known estimate f L p f L p ′ , p ≥ 2, 1/p + 1/p ′ = 1, we easily check that Theorem A.1 holds for (dBO) equation.Finally, we show that Theorem A.1 is sharp.Theorem A.2. Let 1 < α ≤ 2 and s < s α . The the solution map u 0 → u associated with (A.1) (if it exists) is not of class C 2 from H s (R) to C([0, T ]; H s (R)).

  ∼ N 2 ≫ N 3 , we have |ξ 2 -ξ 2 | ∼ N 1 as previously and thus N 2 1 = O(L 2 ), the multiplier vanishes. If N 2 ∼ N 3 ≫ N 1 , then |ξ 2 -ξ 2 | ∼ N 2 and for any γ > 0, we have |ξ 2 -ξ

	N 2 1 and thus |Ω 2 ξ | L	1/2 2	in this region.
	If N 1 2 | ∼ N 1-γ 2 |ξ 2 -ξ 2 | γ . Therefore we see from (3.12) that ξ 2 variable is contained in the union of two intervals of length O(N 1-1/γ 2 N -1/γ 1 L 1/γ 2 ) at worst, and from |ξ 2 -ξ 0 2 | ≪ N min we see that |Ω 2 ξ | N 1/2

we see from (3.12) that ξ 2 variable is contained in the union of two intervals of length O(N -1 1 L 2 ) at worst. But we have L 2 L 1 ∼ min , and (3.10) follows.

  1) dt ′ -t ′ ) -s/α-1+1/2α t ′-2β t ′2β u(t ′ ) 2 L 2/(α-1) dt ′

	t
	0 (t T ν u 2 Y

T

with ν = -s/α + 1/2α -2β > 0. By similar calculations, we get

t β t 0 G α (t -t ′ ) * ∂ x u 2 (t ′ ) L 2/(α-1) dt ′ t β t 0 ∂ x G α (t -t ′ ) L 2/(3-α) u 2 (t ′ ) L 1/(α-1) dt ′

Such spaces were first introduce by Molinet and Ribaud in[START_REF] Molinet | On the low regularity of the Korteweg-de Vries-Burgers equation[END_REF] for the KdV-Burgers equation.

As noticed in[START_REF] Molinet | Ill-posedness issues for the Benjamin-Ono and related equations[END_REF], hN is not a real-valued function but the analysis works as well for ℜe hN instead of hN .

Acknowledgments

The author wishes to express his gratitude to Francis Ribaud for his encouragement and precious advice.

Theorem A.1. Let 1 < α ≤ 2, s > s α and u 0 ∈ H s (R). Then there exist T > 0 and a unique solution u ∈ C([0, T ]; H s (R)) of (A.1) such that

where β = -s/α + (2 -α)/2α. The flow map u 0 → u from H s (R) into the class defined by (A.2)-(A.3) is locally Lipschitz. Moreover, if u 0 H s is small enough, the solution can be extended to any time interval.

Proof. Observe that for any p ∈ [1, ∞] and ρ ≥ 0, we have

We use the Picard iteration theorem to show that the map F defined as

has a fixed point in suitable Banach space. We first consider the case 1 < α ≤ 3/2, and we choose s α < s < 1/2. Set X T = C([0, T ]; H s (R)) endowed with the norm u X T = sup [0,T ] u(t) H s . By Young inequality and (A.4), we have

Using the fractional Leibniz rule, we get

Since 0 < s < 1/2, we can take advantage of the Sobolev embedding

with ν = 1 + s/α -3/2α > 0. Gathering (A.5) and (A.6) we infer


R Haghighi-Khoskoo

Francis Mc Cluskey

R Haghighi-Khoshkhoo

F M J Mccluskey

Air-side fouling of compact heat exchangers for discrete particle size ranges

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Reduced performance due to the fouling of the compact heat exchanger of an industrial vehicle may necessitate it being removed from service for cleaning, a potentially time-consuming process. The more polluted the air is locally, the more often the vehicle is out of action, resulting in increased operating costs. Fouling is a dynamic process that evolves continually until equilibrium is reached and the exchanger needs to be cleaned. This can take hours, weeks, or months, depending on the environment.

Much work has been carried out on the phenomena involved in liquid-side fouling within tubes, and these are now fairly well

The authors would like to thank the French and Iranian governments for cofinancing this project via the SPHERE organization. The work was greatly aided by the donation of compact heat exchangers by Dr. Bruno Hamery of Valeo.

Address correspondence to Professor F. M. J. McCluskey, Laboratoire des Ecoulements Geophysiques et Industriels, Université Joseph Fourier, BP 53, 38041 Grenoble cedex 9, France; E-mail: Francis.Mc-cluskey@ujf-grenoble.fr understood. The term "air-side" fouling refers to the build-up of debris and/or particles on the external surface of the exchanger, most importantly, between the fins. Studies to date generally cover the effects on the heat transfer coefficient of micrometricsized particles; little attention has been paid to the millimetric range, which has no significant effect on it. Cowell & Cross [START_REF] Cowell | Airside Fouling of Internal Combustion Engine Radiators[END_REF] showed that for off-road vehicles, about 7% of the particles involved in fouling were of this larger size. Research on larger particles has also been carried out by Bott and coworkers [START_REF] Bott | Fouling of Heat Exchangers[END_REF], Lankinen et al. [START_REF] Lankinen | The Effect of Air-Side Fouling on the Heat Transfer Efficiency of a Heat Exchanger with a Segmented Finned Tube[END_REF], and Seigel and Nazaroff [START_REF] Seigel | Modeling Particle Deposition on HVAC Heat Exchangers[END_REF]. Cowell & Cross tested 22 different exchangers in a wind tunnel under "oily" conditions to enhance adherence of the particles to the surface. They found that most of the foulant remained on the front of the exchanger, resulting in a high pressure drop due to the supplementary (effective) filter that it formed there. They observed no great modification to the heat transfer measurements at constant air speed. They also defined a fouling coefficient which accounts for the pressure drop due solely to the foulant:

F = ( p total -p exchanger )/1/2ρv 2 .

Air-Side Fouling of Compact Heat Exchangers for Discrete Particle Size Ranges

As fouling increased, F showed little variation with air speed. Similar results and conclusions were found by Bott et al. [START_REF] Bott | Fouling of Heat Exchangers[END_REF] in their wind tunnel tests. However, Lankinen et al. [START_REF] Lankinen | The Effect of Air-Side Fouling on the Heat Transfer Efficiency of a Heat Exchanger with a Segmented Finned Tube[END_REF] found that the thermal performance of the exchanger could be reduced by up to 18% following fouling, a result they did not explain. Seigel and Nazaroff [START_REF] Seigel | Modeling Particle Deposition on HVAC Heat Exchangers[END_REF] examined the front surface of a number of exchangers fouled in a dusty environment by particles of less than 10 µm and found a significant quantity of much larger size particles within the foulant. They concluded that the presence of bigger particles increased the rate of fouling. Once more, the foulant did not penetrate into the heart of the exchanger. In all of the above cited works, the air speed was fixed at values between 1 and 5 m/s.

EXPERIMENTAL PROCEDURE

The test section of the wind tunnel used for the experiments had a 40 cm × 40 cm cross-section and was made of plexiglass, allowing for the visualization of the fouling process. Exchangers were introduced via a lateral slot, which was then insulated from the outside to prevent air leakage. Pressure tappings were drilled into the tunnel floor upstream and downstream of the exchanger and were connected to a differential pressure manometer. In a number of experiments, the exchanger was wetted using a water spray from a side window. Air velocity was measured using a Pitot tube. On the liquid side of the exchanger, thermocouples were set in the entrance and exit flows. A thermostated bath was used for pumping and heating the water for the exchanger.

Particles were introduced into the flow through the test section roof just over 1m upstream of the exchanger. Numerous experiments were carried out for different rates of particle introduction. The particles were introduced in discrete lots rather than continuously due to the technical difficulties and costs of such a setup. A given total mass M of single-size particles was divided equally into n distinct lots and introduced in succession into the wind tunnel. The greater the number n, the closer the procedure approaches the continuous injection case. The measurements showed that for n varying from 3 to 20, the results were very reproducible, showing a variation of less than 5% overall. The conclusion that the rate of introduction had little or no effect on the fouling process apart from duration to saturation is expected, and the mass of foulant on the exchanger does not change from one case to the next. The particles themselves were wood shavings sorted into different size ranges using sieves.

Twoe xchangers were studied, their characteristics shown in Table 1. The particle size ranges were chosen as a func- tion of the fin spacing, it being obvious that particles much larger than this will not contribute to fouling but fall to the tunnel floor following impact. It was also decided initially to discretize over six size ranges for exchanger A; those for B were determined after analyzing the results from exchanger A (see Table 2).

Experiments could be divided into four groups on each exchanger: isothermal wet and dry surfaces, and non-isothermal wet and dry surfaces. As all previous results in wind tunnel tests show that fouling had hardly any effect on heat transfer, the main body of results concern the isothermal case.

Wind velocities were chosen at 2, 2.5, 3, 3.5, 4, 4.5, and 5 m/s. The pressure drop measurement of interest is that due to the foulant only: the evolving total value minus that for the clean exchanger (i.e., p total -p exchanger ). This is proportional to the fouling factor F. Pressure drop is measured continually during particle introduction. Observation shows that the particles either passed through the exchanger, were blocked on the exchanger surface, or rebounded to the tunnel floor upon impact. Each of these components was weighed and compared to the amount introduced. Photographs of the surface were also taken for analysis. Gravitation does affect the passage of particles from the point of introduction 1 m upstream of the exchanger. A small zone on the upper part of the exchanger remains clear of particles. The extent of this "dead zone" is measured for each set of experiments depending on air speed and particle size (or mass) and blocked off. All results refer to the "active" or fouled area of the exchanger surface.

RESULTS

Pressure Drop Due to Foulant

As indicated, pressure measurements refer to the total value across the fouled exchanger minus the clean exchanger value. Typical results for this are shown on Figure 1 for a wind velocity of 4 m/s for different particle sizes (see Table 2, exchanger A).

Referring to the particle size ranges as A1 (smallest) to A6 (largest), it is seen that the A1 particles do not cause any drop in pressure, indicating that they all pass through the exchanger in dry conditions. In fact, several hundred grams of A1-size particles were introduced with the aim of checking whether or not a surface deposit formed on the fins. In such a case the pressure drop measurements would only evolve very slowly. However, no such effect was observed or measured. The rate of foulant buildup increases with particle size from range A2 onwards, with the largest sizes showing a saturation tendency; this suggests that the majority of these particles are blocked by the exchanger and thus contribute to the pressure drop. The main variation occurs between A2 and A4. For each range, the pressure drop varies more or less linearly with the mass of particles introduced. In the presence of humidity, the pressure drops increase slightly, including for the A1 case. This was so while the liquid film was present on the exchanger surface. As it dried, the pressure drop values tended toward the "unwetted" values.

Mass Distribution of Particles

Initial observation showed that of the introduced particles, a proportion remains blocked on the surface contributing to pressure drop, while a further amount rebounded from the surface and dropped to the tunnel floor. Therefore, from the total mass of particles introduced, it is necessary to subtract this mass on the tunnel floor because it makes no contribution to pressure drop. This is removed via the side window while the air velocity is kept constant, and weighed. Following this, the air velocity is reduced to zero, resulting in a fall to the floor of a significant mass of particles from the exchanger surface. This mass is weighed. The particles remaining on the exchanger are then removed (not without difficulty!) and also weighed. The missing mass represents the particles that pass through the exchanger and are lost to the exterior of the system. Clearly, these results are taken at the end of the particle introduction process and correspond only to the final point on the pressure drop curves. It is not possible to take these measurements as the phenomenon evolves; however, it is likely that the tendency is linear and mirrors the pressure drop evolution.

It can be seen in Figure 2 that the percentage of particles blocked by the exchanger with respect to the total mass of particles injected increases with particle size. Beyond the A6 range, 100% of the particles are blocked. The mass of fouling particles (i.e., those on the exchanger contributing to pressure drop) sat-Figure 2 Percentage of introduced particles blocked by exchanger (black), and those contributing to pressure drop (gray), that is, remaining on surface, as a function of particle size range (see Table 2).

urates with particle size, indicating that larger particles rebound more and more from the exchanger surface as it is covered with foulant. This saturation corresponds to a constant pressure drop across the fouled exchanger, a conclusion consistent with the results shown in Figure 1, where the same pressure drop measurement is observed for the A4, A5, and A6 particle ranges. Some extrapolation is necessary to understand this apparent saturation fully. For the case of very small particles, there is no fouling (e.g., A1); however, above some threshold, it will begin and increase with particle size (from A2 on). At the other end of the scale, very large particles all rebound from the surface and fall to the floor. As particle size is decreased, more and more will settle on the surface. There is therefore a range of particles between these two extremes where fouling will reach a maximum level. The results presented in Figure 2 are truncated close to this maximum, giving a false impression of saturation. The overall expected tendency is shown schematically in Figure 3.

As mentioned previously, it was observed that the fouling particles were divided into two distinct groups according to their adherence to the surface of the exchanger: those that fall when the air velocity is zero (external foulant) and those that penetrate into it (sub-surface fouling). Both contribute to pressure drop, butitisthe latter that presents the most problems for any compact heat exchanger. It is necessary to ascertain the size range of these particles and find a relationship between it and the fin geometry of the exchanger.

Interpretation of the results in terms of ratios needs to be carried out rather carefully. By taking the mass of sub-surface particles as a proportion of blocked or fouling particles, the resulting curve is strictly decreasing with rising particle size, giving the impression that the A2 or B2 scale is the most penetrating. However, in referring to Figure 3, it is clear that the quantity M ss /M blocked will be greatest for the smallest (blocked) particle size. It is important to find the size corresponding to the maximum of the sub-surface fouling results. The curve of interest is therefore the ratio of M ss to M total ,asshown in Figures 4 and5. There is a clear maximum value for the A3 range of particles (and for B3 on the second exchanger).

The results were quite similar when the exchanger surface was wetted.

The size ranges A3 and B3 (i.e., those for which penetration into the exchanger is greatest) will be referred to as "critical." Those of a smaller size tend to pass through the exchanger, while larger ones get blocked on the outside surface. The penetration depth is surprisingly small, being only about 3 or 4 mm. This is also consistent with measurements of the pressure drop, the foulant forming a physical screen in front of the exchanger. Observation showed that beyond this initial depth, there was no build-up of particles in the interior. Once any particle went beyond a 4 mm distance, it was transported out the rear side without any apparent hindrance. As the effective surface area through which a particle can pass diminishes with fouling, the constant flow rate of the air passing through the exchanger means that the air velocity through it increases where the surface is clean and drops where it is fouled. It is possible that higher velocities carry particles further (right through the exchanger), and eddying behind the foulant helps a deposit to build up locally, leaving the core free.

These critical particles enter into the inter-fin spacing and effectively reduce the area of passage for other particles that might not otherwise be blocked. They thus form "nucleation" sites for further fouling and will certainly influence the behavior of smaller size particles there. It is clear also that the critical particle size will be related to the fin spacing for any given exchanger.

Particle Size and Fin Spacing

Experiments were initially carried out on exchanger A, with the range of particle sizes in Table 2 chosen without prior knowledge of the possible outcome. Analysis of the results led the authors to attempt to predict the size range of the "critical" particles. The smallest particle that will not automatically pass through the exchanger without hindrance is chosen to have a spherical diameter of half the minimum distance between fins. The largest particle that could penetrate into the exchanger corresponds to the hydraulic diameter of the inter-fin area: d H = 4 × area/perimeter. The measurements showed this to be a very good prediction of what happens. All larger particles either rebounded off the exchanger onto the floor or remained against the outside surface while the velocity was non-zero.

The method of predicting the critical particle size was as follows. Given the fin geometry of the exchanger, it is possible to evaluate the diameter d m of the largest sphere that can be inscribed in the inter-fin spacing (see insets on Figures 4 and5). Most particles smaller than 1/2d m will pass through the exchanger. Two particles of a diameter less than or equal to 1/2d m arriving simultaneously should also pass through. Larger particles will be blocked, though not necessarily in the subsurface region. The limiting value is taken as the diameter of the particles of which the projected area is half that of the circle of diameter d m .Asimilar reasoning can be carried out for several particles arriving simultaneously [START_REF] Graham | Dense Packing of Congruent Circles in a Circle[END_REF]. Based on the case of two particles (the probability of three or more particles arriving simultaneously at the same position is slim), the critical particle range is therefore 1/2d m < d crit < 0.707 d m .T aking the critical value to be (arbitrarily) halfway between these gives d crit = 0.6d m .

Forexchanger A, this gave a diameter of 1.75 mm, which corresponds well with the measurements in Figure 4. This predictive method was then used to fix the ranges of particles to be examined for exchanger B. The critical diameter was found to be 0.78 mm. This is in the upper part of the range B3 (0.6-0.8 mm), suggesting that the range B4 (0.8-1.3 mm) will also contain a significant number of critical particles. This is borne out by the results (see Figure 5), where the maximum level of sub-surface fouling is at the larger end of the B3 range.

The mass of blocked and sub-surface particles relative to the total injected mass, plotted as a function of particle size, is shown in Figure 6 for both exchangers. The curves show similar tendencies. However, when both sets are non-dimensionalized by the appropriate diameter d m , thus effectively eliminating the specific dimensions of the exchanger, the critical particle value becomes very evident (see Figure 7). This corresponds to the maximum value of sub-surface particle penetration and takes a value of d/d m = 0.63, very close to the predicted value of 0.6. The same result would have been found had the critical size been used to non-dimensionalize the results. However, the value of d m is easier to obtain for any exchanger. It is fairly safe to say, even if only on the basis of measurements on two different compact heat exchangers, that once any exchanger geometry is known, it is possible to define the size range of the critical particle for sub-surface fouling.

The curves of Figure 7 are also independent of air speed in the wind tunnel (for velocities between 2 and 5 m/s), indicating that, at least over this range, geometrical effects dominate the fouling process.

The curves for sub-surface fouling correspond well with what is expected: smaller particles (d/d m < 0.2) pass through the exchanger, while larger ones (d/d m > 1.0) remain on the external surface. The difference between the fouling and blocked particles in the figure represents the quantity that rebounds from the exchanger and fall to the tunnel floor. This percentage will evidently increase with particle size.

The percentage of particles that remain on the outer surface of the exchanger (and thus contribute to the pressure drop) appears at about 60% for d/d m > 0.9. This is a more difficult phenomenon to explain physically, particularly because the range of particle sizes tested was limited to d/d m = 1.5. The saturation within this range most likely corresponds to an initial surface build-up of foulant entraining a modification of the air flow in its immediate vicinity. The slight pressure drop that would be felt just downstream of the surface foulant particles within the exchanger is not going to be felt by a second or third layer of particles that deposit on the first. The balancing of the local pressure distribution will cause the later-arriving particles to fall off the surface despite the air flow pushing them toward it (i.e., gravitational effects will overcome local pressure forces).

Fouling Factor

It is now clear from the results that air-side fouling depends on geometrical considerations almost exclusively. The influence of fluid mechanics, thermal gradients, and humidity is very much secondary. The fouling factor F, defined earlier as the nondimensional foulant pressure drop, should reflect this. Figure 8 shows how F varies with air speed V. Above a certain value of V,Fbecomes constant, indicating that the foulant has taken the role of a grid or mechanical filter in front of the exchanger. The weakly decreasing value of F for the lower velocities was noted for all particle sizes below the critical value. Above this, F was constant for all V. The explanation for this weak variation in F is probably that some amount of the smallest particles blocked at the surface are blown through the exchanger as air speed is increased.

F also increases in level as more foulant is added. The constant F values plotted as a function of the mass introduced into the tunnel correspond particularly well with the pressure drop measurements, a typical sample of which is shown in Figure 1.

Non-Isothermal Tests

Several series of tests were carried out to quantify the effects of fouling on heat transfer across the exchanger. The input temperature was fixed at either 60 • Cor70 • C, with the flow rate set by a 3.5 kW thermostated bath. For a given air speed in the wind tunnel, the temperature drop did not change with increasing fouling, although the pressure drop did. A typical set of results are shown in Figure 9. This is easily explained. As the fouling is confined to the front of the exchanger, there is no modification of the surface state in the core (constant exchange coefficient). Secondly, the evacuated heat is proportional to the flow rate of air across the exchanger. This remains constant in a closed wind tunnel test section no matter what the extent of the fouling is. This clearly does not reflect reality. A more appropriate form of testing needs to be defined that takes into account the fact that a fouled heat exchanger will deflect some of the oncoming air flow around it, thus reducing the air flow through the exchanger and, as a consequence, the evacuated heat. The simple expedient of opening a window in the tunnel at the side of the heat exchanger allows for the measurement of the effect of air flow deviation on temperature. Figure 10 shows how the water side temperature drop changes when the air flow is allowed to deviate. Of course, as fouling increases, so does this deviated air flow away from the exchanger, and the temperature drop falls even further. The drop is significant, even at these relatively weak levels of fouling (typically about 8 • -15 • C).

In conclusion, obliging the air flow to pass through the exchanger as in usual wind tunnel experiments will mask the heat transfer problem due to fouling, but it will not influence the pressure drop due to it. The problem of deviated air flow will be all the greater with very closely spaced fins, suggesting the possibility of an optimal spacing.

CONCLUSION

Depending on their relative size, particles approaching a compact heat exchanger either pass through without hindrance, are blocked over a shallow depth at the front, rest on the outer surface while the air speed is non-zero, or rebound from the surface and drop to the wind tunnel floor. Those that penetrate into the sub-surface region are the most detrimental to exchanger performance.

There is a critical particle size for which this penetration is maximal. This is found to be 0.6 times the diameter of the largest sphere that can be inscribed in the fin spacing.

The penetration depth is small, being only about 3 or 4 mm. This is consistent with measurements of the pressure drop, the foulant forming a physical screen in front of the exchanger.

All non-isothermal testing of compact heat exchangers referred to in the literature was carried out with a constant flow rate of air. Fouling will not affect the thermal performance in such circumstances.

A fouled exchanger will deviate the air flow upstream, causing a part of that flow to bypass it. This will automatically have a significant detrimental effect on heat transfer, leading to overheating. Any non-isothermal testing of compact heat exchangers should take this into account. This, along with an analysis of binary mixtures of particles, should be the subject of future research.

Figure 1

 1 Figure 1 Pressure drop due to the foulant as a function of the mass of particles introduced for different size ranges.

Figure 3

 3 Figure 3 Schematic representation of the percentages of particles blocked by the exchanger within the wind tunnel (straight line) and those that foul the exchanger (lower curve) as a function of particle size. The vertical line indicates the cut-off for the histogram of Figure 2.

Figure 4

 4 Figure 4 Percentage of particles penetrating into exchanger (i.e., sub-surface penetration) as a function of their size; exchanger A. Inset: fin geometry.

Figure 5

 5 Figure 5 Percentage of particles penetrating into exchanger (i.e., sub-surface penetration) as a function of their size; exchanger B. Inset: fin geometry.

Figure 6

 6 Figure 6 Masses of recovered particles as a percentage of the total introduced as a function of particle size. The top curves (diamond) represent those blocked, the middle (square) represent those contributing to pressure drop, and the bottom (triangle) represent sub-surface fouling (left: exchanger B, full lines; right: exchanger A, dashed lines).

Figure 7

 7 Figure 7 Results of Figure 6, as a function of non-dimensional particle size.

Figure 8

 8 Figure8Variation of the fouling coefficient F with air speed for the size range B2: evolution for increasing mass. Similar curves are found for all size ranges for both exchangers.

Figure 9

 9 Figure 9 Influence of air-side fouling on the water-side temperature drop across exchanger A as a function of air speed. The curves cover fouled and clean surfaces.

Figure 10

 10 Figure 10 Evolution of the water-side temperature difference as a function of air speed for various levels of fouling-exchanger A with A3 size range foulant. The top curve represents the closed wind tunnel; all others cover the open tunnel case.

Table 1

 1 Relevant exchanger characteristics

		Exchanger A Exchanger B
	Fin density/100 mm	44	95
	Fin spacing at base	3 mm	1.38 mm
	Fin spacing at summit 1.6 mm	0.64 mm
	Fin length	18.5 mm	7.6 mm

Table 2

 2 Particle range sizes for both exchangers

	Exchanger A	Exchanger B
	d < 0.8 mm	d < 0.25 mm
	0.8 < d < 1.25 mm	0.25 < d < 0.5 mm
	1.25 < d < 1.85 mm	0.5 < d < 0.8 mm
	1.85 < d < 2mm	0.8 < d < 1.25 mm
	2 < d < 3.15 mm	1.25 < d < 1.6 mm
	3.15 < d < 4mm