Amandine Michelas

Núria Esteve-Gibert

Sophie Dufour

Running Head: Stress in French word processing On French listeners' ability to use stress during spoken word processing

Keywords: Speech perception, word discrimination, French prosody, primary stress

Previous studies have suggested that French listeners experience difficulties when they have to discriminate between words that differ in stress. A limitation is that these studies used stress patterns that do not respect the rules of stress placement in French. In this study, three stress patterns were tested on bisyllabic words (1) the legal stress pattern in French, namely words that were unstressed compared to words that bore primary stress on their last syllable (/ʒyʁi/-/ʒy'ʁi/), (2) an illegal stress location pattern, namely words that bore primary stress on their first syllable compared to words that bore primary stress on their last syllable (/'ʒyʁi/-/ʒy'ʁi/) and (3) an illegal pattern that involves an unstressed word, namely words that were unstressed compared to words that bore primary stress on their first syllable (/ʒuʁi/-/'ʒuʁi/). In an ABX task, participants heard three items produced by three different speakers and had to indicate whether X was identical to A or B. The stimuli A and B varied in stress (/ʒy'ʁi/-/ʒyʁi/-/ʒy'ʁi/), in one phoneme (/ʒy'ʁi/-/ʒy'ʁɔ /-/ʒy'ʁi/) or in both stress and one phoneme (/ʒy'ʁi/-/ʒyʁɔ /-/ʒy'ʁi/). The results showed that French listeners are fully able to discriminate between two words differing in stress provided that the stress pattern included an unstressed word. More importantly, they suggest that the French listeners' difficulties mainly reside in locating stress within words.

Word recognition from the speech signal is a key aspect of spoken language comprehension. To correctly identify spoken words, one of the first tasks for listeners is to extract both segmental and suprasegmental information from the speech signal. The product of this analysis is then projected onto the phonological representations of words stored in the mental lexicon. The current study focuses on this first step of spoken word recognition, and examines the relative contribution of both segmental and suprasegmental information in the discrimination of French words.

Numerous studies conducted in several languages have already been done to examine the role of segmental and suprasegmental information such as stress 1 in word perception (e.g., [START_REF] Mcqueen | Competition in spoken word recognition: Spotting words in other words[END_REF] for English; [START_REF] Cutler | Voornaam is not (really) a homophone: Lexical prosody and lexical access in Dutch[END_REF] for Dutch; Soto-Faraco, Sebastian-Galles & Cutler, 2001 for Spanish; [START_REF] Sulpizio | Italians use abstract knowledge about lexical stress during spoken-word recognition[END_REF] for Italian; [START_REF] Lee | Lexical tone in spoken word recognition: A view from Mandarin Chinese[END_REF] for Mandarin Chinese; Cutler & Otake, 1999 for Japanese). For example, in a cross-modal priming experiment, [START_REF] Soto-Faraco | Segmental and suprasegmental mismatch in lexical access[END_REF] presented to Spanish listeners sentences ending with word fragments, and asked them to perform a lexical decision task on letter strings (i.e. targets) presented at fragment offset. Fragment primes either fully matched (pa'pi) or mismatched either in stress ('papi) or in one phoneme (pa'ti) the auditory form of the initial portion of target words (papi'lla) 2 . In comparison to a control condition in which there was no overlap between the fragment primes and the target words, the authors reported that fully matching primes facilitated target word processing, while mismatching primes both in stress and phoneme inhibited target word processing. Such results suggest that in Spanish suprasegmental and segmental information similarly influence spoken word perception.

The fact that suprasegmental information contributes as equally as segmental information to word discrimination is not so surprising in languages such as Spanish where stress location is lexically contrastive (e.g., /'bebe/ "s/he drinks" vs. /be'be/ "baby"). But what about languages such as French where it is not possible to find minimal pairs of words differing in stress location?

In French, stress always affects the last syllable of a melodic group, which is a larger unit than the word 3 . As a result, the same word receives stress ([un petit 'CHOU]) "a little cabbage" or not ([un CHOU 'bleu]) "a blue cabbage" depending on its position within a melodic group. As such, several studies indicated that French listeners use stress to segment speech into words, and in particular to locate word onset/offset in continuous speech [START_REF] Rietveld | Word boundaries in the French language[END_REF]Bannel & Bacri, 1994;[START_REF] Christophe | Phonological phrase boundaries constrain lexical access I. Adult data[END_REF][START_REF] Spinelli | Fine-grained access to targets and competitors in phonemically identical spoken sequences: the case of French elision[END_REF][START_REF] Spinelli | An intonational cue to word segmentation in phonemically identical sequences[END_REF]. For example, [START_REF] Christophe | Phonological phrase boundaries constrain lexical access I. Adult data[END_REF] have shown that stress helps French listeners to resolve competition between overlapping lexical candidates. The authors asked participants to detect a target word (e.g., CHAT "cat") in sentences like [Le CHAT grin'cheux] "the grumpy cat" in which only the word "grincheux" is stressed on its last syllable or in sentences like [Le CHAT'] [grim'pait] "the cat climbed up" in which both the words "chat" and "grimpait" are stressed. They found slower detection times in [Le CHAT grin'cheux] than in [Le CHAT'] [grim 'pait]. The slower detection of CHAT in [Le CHAT grin'cheux] has been interpreted as resulting from competition between CHAT and its competitor CHAGRIN "sadness". In this case, since there is no stress on CHAT to indicate its end, the word CHAGRIN remains active and competes with the word CHAT.

Although it clearly appears that French listeners can use stress to resolve lexical ambiguities, the story regarding stress processing in French is rendered more complex than it seems by the fact that stress can also be difficult to perceive by French listeners. In their study, Dupoux and collaborators (Dupoux, Pallier, Sebastian & Mehlher, 1997) compared French and Spanish listeners in the processing of stress. In an ABX task, in which A, B and X were spoken by three different speakers, and in which participants judged whether X was identical to A or B, the authors showed that French listeners had more difficulties than Spanish listeners in discriminating between two nonsense words that differed only in stress position ('fidape, fi'dape,). In addition, French listeners, unlike Spanish listeners, performed worse when the stimuli differed in stress ('fidape, fi'dape) than in phoneme ('fidape, 'lidape).

It was only when the stimuli were produced by a single speaker that French listeners performed equally well on stress ('fidape vs. fi'dape) and phoneme ('fidape vs. 'lidape) contrasts. The authors concluded that stress is likely used at different levels of processing by French and Spanish listeners. Spanish listeners use stress at an abstract level of processing, while French listeners are only capable of using an acoustic representation of stress which consequently impair their performance when different speakers produced the nonsense words.

The French listeners difficulties in using stress has been replicated in several subsequent studies [START_REF] Dupoux | A robust method to study stress "deafness[END_REF]Dupoux, Peperkamp & Sebastian-Galles, 2010;[START_REF] Peperkamp | Perception of predictable stress: A cross linguistic investigation[END_REF] even in French listeners learning of Spanish [START_REF] Dupoux | Persistent stress 'deafness': The case of French learners of Spanish[END_REF], and the terms "stress deafness" have been widely employed to qualify these difficulties.

In this study, we tried to better understand the difficulties encountered by French listeners with stress, and why such difficulties have been reported. An alternative explanation for the difficulty of French listeners in the Dupoux et al. study could be related to the use of stimuli that do not respect the rules of stress placement in French. In the Dupoux et al. study, 1 st /2 nd stressed syllables ('bopelo/bo'pelo) and 2 nd /3 rd stressed syllables (bo'pelo/bope'lo) were used, but in French, a word cannot receive primary stress at any non-final position. We thus re-examined the French listeners ability to use stress during word processing, and tested them on legal and illegal stress patterns in French. More specifically, three stress patterns on bisyllabic French words were tested: (1) the legal stress pattern in French, namely words that were unstressed compared to words that bore primary stress on their last syllable: jury/ju'ry (the 'unstressed/2 nd stressed syllable' pattern), (2) an illegal stress location pattern, namely words that bore primary stress on their first syllable compared to words that bore primary stress on their second syllable: 'jury/ju'ry (the '1 st stressed syllable/2 nd stressed syllable' pattern) and (3) an illegal stress pattern that involves an unstressed word, namely words that were unstressed compared to words that bore primary stress on their first syllable : jury/'jury (the 'unstressed/1 st stressed syllable' pattern). We predicted that if the difficulties reported in the Dupoux et al. study are due to the use of stress patterns that do not respect the rules of stress placement in French, difficulties should be observed for the '1 st stressed syllable/2 nd stressed syllable' pattern ('jury/ju'ry), and that these difficulties would extend to the 'unstressed/1 st stressed syllable' pattern (jury/ 'jury) which is also not encountered in French.

In particular for these two illegal stress patterns in French, French listeners should perform worse on a stress difference ('jury vs. ju'ry; jury vs. 'jury) than on a phoneme difference (e.g. 'jury vs. 'juron). By comparison, good performance is expected for the 'unstressed/2 nd stressed syllable' pattern (jury /ju'ry) because this is precisely THE stress pattern that respects the rules of stress placement in French. In particular, for this last stress pattern, French listeners should use stress difference (jury vs. ju'ry) as they use phonemic difference (e.g. ju 'ry vs. ju'ron) to discriminate between two words and they should perform equally well on a stress and a phoneme difference.

As in Dupoux et al.'s study, we used an ABX task in which participants heard three items produced by three different speakers and had to indicate whether X was identical to A or B. Within the three stress patterns tested, the stimuli A and B varied either in stress (e.g. ju 'ry, jury, ju'ry), or in one phoneme (e.g. ju'ry, ju'ron, ju'ry') so that the relative contribution of phonemic and stress information could be examined for each stress pattern. Finally, similar to [START_REF] Dupoux | A destressing "deafness" in French?[END_REF]'s study, a last condition in which A and B differed both in stress and in one phoneme was tested within each stress pattern. This allowed us to test whether French listeners could benefit from stress information when phonemic information alone is sufficient to differentiate the two members of a minimal pair (e.g., ju 'ry, juron, ju'ry). In particular, a boost in the performance when stress and phonemic information is cumulated could be observed for the 'unstressed/2 nd stressed syllable' pattern, because this precisely for this stress pattern that a good performance in perceiving the difference in stress is expected.

Method

Participants: Twenty-four native speakers of French (18 women) between 18 and 26 years old (mean = 22,5) participated in the experiment. All were students from Aix-Marseille University and all reported having no neurological or hearing impairment. Each participant gave informed consent prior to experimentation and was remunerated for their time with 10 Euros.

Materials: Four pairs of bisyllabic French words, four phonemes in length, differing either in the initial (momie /momi/ "mummy" vs. vomi /vomi/ "vomit", musée /myze/ "museum" vs. rusé /ʁyze/ "clever ") or in the last phoneme (jury /ʒyʁi/ "jury" vs. juron /ʒyʁɔ / "swearword", marin /maʁɛ / "seaman" vs. marrant /maʁɑ / "funny") were selected. Three French speakers (two women, one man) produced the 8 target words (in capital letters in the following example) within carrier sentences in which the target word bore stress on the last syllable or was unstressed depending on its position within the utterance (e.g., [On m'avait parlé] [d'un JURY exi'geant] [qu'il était difficile] [de satisfaire] vs. [On m'avait parlé] [d'un exigeant JU'RY] [qu'il était difficile] [de satisfaire] "I was told about a highly demanding jury that it was difficult to satisfy"). The same carrier sentences as for the unstressed condition were used to obtain 8 target words that bore stress on the first syllable (e.g. [On m'avait parlé] [d'un 'JURY exi'geant] [qu'il était difficile] [de satisfaire]). Because words that bore primary stress on the first syllable are not found in French, speakers were asked to produce stress on the first syllable of the target word by making the target syllable prominent.

To avoid coarticulation effects due to contextualized-speech, each word was extracted from its carrier sentences, and the three speakers heard their own productions of the 8 target words in their stressed (first stressed syllable and second stressed syllable) and unstressed versions in isolation, and were instructed to reproduce the different versions. The sentences and the 8 target words presented in isolation were recorded at a sampling rate of 44 100 Hz. Acoustic analyses using Praat software [START_REF] Boersma | Praat. Doing phonetics by computer[END_REF] were then conducted to ensure that the repeated words were produced with the expected stress patterns. Pre-boundary lengthening and f0 rises, the two main correlates of primary stress in French, were measured (see Table 1). Figure 1 illustrates the words jury /ʒyʁi/ "jury" produced by our male speaker in its unstressed version, with a primary stress on its last syllable and with a primary stress on its first syllable. Statistical analyses performed on the 8 target words produced by the three speakers show that for the unstressed words, the duration and the f0 rise associated with the first syllables were not different from those of the second syllables [syllable duration: t(23) = 0.09, p > .20; f0 rise: t(23) = 0.65; p > .20]. By contrast, for the words that bore stress on the 1 st syllable, the first syllables were longer and associated with a stronger f0 rise than the 2 nd syllables [syllable duration: t(23) = 7.20, p < .0001; f0 rise: t(23) = 13.78; p < .0001]. For the words that bore stress on the 2 nd syllable, the 2 nd syllables were longer and associated with a stronger f0 rise than the 1 st syllables [syllable duration: t(23) = 7.81, p < .0001; f0 rise: t(23) = 33.32; p < .0001]. The root-mean-squared (rms) amplitude levels for all the target words were equated at a level of 70 dB SPL.

< Table 1 about here > < Figure 1 about here > From the recorded words, 576 trials were constructed. Each trial consisted of three stimuli: A, B and X, with the first two spoken by the two females and the third by the male.

Within each stress pattern ('1 st stressed syllable/2 nd stressed syllable', 'unstressed/2 nd stressed syllable', 'unstressed/1 st stressed syllable'), A and B varied either in stress, in one phoneme or both in stress and in one phoneme, leading thus to 9 experimental conditions. The four pairs of words were used in each experimental condition. For each pair of words and within each experimental condition, 16 combinations resulted from the crossing between stress (2 versions), segmental contents (2 words), response type (A or B) and voice (two females).

Each of the experimental conditions is illustrated in Table 2 with the "jury-juron" (/ʒyʁi/-/ʒyʁɔ /) pair of words.

< Table 2 about here > Procedure: Participants were tested in a sound attenuated booth, and stimuli were presented over headphones at a comfortable sound level (60dBA). Each experimental trial presented the three stimuli (A, B and X) separated by an interval of 500ms. Participants were told that the first two stimuli were different, and that the third one was identical to the first or to the second. They were required to press a button on their left or on their right to indicate whether X was identical to A or to B respectively. Each participant heard the 576 trials, which were presented randomly, and a short break was given to the participants after 288 trials. The order of presentation of trials was different for each participant. An inter-trial interval of 1000 ms elapsed between the participant's response and the beginning of the next trial. Reaction times (RTs) were measured from the onset of the target word (stimulus X). The participants began the experiment with a block of 12 practice trials.

Results

Accuracy rates and mean RTs are shown in Table 3.

< Table 3 about here > Accuracy rate analyses: We analyzed accuracy data (1=correct responses, 0= incorrect responses) using a mixed-effects regression model (lme4 package in R-studio statistics Version 0.99.903) with a logistic linking function. The model included mismatch (phoneme mismatch, stress mismatch, phoneme and stress mismatch) and stress pattern (1 st stressed syllable/2 nd stressed syllable, unstressed/2 nd stressed syllable, unstressed/1 st stressed syllable) as fixed effects. The model included participants and items as random intercepts only (random slopes by participants and by items were not included because they did not allow the model to converge). Global effects were obtained using the afex::mixed function. The model revealed a significant main effect of mismatch (X 2 = 181.64, p < .0001). The effect of stress pattern was not significant (X 2 = 1.02, p > .20). Crucially the interaction between mismatch and stress pattern was significant (X 2 = 22.49, p < .001). To further analyse the mismatch x stress pattern interaction, the 3x3 factor design was converted into a 1x9 factor design using the same model structure. Multiple comparisons (Tukey test, p < . 05) were obtained with the glht function from the multcomp package [START_REF] Bretz | Multiple comparisons using R[END_REF] with p-values adjusted by the single step method. For the '1 st stressed syllable/2 nd stressed syllable' pattern (e.g., 'jury/ju'ry), the model revealed more correct responses in the phoneme mismatch condition compared to the stress mismatch condition (z = 6.53, p < .001). There were also more correct responses in the phoneme and stress mismatch condition compared to the stress mismatch condition (z = 8.89, p < .001). However, the difference between the phoneme and stress mismatch condition compared to the phoneme mismatch condition failed to reach significance (z = 2.82, p = .11). For the 'unstressed/2 nd stressed syllable' pattern (e.g., jury/ju'ry), no significant difference was found between the phoneme mismatch condition and the stress mismatch condition (z = 1.00, p > .20) while participants gave more correct responses in the phoneme and stress mismatch condition compared to both the phoneme mismatch condition (z = 6.83, p < .001) and the stress mismatch condition (z = 7.68, p < .001). Similarly, for the 'unstressed/1 st stressed syllable' pattern (e.g., jury/'jury), no significant difference was found between the phoneme mismatch and the stress mismatch conditions (z = 1.90, p > .20) while participants gave more correct responses in the phoneme and stress mismatch condition compared to both the phoneme mismatch (z = 3.99, p < .01) and the stress mismatch condition (z = 5.76, p < .001).

RTs analyses: These analyses were performed only on correct responses. One absurd data (RTs = 13028ms) was excluded. A linear mixed-effects regression model was used on the log-transformed RTs (fixed effects: mismatch and stress pattern; random intercepts: participants and items; no random slopes because they did not allow the model to converge).

The model revealed both a significant main effect of mismatch (X 2 = 64.20, p < .0001), and a significant main effect of stress pattern (X 2 = 120.76, p < .0001). Crucially, the interaction between mismatch and stress pattern was significant (X 2 = 30.70, p < .0001). To further analyse the mismatch x stress pattern interaction, the 3x3 factor design was converted into a 1x9 factor design using the same model structure. Multiple comparisons (Tukey test, p <. 05) were obtained with the glht function from the multcomp package [START_REF] Bretz | Multiple comparisons using R[END_REF] with pvalues adjusted by the single step method. For the '1 st stressed syllable/2 nd stressed syllable' pattern (e.g., 'jury / ju'ry), participants responded faster in the phoneme mismatch condition than in the stress mismatch condition (z = 6.65, p < .0001). They were also faster in the phoneme and stress mismatch condition compared to the stress mismatch condition (z = 8.02, p < .0001). No significant difference was found between the phoneme and stress mismatch and the phoneme mismatch conditions (z = 1.36, p = 0.17). For the 'unstressed syllable/2 nd stressed syllable' pattern (e.g., jury/ju'ry), no significant difference was found between the phoneme mismatch and the stress mismatch conditions (z = 1.48, p = .14). Participants were faster in the phoneme and stress mismatch condition compared to both the phoneme mismatch (z = 2.05, p < .05) and the stress mismatch conditions (z = 3.55, p < .001). Similarly, for the 'unstressed/1 st stressed syllable' pattern (e.g., jury/'jury), no significant difference was found between the phoneme mismatch condition compared to the stress mismatch condition (z = -0.452, p > .20). Participants were faster in the phoneme and stress mismatch condition than both in the stress mismatch condition (z = 2.28, p < .05) and in the phoneme mismatch condition (z = 2.75, p < .01).

Discussion

Our study examined the contribution of stress and segmental information in the discrimination of French words. In light of the results of [START_REF] Dupoux | A destressing "deafness" in French?[END_REF]'s study, we hypothesized that French listeners would experience difficulties in using stress during word discrimination, when they are faced with stress patterns that do not respect the rules of stress placement in French. In accordance with our prediction, French listeners performed worse on stress differences ('jury vs. ju'ry) than on phoneme differences ('jury vs. 'juron) when the stress pattern involved two words stressed either in their first or second syllable. Hence, we have replicated Dupoux et al.'s results for stress patterns that are prohibited in French. In accordance with our prediction, our results indicated that French listeners perform equally well on stress (ju'ry vs. jury) and phoneme differences (ju'ry, ju'ron), for the unique legal stress pattern in French, namely for words that were unstressed or bore stress on their last syllable. Unexpected however was the observation that French listeners also performed equally well on stress ('jury vs. jury) and phoneme differences ('jury vs. 'juron), when the stress pattern involved two words that were either unstressed or stressed on their first syllable, and so for a stress pattern which is not found in French.

What our results reveal is that French listeners experience difficulties with stress when they have to discriminate between two words that differ in stress location. No difficulty is observable when French listeners have to discriminate between an unstressed word and a stressed word, and this, regardless of the position of the stress (first/second syllable) in the stressed words. Such findings are particularly important since they suggest that French listeners are fully capable of discriminating an unstressed word from its stressed counterpart even if the stress pattern is not legal in their language. French listeners thus use the presence versus absence of stress information as they use phonemic information to discriminate between words. An alternative explanation to our findings could be that the comparison between stressed and unstressed words is a simpler task than the comparison between words that differ in stress location. Although more studies are required to definitively discard this possibility, we believe that this is rather unlikely mainly because this is not in these particular stress mismatch conditions that the better performances were observed.

Crucially, our study shows that the difficulties of French listeners with stress mainly reside in locating stress within words. Nonetheless, it seems important to underline that French listeners are not deaf to differences in stress location. As it was already the case in [START_REF] Dupoux | A destressing "deafness" in French?[END_REF]'s study, French listeners showed performance above the chance level in the 1 st stressed syllable/2 nd stressed syllable pattern, thus corroborating the claim that French listeners indeed use stress information. Nonetheless, for this stress pattern, no benefit in the discrimination of words was observed when the difference between words is carried out by stress difference in addition to phonemic difference. Hence, for stress patterns which are difficult to process, French listeners appear to ignore stress information when word discrimination can be made on phonemic information. In contrast, when French listeners have no difficulty in processing stress information -when stress patterns included an unstressed words -differences between words on both phonemic and stress information are the easiest to process, thus suggesting that French listeners under some circumstances use stress information together with phonemic information to discriminate between words.

The present results converge in showing that French listeners are able to categorize a word as stressed or unstressed but experience difficulties to establish categories based on stress location. Similarly to what it has been observed in Italian listeners [START_REF] Sulpizio | Italians use abstract knowledge about lexical stress during spoken-word recognition[END_REF], an explanation to our findings could be in terms of abstract prosodic knowledge in French listeners. According to [START_REF] Sulpizio | Italians use abstract knowledge about lexical stress during spoken-word recognition[END_REF], Italian listeners have abstracted the knowledge that in their language, trisyllabic words usually have penultimate stress, and they use this knowledge during word recognition. Because in the French language, primary stress falls on the last syllable of the melodic group, French listeners inevitably encounter unstressed words more frequently than stressed words in everyday communication [START_REF] Jun | A phonological model of French intonation[END_REF], and they have likely abstracted the knowledge that French words are more often unstressed. As a consequence, their performance is optimized when they encounter these default unstressed words, and they experience no difficulty in discriminating between two words differing in their accentual pattern provided that the stress pattern is made-up of an unstressed word.

To conclude, the present research tempers the idea of so-called "stress deafness" in French. French listeners appear to use stress information in the same way as they use phonemic information to discriminate between words, provided that the stress pattern is made up of unstressed words. The difficulties that French listeners can experience mainly reside in the location of stress within a word. Further research is now required to examine how prosodic information is integrated into lexical representations, and whether the unstressed version of words is stored by default in the French mental lexicon. Footnotes 1. We use the term "stress" as a synonym of "accent" to refer to the phonetic/perceptual prominence of a syllable which affects words or units larger than the words.

2. We use the IPA stress mark "ˈ" before the stressed syllable to indicate stress. For more clarity, the stressed syllable is also in bold.

3. While the obligatory stress (i.e. called primary stress) can never occur on the first syllable of words in French, an optional secondary stress can affect it. This secondary stress is rather unfrequent on bisyllabic words [START_REF] Michelas | When syntax meets prosody: Tonal and duration variability in French Accentual Phrases[END_REF]. It is more likely to occur with words that contain more than two syllables, when the speaking rate is slow and in case of specific speech style such as conference and journalistic style [START_REF] Jun | A phonological model of French intonation[END_REF][START_REF] Welby | French intonational structure: Evidence from tonal alignment[END_REF]. For instance, the word diffamation "defamation" tends to be produced with a secondary stress on its first syllable and a primary stress on its final syllable at a slow speech rate (diffama'tion). Importantly, this secondary stress does not have the same acoustic properties as the primary stress. It is cued by an initial f0 rise realized on the syllable which is not accompanied by syllabic lengthening, unlike for primary stress [START_REF] Jun | A phonological model of French intonation[END_REF][START_REF] Welby | French intonational structure: Evidence from tonal alignment[END_REF].

Table 3. Accuracy rates and mean RTs (in ms) for the three stress patterns and the three mismatch conditions. Standard deviations are given in parenthesis.

Phoneme mismatch

FigureFigure 1 .

 1 Figure

Stress mismatch Phoneme and stress mismatch 1 st stressed syllable/2 nd stressed syllable

	% of correct responses	93	85	95
	RTs	1090 (256)	1203 (311)	1064 (240)
	Unstressed/2 nd stressed
	syllable
	% of correct responses	89	88	96
	RTs	1056 (243)	1071 (271)	1027 (249)
	Unstressed/1st stressed
	syllable
	% of correct responses	91	90	95
	RTs	1052 (241)	1055 (289)	1012 (256)

Acknowledgments

This work was supported by the Labex Brain and Language Research Institute (ANR-11-LABX-0036) and has benefited from support from the French National Agency of Research (ANR), under the project title "Investments of the Future" A*MIDEX (ANR-11-IDEX-0001-02). We thank Zeynep Özyurt for her help in running the experiment.

Table 1. Acoustic properties of target words in their unstressed and stressed versions (1 st and 2 nd stressed syllable) for the three speakers (averaged on the eight target words).

Stress


Lorenzo Brandolese

Sylvie Monniaux

Well-posedness for the Boussinesq system in critical spaces via maximal regularity

We establish the existence and the uniqueness for the Boussinesq system in R 3 in the critical space C ([0, T ], L 3 (R 3 ) 3 ) × L 2 (0, T ; L 3/2 (R 3 )).

Introduction

We consider the Cauchy problem associated with the Boussinesq system in R3 :

∂ t u -∆u + ∇π + ∇ • (u ⊗ u) = θe 3 in (0, T ) × R 3 div u = 0 in (0, T ) × R 3 ∂ t θ -∆θ + u • ∇θ = 0 in (0, T ) × R 3 , (B)
where u denotes the velocity of the fluid, π the pressure, θ the temperature and e 3 the third vector of the canonical basis in R 3 . The given initial velocity and temperature are denoted respectively u 0 and θ 0 . The initial velocity will be always assumed to satisfy the condition div u 0 = 0. The system (B) appears in the study of the motion of incompressible viscous flows when one takes into account buoyancy effects arising from temperature variations inside the fluid. When the latter are neglected (θ ≡ 0), the system boils down to the classical Navier-Stokes equations.

The natural scaling leaving the Boussinesq system invariant is λ → (u λ , θ λ ) with u λ (t, x) = λu(λ2 t, λx) and θ λ (t, x) = λ 3 θ(λ 2 t, λx).

This motivates the study of (B) in function spaces that are left invariant by the above scaling. Assuming (u 0 , θ 0 ) ∈ L 3 (R 3 ) 3 × L 1 (R 3 ), an adaptation of Kato's L p -theory on strong solutions of Navier-Stokes [START_REF] Kato | Strong L p -solutions of the Navier-Stokes equation in R m , with applications to weak solutions[END_REF], yields the local-in-time existence and the uniqueness in appropriate scaleinvariant function spaces where the fixed point argument applies. But, as discussed in [2], the uniqueness problem in the natural space C ([0, T ], L 3 (R 3 ) 3 ) × C ([0, T ], L 1 (R 3 )) seems to be out of reach, due to the lack of regularity results in this class, and to the difficulty of giving a meaning, in the distributional sense, to the nonlinearity ∇ • (uθ) (and, of course, to u • ∇θ) in the last equation in (B). To circumvent this difficulty, the uniqueness for the Cauchy problem, in [2], was established in a smaller space for the temperature, namely C ([0, T ], L 1 (R 3 ) ∩ L ∞ loc (0, T ; L q,∞ (R 3 )), for some q > 3/2. This restriction on the temperature, however, is a bit artificial: the excluded borderline case, q = 3/2, is precisely the most interesting one, as it corresponds to the minimal regularity to be imposed on the temperature, when the velocity is in the natural space L 3 (R 3 ) 3 , to give a sense to the nonlinearity.

The scaling relations then lead us to consider solutions such that t → θ(t) L 3/2 is in L 2 . Therefore, it seems natural to address the uniqueness problem (and the existence) in

C ([0, T ], L 3 (R 3 ) 3 ) × L 2 (0, T ; L
As the Lorentz-space approach of [START_REF]Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces[END_REF][START_REF] Kozono | Local and global unique solvability of the Navier-Stokes exterior problem with Cauchy data in the space L n,∞[END_REF][START_REF] Meyer | Wavelets, paraproducts, and Navier-Stokes equations[END_REF], applied in [2], fails when q = 3/2, we have to adopt a different strategy. Our main tools will be maximal regularity estimates. The idea of using the maximal regularity in uniqueness problems goes back to [START_REF] Monniaux | Uniqueness of mild solutions of the Navier-Stokes equation and maximal L p -regularity[END_REF], where the second author gave a short proof of celebrated Furioli, Lemarié and Terraneo's uniqueness theorem [START_REF] Furioli | Unicité dans L 3 (R 3 ) et d'autres espaces fonctionnels limites pour Navier-Stokes[END_REF] of mild solutions of the Navier-Stokes equations in C ([0, T ], L 3 (R 3 )). In the present paper, we will need to use the maximal regularity in an original way, in order to make it applicable despite the product uθ a priori just belongs to L 1 (R 3 ) 3 .

In fact, our approach allows us to obtain the uniqueness, and then the regularity as a byproduct of the existence theory, in a larger class, namely

C ([0, T ], L 3 (R 3 ) 3 ) + r L ∞ (0, T ; L 3 (R 3 ) 3 ) × L 2 (0, T ; L 3 2 (R 3 )) (1.1)
for some small enough r > 0.

One could speculate that the smallness condition on the parameter r may be unessential and that the uniqueness and the regularity could be true in the larger space L ∞ (0, T ;

L 3 (R 3 ) 3 ) × L 2 (0, T ; L 3 2 (R 3 )
). This would be a nontrivial generalization for the system (B) of the deep result of Escauriaza, Seregin and Šverák [START_REF] Iskauriaza | L3,∞-solutions of Navier-Stokes equations and backward uniqueness[END_REF], about endpoint Serrin regularity criteria for the Navier-Stokes equations. Establishing such a result would probably require various ingredients (backward uniqueness, profile decompositions, [START_REF] Gallagher | A profile decomposition approach to the L ∞ t (L 3 x ) Navier-Stokes regularity criterion[END_REF][START_REF] Iskauriaza | L3,∞-solutions of Navier-Stokes equations and backward uniqueness[END_REF][START_REF] Phuc | The Navier-Stokes equations in nonendpoint borderline Lorentz spaces[END_REF], etc.). Whether or not such a stronger statement is true, we feel that our main theorem would remain of interest because of its attractive proof, entirely based on maximal regularity estimates.

Statement of the main results

Let T > 0 and r > 0. Let S (R 3 ) denote the dual of Schwartz space. In order to state our uniqueness result in the class

X T,r := C ([0, T ], L 3 (R 3 ) 3 ) + r L ∞ (0, T ; L 3 (R 3 ) 3 ) × L 2 (0, T ; L 3 2 (R 3 )),
we first clarify what we mean by solution of (B). By definition, a mild solution of (B) with inital data (u 0 , θ 0 ) ∈ S (R 3 ) 3 × S (R 3 ), and div u 0 = 0, is a couple (u, θ) ∈ X T,r solving the Boussinesq system written in its integral form (2.1) below:

u = a + B(u, u) + L(θ) θ = b + C(u, θ) (2.1)
with a(t) = e t∆ u 0 , b(t) = e t∆ θ 0 .

The operators B, C and L are defined, for t ∈ [0, T ], and (u, θ) ∈ X T,r , by

B(u, v)(t) = - ˆt 0 e (t-s)∆ P ∇ • u(s) ⊗ v(s) ds, (2.2)
C(u, θ)(t) = - ˆt 0 e (t-s)∆ div θ(s)u(s) ds, (2.3)
and L(θ)(t) = ˆt 0 e (t-s)∆ P(θ(s)e 3 ) ds.

(2.4)

Here P denotes Leray's projector onto divergence-free vector fields and e t∆ t≥0 is the heat semigroup.

Our main result then is stated as follows:

Theorem 2.1. There is an absolute constant r 0 > 0 such that, if (u 1 , θ 1 ) and (u 2 , θ 2 ) are two mild solutions to (B) in X T,r , with the same initial data (u 0 , θ 0 ) ∈ S (R 3 ) 3 × S (R 3 ), div u 0 = 0 and 0 ≤ r < r 0 , then (u 1 , θ 1 ) = (u 2 , θ 2 ).

As we will see, any solution (u, θ) as in Theorem 2.1 must belong to C ([0, T ], S (R 3 ) 3 × S (R 3 )), and the initial data must belong more precisely to L 3 (R 3 ) 3 × B -1 3/2,2 (R 3 ). The above theorem is then completed by the corresponding existence result:

Theorem 2.2. i) Let (u 0 , θ 0 ) ∈ L 3 (R 3 ) 3 × B -1
3/2,2 (R 3 ), with div u 0 = 0. Then there exists T > 0 and a solution of

(2.1) (u, θ) ∈ C ([0, T ], L 3 (R 3 ) 3 ) × L 2 (0, T ; L 3 2 (R 3 )).
ii) If θ 0 belongs to the smaller homogeneous Besov space Ḃ-1 3/2,2 (R 3 ) and if

u 0 L 3 + θ 0 Ḃ-1 3/2,2
is small enough, than such solution is global and

(u, θ) ∈ C b (0, ∞; L 3 (R 3 ) 3 )×L 2 (0, ∞; L 3 2 (R 3 )).
For some other existence results for the Boussinesq system in different functional setting we refer, e.g., to [START_REF] Danchin | Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux[END_REF][START_REF]Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces[END_REF][START_REF] Karch | Self-similarity in viscous Boussinesq equations[END_REF].

Applications of the maximal regularity

The purpose of this section is to study the properties of the operators B, C and L, respectively defined by (2.2), (2.3) and (2.4), by means of the following maximal regularity result. The theorem below is classical. See [START_REF] De Simon | Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine[END_REF] for the case p ∈ (1, ∞) and q = 2. The general case was first proved in [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF], Chapter IV, §3. The proof of the estimates for the mixed fractional space-time derivatives goes back to [START_REF] Sobolevskiȋ | Fractional powers of coercive-positive sums of operators[END_REF]Theorem 6]. See also [START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF]Theorem 7.3] for a modern proof.

Theorem 3.1 (Maximal regularity). Let 1 < p, q < ∞. Let R be the operator defined for f ∈ L 1 loc (0, ∞; S (R d )), d ≥ 1, by
Rf (t) = ˆt 0 e (t-s)∆ f (s) ds, t > 0. (3.1)
Such operator R is bounded from L p 0, ∞; L q (R d ) to Ẇ 1,p 0, ∞; L q (R d ) ∩ L p 0, ∞; Ẇ 2,q (R d ) .
In other words, d dt R, ∆R, and

(-∆) α d dt 1-α R, for any 0 < α < 1, are bounded operators in L p 0, ∞; L q (R d ) . Moreover, there exists a constant C p,q such that d dt Rf L p (L q ) + ∆Rf L p (L q ) + (-∆) α d dt 1-α Rf L p (L q ) ≤ C p,q f L p (L q ) ,
for all α ∈ (0, 1).

To establish Theorem 2.1, we assume that we have two mild solutions (u 1 , θ 1 ) ∈ X T,r and (u 2 , θ 2 ) ∈ X T,r of (B), arising from the same initial datum (u

0 , θ 0 ) ∈ S (R 3 ) 3 × S (R 3 ). Letting u = u 1 -u 2 and θ = θ 1 -θ 2 , we then obtain (u, θ) ∈ X T,r and u = B(u, u 1 ) + B(u 2 , u) + L(θ), θ = C(u 1 , θ) + C(u, θ 2 ). (3.2)
The maximal regularity theorem allows us to obtain all the relevant estimates for the operators B and C and L. is bounded, with operator norm less than ε.

Proof. Let r > 0, to be chosen later. For v, w ∈ C ([0, T ], L 3 (R 3 ) 3 ) + r L ∞ (0, T ; L 3 (R 3 ) 3 ), we can find v r , w r ∈ C c ([0, T ] × R 3 ) such that ess sup [0,T ] v -v r L 3 + ess sup [0,T ] w -w r L 3 ≤ 3r. (3.5)
Let us introduce the functions f and g defined by

f (s) = (-∆) -1 P ∇ • u(s) ⊗ (v -v r )(s) + (w -w r )(s) ⊗ u(s) , s ∈ [0, T ] (3.6) and g(s) = -(-∆) -3/4 P ∇ • u(s) ⊗ v r (s) + w r (s) ⊗ u(s) , s ∈ [0, T ].
We have

B(u, v) + B(w, u) = ∆Rf + (-∆) 3/4 Rg, (3.7)
where R is the vector-valued analogue of the scalar operator defined in Theorem 3.1.

(i) Let us first consider (3.3). We easily see that the norm of

f in L 4 (0, τ ; L 6 (R 3 ) 3 ) is bounded by the norm of u ⊗ (v -v r ) + (w -w r ) ⊗ u in L 4 (0, τ ; L 2 (R 3 ) 3×3 ). Indeed, the operator (-∆) -1 P∇• is bounded from L 2 (R 3 ) 3×3 to L 6 (R 3 ) 3 . Hence, ∆Rf L 4 (0,τ ;L 6 (R 3 ) 3 ) ≤ C 4,6 u ⊗ (v -v r ) + (w -w r ) ⊗ u L 4 (0,τ ;L 2 (R 3 ) 3×3 ) ≤ 3rC 4,6 u L 4 (0,τ ;L 6 (R 3 ) 3 ) .
The norm of g in L 4 (0, τ ; L 6 (R 3 ) 3 ) is bounded by the norm of u⊗v r +w r ⊗u in L 4 (0, τ ; L 3 (R 3 ) 3×3 ).

To see this, first observe that the operator (-∆)

-3/4 P∇• is bounded from L 3 (R 3 ) 3×3 to L 6 (R 3 ) 3 . Moreover, (-∆) 3/4 e t∆ L (L 6 (R 3 ) 3 ) t -3/4 . As (-∆) 3/4 R is a convolution opera- tor, we have (-∆) 3/4 Rg L 4 (0,τ ;L 6 (R 3 ) 3 ) ≤ c t → (-∆) 3/4 e t∆ L 1 (0,τ ;L (L 6 (R 3 ) 3 )) g L 4 (0,τ ;L 6 (R 3 ) 3 ) ≤ c τ 1 4 u ⊗ v r + w r ⊗ u L 4 (0,τ ;L 3 (R 3 ) 3×3 ) ≤ c τ 1 4 v r L ∞ ((0,τ )×R 3 ) 3 ) + w r L ∞ ((0,τ )×R 3 ) 3 ) u L 4 (0,τ ;L 6 (R 3 ) 3 ) . We first choose r > 0, such that 3rC 4,6 ≤ ε 2 , next v r and w r in C c ([0, T ]×R 3 ), satisfying (3.5) and last τ > 0 such that c τ 1 4 v r L ∞ ((0,τ )×R 3 ) 3 ) + w r L ∞ ((0,τ )×R 3 ) 3 ) ≤ ε 2 .
This finally establishes (3.3).

(ii) Let us now consider assertion (3.4). Let 1 < p < ∞. We slightly modify the expression of B(•, v) + B(w, •) given by (3.7):

B(u, v) + B(w, u) = ∆Rf + (-∆) 1/2 Rg, (3.8)
where we set

g(s) = -(-∆) -1/2 P ∇ • u(s) ⊗ v r (s) + w r (s) ⊗ u(s) , s ∈ [0, T ] (3.9)
The function f defined by (3.6) is bounded in L p (0, τ ; L3 (R 3 ) 3 ) by 3r u L p (0,τ ;L 3 (R 3 ) 3 ) , up to a multiplicative constant involving C p,3 and the norm of bounded operator (-∆)

-1 P∇• from L 3 2 (R 3 ) 3×3 to L 3 (R 3 ) 3 . The norm of g in L p (0, τ ; L 3 (R 3 ) 3 ) is bounded by the norm of u ⊗ v r + w r ⊗ u in L p (0, τ ; L 3 (R 3 ) 3×3 ). Indeed, the operator (-∆) -1 2 P∇• is bounded from L 3 (R 3 ) 3×3 in L 3 (R 3 ) 3 and so (-∆) 1/2 Rg L p (0,τ ;L 3 ) ≤ c t → (-∆) 1/2 e t∆ L 1 (0,τ ;L (L 3 )) g L p (0,τ ;L 3 ) √ τ v r L ∞ ((0,τ )×R 3 ) 3 + w r L ∞ ((0,τ )×R 3 ) 3 u L p (0,τ ;L 3 ) .
Proceeding as in item (i) settles (3.4).

This establishes Proposition 3.2.

Remark 3.3. Notice that if one assumes that v and w belong to the larger space L ∞ (0, T ; L 3 (R 3 ) 3 ), and if r > 0 is fixed, then in general one cannot ensure the existence of v r and w r in L ∞ ((0, T )×R 3 ) such that ess sup t∈(0,T ) v(t) -v r (t) L 3 + ess sup t∈(0,T ) w(t) -w r (t) L 3 < 3r. This is the case if, for example, v or w are of the form t -1 φ(•/t) with φ ∈ L 3 (R 3 ) and r is small with respect to φ L 3 .

Proposition 3.4.

1. For all ε > 0, there exists

r > 0 such that all v ∈ C ([0, T ], L 3 (R 3 ) 3 ) + r L ∞ (0, T ; L 3 (R 3 ) 3 ), there exists τ = τ (ε, v) > 0 such that C(v, •) : L 4 3 (0, τ ; L 2 (R 3 )) -→ L 4 3 (0, τ ; L 2 (R 3
)) is bounded with norm less than ε.

2. For all ε > 0 there exists r > 0 such that, for all v ∈ C ([0, T ], L 3 (R 3 ) 3 )+r L ∞ (0, T ; L 3 (R 3 ) 3 ), there exists τ = τ (ε, v) > 0 such that

C(v, •) : L 2 (0, τ ; L 3 2 (R 3 )) ∩ L 4 3 (0, τ ; L 2 (R 3 )) -→ L 2 (0, τ ; L 3 2 (R 3
)) is bounded with norm less than ε.

3. For all ε > 0 all v ∈ L 4 (0, T ; L 6 (R 3 )) 3 , there exists τ = τ (ε, v) > 0 such that

C(v, •) : L 2 (0, τ ; L 3 2 (R 3 )) -→ L 2 (0, τ ; L 3 2 (R 3
)) is bounded with norm less than ε.

For all

ε > 0, all ϑ ∈ L 2 (0, T ; L 3 2 (R 3 )), there exists τ = τ (ε, ϑ) > 0 such that C(•, ϑ) : L 4 (0, τ ; L 6 (R 3 ) 3 ) -→ L 4 3 (0, τ ; L 2 (R 3
)) is bounded with norm less than ε.

Proof. We proceed as in the previous proposition. For r > 0, let us choose where, now, we set f (s) = (-∆) -1 div θ(s)(v(s) -v r (s)) and g(s) = -(-∆) -1/2 div θ(s)v r (s) .

v r ∈ C c ([0, T ] × R 3 ) 3 such that ess sup [0,T ] v -v r L 3 ≤ 2r. ( 3
As in the proof of previous proposition we see that the norm of f in L

4 3 (0, τ ; L 2 (R 3 )) is bounded by the norm of θ(v -v r ) in L 4 3 (0, τ ; L 6 5 (R 3 ) 3 ) (because of the Sobolev embedding Ẇ 1, 6 5 → L 2 in dimension 3). As ∆R is a bounded operator in L 4 3 (0, τ ; L 2 (R 3 )), we have (-∆)Rf L 4 3 (0,τ ;L 2 (R 3 )) ≤ C 4 3 ,2 C θ(v -v r ) L 4 3 (0,τ ;L 6 5 (R 3 ) 3 ) ≤ 2rC 4 3 ,2 C θ L 4 3 (0,τ ;L 2 (R 3 ))
where C is the constant arising from the Sobolev embedding Ẇ 1, 6 5 → L 2 in dimension 3 and r comes from the choice of v r .

The norm of g in L

4 3 (0, τ ; L 2 (R 3 )) is bounded by the norm of θ v r dans L 4 3 (0, τ ; L 2 (R 3 ) 3 ), because (-∆) -1 2 div is a bounded operator in L 2 (R 3 ) 3 . Moreover, (-∆) 1/2 e t∆ L (L 2 (R 3 ))
t -1/2 . Then, viewing as before (-∆) 1/2 R as a convolution operator, we get

(-∆) 1/2 Rg L 4 3 (0,τ ;L 2 (R 3 )) ≤ c t → (-∆) 1/2 e t∆ L 1 (0,τ ;L (L 2 (R 3 ))) g L 4 3 (0,τ ;L 2 (R 3 )) ≤ c τ 1 2 θ v r L 4 3 (0,τ ;L 2 (R 3 ) 3 ) ≤ c τ 1 2 v r L ∞ ((0,τ )×R 3 ) 3 ) θ L 4 3 (0,τ ;L 2 (R 3 ))
.

We then choose r > 0 such that 2rC 4

3 ,2 C ≤ ε 2 , next v r ∈ C c ([0, T ] × R 3 ) 3 satisfying (3.10) and last θ > 0 such that c τ 1 2 v r L ∞ (((0,τ )×R 3 ) 3 ) ≤ ε 2 .
This establishes the first assertion of the proposition. To prove the second assertion we proceed as before: for r > 0, we choose v r ∈ C ([0, T ] × R 3 ) 3 such that (3.10) holds. Then for all θ ∈ L 2 (0, τ ;

L 3 2 (R 3 )), C(v, θ) = C(v -v r , θ) + C(v r , θ) = (-∆) 3 4 Rf + (-∆) 1/2 Rg (3.11)
where f (s) = -(-∆) -3 4 div θ(s)(v(s) -v r (s)) and g(s) = -(-∆) -1/2 div θ(s)v r (s) . We easily see that the norm of f in L

3 4 R is a bounded operator in L 4 3 (0, τ ; L 3 2 (R 3 )), we have (-∆) 3 4 Rf L 2 (0,τ ;L 3 2 (R 3 )) ≤ C d dt 1 4 (-∆) 3 4 Rf L 4 3 (0,τ ;L 3 2 ) ≤ CC 4 3 , 3 2 C θ(v -v r ) L 4 3 (0,τ ;L 6 5 (R 3 ) 3 ) ≤ CC 4 3 , 3 2 C (r + η) θ L 4 3 (0,τ ;L 2 (R 3 )) . (3.12)
Here C is the constant coming from the Sobolev embedding

Ẇ 1 4 , 4 3 → L 2 in dimension 1, C is the constant of the Sobolev embedding Ẇ 1 2 , 6 5 → L 3 2 in dimension 3 and r comes from the choice of v r . The norm of g in L 2 (0, τ ; L 3 2 (R 3 )) is controlled by the norm of θ v r in L 2 (0, τ ; L 3 2 (R 3 ) 3 ), because (-∆) -1 2 div is a bounded operator from L 3 2 (R 3 ) 3 to L 3 2 (R 3 ). As (-∆) 1 2
R is a convolution operator, we can write (-∆)

1 2 Rg L 2 (0,τ ;L 3 2 (R 3 )) ≤ c t → (-∆) 1 2 e t∆ L 1 (0,τ ;L (L 3 2 (R 3 ))) g L 2 (0,τ ;L 3 2 (R 3 )) ≤ c τ 1 2 θ v r L 2 (0,τ ;L 3 2 (R 3 ) 3 ) ≤ c τ 1 2 v r L ∞ ((0,τ )×R 3 ) 3 ) θ L 2 (0,τ ;L 3 2 (R 3 )) . (3.13) It just remains to choose r, η > 0 such that 2r CC 4 3 , 3 2 C ≤ ε 2 , next v r ∈ L ∞ ((0, T ) × R 3
) 3 such that (3.10) holds and finally θ > 0 such that c τ

1 2 v r L ∞ ((0,τ )×R 3 ) 3 ) ≤ ε
2 . This establishes the second assertion of the proposition.

Let us prove the third assertion. As v ∈ L 4 (0, T ; L 6 (R 3 ) 3 ), for an arbitrary r > 0 we can choose now v r ∈ L ∞ ((0, T ) × R 3 ) 3 such that v -v r L 4 (0,T ;L 6 (R 3 ) 3 ) < r.

(3.14)

If τ > 0 and θ ∈ L 2 (0, τ ; L 3 2 (R 3 )), then (v -v r )θ ∈ L 4 3 (0, τ ; L 6 5 (R 3 ) 3
) by Hölder inequality. Therefore, splitting C(v, θ) as in (3.11), the above computations (3.12)-(3.13) can be reproduced: the only change that needs to be done is the application of (3.14) instead of (3.10). We get in this way (-∆)

3 4 Rf L 2 (0,τ ;L 3 2 (R 3 )) ≤ CC 4 3 , 3 2 C r θ L 2 (0,τ ;L 3 2 (R 3 ))
. This, combined with (3.13) proves our third assertion.

The proof of the fourth assertion follows the same scheme. For r > 0, choose

ϑ r ∈ C c ([0, T ]×R 3 ) such that ϑ -ϑ r L 2 (0,T ;L 3 2 (R 3 )) ≤ r. (3.15)
Then, for all v ∈ L 4 (0, τ ; L 6 (R 3 ) 3 ), we have

C(v, ϑ) = C(v, ϑ -ϑ r ) + C(v, ϑ r ) = ∆Rf + (-∆) 1/2 Rg with f (s) = (-∆) -1 div (ϑ(s) -ϑ r (s))v(s) and g(s) = -(-∆) -1 2 div ϑ r (s)v(s) .
One easily shows that the norm of f in L 4 3 (0, τ ; L 2 (R 3 )) is bounded by Cr v L 4 (0,τ ;L 6 (R 3 ) 3 ) , where C is the norm of (-∆) -1 div : L 6/5 (R 3 ) 3 → L 2 (R 3 ). Thus, as the operateur ∆R is bounded on

L 4 3 (0, τ ; L 2 (R 3 )) by C 4 3 ,2 = C C 4 3 ,2 , we have ∆Rf L 4 3 (0,τ ;L 2 (R 3 )) ≤ C 4 3 ,2 r v L 4 (0,τ ;L 6 (R 3 ) 3 ) .
The norm of g in L 4 3 (0, τ ; L 2 (R 3 )) is bounded by the norm of ϑ r in L 2 (0, τ ; L 3 (R 3 )) and the norm of v in L 4 (0, τ ; L 6 (R 3 ) 3 ). As (-∆) 1/2 R is a convolution operator, we deduce that (-∆)

1 2 Rg L 4 3 (0,τ ;L 2 (R 3 )) ≤ t → (-∆) 1/2 e t∆ L 1 (0,τ ;L (L 2 (R 3 ))) g L 4 3 (0,τ ;L 2 (R 3 )) ≤ c τ 1 2 ϑ r L 2 (0,τ ;L 3 (R 3 )) v L 4 (0,τ ;L 6 (R 3 ) 3 ) .
We then choose r > 0 such that

C 4 3 ,2 r ≤ ε 2 , next ϑ r ∈ C c ([0, T ] × R 3 ) satisfying (3.15) and last τ > 0 such that c τ 1 2 ϑ r L 2 (0,τ ;L 3 (R 3 )) ≤ ε 2 .
We thus get the last assertion of the proposition.

Proposition 3.5. For all τ > 0, the operator L defined by (2.4) is linear and bounded from

L 2 (0, τ ; L 3 2 (R 3 )) to L 4 (0, τ ; L 6 (R 3 ) 3 and from L 4 3 (0, τ ; L 2 (R 3 )) to L 4 (0, τ ; L 6 (R 3 ) 3 ), with operator norms independent on τ . Moreover, for all p ∈ [1, ∞), L is bounded from L 2 (0, τ ; L 3 2 (R 3 )) to L p (0, τ ; L 3 (R 3 )), with norm of order τ 1/p . Proof. For θ ∈ L 2 (0, τ ; L 3 2 (R 3 )), we write L(θ) = d dt -1 4 d dt 1 4 (-∆) 3 4 Rϕ , 7 where ϕ(s) = (-∆) -3 4 P θ(s)e 3 . Observe that ϕ ∈ L 2 (0, τ ; L 6 (R 3 ) 3 ), because of the Sobolev embedding (-∆) -3 4 (L 3
2 ) → L 6 (in dimension 3), with norm bounded by the norm of θ in L 2 (0, τ ; L 3 2 (R 3 )). By Theorem 3.1, we deduce that L(θ)

∈ d dt -1 4 L 2 (0, τ ; L 6 (R 3 ) 3 ) → L 4 (0, τ ; L 6 (R 3 ) 3 ),
the last inclusion arising from the Sobolev embedding d dt -1 4 (L 2 ) → L 4 (in dimension 1). This establishes the first assertion of the proposition.

When θ ∈ L 4 3 (0, τ ; L 2 (R 3 )), we write

L(θ) = d dt -1 2 d dt 1 2 (-∆) 1 2 Rψ , (3.16)
with ψ(s) = (-∆) -1 2 P θ(s)e 3 . Notice that ψ ∈ L 4 3 (0, τ ; L 6 (R 3 ) 3 ), because of the Sobolev embedding (-∆) -1 2 L 2 → L 6 (in dimension 3), with norm bounded by the norm of θ in L

4 3 (0, τ ; L 2 (R 3 )). Applying Theorem 3.1 with α = 1 2 , we get L(θ) ∈ d dt -1 2 L 4 3 (0, τ ; L 6 (R 3 ) 3 ) → L 4 (0, τ ; L 6 (R 3 ) 3 ).
The last inclusion comes from the Sobolev embedding d dt

-1 2 (L 4
3 ) → L 4 (in dimension 1). The second assertion of the proposition follows.

Next, for θ ∈ L 2 (0, τ ; L 3 2 (R 3 )), let us write L(θ) as before in (3.16). By Sobolev embedding (-∆) -1 2 (L

2 ) → L 3 in dimension 3, we have ψ ∈ L 2 (0, τ ; L 3 ) with norm bounded by θ

L 2 (0,τ ;L 3 2 )
.

By Theorem 3.1 with α = 1 2 , we deduce that L(θ) ∈ d dt -1 2 L 2 (0, τ ; L 3 ) → L p (0, τ ; L 3 ) for all 1 ≤ p < ∞. The last inclusion follows, for 2 < p < ∞, from the Hölder injection L 2 ((0, τ )) → L q (0, τ ) for q ∈ [1, 2] (with norm τ 1/q-1/2 ) and Hardy-Littlewood-Sobolev inequality d dt -1 2 (L q ) → L p (in dimension 1), for all p ∈ (2, ∞) and 1 p = 1 q -1 2 . For 1 ≤ p ≤ 2 it is sufficient to apply once more Hölder inequality.

The proof of the uniqueness

We need a few lemmas before proving Theorem 2.1. Lemma 4.1. Let (u 0 , θ 0 ) ∈ S (R 3 ) 3 ×S (R 3 ) with div u 0 = 0, and let (u, θ) ∈ L ∞ (0, T ; L 3 (R 3 ) 3 )× L 2 (0, T ; L 3 2 (R 3 )) be a mild solution of (2.1) with initial data (u 0 , θ 0 ). Then

(u, θ) ∈ C ([0, T ], S (R 3 ) 3 × S (R 3 )).
Moreover, we have u 0 ∈ L 3 (R 3 ) 3 and for every t ∈ [0, T ], u(t) does also belong to L 3 (R 3 ) 3 .

Proof. Let us denote by F (t, x) the kernel of the operator e t∆ P∇•. It is well known, and easy to check, that F satisfies the scaling relations 3×3 . From these properties and the dominated convergence theorem one deduces that, for all

F (t, x) = t -3 2 F (1, x/ √ t), with F (1, •) ∈ (L 1 (R 3 ) ∩ C 0 (R 3 ))
1 ≤ p ≤ ∞, that F ∈ C (0, ∞; L p (R 3 )). Moreover, F (t, •) 1 = t -1 2 F (1, •) 1 . Now, if (u, θ) ∈ X T,r , then u ⊗ u ∈ L ∞ (0, T ; L 3/2 (R 3 ) 3×3 ).
Then, recalling the definition of the bilinear operator B and applying the above properties of F with p = 1, next applying the L 1 -L 3/2 convolution inequality, shows that the map t → B(u, u)(t) is continuous from (0, T ] to L 3/2 (R 3 ) 3 . Moreover, B(u, u)(t) L 3/2 → 0 as t → 0. Hence, the map t → B(u, u)(t) is continuous from [0, T ] to L 3/2 (R 3 ) 3 with value 0 at t = 0.

Let us now consider L(θ). Using the fact that the heat kernel e -|x| 2 /(4t) /(4π t) 3/2 is in C b (0, ∞; L 1 (R 3 )), we readily see that L(θ) ∈ C ((0, T ]; L 3/2 (R 3 )). To study the behavior of L(θ) near t = 0 we consider ϕ ∈ S (R 3 ) and observe, computing the Fourier transform of Pθe 3 with respect to the space variable, that t → h(t, •) = Pθe 3 (t, •) belongs to L 2 (0, T ; L 3 (R 3 )) by the Hausdorff-Young theorem. Then we have

| L(θ)(t), ϕ | ≤ ˆt 0 | h(s), e -(t-s)|•| 2 ϕ | ds ≤ ˆt 0 h(s) L 3 ϕ L 3/2 ds ≤ ϕ L 3/2 ˆt 0 θ(s) L 3/2 ds ≤ C ϕ θ L 2 (0,T ;L 3/2 (R 3 )) √ t.
Therefore, L(θ)(t) → 0 as t → 0 in S (R 3 ) and we deduce that L(θ) ∈ C ([0, T ], S (R 3 ) 3 ), with value 0 at t = 0.

Let us now consider C(u, θ). We have uθ ∈ L 2 (0, T ; L 1 (R 3 ) 3 ). Moreover, the kernel of the operator e t∆ ∇• has the same scaling properties as F . Therefore, proceeding as for B(u, u) we see on the one hand that C(u, θ) ∈ C ((0, T ]; L 1 (R 3 ) 3 ). On the other hand, we can also write

C(u, θ)(t) = div
ˆt 0 e (t-s)∆ (uθ) ds.

But the L 1 (R 3 )-norm of ´t 0 e (t-s)∆ (uθ) ds is bounded by √ t uθ L 2 (0,T ;L 1 (R 3 )) that goes to zero as t → 0. Hence, C(u, θ)(t) ---→ t→0 0 in S (R 3 ) 3 , by the continuity of the divergence operator from L 1 to S .

For the linear terms a and b it is obvious that they are both in C ([0, T ], S (R 3 )), with values at t = 0 given by u 0 and θ 0 , respectively.

Summarising, from the equation (2.1) we see that (u, θ) ∈ C ([0, T ], S (R 3 ) 3 × S (R 3 )), with values at t = 0 given by (u 0 , θ 0 ). But u ∈ L ∞ (0, T ; L 3 (R 3 ) 3 ), hence, for all 0 ≤ t ≤ T , we can find a sequence t n ----→ n→∞ t, contained in [0, T ], such that u(t n ) ∈ L 3 (R 3 ) 3 for all n ∈ N, with L 3 -norm uniformly bounded by u L ∞ (0,T ;L 3 (R 3 ) 3 ) , and u(t n ) ----→ n→∞ u(t) in S (R 3 ) 3 . By duality we deduce that u(t) ∈ L 3 (R 3 ) for every t ∈ [0, T ]. In particular, the initial velocity u 0 must belong to L 3 (R 3 ). Lemma 4.2. There exists an absolute constant r 0 > 0 such that if 0 ≤ r < r 0 and (u, θ) ∈ X T,r is a solution of (2.1), with (u 0 , θ 0 ) ∈ S (R 3 ) 3 × S (R 3 ) and div u 0 = 0, then there exists τ > 0 such that u ∈ L 4 (0, τ ; L 6 (R 3 ) 3 ).

Proof. Let us take p = 2 throughout this proof (any other choice 1 < p < ∞ would do: a different choice of p would just affect the value of r 0 and τ ). We know, by Proposition 3.2, that there exists r 0 and τ > 0 such that if (u, θ) ∈ X T,r with 0 ≤ r < r 0 , then the norm of the linear operator B(•, u) from L 4 (0, τ ; L 6 (R 3 ) 3 ) ∩ L p (0, τ ; L 3 (R 3 ) 3 ) to itself is bounded, with norm smaller than 1 2 . This shows that Id -B(•, u) is invertible in L 4 (0, τ ; L 6 (R 3 ) 3 ) ∩ L p (0, τ ; L 3 (R 3 ) 3 ).

Moreover, as θ ∈ L 2 (0, τ ; L 3 2 (R 3 )) by our assumption, we get from Proposition 3.5 that L(θ) ∈ L 4 (0, τ ; L 6 (R 3 ) 3 ) ∩ L p (0, τ ; L 3 (R 3 ) 3 ).

As observed in the previous Lemma, we have u 0 ∈ L 3 (R 3 ). Moreover, L 3 (R 3 ) ⊂ Ḃ0 3,3 (R 3 ) ⊂ Ḃ-1/2 6,3 (R 3 ) ⊂ Ḃ-1/2 6,4 (R 3 ). See [1, Chapt. 2] for generalities on Besov spaces. The characterisation of Besov spaces through the heat kernel (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]Theorem 2.34]) then implies that t → e t∆ u 0 ∈ L 4 (0, τ ; L 6 (R 3 ) 3 ). Since we have also that t → e t∆ u 0 ∈ C ([0, τ ]; L 3 (R 3 ) 3 ), we obtain a ∈ L 4 (0, τ ; L 6 (R 3 ) 3 ) ∩ L p (0, τ ; L 3 (R 3 ) 3 ), for all 1 < p < ∞.

These considerations allow us to define ũ = Id -B(•, u) -1 a + L(θ) .

. 10 )

 10 Then we have, for all θ ∈ L

4 3 3 2 4 3 6 5

 4346 (0, τ ; L (R 3 )) is controlled by the norm of θ(v -v r ) in L (0, τ ; L (R 3 ) 3 ) (owing to the Sobolev embedding Ẇ

(R

)).

(0, τ ; L 2 (R 3 )), C(v, θ) = C(v -v r , θ) + C(v r , θ) = ∆Rf + (-∆) 1/2 Rg

(R

)), where κ > 0 is some constant independent on T , u and θ.

We would like to show that u = ũ. By the assumption on u and the construction of ũ, these two functions satisfy u = B(u, u) + a + L(θ) and ũ = B(ũ, u) + a + L(θ).

Moreover, ũ ∈ L 4 (0, τ ; L 6 (R 3 ) 3 ) ∩ L p (0, τ ; L 3 (R 3 ) 3 ). Their difference v := u -ũ satisfies v ∈ L p (0, τ ; L 3 (R 3 ) 3 ) and v = B(v, u).

Reducing (if necessary) the value of τ , we deduce from the last point of Proposition 3.2 that v L p (0,τ,L 3 ) ≤ 1 2 v L p (0,τ ;L 3 (R 3 ) 3 ) .

This implies that v = 0 in L p (0, τ ; L 3 (R 3 ) 3 ). In particular, u = ũ ∈ L 4 (0, τ ; L 6 (R 3 ) 3 ).

Remark 4.3. Under the conditions of Lemma 4.1, the initial temperature θ 0 must belong to the inhomogeneous Besov space B -1 3/2,2 (R 3 ). Indeed, θ ∈ L 2 (0, τ ; L 3 2 (R 3 )), and C(u, θ) then belongs to this same space by the third claim of Proposition 3.4 and the previous lemma, for τ > 0 small enough. Then, by the second equation of (2.1), we obtain b ∈ L 2 (0, τ ; L 3 2 (R 3 )). The characterisation of inhomogeneous Besov spaces with negative regularity (see [START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF]Theorem 5.3]) then immediately gives

. By Lemma 4.2 we know that there exists τ 0 > 0 such that u 1 , u 2 ∈ L 4 (0, τ 0 ; L 6 (R 3 ) 3 ). Applying the last two assertions of Proposition 3.4 we get C(u, θ 2 ) ∈ L 4 3 (0, τ 0 ; L 2 (R 3 )) ∩ L 2 (0, τ 0 ; L 3 2 (R 3 )). The first and the second assertions of Proposition 3.4 ensure the existence of τ > 0 (we can assume τ ≤ τ 0 ) such that

)) to itself, with norm less than 1 2 . Therefore we can define θ = Id -C(u 1 , •) -1 (C(u, θ 2 )).

We see that θ ∈ L

)), and moreover θ = C(u 1 , θ) + C(u, θ 2 ). Let ψ = θ -θ. Then, subtracting the second equation in (3.2), we obtain

But u 1 ∈ L 4 (0, τ ; L 6 (R 3 ) 3 ) by Lemma 4.2. Hence, applying the third assertion of Proposition 3.4 we get ψ = 0 and so θ = θ. The latter equality implies that θ ∈ L

Proof of Theorem 2.1. Let r 0 > 0 be the absolute constant determined in Lemma 4.2. Assume that (u 1 , θ 1 ) and (u 2 , θ 2 ) are two mild solutions in X T,r of (B), with 0 ≤ r < r 0 , starting from (u 0 , θ 0 ) ∈ S (R 3 ) 3 × S (R 3 ). In fact, by Lemma 4.1, there is no restriction in assuming that

)) by our assumption and the first item of Proposition 3.5, we know that L(θ) ∈ L 4 (0, T ; L 6 (R 3 ) 3 ). Applying Proposition 3.2, we know that there exists τ > 0 such that B(u, u

. This allows us to show, applying Lemma 4.2, next using the first equation in (3.2), that

10

Applying the first assertion of Proposition 3.4, with v = u 1 and the last assertion of Proposition 3.4 with ϑ = θ 2 , we deduce from the second equality in (3.2) that, for all ε > 0, there exists 0 < τ ≤ τ such that θ

The second item of Proposition 3.5 allows us to take ε > 0 such that

We conclude that u = 0 in L 4 (0, τ ; L 6 (R 3 ) 3 ). This implies that θ = 0 a.e. on (0, τ ). The uniqueness is thus established at least during a short time interval [0, τ ), for a suitable 0 < τ ≤ T .

A standard argument now allows us to deduce that the uniqueness holds, in fact, in the whole interval [0, T ]: let τ * be the supremum of the times t 0 ∈ [0, T ] such that (u 1 , θ 1 ) = (u 2 , θ 2 ) in X t0,r . Let us show that τ * = T . Indeed, otherwise, by the continuity of (u 1 , θ 1 ) and (u

, with initial data given by (4.1). Therefore, applying the uniqueness result in short-time intervals established before, we see that there exists τ , such that 0 < τ < T -τ * , and (u

Existence

Let us prove Theorem 2.2, that ensures the existence of solution in the space where we obtained the uniqueness. In fact, an existence theorem for solutions to the Boussinesq system was established in [2], under assumptions more general than that of Theorem 2.2. However, the solution constructed in [2] a priori does not satisfy the required condition on the temperature, θ ∈ L 2 (0, T ; L 3 2 (R 3 )). Therefore, what remains to do in order to establish Theorem 2.2, is to show that the solution constructed in [2] does satisfy such condition, as soon as the initial temperature does belong to B -1 3/2,2 (R 3 ). For this, let us introduce some useful function spaces: For 1 ≤ p ≤ ∞ and 0 < T ≤ ∞, we define Z p,T to be the subspace of all vector fields u ∈ L 1 loc (0, T ; L p (R 3 ) 3 ) such that

In the same way, let Y q,T be the subspace of all the functions θ ∈ L 1 loc (0, T ; L q (R 3 )) such that θ Y q,T = ess sup t∈(0,T ) t

We will need the following bilinear estimate:

Proposition 5.1. For all u ∈ Z 6,T and θ ∈ L 2 (0, T ;

Here c is the L 6 5 (R 3 )-norm of the kernel of e ∆ div . But f ∈ L 4 3 ,2 (R) by Hölder inequality in Lorentz spaces, with norm controlled by the norm of θ in L 2 (0, T ;

Let us now recall the local existence theorem in [2, Theorem 2.4].

Theorem 5.2 (See [2]). If 3 < p < ∞, 3 2 < q < 3 and 2 3 < 1 p + 1 q , and if (u 0 , θ 0 ) belongs to the closure of the Schwartz class

with div u 0 = 0, then there exists T > 0 and a solution (u, θ) to (B) such that

Let us observe that the perturbation method used in [2] to establish Theorem 5.2 provides the well-posedness only in the space where the solution is constructed.

Proof of Theorem 2.2. Under the assumptions of the first item of Theorem 2.2, we have u 0 ∈ L 3 (R 3 ) 3 and θ 0 ∈ B -1 3 2 ,2 (R 3 ), which is continuously embedded in B -3(1-1/q) q,∞ , for all q > 3/2. Moreover, the Schwartz class is dense both in L 3 and in Besov spaces with finite third index. Therefore we may apply Theorem 5.2. Choosing, for example, p = 6 and q = 2 we obtain the existence, for some T > 0, of a solution (u, θ) ∈ Z 6,T × Y 2,T , such that u ∈ C ([0, T ], L 3 (R 3 ) 3 ) and

). Now, reducing if necessary the length of time interval where the solution is considered, we can assume that T is such that κ u Z 6,T < 1. Hence, by Proposition 5.1, we see that the linear operator C(•, u) :

Let us prove the second assertion of Theorem 2.2. If θ 0 belongs to the homogeneous Besov space Ḃ-1 3/2,2 (R 3 ), then b ∈ L 2 (0, ∞; L (and conversely). If u 0 L 3 + θ 0 Ḃ-1 3/2,2 (R 3 ) is smaller than a suitable absolute constant (or, more in general, if u 0 Ḃ-(1-3/p) p,∞ (R 3 ) + θ 0 Ḃ-3(1-1/q) q,∞ (R 3 ) is smaller than a constant depending only on p and q, where p and q are as in Theorem 5.2), then the estimates in [2] provide the global existence of the solution, with u ∈ C b (0, ∞; L 3 (R 3 )). Moreover, u Z6,∞ is controlled by the size of the initial data (u 0 , θ 0 ) in L 3 (R 3 ) 3 × Ḃ-1 3/2,2 (R 3 ). Therefore, κ u Z6,∞ can be assumed to be smaller than 1. Then the above argument applies with T = +∞. This completely establishes Theorem 2.2.


P Sandhya

M Sreekala

Abderrahim Boudenne

Bertrand Garnier

Didier Rouxel

Moothetty Padmanabhan

Nandakumar Kalarikkal

Sabu Thomas

Thermal and electrical properties of phenol formaldehyde foams reinforcing with reduced graphene oxide

Keywords: Phenol formaldehyde, reduced graphene oxide, dielectric properties, thermal conductivity Graphical

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

Introduction

Among the polymeric foams, phenolic foam (PF) is readily available and relatively cheap and at the same time it has the greatest FST (Fire, Smoke and less Toxic) properties. Phenolic foams have attained great attention as an insulating material because of its high self-ignition temperature (480 o C) and high thermal stability over a broad temperature range (from -196 o C to 200 o C) [1].

During combustion, it has excellent flame resistance and low generation of toxic gases. Since the phenolic resins contain various solvents, there is no control over the cell morphology during foaming and curing. The cell density and cell uniformity are the factors that affect the physical performance of different polymer foams. The interesting properties of phenolic foams are excellent flame retardancy, low smoke density, no dripping of molten plastic during combustion, high resistance to chemicals and solvents etc. PF foams suffer from lower mechanical properties such as compressive and flexural strengths and high friability and due to these reasons numerous studies have focused on improving these properties [2][3][4]. Several works have been reported in this field regarding the improvement in the mechanical properties of PF foams [5][6][7][8][START_REF] Wang | High Perform[END_REF]. One of the main drawbacks of PF foam is its intrinsic brittle nature due to the structure of the molecular chain, which leads to dust pollution in the work place, susceptibility to damage during handling and skin debonding in sandwich structure [10]. The factor that restricts the application of phenolic foam is its brittle nature and powdering [11]. Due to the high friability and low mechanical strength, great efforts have devoted for improving its properties. Improvement in these properties can be achieved by chemical modification, the addition of inert fillers, and fiber reinforcement [START_REF] Xu | High Perform[END_REF]. Among the different methods, the chemical modification method has gained more attention for its notable toughening effect. Highly flammable foams such as polyurethane and polystyrene can be replaced with phenolic foam for building thermal insulation materials. Toughness is a problem that exists in the field of phenolic foams and it can be improved by adding polyethylene glycol, polyvinyl alcohol, and polyurethane pre-polymer but the flame retardancy is deteriorated because of their flammability [13,14]. So, to extend the applications of PF foams, it is important to do some sort of research intensely on toughening agents with improved flame retardancy. Studies on PF foams with aramid fibers [15], glass fibers [16] cardanol [17], boron containing toughening agent [3] were also reported. Traditional PF foams exhibit thermal and electrical insulating property with the closed-cell structure that restricts applications [18]. As a result, it is important to develop PF foams with excellent electrical and electromagnetic interference shielding property by reinforcing conductive nanofillers into the PF matrix.

In the field of material science and condensed-matter physics, graphene is considered as a rising star because of its high electrical conductivity, high surface area, high flexibility, and superior mechanical strength. This two-dimensional (2D) single layer or few layers sp 2 -bonded graphene has attracted a great deal of interest in recent years owing to its unique structure, cost effective, and extraordinary electronic properties. The enhancement in the texture and homogeneity of foam could be achieved by the addition of inert fillers like talc, mica, clay, glass, and ceramic micro balloons, fly ash etc. This process increases the viscosity of the resin before foaming, and thus makes foams heavier. Increase in the foam strength, stiffness, resistance to abrasion and dimensional stability could be obtained by the addition of chopped fibers, including glass fiber, aramid fibers (e.g., Nomex and Kevlar) and natural fibers (bamboo and cellulose fiber).

The addition of these fibers affects the mixing and foaming process leading to increased density and poor surface quality. But these results in an enhancement in the mechanical properties of phenolic foam [10].

Seung and colleagues [1] prepared phenolic foams with multi-wall carbon nanotube and graphene and they could improve the mechanical and thermal performance by a short-period microwave foaming process. From the results, they found that 0.5% MWCNT and 1.0% graphene (GP) samples show high specific compressive strength with improvement of 71.8% and 157.3%, respectively. The thermal conductivity of GP 1.0, which has the highest cell density and cell uniformity, decreased by 37.4% compared with that of the neat phenolic foam and GP 1.0 showed the highest thermal stability from 40 to 390 o C due to its low thermal conductivity.

MWCNTs/phenolic composite foams with three different types of MWCNTs, including pristine, carboxyl and amino MWCNTs were successfully manufactured with the aid of ultrasonic cavitation by Yang et al. [10]. About 31% increase is observed in the specific strength of foam by the addition of 0.05 wt % MWCNTs-COOH. It is also observed that a marginal enhancement in thermal stability and flame-retardant property of the composite foam by the addition of various MWCNTs.

Li and colleagues [19] prepared α-zirconium phosphate/graphene oxide and reinforced in phenolic foam and studied mechanical strengths, pulverization ratio, flame retardancy and thermal stability. From the analysis, it was found that there is an increase in flexural and compressive strength and the friability gets decreased. Li and co-workers [20] synthesized silica nanospheres/graphene oxide hybrid and through immobilization of SiO2 nanospheres on the surface of GO sheets (SGO) and incorporated into PF foams to study its effect on the mechanical strengths, thermal stability, pulverization ratio, and flame retardancy. PF foam with 1.5 phr SGO showed a 36.2% increase in compressive, a 31.6% increase in the flexural strength and 70% decrease in the pulverization ratio. Luan et al. [21] studied the modification of phenolic foam by graphene oxide. From the results, they found that very low percentage of filler can decreased the thermal conductivity of phenolic foam. Zhou and co-workers [22] investigated the effect of graphene oxide on the mechanical properties of phenolic foam, observed an improvement in friability and 68.2% increase of the gel time by using 0.5% graphene oxide. Li and co-workers [23] focused on increasing the electrical conductivity as well as the mechanical and thermal insulating properties by introducing the MWCNTs into the PF via in-situ polymerization. From the results, it was found that PF/2 wt% MWCNTs nanocomposites showed an increase in compressive strength (241.9%), compressive modulus (166.7%), tensile strength (264.5%) and tensile modulus (125.7%) as compared with that of pure PF.

Saz-Orozco et al. [24] incorporated viscose cellulose fibers into the phenolic foam to improve its compressive mechanical properties. The highest compressive modulus and strength was observed for 2 wt % cellulose fiber-reinforced phenolic foam and showed an increase of 21% and 18%, respectively, compared to the unreinforced material. The thermal stability of the phenolic foam gets decreased slightly by the incorporation of cellulose fibers. Liu and co-workers [8] investigated the physical, thermal, flame-retardant and smoke suppression properties of PF foams reinforcing with graphene oxide (obtained by reducing and functionalized with magnesium aminotris-(methylenephophonate). They observed that the pulverization ratio of the samples reduced and enhancement was observed in mechanical strength and flame retardancy. Agrawal et al. [25] developed PF/RGO composite foams by a simple and cost-effective template method.

RGO contents in PF foams improved not only the electromagnetic interference shielding effectiveness of composite foams (-23.2 to -50.7 dB in the X-band frequency region with 1 wt% RGO) but also thermal (1.2 W/(mK) and electrical properties. Li et al. [26] prepared silica nanoparticle reinforced PF foams using a chemical foaming method to improve their mechanical and thermal properties. The excellent cell morphology and stronger chemical cross-linked network structure of 2% silica nanoparticle reinforced PF foams showed improvement in compression strength (47.4%), compression modulus (38.6%) and tensile strength (42.9%) compared with the reference PF. TGA results showed that the nanocomposite foams were thermally stable better than pure PF foam. Wang et al. [START_REF] Wang | High Perform[END_REF] fabricated PF foams with CoAl-layered double hydroxides (LDH)/GO nanohybrid and the resultant samples exhibited an increase in compressive strength (54.8%), flexural strength (35.2%) with enhanced thermal stability.

In the present work, in order to avoid the use of a poisonous reducing agent such as hydrazine, an eco-friendly natural reducing agent starch potato was used for the reduction of graphene oxide. The obtained reduced graphene oxide (RGO) was used as a filler in PF for the production of PF/ RGO foam. Here, results from various studies such as morphology, thermal stability, dielectric properties, and thermal conductivity of PF/RGO foams are discussed in detail and finally, morphology-property correlations are presented. Hence the applicability of the prepared nanocomposites as a multifunctional material in different fields is exposed.

Experimental

Materials

Graphite powder, NaNO3, Conc. H2SO4, KMnO4, hydrogen peroxide, ammonia, phosphoric acid (H3PO4), Tween 80 (polysorbate 80), p-toluenesulfonic acid, and n-pentane were all commercial products with analytical grades and were purchased from Merck, India. The starch potato was obtained from LOBA Chemie. Pvt. Ltd. Phenol formaldehyde resole resin was provided by Polyformalin (Ernakulam, Kerala).

Preparation of RGO via graphene oxide synthesis

Reduced graphene oxide was synthesized by reducing graphene oxide (GO) with potato starch. The detailed procedure for the preparation of RGO and its characterization were discussed in our research article published elsewhere [27,28]. The reduction was carried out by refluxing GO solution using potato starch at 80 o C for 3 h by maintaining the pH in the range of 9-10 through the addition 25% ammonia solution. The resulting solution washed several times with distilled water and dried at 30 o C for further use.

Preparation of PF/RGO foam nanocomposites

The weight percentage of RGO for the preparation of PF/RGO foams was taken on the basis of the solid content of the PF resin. For good dispersion of RGO into PF, acetone is used as the solvent and the mixture is stirred in a mechanical stirrer with a speed of 4000 rotations/min for 2 h and then sonicated for one hour at 60 o C. For 100 g of PF, n-pentane (blowing agent, 8 g), Tween 80 (surfactant, 5 g), p-toluenesulfonic acid/phosphorous acid/distilled water in the ratio 2:1:2 by weight (curing agent,12 g) were mixed rapidly at room temperature and is transferred into a paper mould. It was then kept in a pre-heated air oven at 70 o C for 12 h. The obtained PF and PF/ RGO foams were cut in desired dimensions for further analysis [19]. A schematic representation of the preparation of PF/RGO foam is shown in Figure 1. The thermal stability of the PF foam samples was studied by thermogravimetric analysis (TGA) using Perkin Elmer STA 6000 (Simultaneous Thermal Analyzer) at a heating rate of 20 o C per minute under nitrogen atmosphere. The weight of the samples used for the analysis was between 5 and 10 mg and the thermograms were recorded from room temperature to 700 o C.

The dielectric properties of all the samples (10 x10x 3 mm 3 ) were measured using Wayne Kerr 600B, precision LCR meter with frequency ranging from 20 Hz to 30 MHz. The testing procedure was repeated with four different samples and averaged value was used.

Thermal conductivity (λ) was measured at room temperature using a laboratory made experimental device which was designed for the measurement of low and average thermal conductivity from 0.1 to 5 W.m -1 .K -1 and is suitable for our sample size (10x10 mm 2 section and 3 mm of thickness). The experimental procedure represents an extension of the normalized hot guarded plate (HGP) method with a single-sided specimen state [29,30]. A heat source and a heat sink (an electrical heater and two heat exchangers with thermostat baths) are located respectively on the bottom and top surfaces of the sample. This setup generates a temperature gradient in the thickness direction of the sample. The main characteristic of the experimental concerns the use of an active guard under the heater that allows the heat flux, dissipated by the electric heater, to flow entirely through the sample. In fact, a temperature controller cancels the heat flux from the electric heater to the heat source. Heat exchanges by adjusting the electric heating in order to obtain its temperature as close as possible to the one of the heat source. The heat exchanger is thus ensuring an adiabatic environment under the heater (Figure 2). The parallelism between upper and lower plates of the experimental setup was achieved using guides for vertical displacement that was controlled by a pneumatic jack in order to ensure an adjustable pressure P on both sides of the sample (typically 4 bar). Thermal resistance between sample and heat source and, between sample and heat sink, were minimized by spreading thermal grease on the top and bottom sides of the sample, the effect of thermal grease on thermal conductivity measurement being previously studied [29,30]. When the steady state regime is reached, the measured electric power with respect to the sample surface and the temperatures on both sides of the heater is used to find the thermal conductivity using Fourier's law. This setup which requires only one sample provides thermal conductivity measurement with accuracy smaller than 5% [30].

Density was found using the sample dimensions and weight, the latter being measured with an A340 Mettler precision balance. The specific heat (cp) measurements were performed using a Perkin Elmer DSC Pyrismicrocalorimeter with a -70°C cryostat. Before heat capacity measurement, the DSC device was calibrated with a 32 mg sapphire reference sample [31].

Results and discussions

Characterization of graphene derivatives

The Raman spectra patterns of graphite powder, GO, and RGO is represented in Figure 3.

Raman spectroscopy analysis was helpful to evaluate the structural properties of graphene based materials. Weak D band observed at 1361 cm -1 in graphite changes to 1369 cm -1 indicating the reduction in the size of the sp 2 domains due to the extensive oxidation. After reduction with potato starch, the obtained RGO shows a D band at 1361 cm -1 . G band is observed at 1581 cm -1 which corresponds to the vibrations of sp 2 carbon atom domains of graphite [32]. If the G band is present in the Raman spectra, it can be articulated that the sample contains sp 2 carbon networks [33]. In the case of GO, the broad G band at 1593 cm -1 is due to the presence of isolated double bonds that resonate at much higher frequencies than the G band of graphite and in the case of RGO it is observed at 1589 cm -1 . The ratio of the peak intensities of D and G bands (ID/IG) provides information about the extent of defects on the graphene based materials. The small increase in the D/G intensity ratio of GO (0.9836) and that of RGO (0.9865) indicating that most of the Figure 2. Thermal conductivity experimental setup oxygenated groups would have been removed during reduction and partially ordered crystal structure of graphene is attained. Similar results were also reported elsewhere [34][35][36]. The EDX spectrum of RGO is shown in Figure 4(b) and it is clear that carbon and oxygen is present in it. Moreover, the spectrum corresponding to sulphur is absent in the EDX spectrum of RGO.

The increase in the weight percentage of oxygen in RGO (42.06%) as compared to GO (28.56%) is due to the presence of oxygen containing functional groups in potato starch. This also confirms that GO is reduced to RGO by potato starch. obtained from the AFM images was found to be 1 nm or less than 1 nm. This may be due to the molecules present on the surface of the graphene sheet, the imperfect interface between the silicon substrate and graphene, and the different attractive forces between AFM probes as compared to the silicon substrate. The thickness of graphene sheets is 2 nm indicating that it exists as two or three layers. The 3-dimensional AFM image of RGO is obtained by drop casting the dispersion of RGO in water after deposition on a silicon wafer and is represented in Figure 5. From the image, the thickness of RGO sheets is about 0.67 nm which corresponds to the thickness of single layer graphene sheets [34,37,38]. The TEM images of RGO sheets were already published in our previous studies, in which the RGO sheets are irregular and layer structured with wrinkles [28,39].

Scanning electron microscopic images of PF/RGO foam nanocomposites

The cell geometry and the effect of RGO on the microstructure of the PF foams were studied using the SEM analysis. As shown in Figure 6(a), in the case of neat PF, mostly closed cells with large sizes were observed. The walls of some of the cells were broken. With the addition of RGO into the PF matrix, an increase in the number of clear cells was observed in the SEM images as shown in Figure 6(b)-(c). In addition, the shape and size of the cells were almost spherical and the walls of the cells were more rigid than the ones of neat PF. Figure 6(b) shows cells of small and big size distributed randomly with some voids. In Figure 6(c) one can observe clear cells of almost equal shape and size distributed uniformly. This is due to the presence of RGO sheets that embed in the cell walls made them thick, which leads the cells to grow almost similar in size [40]. But on increasing the RGO content, the chances of agglomeration of the RGO sheets increase and the walls of the cells get ruptured as shown in Figure 6(d). Indeed, the SEM image of 0.15 wt% RGO (Figure 6(d)) shows some collapse or collisions that occurred in the cell.

Similar results were reported in studies of Li et al. [40] and Wang et al. [START_REF] Wang | High Perform[END_REF]. It is observed from the SEM images (Figure 6(a)-(d)) that there is a large numbers of voids present on the surface of samples. This is due to the evaporation of water during the process of foaming and curing. The better interaction of PF with RGO sheets resulted in the formation of almost uniform cells with rigid cell walls.

Electrical properties of phenolic foam nanocomposites

The electrical property of phenol formaldehyde foam with different wt% of RGO is shown in Figure 8 and it is observed that the dielectric permittivity, dielectric loss and AC conductivity of PF/RGO foams gradually increased with increase in the wt% of RGO. Here, only a small increase in conductivity is observed at lower wt% RGO. From Figure 8(a), it can be seen that the dielectric constant of PF/RGO foams increased dramatically at 0.15 wt% RGO because, at higher concentration of RGO the possibility for the formation of conductive networks increases as a result of the accumulation of a large number of graphene sheets in the cell walls of the foam. Here, foams exhibit a significant percolation property when the RGO content reach 0.15 wt%, due to the formation of the conductive path of the interconnected RGO sheets. Narasimman et al. studied the electrical conductivity in GO based carbon foams and found that higher wt% of GO (1.25%) showed an increase in electrical conductivity (4.8 S cm -1 ) [41]. Figure 9 represents the conductivity network formed by RGO sheets in PF foam which cause electrical conductivity. Figure 9. PF foam with RGO sheets

Thermophysical properties of PF/RGO foam nanocomposites

The investigation of thermal conductivity of PF/RGO foams is essential when it plays a very important role in its application. In the case of foams, the heat transfer mainly occurred through the solid polymer than through the gaseous phase. The density, size, and distribution of the cells, the thermal conductivity of the polymer etc. are some of the factors that affect the thermal conductivity of foams [29]. The relationship between the thermal conductivity of the resulting foams and RGO volume content is shown in Figure 10. 2 between densities with various RGO contents. Therefore the observed effects of RGO content on density are due to some phenomenon. Indeed, the addition of 0.08wt% RGO decreases the size of the cell foam as observed in section 3.2 dedicated to SEM observations. Therefore, the density increases with RGO content and also because of microstructure change of the foam. As further amount of RGO is added, some destruction of cell walls is observed (section 3.2) which could explain the decrease of density with the 0.12 and 0.15wt% of RGO. The heat capacity of PF/RGO is affected by the addition of RGO but not by the microstructure change of the foam as it is a thermal property related to the mass. The change of microstructure of the foam could affect the thermal conductivity however this effect is hidden by the increase in effective thermal conductivity due to the addition of RGO material which has a much higher thermal conductivity (typically about 6 W.m -1. K -1 ) than PF material.

0

Conclusions

In our work, PF foams were reinforced using RGO and resulting morphology, thermal stability, dielectric property and thermal conductivity of the foams were studied. The SEM images of PF/RGO foams show changes in the size and distribution of the cells with an increase in RGO content. PF foams with 0.12 wt% RGO shows cells of uniform size and shape. From the TGA results, it is found that there is an improvement in the thermal stability of PF foams. The dielectric properties of the PF foams increase with an increase in wt% of RGO. The thermal conductivity study of the prepared foams shows that the introduction of RGO into the PF matrix increases slightly the thermal conductivity of the foams. Here, the samples remained as thermal insulators.

On analyzing the thermal conductivity and dielectric properties of PF foams, one could expect applications of this material when relatively electrical conductor and thermal insulating properties are both required.

Figure 1 .

 1 Figure 1. Preparation of PF/RGO foam

Figure 3 .Figure 4

 34 Figure 3. Raman spectra of graphite, GO, and RGO

Figure 4 .

 4 Figure 4. EDX spectra of (a) GO (b) RGO

Figure 5 . 3 -

 53 Figure 5. 3-dimensional AFM image of RGO

Figure 6 . 3 . 2 .

 632 Figure 6. SEM images of (a) neat PF foam (0% RGO) (b) 0.08 wt% RGO (c) 0.12 wt% RGO and (d) 0.15 wt% RGO

Figure 7 .

 7 Figure 7. (a) TGA and (b) DTG of PF/RGO foams

Figure 8 .

 8 Figure 8. Dielectric properties of PF/RGO foams

Table 1 .

 1 As compared to neat PF, a slight increase in degradation temperature is observed for PF/RGO foams. The char residue at 700 o C of PF/RGO foams gradually increased with increasing the RGO content as evident from Table1. The char residue obtained at 700 o C for 0.15% RGO shows 16.14% increase as compared to neat PF. PF/RGO foam with 0.15% shows the highest value for T-5%, T-10% and highest char residue at 700 o C indicating the highest thermal stability. The network of RGO sheets in the PF matrix affects the heat and mass transfer in phenolic foams. From these results, it is clear that there is an improvement in the thermal stability of the PF foams with the incorporation of RGO into the PF matrix.

Table 1 .

 1 Degradation temperatures of PF and PF/RGO foams

	Sample code	T -5% ( o C)	T -10% ( o C)	T max ( o C)	Char residue
					(%)
					at 700 o C
	0 % RGO	177.83	254.86	482.82	51.99
	0.08 % RGO	225.54	352.97	480.70	57.99
	0.12 % RGO	225.41	359.94	502.17	58.73
	0.15 % RGO	350.45	440.75	483.22	60.38

0.14 0.16 0.18 Thermal conductivity (W.m -1 .K -1 ) RGO content (vol%)

  Figure 10. Thermal conductivity of PF/RGO foamsThe evolution of the thermal conductivity of composite materials is almost presented in vol% because of the predicted and theoretical models are expressed in volume content. From figure, it is clear that the thermal conductivity of PF/RGO foams shows a slight increase in thermal conductivity with an increase in vol% of RGO. The thermal conductivity of neat PF foam is 0.089 W.m -1 .K -1 and the maximum value (0.113 W.m -1 .K -1 ) is observed for PF/RGO foam with 0.9 vol% RGO which corresponds to 0.15 wt% RGO. The slight increase in thermal conductivity of PF/RGO foams is due to the presence of RGO which creates thermal paths through the solid PF. From the results, it can be conclude that the addition of RGO in phenolic foams did not produce any dramatic change in the insulating property of PF foam.

		2.2
		2.0
	Cp (J.g -1 .K -1 )	1.4 1.6 1.8
		1.2
		1.0
		0.00	0.04	0.08	0.12	0.16
			RGO content (wt%)
	0.12
		Figure 11. Specific heat of PF/RGO foams
	0.10
	Figure 11 shows the specific heat of PF/RGO foams. According to the results, the specific
	0.08 heat of PF/RGO foams decreases with an increase in RGO content. The specific heat of neat PF is
	0.06 1.848 J.g -1 .K -1 , and decreases with the adding of RGO due to the smaller heat capacity of RGO
	0.04 (around 720 J.kg-1.K-1) therefore the minimum value of specific heat is observed for 0.15 wt%
	0.02
		.00	0.02	0.04	0.06	0.08	0.10

RGO. Here, the reinforcement of RGO causes a small change in the heat capacity of PF/RGO foams. The thermophysical measurements of PF/RGO foams with varying wt% RGO are summarized in Table

2

.

Table 2 .

 2 Thermophysical properties of PF/RGO foams (at 25°C) Each density value from Table2is the average of four measurements on four different samples showing relative standard deviations less than 9% which are smaller than the discrepancies shown in the Table

	Sample	Thermal conductivity λ (W.m -1 .K -1 )	Heat capacity Cp (J.g -1 .K -1 )	Density (kg.m -3 )
	0% RGO	0.089	1.848	58.80
	0.08% RGO	0.107	1.503	84.90
	0.12% RGO	0.108	1.420	60.00
	0.15% RGO	0.113	1.387	70.00

Acknowledgements

The authors are grateful to the financial support under DST-FIST (No. 487/DST/FIST/15-16) New Delhi to Sree Sankara College, Kalady and Mahatma Gandhi University, Kottayam, India for the award of Junior Research Fellowship for Sandhya P K.