C Bouget

A Brin

L Valladares
email: lionel.valladares@esa-purpan.fr

Evaluation of window flight traps for effectiveness at monitoring dead wood associated beetles: the effect of ethanol lure under contrasting environmental conditions

Keywords: optimization, sampling methods, efficiency, monitoring, alcohol bait 26 27

1. Since the species-rich group of saproxylic beetles has been proposed to be used as a factor in forest management, more explicit knowledge about the efficiency and selective properties of beetle sampling methods is needed.

2. We compared saproxylic beetle assemblages caught by alcohol-baited or unbaited window traps in different forest contexts. Considering that trap attractiveness depends on kairomone concentrations, we appraised whether the trap efficiency was influenced by trap environment (openness and local supply of fresh dead wood). 3. Saproxylic beetles were sampled using 48 cross-vane window flight traps, arranged in paired designs (alcohol-baited/unbaited), in 8 ancient and 8 recent gaps (open stands), and 8 closed-canopy control stands in an upland beech forest in the French Pyrenees.

4. Baited traps were more efficient than unbaited devices in terms of abundance and richness in our deciduous forests. The ethanol lure did not have any repellent effect on the individual response of saproxylic taxa.

The influence of local environmental conditions on trap attractiveness was

observed. The effects of variations in openness were actually moderate, whereas trap attractiveness appeared to be reduced in the alcohol-saturated environment of recent gaps due to a disruption by local fresh dead-wood concentrations of the kairomonal response of saproxylic beetles to baited traps ('alcohol disruption').

6. Since the ethanol lure globally enhances species detection probability (no repellent effect, many individual attractive effects), it may be extensively used in programs of early-warning surveillance, monitoring and control of wood borers. We recommend to account for the slight influences of local conditions on baited trap efficiency while using them for beetle biodiversity monitoring.

Introduction

Forest management practices currently include many nature conservation measures in favour of fauna and flora. Nonetheless, in managed forest ecosystems, the dead wood component has been severely reduced by intensive forestry and still is underrepresented [START_REF] Siitonen | Forest management, coarse woody debris and saproxylic organisms : Fennoscandian boreal forests as an example[END_REF].

Since considerable effort is devoted to preserving dead wood and the associated fungi and fauna involved in the wood decaying process, we need an explicit ecological assessment of the performance of forestry measures designed to increase deadwood.

Saproxylic beetles have been widely studied over the past 20 years, especially in northern and central Europe, and have been proposed as indicator species of forest integrity [START_REF] Speight | Les invertébrés saproxyliques et leur protection[END_REF][START_REF] Nilsson | Biodiversity and its assessment in boreal and nemoral forests[END_REF] because they are highly dependent on dead wood and have been particularly affected by forest management practices (e.g. Siitonen, 2001 and references therein). They make up one of the largest groups of red-listed species, and they represent many different functional groups [START_REF] Siitonen | Forest management, coarse woody debris and saproxylic organisms : Fennoscandian boreal forests as an example[END_REF].

As saproxylic beetles are diverse, and mainly small and cryptic, their study is very challenging. If we want to use saproxylic beetles as a management tool in forestry (evaluation of practices, biodiversity monitoring schemes), we need much more explicit knowledge about the efficiency and selective properties of sampling methods.

Several different methods are generally used to collect saproxylic beetles. These are (i) direct active hand-collecting techniques, including peeling, sifting the bark of dead trees and beating dead wood, (ii) rearing techniques (log emergence traps, Owen traps, eclectors, extraction cylinders) and (iii) trapping methods such as interception or attractive (coloured, silhouette, chemo-attractive) traps [START_REF] Leather | Insect Sampling in Forest Ecosystems[END_REF]. Window-flight trapping (WFT), also called Flight-Intercept Trapping (FIT), window/ barrier trapping or collision trapping), developed by Chapman & Kinghorn Peck & Davies, 1980), is currently the most frequently used technique for catching active flying saproxylic beetles [START_REF] Økland | A comparison of three methods of trapping saproxylic beetles[END_REF][START_REF] Wikars | A comparison of three methods to estimate species richness of saproxylic beetles (Coleoptera) in logs and high stumps of Norway spruce[END_REF][START_REF] Alinvi | Sampling the saproxylic beetle community in dead wood logs: comparing window and eclector traps to traditional bark sieving and a refinement[END_REF]. Window flight traps consist of a vertical barrier to insect flight that is considered invisible to the insect. On hitting the barrier, most beetles drop down and fall into a collection container with liquid preservatives.

Although interception traps do not give accurate information about the local habitat, they have proved to be satisfactory in many respects. Compared with extraction methods they offer reliable means of replication and standardization, and give a representative picture of saproxylic beetle fauna [START_REF] Siitonen | Decaying wood and saproxylic Coleoptera in two old spruce forests: a comparison based on two sampling methods[END_REF].

From the original simple model of flight-interception trap, modifications have been proposed concerning the dimensions, colour, shape, bait… For instance, windowflight traps may be fitted with an attractant dispenser to increase catches of dead wood associated beetles [START_REF] Brustel | Polytrap™" a window flight trap for saproxylic beetles[END_REF].

Dead wood associated species have evolved behavioral responses to volatile hostplant chemicals that indicate the presence of a suitable host (i.e. kairomones). It is well known that ethanol, a volatile compound released by micro-organisms in decaying woody tissue (mainly in fresh tissues ; [START_REF] Cade | Gnathotrichus sulcatus (Coleoptera: Scolytidae): identification of a primary attractant[END_REF] and stressed plants [START_REF] Kimmerer | Ethylene, ethane, acetaldehyde and ethanol production by plants under stress[END_REF], acts as a foraging kairomone (used in the context of food location ; [START_REF] Döring | Zur Biologie des grossen Eichenbockkäfers (Cerambyx cerdo L.) unter besonderer Berücksichtigung der Populationsbewegungen im Areal[END_REF][START_REF] Ruther | Rich in phenomena-lacking in terms. A classification of kairomones[END_REF]. It is therefore attractive to a wide variety of species of dead wood associated beetles [START_REF] Montgomery | Ethanol and other host-derived volatiles as attractants to beetles that bore into hardwoods[END_REF][START_REF] Lindelöw | Attraction during flight of scolytids and other bark-and wood-dwelling beetles to volatiles from fresh and stored spruce wood[END_REF], including early- [START_REF] Nordlander | Orientation of the pine weevil Hylobius abietis to underground sources of host volatiles[END_REF][START_REF] Byers | Chemical ecology of bark beetles[END_REF] and late-successional saproxylic species [START_REF] Jonsell | The saproxylic beetle Bolitophagus reticulatus: its frequency in managed forests, attraction to volatiles and flight period[END_REF], and beetles associated with dead wood from deciduous [START_REF] Roling | Seasonal flight and vertical distribution of Scolytidae attracted to ethanol in an oak-hickory forest in Missouri[END_REF] or conifer trees [START_REF] Magema | Efficacité de l'éthanol dans le piégeage du scolyte Trypodendron lineatum (Olivier, 1795)(Coleoptera, Scolytidae) et rôle des constituants terpéniques de l'épicéa[END_REF][START_REF] Chénier | Field responses of certain forest Coleoptera to conifer monoterpenes and ethanol[END_REF].

Previous studies have already demonstrated the general effects of ethanol as a lure in attractive window flight traps [START_REF] Berti | A model of ethanol trap to collect Scolytidae and Platypodidae (Insecta, Coleoptera)[END_REF][START_REF] Shibata | Cerambycid beetles (Coleoptera) lured to chemicals in forests of Nara prefecture, Central Japan[END_REF]. In this study comparing two sampling techniques in paired designs, we first intended to confirm and detail the differences in catches between alcohol-baited and unbaited traps under field conditions.

Secondly, we wanted to assess whether the catches of dead wood associated beetles in alcohol-baited or unbaited traps are influenced by local environmental conditions. Indeed, we need standardized unbiased methods to study the effects of forest management practices and compare open (e.g. felling areas) and closed-canopy stands; ideally saproxylic beetle detectability should not change with habitat structure. We thus asked the two following questions.

(i) Did the difference between the two methods (alcohol-baited and unbaited traps)

remain constant in open gaps or closed-canopy stands? From [START_REF] Schroeder | Attraction of scolytids and associated beetles by different amounts and proportions of alpha pinene and ethanol[END_REF], release rates of chemicals are known to affect the attraction. We may assume that kairomone volatility, and therefore trap attractiveness, increases with stand openness. In this perspective, the difference between baited and unbaited traps would be expected to be higher in open areas than in closed-canopy stands. This bias variation between stands would call into question the between-stand comparison using alcohol-baited traps.

(ii) Did the difference between the two methods remain constant whether or not the close trap surroundings were poor or rich in naturally emitted ethanol from fresh or decaying dead wood?

We may assume that trap attractiveness is influenced by the alcohol concentration in the atmosphere surrounding the trap. In recent sun-exposed gaps, the high volumes of fresh dead wood release a large amount of ethanol, which may lead to a saturated olfactory landscape and therefore to a reduction in the capture efficiency of the trap attractant. Due to an alcohol disruption, we hypothesize that the differences between paired baited and unbaited traps would be lower in recent gaps than in both old gaps and forest controls where the local atmosphere is not alcohol-saturated.

Material and methods

Study areas and sampling designs

Our investigations were carried out in an upland beech forest in the central French Pyrenees, in the National Orlu Reserve, about 150 km south of Toulouse. Altitude ranged from 1600 m to 1750 m. The selected plots were 100 to 140 year-old beech stands, and had previously been exploited for charcoal by coppicing.

The balanced sampling design included three types of stands: 8 ancient gaps, 8 recent gaps (open stands) and 8 forest controls (closed-canopy stands). The 16 gaps were originally dedicated to the monitoring of conservation measures in favour of the Western capercaillie (Tetrao urogallus, L. 1758). Ancient and recent gaps were created in 1999 and 2004 respectively; they were circular in shape and ca 0.05 ha in area. Distance between gaps varied from 80 m to 1000 m. Cut trees were left in all gaps but only recent gaps had high amounts of recent (freshly-cut) dead wood. In parallel, a transect of 8 forest control plots was set up in closed-canopy beech stands, where the cover was quite dense (basal area = 23 m²/ha, stem density = 300 stems/ha, mean dbh = 25cm, mean height=15m) and the understory very poor.

Beetle sampling

Beetles were sampled with a multidirectional Polytrap TM [START_REF] Brustel | Polytrap™" a window flight trap for saproxylic beetles[END_REF], a crossvane window flight trap with a cumulative panel area of 1 m² ((Fig. 1). For the preservation of insects, containers were half-filled with a salt mixture. A detergent was added to reduce surface tension. Two traps were set in each plot, one baited with an alcohol lure (1:10 ethanol released from a dispenser) and one unbaited trap.

Alcohol lure is known to attract hardwood-feeding species (Mongomery & Wargo, 1983).

Overall, 24 pairs of Polytraps TM were set up in 2004, from May 15 to August 30 in recent and ancient gaps and closed-canopy beech stands. The traps were emptied every two weeks.

Collected specimens were stored in alcohol and we identified them to species level.

Several amateur entomologists helped with the identification on the following families: Anobiidae, Curculionidae, Dasytidae, Melyridae, Mordellidae, Pselaphidae, Scolytinae, Scraptiidae, Staphylinidae. The nomenclature follows the online FaunaEuropaea database (www.faunaeur.org). We pooled the data for the entire sampling period.

Data processing

The two sampling methods were compared in terms of total saproxylic beetle abundance, cumulative species number per trap, family and species abundance per trap (cumulated over the trapping periods).

Fig. 1

Linear mixed-model ANOVA tests [START_REF] Pinheiro | Mixed-effects models in S and S-PLUS[END_REF] were carried out to check for differences in abundance and species richness between the two trap types in each paired comparison. The plot variable was classified as a spatial random effect to take the pairing pattern of the sampling design into account. Some differences were exceptionally tested by Wilcoxon signed-rank tests. Species and families represented by fewer than 30 individuals were not tested.

The effects of the interaction between bait and two factors on total abundance and abundance of alcohol-attracted taxa (in our data), species richness were assessed. The three stand types of the study were clustered using two different grouping factors. Non-Metric Multidimensional Scaling (NMDS) based on the Bray-Curtis dissimilarity was used for pattern recognition in species composition and the ANOSIM procedure was carried out to test for differences in assemblage composition between trap types [START_REF] Clarke | Non-parametric multivariate analyses of changes in community structure[END_REF]. Before calculation of the Bray-Curtis distance matrix, species represented by only 1 individual were discarded and abundance data were (ln+1) transformed.

ANOVA, NMDS and ANOSIM were performed using S.Plus 7.0.

RESULTS

When baited with alcohol, window flight traps were more selective: the ratio (number of saproxylic beetles/number of beetles) reached 85.8% in baited traps but was only 69.4% in unbaited traps. A total of 12,211 saproxylic beetles were identified during the study, including 196 species.

Comparison of effectiveness between alcohol-baited and unbaited traps

Alcohol-baited or unbaited traps significantly differed in effectiveness. The catches of saproxylic beetles were significantly affected by the alcohol lure (mixed-model ANOVA F (1,23) = 66.48, p<0.0001; Table 1). Overall, more than twice as many saproxylic beetle individuals were captured in alcohol-baited traps as in unbaited traps (Table 1).

Similarly, significantly more saproxylic species were caught in alcohol-lured traps than in unbaited (ANOVA F (1,23) = 68.65, p<.0001 ; Table 1). The richness per trap was 40% higher in baited traps than in unbaited traps (Table 1). From the ordination plot of the Bray-Curtis distance matrix between samples (Fig. 2), the species composition of saproxylic beetle assemblages caught with or without an alcohol lure clearly differed. Samples from baited or unbaited traps were obviously separated in ordination space. This distinction was consistent with the results of the ANOSIM test (10 000 permutations, R=0.65, p<0.0001). The average Bray-Curtis dissimilarity between baited and unbaited traps reached a value of 44%.

Tab. 1 Fig. 2 In accordance with this global dissimilarity, important specific differences were noticed. For several families and species (Cryptophagidae (Cryptophagus spp.), Staphylinidae (Paraphloeostiba gayndahensis, Aleochara sparsa), Salpingidae (Rhinosimus ruficollis, R. planirostris, Rabocerus foveolatus)), significantly more individuals were caught in baited traps than in unbaited traps (mixed-model ANOVA; Table 2). Rhizophagus spp. (Monotomidae) was more abundant in baited traps compared with unbaited traps, whereas Cleridae were not affected by trap bait.

Amongst xylophagous taxa, Scolytidae (Ernoporicus fagi, Xyloterus signatus, X. domesticus, Xyleborus saxesenii) and Lymexylidae (Hylecoetus dermestoides) were more abundantly caught in baited traps, whereas Cerambycidae, Anobiidae, Melyridae were not sensitive to alcohol bait.

No species or family was significantly more abundant in unbaited traps.

Comparison of effectiveness between alcohol-baited and unbaited traps in open or closed-canopy stands

The difference between alcohol-baited and unbaited traps in open (gaps) or closedcanopy stands may be interpreted first in terms of abundance or species richness. As shown in figure 3, saproxylic beetle abundance in both baited and unbaited traps was higher in open sites than in closed-canopy controls. The difference in abundance between baited and unbaited devices was only slightly higher in open plots; in other words, the bait-openness interaction effect was quite moderate (mixed-model ANOVA test F 2,21 = 5.17, p=0.015).

The difference in species richness between baited and unbaited traps was not significantly influenced by the openness of trap environment (Fig. 3: mixed-model ANOVA test of the bait-openness interaction effect: F 2,21 = 3.23, p=0.060).

Tab. 2

Fig.

For several taxa noticed to be sensitive to the alcohol attractant (see above), i.e.

Monotomidae, Scolytidae and Lymexylidae as a whole, the difference baitedunbaited was not affected by the environment closure. Two families (Cryptophagidae, i.e. Cryptophagus spp., and Salpingidae), and 4 species (Rabocerus foveolatus, Rhinosimus ruficollis, Rhinosimus planirostris, Aleochara sparsa) showed a significant difference baited-unbaited, but not in compliance with the hypothesis of a bait-openness interaction. The difference was indeed lower in open stands than in closed-canopy plots (fig. 4). Only the bark beetle species Xyloterus domesticus showed a significant and higher difference baited-unbaited in open stands.

In conclusion, the difference between alcohol-baited and unbaited traps was not increased greatly by the openness of trap environment.

Comparison of effectiveness between alcohol-baited and unbaited traps in fresh dead wood poor or rich stands

Regarding this potential bait-environment interaction effect, the difference between alcohol-baited and unbaited traps was only slightly influenced by the local volume of fresh woody debris releasing alcohol kairomones. The baited-unbaited difference in fresh dead wood poor (ancient gaps and controls) or rich stands (recent gaps) was examined in terms of total abundance, species richness, family and species abundances.

In figure 5, the difference between baited and unbaited lures in terms of abundance and species richness seems to be slightly lower in gaps rich in recent fresh dead wood. Nevertheless, this bait-environment interaction effect was only slightly Fig. 4 Fig. 5 significant on saproxylic beetle abundance (mixed-model ANOVA, F 2,35 = 4.84, p = 0.014). The only significant pairwise difference was observed between ancient and recent gaps (Multiple comparison Tukey test), the difference being larger in ancient gaps. This bait-environment interaction did not significantly affect species richness in traps (mixed-model ANOVA, F 2,35 = 1.66, p = 0.204).

A bait-environment interaction effect was observed on the abundance of several families and species known to be sensitive to the ethanol attractant (fig. 6). The baited-unbaited differences were lower in recent gaps than in ancient gaps and forest controls for the abundance of Salpingidae (and the species Rhinosimus ruficollis in particular) (multiple comparison Tukey test). Similarly, we observed a smaller yet still significant baited-unbaited difference for the abundance of Cryptophagidae and Rhinosimus planirostris in recent gaps compared with closed-canopy stands. For Rabocerus foveolatus, Aleochara sparsa, Xyloterus signatus, the difference was less in recent gaps than in closed-canopy controls (but the difference between ancient or recent gaps was not significant). No difference was measured for Scolytidae as a whole (and Xyloterus domesticus, Ernoporicus caucasicus, Xyleborus saxesenii in particular), Lymexylonidae, Staphylinidae (peculiarly Paraphloeostiba gayndahense) and Monotomidae (Rhizophagus spp. in particular).

Discussion

Alcohol lure effect

The results from our study confirm that ethyl alcohol, acting as a kairomone released from decaying woody tissues, strongly attracts some beetles associated with dead hardwood [START_REF] Montgomery | Ethanol and other host-derived volatiles as attractants to beetles that bore into hardwoods[END_REF]) and suggests that an ethanol lure Fig. 6 strengthens the effectiveness of window-flight traps in deciduous forests. Indeed, more than twice as many saproxylic beetles, representing a 40% higher species richness, were captured in alcohol-baited traps as in unbaited traps. Assemblages were globally dissimilar between baited and unbaited devices, mainly because more species were caught in baited traps. It is very important to stress that we did not detect any species or family significantly more abundant in unbaited traps. We may therefore infer that the ethanol lure did not have any inhibiting or repellent effect.

In our methodological approach in beech forests, the individual responses of deadwood taxa to alcohol lure were actually heterogeneous. As previously demonstrated by [START_REF] Byers | Attraction of bark beetles, Tomicus piniperda, Hylurgops palliatus, and Trypodendron domesticum and other insects to short chain alcohols and monoterpenes[END_REF], significantly more individuals of Rhizophagus spp. (Monotomidae) were caught in baited traps than in unbaited traps. Host tree chemicals are known to be important to Rhizophagus species in finding their habitat (scolytid galleries).

Conversely, natural enemies such as Cleridae were not affected by trap bait in our study. Likewise, Schroeder (2003) pointed out that two Thanasimus clerid species preying on several bark beetle species differ in their response to α-pinene and ethanol. Our results showed that the abundance of ambrosia and bark beetles was about 4 times as high in alcohol-baited traps, in agreement with previous findings [START_REF] Magema | Efficacité de l'éthanol dans le piégeage du scolyte Trypodendron lineatum (Olivier, 1795)(Coleoptera, Scolytidae) et rôle des constituants terpéniques de l'épicéa[END_REF][START_REF] Berti | A model of ethanol trap to collect Scolytidae and Platypodidae (Insecta, Coleoptera)[END_REF][START_REF] Poland | Evaluation of three trap types and five lures for monitoring Hylurgus ligniperda (Coleoptera: Scolytidae) and other local scolytids in New York[END_REF]). Amongst the other wood-feeding taxa, Cerambycidae were not sensitive to alcohol bait, which had been shown previously by [START_REF] Sweeney | Host volatile attractants and traps for detection of Tetropium fuscum (F.), Tetropium castaneum L., and other longhorned Beetles (Coleoptera : Cerambycidae)[END_REF]. Conversely, [START_REF] Döring | Zur Biologie des grossen Eichenbockkäfers (Cerambyx cerdo L.) unter besonderer Berücksichtigung der Populationsbewegungen im Areal[END_REF] demonstrated that ethanol acts as a kairomone for the longhorned beetle Cerambyx cerdo. Among the fungus feeders, unlike Tetratomidae, Melandryidae and Ciidae, only Cryptophagidae were significantly attracted by the alcohol lure, as was the tenebrionid studied by [START_REF] Jonsell | The saproxylic beetle Bolitophagus reticulatus: its frequency in managed forests, attraction to volatiles and flight period[END_REF].

Interaction effects between trap attractant and trap environment Some results did not comply with the hypothesized bait-openness interaction. Five species showed a significant lower baited-unbaited difference in open areas.

Moreover, the baited-unbaited difference in species richness and in abundance of the other alcohol-attracted taxa was not significantly influenced by the openness of trap environment. Only the total saproxylic beetle abundance and the abundance of the bark beetle Xyloterus domesticus were in accordance with the hypothesis of an increased baited-unbaited difference in open areas compared with closed-canopy stands. In other words, trap attractiveness appeared to increase only slightly, but not significantly, with stand openness, even though canopy cover may strongly influence kairomone volatility. However, such an interaction effect was reported by [START_REF] Honek | The effect of crop density and microclimate on pitfall trap catches of Carabidae, Staphylinidae (Coleoptera) and Lycosidae (Araneae) in cereal fields[END_REF], between crop density and an attractant in pitfall traps, on catches of epigaeic arthropods.

Apart from openness, another environmental factor separated the plots into groups. Indeed, recent gaps, ancient gaps and forest controls differed in terms of local volume of fresh dead wood. We hypothesized that the high volume of fresh dead wood in recent sun-exposed gaps would release a large amount of ethanol, which could saturate the beetle's olfactory landscape around the traps and therefore reduce trap attractiveness. In accordance with this hypothesis, the baited-unbaited difference was lower in recent and fresh dead wood rich gaps for some assemblage parameters.

Though this bait-environment interaction was slight, it significantly affected the total abundance of saproxylic beetles, the abundance of several families and species known to be sensitive to the ethanol attractant (Salpingidae, Rhinosimus ruficollis and Rhinosimus planirostris in particular, Cryptophagidae). Conversely, the response of species richness and abundance of the other families (such as Scolytidae) to traps was not influenced by the local volume of fresh dead wood.

The alternative hypothesis of a "mass effect", i.e. a stronger attractiveness of baited traps in the alcohol-rich atmosphere of recent gaps, was thus invalidated. These results may be related to a phenomenon that could be called 'alcohol disruption' (in the spirit of the 'mating disruption'; [START_REF] Ruther | Rich in phenomena-lacking in terms. A classification of kairomones[END_REF], i.e. the disruption by local fresh dead-wood concentrations of the kairomonal response of saproxylic beetles to baited traps. Nonetheless, even though trap attractiveness appeared to be reduced in an alcohol-saturated environment, baited traps remain more efficient than unbaited devices in terms of abundances and richness. They have the capacity to sample the pool of saproxylic beetles in recent gaps.

Conclusion -Implications for saproxylic beetle sampling

Even though the extrapolation of our findings from upland forests to other climatic contexts should be done with care, their interpretation may help with the optimization of the two types of forest beetle monitoring.

The first type, i.e. monitoring and control of forest pest populations and earlywarning surveillance programs to detect invasive wood borers, aim at maximizing detection probability. We demonstrated that the ethanol lure does not have any repellent effect but significantly attracts several wood feeders. Like the New Zealand monitoring programme for invasive wood-feeders (Brockerhoff et al., 2006), effective sampling designs may use the large spectrum of ethanol attractiveness.

The second type, i.e. biodiversity surveys of saproxylic beetle diversity in monitoring networks or evaluation studies of forest management practices, implies distinct constraints. To compare the biodiversity between stands or to assess trends in saproxylic beetle diversity over broader landscapes, ideally saproxylic beetle detectability should not change with habitat structure. A variation in trap efficiency as a function of any stand environmental parameter would call into question an intersite comparison. For instance, a relation between trap efficacy and stand openness would prevent the comparison of open (e.g. felling areas) and closed-canopy stands with alcohol-baited traps, or studies on the influence of shading for the attractiveness of dead wood (Hjältèn et al., 2007). Our results show slight interaction effects between trap attractant and trap environment. A very moderate bait-openness interaction effect was measured, whereas trap attractiveness appeared to be reduced in an alcohol-saturated environment, through a disruption by local fresh dead-wood concentrations of the kairomonal response of saproxylic beetles to baited traps ('alcohol disruption').

In accordance with these results and although alcohol-baited traps appeared to be more efficient than unbaited devices, we recommend to take care to use baited brine or glycol solution used as a preservative fluid does not vary with forest stands.

Beside the attractant, other trap parameters, such as colour and shape, should be assessed for optimization and standardization and have been partly examined in another study (Bouget et al., in press).

  First, to study the influence of stand openness on trap attractiveness, we compared the difference between paired baited and unbaited traps in open (recent and ancient gaps) or closed-canopy stands. We analysed this difference through the interaction effect in linear mixed-model ANOVAs (bait*openness). Second, to study the influence of local fresh dead wood concentrations on trap attractiveness, we examined the difference (baited -unbaited) in fresh dead-wood rich (recent gaps) or poor (ancient gaps and forest controls) stands. The F-test of the interaction effect (bait*stand type) in linear mixed-model ANOVAs was followed by a multiple comparison post hoc Tukey test of the mean value of this difference.Graphs include multipanel boxplots displaying the distribution of data according to the two factors bait*environment, or simple boxplots showing the numerical difference (baited-unbaited) for different environment types.

Figure 1

 1 Figure 1

Figure 3

 3 Figure3

Figure 4

 4 Figure 4

Figure 5

 5 Figure5

Figure 6

 6 Figure 6

  FiguresFigure1

  Differences between baited (AL) and unbaited (NAL) traps, in terms of saproxylic beetle abundance, species richness. Test significance: ** p<0.01, * 0.01<p<0.05, ns p>0.05. Differences were tested by linear mixed-model ANOVA F-tests. Mean values are expressed +/-Standard Deviation.

	Tables
	Table 1
	Taxa	Alcohol-baited traps	Unbaited traps
	Abundance	331,417 +/-156,330	143,750 +/-55,155	F (1,23) =66,482 **
	Species richness	43,125 +/-9,768	31,458 +/-8,827	F (1,23) =68,649 **

Table 2

 2 Mean abundance per trap of saproxylic beetle species and families in baited (AL) and unbaited (NAL) traps. Linear mixed-model ANOVA test significance: ** p<0.01, * 0.01<p<0.05, ns p>0.05. Mean values are expressed +/-Standard Deviation.

	Taxa	Alcohol-baited traps Unbaited traps	F (1,23)	p
	LEIODIDAE	1,125 +/-1,424	1,625 +/-2,123	F=1,274

Acknowledgements

We would like to thank the French National Forest Service (ONF) for funding a part of the analytical work. Victoria Moore and 3 anonymous reviewers deeply improved the first drafts of the manuscript. Many thanks to Hélène Barbareau for technical assistance. We are also grateful to several amateur entomologists who helped with beetle identification. This research was mainly funded by the European Regional Development Fund (ERDF) and the French National Hunting and Wildlife Office (ONCFS).

ns STAPHYLINIDAE 49,458 +/-28,290 34,667 +/-19,325 F=15,243 ** Aleochara sparsa 1,833 +/-1,711 0,167 +/-0,637 F=27,688 ** Eusphalerum angustum 29,542 +/-26,132 30,500 +/-19,638 F=1,908 ns Paraphloeostiba gayndahense 9,958 +/-8,175 0,375 +/-0,576 F=198,854 ** Leptusa pulchella 0,958 +/-1,301 0,500 +/-0,780 F=1,929 ns LUCANIDAE 0,625 +/-0,824 0,833 +/-1,007 F=0,512 ns Sinodendron cylindricum 0,542 +/-0,658 0,750 +/-0,847 F=0,682 ns EUCNEMIDAE 1,250 +/-2,345 0,667 +/-1,049 F=0,414 ns Melasis buprestoides 1,250 +/-2,345 0,667 +/-1,049 F=0,467 ns ELATERIDAE 1,667 +/-1,903 2,292 +/-1,944 F=1,422 ns Idolus picipennis 0,917 +/-1,501 1,375 +/-1,610 F=2,293 ns ANOBIIDAE 26,500 +/-22,043 24,042 +/-22,160 F=0,141 ns Grynobius planus 0,917 +/-1,558 1,167 +/-2,160 F=0,012 ns Hemicoelus costatus 7,292 +/-8,961 7,250 +/-7,731 F=0,000 ns Ptilinus pectinicornis 16,500 +/-20,061 12,458 +/-15,704 F=0,280 ns Ptinomorphus imperialis 0,458 +/-0,932 0,958 +/-3,458 F=0,043 ns Xestobium plumbeum 1,333 +/-1,494 2,083 +/-5,579 F=0,209 ns LYMEXYLIDAE 6,333 +/-10,639 0,375 +/-1,279 F=38,526 ** Hylecoetus dermestoides 6,333 +/-10,639 0,375 +/-1,279 F=41,528 ** CLERIDAE 0,917 +/-1,283 1,042 +/-1,459 F=0,053 ns Tillus elongatus 0,917 +/-1,283 1,042 +/-1,459 F=0,125 ns MELYRIDAE 12,792 +/-12,646 8,875 +/-8,295 F=0,617 ns MALACHIIDAE 2,000 +/-2,537 3,292 +/-3,196 F=3,651 ns MONOTOMIDAE 2,417 +/-2,263 0,250 +/-0,442 F=60,460 ** Rhizophagus spp. 2,375 +/-2,281 0,167 +/-0,381 F=57,873 ** CRYPTOPHAGIDAE 14,625 +/-14,984 5,917 +/-5,823 F=17,544 ** Cryptophagus spp. 14,125 +/-15,103 5,833 +/-5,880 F=14,091 ** LATRIDIIDAE 8,625 +/-6,851 6,000 +/-5,703 F=3,009 ns MYCETOPHAGIDAE 1,750 +/-1,539 1,042 +/-1,083 F=2,757 ns CIIDAE 5,667 +/-5,346 5,208 +/-5,073 F=0,143 ns TETRATOMIDAE 3,583 +/-3,202 2,625 +/-3,609 F=4,037 ns Tetratoma ancora 3,583 +/-3,202 2,625 +/-3,609 F=4,037 ns MELANDRYIDAE 2,250 +/-2,592 1,542 +/-1,615 F=1,850 ns Melandrya caraboides 0,958 +/-1,398 0,792 +/-1,285 F=0,456 ns MORDELLIDAE 1,333 +/-3,199 1,083 +/-1,909 F=0,337 ns Tomoxia bucephala 0,917 +/-2,586 0,917 +/-1,886 F=0,316 ns SALPINGIDAE 89,417 +/-44,071 3,417 +/-2,165 F=162,552 ** Rabocerus foveolatus 4,750 +/-3,650 0,208 +/-0,588 F=82,129 ** Rhinosimus planirostris 53,167 +/-25,426 1,583 +/-1,248 F=486,084 ** Rhinosimus ruficollis 31,458 +/-18,967 1,583 +/-1,472 F=201,812 ** SCRAPTIIDAE 2,292 +/-2,510 2,458 +/-1,719 F=1,540 ns Anaspis rufilabris 1,417 +/-1,213 1,792 +/-1,587 F=0,428 ns CERAMBYCIDAE 10,375 +/-7,471 9,792 +/-7,396 F=0,042 ns Clytus arietis 1,625 +/-2,143 1,042 +/-1,654 F=2,878 ns Oxymirus cursor 1,625 +/-2,018 1,542 +/-1,250 F=0,065 ns Rhagium bifasciatum 3,542 +/-3,176 3,958 +/-3,862 F=0,007 ns Rhagium mordax 1,417 +/-1,349 1,667 +/-2,160 F=0,005 ns SCOLYTIDAE 81,417 +/-127,180 22,583 +/-21,040 F=23,698 ** Ernoporicus caucasicus 18,667 +/-37,632 4,417 +/-4,452 F=10,469 ** Taphrorychus bicolor 53,083 +/-125,971 17,208 +/-20,121 F=0,586 ns Xyleborus saxesenii 4,792 +/-3,683 0,042 +/-0,204 F=107,812 ** Xyloterus domesticus 2,000 +/-2,284 0,125 +/-0,448 F=22,674 ** Xyloterus signatus 1,792 +/-2,021 0,000 +/-0,000 F=32,427 **


Daniel Han-Kwan
email: hankwan@dma.ens.fr

The three-dimensional finite Larmor radius approximation

Keywords:

Following Frénod and Sonnendrücker ([12]), we consider the finite Larmor radius regime for a plasma submitted to a large magnetic field and take into account both the quasineutrality and the local thermodynamic equilibrium of the electrons. We then rigorously establish the asymptotic gyrokinetic limit of the rescaled and modified Vlasov-Poisson system in a three-dimensional setting with the help of an averaging lemma.

1 Introduction and main results

Physical motivation

We are interested in the behaviour of a plasma (id est a gaz made of ions with individual charge Ze and mass m i and electrons with individual charge -e and mass m e , with m i >> m e ) which is submitted to a large external magnetic field. It is "well-known" that such a field induces fast small oscillations for the particles and consequently introduces a new small time scale which is very restrictive and inconvenient from the numerical point of view. The simulation of such plasmas appears to be primordial since the model can be applied to tokamak plasmas from magnetic confinement fusion (like for the ITER project).

Heuristic study

Let us give some heuristic formal arguments to investigate the behaviour of the plasma: if we consider the motion of one particle (of charge q > 0, mass m, position x and velocity v) submitted to an external constant field B, the fundamental principle of mechanics gives that:

dx dt = v, dv dt = q m (v ∧ B) (1.1)
Straightforward calculations show first of all that the parallel velocity, denoted by v (that is to say the component of the velocity in the direction of the magnetic field) is conserved and thanks to the conservation of the kinetic energy, so is the norm of the perpendicular velocity v ⊥ (the component of the velocity in the perpendicular plane). Actually, we can see that the particle moves on a helix whose axis is the direction of the magnetic field.

The rotation period (around the axis) is the inverse of the cyclotron frequency Ω:

Ω = |q||B| m (1.2)
and the radius is the so-called Larmor radius:

r L = |v ⊥ | Ω (1.3)
In the case where the magnetic field is very strong, Ω tends to infinity whereas r L tends to zero. More precisely, if we take |B| ∼ 1 ǫ (with ǫ → 0) we have:

Ω ∼ 1 ǫ r L ∼ ǫ
The approximation which consists in considering r L = 0 is the classical guiding center approximation ( [START_REF] Grad | The guiding center plasma[END_REF]). This means that each particle is assimilated to its "guiding center" (in other words its "instantaneous rotation center"), which is equivalent to neglect the very fast rotation of the particle around the axis.

If one also applies some external constant electric field E, a similar computation shows that there appears:

1. an acceleration E.B |B| in the direction of B. If we consider E ∼ 1, then:

E.B |B| ∼ 1 (1.4)
2. a drift E∧B |B| 2 in the orthogonal plane. We have:

E ∧ B |B| 2 ∼ ǫ (1.5)
This drift, usually called the electric drift is problematic as regards to the issue of plasma confinement. It is negligible compared to the acceleration in the direction of B, but in the time scale for plasma fusion which is expected to be very long, one can not neglect this small drift, since it creates a displacement of order ǫt (t represents the time). At last, note also that if the fields are not constant, various other drifts may appear, whose order in ǫ is higher than those of the electric drift.

Actually, the fields considered are neither constant, nor external, but self-induced by the plasma itself. The effects we would like to describe are due to the non-linear interaction between the particles and the electromagnetic field.

The mathematical model

In all the sequel, we assume that the magnetic field is external and constant and we suppose that the speed of particles is small compared to the speed of light, so that we can use the electrostatic approximation which consists in reducing the Maxwell equations to the Poisson equation. Finally, we decide to opt for a kinetic description for the ions: in other words, the time and space scales considered here are such that ions are not at a thermodynamic equilibrium and their density is governed by a kinetic equation.

The basic model usually considered for the ions is the following Vlasov-Poisson system:

       ∂ t f + v.∇ x f + (E + v ∧ B).∇ v f = 0 E = -∇ x V -∆ x V = f dv f t=0 = f 0 where f (t, x, v) is the density of ions, with t ∈ R + , x ∈ R d or R d /Z d , v ∈ R d (usually d = 2 or 3), meaning that f (t, x, v)dxdv gives the number of ions in the infinitesimal volume [x, x + dx] × [v, v + dv]
at time t (note that in this model, electrons are for the moment neglected).

The gyrokinetic approximation

It is important from a numerical point of view to establish the asymptotic equation when |B| tends to infinity. Indeed, we expect the asymptotic equation to be "easier" to handle: only one time and space scale, perhaps less variables in the phase space to deal with... The derivation of such equations is usually referred to in the mathematic literature as "gyrokinetic approximation".

Rigorous justifications of these derivations with various time and space observation scales have only appeared at the end of the nineties. We refer for instance to the works of Brenier ([5]), Frénod and Sonnendrücker ([11]- [START_REF] Frénod | The Finite Larmor Radius Approximation[END_REF]), Frénod, Raviart and Sonnendrücker ( [START_REF] Frénod | Asymptotic expansion of the Vlasov equation in a large external magnetic field[END_REF]), Golse and Saint-Raymond ( [START_REF] Golse | The Vlasov-Poisson system with strong magnetic field[END_REF]- [START_REF] Golse | The Vlasov-Poisson system with strong magnetic field in quasineutral regime[END_REF]), Saint-Raymond ([21]- [START_REF] Saint-Raymond | Control of large velocities in the two-dimensional gyrokinetic approximation[END_REF]).

The classical "guiding center approximation" corresponds to the following scaling for the Vlasov-Poisson system (from now on and until the end of the paper, B is a constant vector, say for instance B = 1 ǫ e z ):

       ∂ t f ǫ + v.∇ x f ǫ + (E ǫ + v∧ez ǫ ).∇ v f ǫ = 0 E ǫ = -∇ x V ǫ -∆ x V ǫ = f ǫ dv f ǫ,t=0 = f 0 (1.6)
The articles [START_REF] Frénod | Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field[END_REF] and [START_REF] Golse | The Vlasov-Poisson system with strong magnetic field[END_REF] show that when ǫ → 0, this leads to a one-dimensional kinetic equation in the direction of B:

       ∂ t f + v .∇ x f + E .∇ v f = 0 E = -∇ x V -∆ x V = f dv f t=0 = f 0 (1.7)
Notice that the electric drift does not appear; this was expected since we have seen in the formal analysis that this drift was of higher order in ǫ than the other effects. This shows in particular that this approximation is not sufficient for the numerical simulation of tokamaks. In order to make this drift appear, there exists to our knowledge two main possibilities:

1. one consists in restricting to a 2D problem in the plane orthogonal to B ([15]), 2. the other consists in rescaling the orthogonal scales in order to get both transport and electric drift at the same order ( [START_REF] Frénod | The Finite Larmor Radius Approximation[END_REF]).

This work directly follows the articles [START_REF] Frénod | The Finite Larmor Radius Approximation[END_REF] and [START_REF] Frénod | Asymptotic expansion of the Vlasov equation in a large external magnetic field[END_REF] where the authors considered the "finite Larmor radius approximation". This means that the spatial observation scale in the plane orthogonal to B is chosen smaller than the one in the parallel direction, more precisely with the same order as the Larmor radius r L , so that one can expect the electric drift to appear in the asymptotic equation.

In some sense, having such a scaling allows the electric field to significantly vary across a Larmor radius, which is not the case for instance in (1.6). Moreover, in this situation, the positions of the particles are no longer assimilated to the position of their "guiding center" and we will have to perform an average over one fast oscillation period (the so-called gyroaverage) in order to get a sort of averaged number density.

Scaling and existing results

The system we are going to study is based on the "finite Larmor radius scaling" and takes into account the quasineutrality of the plasma.

The (refined) mathematical model

We refer to [START_REF] Frénod | The Finite Larmor Radius Approximation[END_REF] for a complete discussion on the scaling. Let us recall briefly and quite crudely how it works.

Let L be the characteristic length in the direction of the magnetic field and L ⊥ be the characteristic length in the perpendicular plane. We consider that L ∼ 1 and L ⊥ ∼ ǫ and define the dimensionless variables

x ′ = x L and x ′ ⊥ = x ⊥ L ⊥ .
In the same fashion we also define the dimensionless variables t ′ and v ′ with characteristic time and velocities with the same order as L and introduce the new number density f ′ defined by f f ′ (t ′ , x ′ , v ′ ) = f (t, x, v) (and we define likewise the new electric field and potential ĒE ′ (t ′ , x ′ , v ′ ) = E(t, x, v) and V V ′ (t ′ , x ′ , v ′ ) = V (t, x, v)). We consider the scaling f , Ē ∼ 1 and V ∼ ǫ. At last, we introduce the Debye length of the plasma λ D , which appears in the Poisson equation. In order to take into account the quasineutrality of the plasma, we take from now on λ D ∼ √ ǫ.

The Poisson equation states in this scaling:

-ǫ∆ x ′ V ′ ǫ - 1 ǫ ∆ x ′ ⊥ V ′ ǫ = 1 ǫ n i ǫ -n e ǫ (1.8)
where n i ǫ = f ′ ǫ dv ′ is the density of ions and n e ǫ the density of electrons. The density distribution of ions is normalized so that f ′ 0 dv ′ dx ′ = 1. The main difference between Frénod and Sonnendrücker's model and ours lies in the following. Instead of considering a fixed background of electrons, and since me m i << 1, we make the usual assumption that the (adiabatic) electrons are instantaneously at a local thermodynamic equilibrium, so that their density follows a Boltzmann-Maxwell distribution:

n e ǫ (x, t) = exp eV ′ ǫ k B T e (1.9)
where k B is the Boltzmann constant, -e the charge and T e the temperature of the electrons. We consider that e k B Te ∼ 1. We make the assumption that we are not far from a fixed background of electrons, so that we can linearize this expression:

n e ǫ (x, t) = 1 + V ′ ǫ (1.10)
We are obviously aware that this assumption is not really satisfactory both from a mathematical and physical point of view; let us just consider this as a plain mathematical model which allows us to give rigorous justifications in this work. The problem of a fixed background of electrons, i.e. n e ǫ = 1, brings actually more interesting formal results; this point will be discussed in the last section.

The Poisson equation can now be written:

V ′ ǫ -ǫ 2 ∆ x ′ V ′ ǫ -∆ x ′ ⊥ V ′ ǫ = f ǫ dv ′ -f 0 dv ′ dx ′ (1.
11)

The dimensionless system (1.6) becomes (for the sake of simplicity, we forget the primes):

       ∂ t f ǫ + v ⊥ ǫ .∇ x f ǫ + v .∇ x f ǫ + (E ǫ + v∧B ǫ ).∇ v f ǫ = 0 E ǫ = (-∇ x ⊥ V ǫ , -ǫ∇ x V ǫ ) V ǫ -ǫ 2 ∆ x V ǫ -∆ x ⊥ V ǫ = f ǫ dv -f 0 dvdx f ǫ,t=0 = f ǫ,0 (1.12)
with the notation ∆ x = ∂ 2

x and ∆ x ⊥ = ∆ -∆ x , the problem being posed for (x ⊥ , x , v) ∈ T 2 × T × R 3 (with T = R/Z equipped with the restriction of the Lebesgue measure to [0, 1[).

State of the art about the Finite Larmor Radius Approximation

Using homogenization arguments, Frénod and Sonnendrücker established the convergence in some weak sense of sequences of solutions (f ǫ ) ǫ≥0 of similar systems, in two cases, namely in some pseudo 2D case (assuming that nothing depends on x and v ) and in a 3D case when the electric field is external. The main tool used to establish the convergence is the "2-scale convergence" introduced by Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF] and Allaire [START_REF] Allaire | Homogenization and Two-scale Convergence[END_REF] that we will recall later on.

The 3D case:

Assume that we deal with an external electric field

E ǫ = E ∈ C 1 (R × R 3 ): ∂ t f ǫ + v .∇ x f ǫ + v ⊥ ǫ .∇ x f ǫ + E + v∧ez ǫ .∇ v f ǫ = 0 f t=0 = f 0
Frénod and Sonnendrücker proved the following theorem: Theorem 1.1. For each ǫ, let f ǫ be the unique solution of the scaled Vlasov equation in

L ∞ t (L 1 x,v ∩ L 2 x,v ).
Then the following convergence holds as ǫ tends to 0:

f ǫ ⇀ f weak-* L ∞ t (L 2 x,v ) (1.13) where f ∈ L ∞ t (L 2 x,v
) is the unique solution to:

∂ t f + v .∇ x f + 1 2π 2π 0 R(τ )E(t, x + R(-τ )v)dτ .∇ x f + 1 2π 2π 0 R(τ )E(t, x + R(-τ )v)dτ .∇ v f = 0 f |t=0 = 1 2π 2π 0 f 0 (x + R(τ )v, R(τ )v)dτ
denoting by R and R the linear operators defined by:

R(τ ) =   cos τ -sin τ 0 sin τ cos τ 0 0 0 1   , R(τ ) = (-R(-π/2) + R(-π/2 + τ ))
2. The pseudo 2D case:

The Vlasov-Poisson system considered in this case is the following 2D system:

∂ t f ǫ + v ǫ .∇ x f ǫ + E ǫ + v ⊥ ǫ .∇ v f ǫ = 0 (1.14) f ǫ|t=0 = f 0 (1.15) E ǫ = -∇V ǫ , -∆ x V ǫ = ρ ǫ (1.16
)

ρ ǫ = f ǫ dv (1.17) If v = (v x , v y ), v ⊥ is defined by (v y , -v x ).
We recall that there exist global weak solutions of Vlasov-Poisson systems in the sense of Arsenev ([3]).

Assuming here that

f 0 ≥ 0, f 0 ∈ L 1 x,v ∩ L p
x,v (for some p > 2) and that the initial energy is bounded, Frénod and Sonnendrücker proved the following theorem (we voluntarily write an unprecise meta-version of the result) Theorem 1.2. For each ǫ, let (f ǫ , E ǫ ) be a solution in the sense of Arsenev to (1.14)- (1.17).

Then, up to a subsequence, f ǫ weakly converges to a function f Moreover, there exists a function G such that :

f = 2π 0 G(t, x + R(τ )v, R(τ )v)dτ (1.18)
and G satisfies :

∂ t G + 1 2π 2π 0 R(τ )E(t, τ, x + R(-τ )v)dτ .∇ x G + 1 2π 2π 0 R(τ )E(t, τ, x + R(-τ )v)dτ .∇ v G = 0 G |t=0 = f 0 E = -∇Φ, -∆Φ = G(t, x + R(τ )v, R(τ )v)dv
denoting by R and R the linear operators defined by :

R(τ ) = cos τ -sin τ sin τ cos τ , R(τ ) = (R(-π/2) -R(-π/2 + τ ))
In this case, we have to introduce an additional variable, the "fast-time" variable τ which comes from the fact that we need to precisely describe the oscillations in order to study the limit in non-linear terms.

Note that the authors actually developped a generic framework that allows them to deal with different scalings and to give a precise approximation at any order. We do not wish to do so in our study.

A bit of homogenization theory and some useful definitions

Let us now precisely state the "2-scale" convergence tools used in this paper.

Definition. Let X be a separable Banach space, X ′ be its topological dual space and (., .) the duality bracket between X ′ and X. For all α > 0, denote by C α (R, X) (respectively

L q ′ α (R; X ′ )) the space of α-periodic continuous (respectively L q ′ ) functions on R with values in X. Let q ∈ [1; ∞[.
Given a sequence (u ǫ ) of functions belonging to the space L q ′ (0, t; X ′ ) and a function

U 0 (t, θ) ∈ L q ′ (0, T ; L q ′ α (R; X ′ )) we say that u ǫ 2-scale converges to U 0 if for any function Ψ ∈ L q (0, T ; C α (R, X))
we have:

lim ǫ→0 T 0 u ǫ (t), Ψ t, t ǫ dt = 1 α T 0 α 0 U 0 (t, τ ), Ψ(t, τ ) dτ dt (1.19) Theorem 1.3. Given a sequence (u ǫ ) bounded in L q ′ (0, t; X ′ ), there exists for all α > 0 a function U 0 α ∈ L q ′ (0, T ; L q ′ α (R; X ′ )) such that up to a subsequence, u ǫ 2-scale converges to U 0 α
The profile U 0 α is called the α-periodic two scale limit of u ǫ and the link between U 0 α and the weak-* limit u of u ǫ is given by:

1 α α 0 U 0 dτ = u (1.20)
We also introduce some notations:

Notations. We define for all p ∈ [1; ∞] the space L p x,v :=L p x (T d , (L p v (R d ))
). In the same fashion, we define the spaces L p t,x , L p t,x,v ... Let L p 2π,τ be the space of 2π-periodic functions of τ which are in L p τ . Let L p x,loc be the space of functions f such that for all infinitely differentiable cut-off functions ϕ ∈ C ∞ c , ϕf belongs to L p x . We will say that a sequence (f ǫ ) is uniformly bounded in L p x,loc if for each compact set K, the sequence of the restrictions to K is uniformly bounded in L p

x with respect to ǫ (but this bound can depend on K). We will also use the same notations for Sobolev spaces W s,p (s ∈ R).

Statement of the result

In this paper we prove that the 2-scale convergence established in the previous 2D case is also true in our 3D framework. The difficulty comes from the fact that there is no uniform elliptic regularity for the electric field because of the factor ǫ 2 in front of ∆ x in the Poisson equation:

V ǫ -ǫ 2 ∆ x V ǫ -∆ x ⊥ V ǫ = f ǫ dv -f ǫ dvdx
In particular there is no a priori regularity on x and therefore no strong compactness. Nevertheless, we actually prove that due to the particular form of the asymptotic equation, the moments of the solution with respect to v are more regular in x than the solution itself. We can then easily pass to the weak limit.

The reason why we have opted for this strange Poisson equation instead of the usual one will appear at the end of the next section and especially in the last one. Roughly speaking it allows us to "kill" plasma waves which appear in the parallel direction due to the quasineutrality.

Notice that this result is in the same spirit as the proof of the weak stability of the Vlasov-Maxwell system by DiPerna and Lions ( [START_REF] Diperna | Global weak solutions of Vlasov-Maxwell systems[END_REF]), where the authors have regularity on moments, by opposition to the proof of the weak stability of the Vlasov-Poisson system by Arsenev ([3]), where the author has compactness on the electric field. Actually our result is a kind of a hybrid one, since we get on one hand regularity with respect to x ⊥ by elliptic regularity and in the other hand regularity with respect to x by averaging.

We assume here that the initial data (f ǫ,0 ) ǫ>0 satisfy the following conditions:

• f ǫ,0 ≥ 0 (positivity)

• (f ǫ,0 ) ǫ>0 is uniformly bounded with respect to ǫ in L 1 x,v ∩ L p
x,v (for some p > 3) and for each ǫ, f ǫ,0 dxdv = 1.

• The initial energy is uniformly bounded with respect to ǫ:

f ǫ,0 |v| 2 dvdx + ǫ V 2 ǫ,0 dx + ǫ |∇ x ⊥ V ǫ,0 | 2 dx + ǫ 3 |∇ x V ǫ,0 | 2 dx ≤ C Theorem 1.4. For each ǫ, let (f ǫ , E ǫ ) in L ∞ t (L 1 x,v ∩ L p x,v ) × L ∞ t (L 2
x ) be a global weak solution in the sense of Arsenev to (1.12). Then up to a subsequence we have the following convergence as ǫ tends to 0:

f ǫ,0 weakly-* converges to f 0 ∈ L p x,v (1.21) f ǫ 2-scale converges to F ∈ L ∞ t (L ∞ 2π,τ (L 1 x,v ∩ L p x,v )) (1.22) E ǫ 2-scale converges to E ∈ L ∞ t (L ∞ 2π,τ (L 3/2 x (W 1, 3 2 x ⊥ ))) (1.23) Moreover, there exists a function G ∈ L ∞ t (L 1 x,v ∩ L p x,v
) such that:

F (t, τ, x, v) = G(t, x + R(τ )v, R(τ )v) (1.24)
and (G, E) is solution to:

∂ t G + v .∇ x G + 1 2π 2π 0 R(τ )E(t, τ, x + R(-τ )v)dτ .∇ x G + 1 2π 2π 0 R(τ )E(t, τ, x + R(-τ )v)dτ .∇ v G = 0 G |t=0 = f 0 E = (-∇ ⊥ V, 0), V -∆ ⊥ V = G(t, x + R(τ )v, R(τ )v)dv -f 0 dvdx
denoting by R and R the linear operators defined by:

R(τ ) =   cos τ -sin τ 0 sin τ cos τ 0 0 0 1   , R(τ ) = (R(-π/2) -R(-π/2 + τ ))
As it has been said, for the proof of this theorem, we will first prove a proposition which gives the regularity of moments in v of the solution. For this, we use an averaging lemma. The beginning of the proof is very similar to the proof in the 2D case, but we will give it again for the sake of completeness.

Remarks.

1. The assumption on the initial energy may, at first sight, look a bit restrictive but in the "usual" Vlasov-Poisson scaling, it only means that the inital electric potential and field are bounded in L 2 .

2. The constant q = 3 will come quite naturally from Lemma 2.3 and Proposition 3.1.

3. This theorem implies that for a given non-negative initial data

G |t=0 = G 0 in L 1 x,v ∩ L p
x,v (with p > 3) and satisfying the energy bound, the asymptotic system admits at least one global weak solution

G ∈ L ∞ t (L 1 x,v ∩L p x,v
). With the additional assumptions on the inital data:

G 0 ∈ W 1,1 x,v , (1 + |v| 4 )G 0 L ∞ x,v < ∞, (1 + |v| 4 )DG 0 L ∞ x,v < ∞.
we are actually able to prove the uniqueness of the solution, using the same ideas than Degond in [START_REF] Degond | Global existence of smooth solutions for the Vlasov-Fokker-Planck equations in 1 and 2 space dimensions[END_REF] (and also used afterwards by Saint-Raymond in a gyrokinetic context ( [START_REF] Saint-Raymond | The gyrokinetic approximation for the Vlasov-Poisson system[END_REF])). Hence, it means that if the whole sequence (f ǫ,0 ) satisfies the same additional estimates, uniformly with respect to ǫ, and weak * converges to some f 0 then there is also convergence for the whole sequence (f ǫ ).

2 A priori uniform estimates for the scaled Vlasov-Poisson system 2.1 Conservation of L p norms and energy for the scaled system

In this section we give a priori estimates which are very classical for the Vlasov-Poisson system (used for example in [START_REF] Frénod | Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field[END_REF], [START_REF] Frénod | The Finite Larmor Radius Approximation[END_REF], [START_REF] Golse | The Vlasov-Poisson system with strong magnetic field[END_REF]). In order to recall how one can get them, we will give some formal computations. If one wants to have rigorous proofs, one should deal with smooth and compactly supported functions, namely with a sequence (f n ǫ ) n≥0 of solutions of some regularized Vlasov-Poisson equations then pass to the limit (that is the way one can clasically build a global weak solution in the sense of Arsenev ([3])).

First, as usual for such Vlasov equations, L p norms are conserved (we work here at a fixed ǫ):

Lemma 2.1. For all 1 ≤ p ≤ ∞, ∀t ≥ 0, f (t) L p x,v ≤ f (0) L p x,v (2.1)
Moreover, f 0 ≥ 0 if and only if ∀t ≥ 0, f (t) ≥ 0 (referred to as the maximum principle)

That precisely means that if f 0 ∈ L p x,v , then f ∈ L ∞ t (L p x,v
). Let us now compute the energy for the scaled system: Lemma 2.2. We have the estimate:

E ǫ (t) = f ǫ |v| 2 dvdx + ǫ V 2 ǫ dx + ǫ |∇ x ⊥ V ǫ | 2 dx + ǫ 3 |∇ x V ǫ | 2 dx ≤ E ǫ (0) (2.2)
In particular if there exists C > 0 independent of ǫ such that E ǫ (0) ≤ C, then:

f ǫ |v| 2 dvdx ≤ C (2.3)
Formal proof. We multiply the scaled Vlasov equation by |v| 2 and integrate with respect to x and v.

∂ t f ǫ |v| 2 dvdx + E ǫ .∇ v f ǫ |v| 2 dvdx = d dt f ǫ |v| 2 dvdx -2 E ǫ (x).vf ǫ dvdx = 0
We then integrate the Vlasov equation with respect to v. We get the so called conservation of charge:

d dt f dv + ∇ x . f v dv + ∇ x ⊥ ǫ . f v ⊥ dv = 0 (2.4)
Therefore, we have:

E ǫ (x).vf ǫ dvdx = -(∇ x ⊥ V ǫ , ǫ∇ x V ǫ ).vf ǫ dvdx = V ǫ ∇ x ⊥ .(f ǫ v ⊥ ) + ǫ∇ x .(f ǫ v ) dvdx = -ǫ V ǫ ∂ t f ǫ dvdx
Finally, using the Poisson equation, we get:

-ǫ V ǫ ∂ t f ǫ dvdx = -ǫ V ǫ ∂ t V ǫ -ǫ 2 ∆ x V ǫ -∆ x ⊥ V ǫ dx = -ǫ V ǫ ∂ t V ǫ dx + ∇ x ⊥ V ǫ ∂ t ∇ x ⊥ V ǫ dx + ǫ 2 ∇ x ∂ t V ǫ ∇ x V ǫ dx = -ǫ 1 2 d dt V 2 ǫ dx + |∇ x ⊥ V ǫ | 2 dx + ǫ 2 |∇ x V ǫ | 2 dx
Thus it comes:

d dt f ǫ |v| 2 dvdx + ǫ V 2 ǫ dx + ǫ |∇ x ⊥ V ǫ | 2 dx + ǫ 3 |∇ x V ǫ | 2 dx = 0 (2.5)

Regularity of the electric field

Let us recall a classical lemma obtained by a standard real interpolation argument:

Lemma 2.3. Let f (x, v) be a mesurable positive function on R 3 × R 3 . Then: f (x, v)dv 3/2 dx ≤ C f 3/4 L 3 x,v |v| 2 f dxdv 3/4
(2.6)

Proof. For any R > 0, we can write the following decomposition:

f ǫ dv = |v|≤R f dv + |v|>R f dv ≤ CR 2 f L 3 v + 1 R 2 |v| 2 f dv Then we can take R such that R 2 f L 3 v = 1 R 2
|v| 2 f dv so that we get:

f dv ≤ C f 3 dv 1/6 |v| 2 f dv 1/2
(2.7)

We then raise the quantities to the power 3/2, integrate with respect to x and use Hölder's inequality which gives the estimate.

By conservation of the L 3 norm and the uniform bound on the intial energy, Lemmas 2.2 and 2.3 entail that:

ρ ǫ ∈ L ∞ t (L 3/2 x ) (2.8)
and the norm is bounded uniformly with respect to ǫ. We now use the Poisson equation to compute the regularity of the electric field. Let us recall that:

E ǫ = -ǫ∇ x V ǫ , -∇ x ⊥ V ǫ V ǫ -ǫ 2 ∆ x V ǫ -∆ x ⊥ V ǫ = ρ ǫ -ρ 0 dx Lemma 2.

With the above notations and assumptions:

E ǫ is uniformly bounded with respect to ǫ in L ∞ t (L 3/2

x (W 1,3/2

x ⊥ )) Proof. Let ǫ > 0 and t > 0 be fixed. For the sake of simplicity we write V instead of V ǫ and E instead of E ǫ .

For any function f (x , x ⊥ ), define the rescaled function f (z, x ⊥ ) by

f x ǫ , x ⊥ = ǫ 2 3 f (x , x ⊥ ) so that: f (z, x ⊥ ) L 3/2 z = f (x , x ⊥ ) L 3/2 x (2.9)
The Poisson equation becomes:

Ṽ -∆ z Ṽ -∆ x ⊥ Ṽ = ρ -ǫ 2 3
ρ 0 dx and the scaled electric field is given by:

Ẽ = -∇ z Ṽ , -∇ x ⊥ Ṽ
Since ρ ǫ (t, ., .) and V ǫ are uniformly bounded in L 3/2

x , standard results of elliptic regularity on the torus T 2 × 1 ǫ T show that there exists C > 0 independent of ǫ such that:

Ṽ W 2,3/2 z,x ⊥ ≤ C ρ -ǫ 2 3 ρ 0 dx L 3/2 z,x ⊥
Remark. Notice here that due to the dilatation of order 1 ǫ in the parallel direction, being periodic in this direction does not make things easier.

Thanks to (2.9) we get:

Ṽ W 2,3/2 z,x ⊥ ≤ C ρ -ρ 0 dx L ∞ t (L 3/2 x )
≤ C 0 with C 0 independent of ǫ. Consequently, we have:

Ẽ L 3/2 z (W 1,3/2 x ⊥ ) ≤ Ẽ W 1,3/2 z,x ⊥ ≤ C 0
Finally from (2.9) we get

E ǫ L ∞ t (L 3/2 x (W 1,3/2 x ⊥ )) ≤ C 0
We can see as expected that the regularity of the electric field with respect to the x variable is not sufficient to get some strong compactness.

Remarks.

1. We can write the identity:

-∆ x ⊥ V ǫ = -∆ x ⊥ (Id -ǫ 2 ∆ x -∆ x ⊥ ) -1 ρ ǫ -ρ ǫ dx (2.10)
so that, thanks to elliptic estimates on the torus

T 2 , V ǫ ∈ L 3/2 x (W 2,3/2 x ⊥ ). Conse- quently, ∂ x V ǫ is bounded in L 3/2 x ⊥ (W -1,3/2 x
). This implies that E ǫ, = -ǫ∂ x V ǫ tends to zero in the sense of distributions.

2. A typical function ϕ ǫ such that ϕ ǫ is bounded in L p and 1 ǫ ϕ ǫ is bounded in W -1,p is the oscillating function cos( 1 ǫ x). This indicates that E ǫ, oscillates with a frequency of order 1 ǫ in the parallel direction.

If we work with the usual Poisson equation

-ǫ 2 ∆ x V ǫ -∆ x ⊥ V ǫ = ρ ǫ -ρ ǫ dx
we only get homogeneous estimates for V ǫ and we have not been able to deal with such anisotropic estimates in the following of the paper (namely in the estimates of Proposition 3.1). Roughly speaking, if V is a solution of the Poisson equation -∆V = ρ with ρ ∈ L 3/2 (R 3 ), we can only say that V ∈ Ẇ 2,3/2 (R 3 ) (the homogeneous Sobolev space) and not W 2,3/2 . 4. This difficulty seems to be not only a technical one, but appears to be linked to the existence of plasma waves (with frequence and magnitude of order 1 √ ǫ ) in the parallel direction which prevents us from passing directly to the limit ǫ → 0 (see [START_REF] Grenier | Oscillations in quasineutral plasmas[END_REF] and last section).

Proof of Theorem 1.4

Proof. The first two steps are identical to the one given in [START_REF] Frénod | The Finite Larmor Radius Approximation[END_REF]. For the sake of completeness we recall here the main arguments and refer to [START_REF] Frénod | The Finite Larmor Radius Approximation[END_REF] for the details.

Step 1: Deriving the constraint equation

First of all, since (f ǫ ) is bounded in L ∞ t (L 1 x,v ∩ L p x,v
), Theorem 1.3 shows that for all α > 0:

f ǫ 2-scale converges to F α ∈ L ∞ (0, T ; L ∞ α (R; L p x,v ))
Let Ψ(t, τ, x, v) be an α-periodic oscillating test function in τ and define:

Ψ ǫ ≡ Ψ(t, t ǫ , x, v)
We start by writing the weak formulation of the scaled Vlasov equation against Ψ ǫ . Since

∇ x .v = ∇ x ⊥ .v ⊥ = div v E ǫ + v ∧ e z ǫ = 0,
we get the following equation:

f ǫ (∂ t Ψ) ǫ + 1 ǫ (∂ τ Ψ) ǫ + v .(∇ x Ψ) ǫ + v ⊥ ǫ .(∇ x Ψ) ǫ + E ǫ + v ∧ e z ǫ .(∇ v Ψ) ǫ dtdxdv = -f 0 Ψ(0, 0, x, v)dxdv
Multiply then by ǫ and pass up to a subsequence to the (2-scale) limit. We get the so called constraint equation for the α-periodic profile F α :

∂ τ F α + v ⊥ .∇ x F α + v ∧ e z .∇ v F α = 0, (3.1)
which means that F α is constant along the characteristics:

dV dτ = V ∧ e z (3.2) dX dτ = V ⊥ (3.3)
A straightforward calculation therefore shows that there exists

F 0 α ∈ L ∞ (0, T, L p x,v ) such that: F α (t, τ, x, v) = F 0 α (t, x + R(τ )v, R(τ )v) (3.4)
with:

R(τ ) =   cos τ -sin τ 0 sin τ cos τ 0 0 0 1   , R(τ ) = (-R(-π/2) + R(-π/2 + τ )) i.e. R(τ ) =   sin τ cos τ -1 0 1 -cos τ sin τ 0 0 0 0   .
Since R and R are 2π-periodic, we will consider the 2π profile: indeed if α and 2π were incommensurable, F α could not depend on τ and consequently we would have no information on the oscillations.

Step 2: Filtering the essential oscillation

We now look for the equation satisfied by F 0 2π := G; we introduce the filtered function g ǫ :

g ǫ (t, x, v) = f ǫ (t, x + R(-t/ǫ)v, R(-t/ǫ)v) (3.5)
(meaning that we have removed the oscillations) We easily compute the equation satisfied by g ǫ :

∂ t g ǫ + v .∇ x g ǫ + R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ x g ǫ (3.6) +R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ v g ǫ = 0
Remark. Note here that g ǫ 2-scale converges to G, and since it does not depend on τ , it also weakly converges to G.

Step 3: Getting some regularity on moments

From now on, the goal is to get some compactness for the moments of g ǫ with respect to v . The main tool we have in mind is the following averaging lemma proved by Bézard in [START_REF] Bézard | Régularité L p précisée des moyennes dans les équations de transport[END_REF], which is a refined version of the fundamental result of DiPerna, Lions and Meyer ( [START_REF] Diperna | L p regularity of velocity averages[END_REF]):

Theorem 3.1. Let 1 < p ≤ 2. Let f, g ∈ L p (dt ⊗ dx ⊗ dv) be solutions of the following transport equation ∂ t f + v.∇ x f = (I -∆ t,x ) τ /2 (I -∆ v ) m/2 g (3.7) with m ∈ R + , τ ∈ [0, 1[. Then ∀Ψ ∈ C ∞ c (R d ), ρ Ψ (t, x) = f (t, x, v)Ψ(v)dv ∈ W s,p t,x (R × R d ) where s = 1 -τ (1 + m)p ′ (3.8) Moreover, ρ Ψ W s,p t,x (R×R d ) ≤ C f L p (dt⊗dx⊗dv) + g L p (dt⊗dx⊗dv) (3.9)
(C is a positive constant independent of f and g)

Averaging lemmas are an important feature of transport equations: since the transport equation (3.7) is hyperbolic, one can obviously not expect the solution f to be more regular than the right hand side or the inital data. Nevertheless, if one considers the averaged quantity ρ Ψ , one can actually notice a gain of regularity. This phenomenon was first observed independently by Golse, Perthame and Sentis ( [START_REF] Golse | Un résultat de compacité pour les equations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport[END_REF]) and Agoshkov ([1]) then was formulated in a precise way for the first time by Golse, Lions, Perthame and Sentis (see [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF]); it is referred to as "velocity averaging". There exists many refined versions of these results and numerous interesting applications in kinetic theory, but we shall not dwell on that. We simply point out that this tool has been successfully applied to Vlasov equations, for instance to prove the existence of global weak solutions to the Vlasov-Maxwell system as it has been done by DiPerna and Lions ( [START_REF] Diperna | Global weak solutions of Vlasov-Maxwell systems[END_REF]).

These results have been proved for functions with values in R. Here, for our purpose, we need a new version of L p averaging lemma for functions with values in some Sobolev space W λ,p (R k ) (k ∈ N * ). We prove the following result, which is sufficient in our case (probably an analogous of Bezard's optimal result is also true):

Lemma 3.1. Let 1 < p < +∞ and λ ∈ R. Let f, g ∈ L p
t,x,v (W λ,p y ) be solutions of the following transport equation

∂ t f + v.∇ x f = (I -∆ v ) m/2 g (3.10) with m ∈ R + . Then ∀Ψ ∈ C ∞ c (R d ), ρ Ψ (t, x) = f (t, x, v)Ψ(v)dv ∈ W s,p t,x (W λ,p y ) for any s such that s ≤ s 2 = 1 2(1 + m) for p = 2 (3.11) and s < s p = 1 (1 + m)p ′ for p = 2 (3.12) Moreover, ρ Ψ W s,p t,x (W λ,p y ) ≤ C f L p t,x,v (W λ,p y ) + g L p t,x,v (W λ,p y ) (3.13)
(C is a positive constant independent of f and g) Sketch of proof. We prove the result in the stationnary case only:

v.∇ x f = (I -∆ v ) m/2 g (3.14)
By standard arguments (see [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF]) the general case then follows.

The following estimate is obvious for q = 1 or q = +∞ (and actually we can not expect any smoothing effect) :

ρ Ψ L q x (W λ,q y ) ≤ C f L q x,v (W λ,q y ) + g L q x,v (W λ,q y ) (3.15)
For p = 2, we prove the result as in Golse-Lions-Perthame-Sentis [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF]. We denote by ξ (resp. η) the Fourier variable associated to x (resp. y).

The only point is to notice (using Fubini's inequality):

ρ Ψ 2 H s x (H λ y ) = (1 + |ξ| 2 ) s/2 (1 + |η| 2 ) λ/2 F ξ,η f Ψ(v)dv 2 dηdξ
The proof is then identical and we get for s = 1 2(1+m) :

ρ Ψ H s x (H λ y ) ≤ C f L 2 x,v (H λ y ) + g L 2 x,v (H λ y ) (3.16)
Finally the general case 1 < p < +∞ is obtained by complex interpolation [START_REF] Bergh | Interpolation Spaces. An introduction[END_REF].

Equipped with this tool, we can now prove that moments in v are more regular with respect to t and x than the solution itself. Proposition 3.1. For each ǫ > 0, let g ǫ be a function in L 1

x,v ∩ L p x,v (with p > 3) bounded uniformly with respect to ǫ and satisfying:

∂ t g ǫ + v .∇ x g ǫ + R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ x g ǫ +R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ v g ǫ = 0 with E ǫ the electric field uniformly bounded in L ∞ t (L 3/2 x ). Let Ψ ∈ D(R). Define η ǫ (t, x, v ⊥ ) = g ǫ (t, x, v)Ψ(v )dv Then, η ǫ is uniformly bounded in W s,γ t,x ,loc (W -1,γ x ⊥ ,v ⊥ ,loc ) (3.17) for γ ∈]1; 2[ defined by 1 γ = 2 3 + 1 p and some s ∈]0; 1[ (depending on γ)
Proof.

• The first step is to localize the equation. Let K be the cartesian product of compact sets:

K = [0, T ] × K x × K x ⊥ × K v × K v ⊥
We now consider some positive smooth function Φ(t, x , x ⊥ , v , v ⊥ ) which is C ∞ c and which satisfies the condition:

Φ ≡ 0 outside K (3.18)
Noticing that:

div x (R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v)) + div v (R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v)) = 0
The equation satisfied by g ǫ Φ is the following one:

∂ t (g ǫ Φ) + v .∇ x (g ǫ Φ) = -∇ x .(R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v)g ǫ Φ) (1)
-∇ v .(R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v)g ǫ Φ) (2)
-∂ t (Φ)g ǫ -v .∇ x (Φ)g ǫ +R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ x (Φ)g ǫ + R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ v (Φ)g ǫ
The idea is now to consider this equation as a kinetic equation with respect to the variables (t, x , v ) and with values in an abstract Banach space (which will be

W -1,γ x ⊥ ,v ⊥ ).
We then only study the first two terms of the right-hand side (noticing that the other terms have more regularity than these ones).

From now on, for the sake of simplicity and readability, we will write L p and W s,p norms without always specifying that they are taken on the compact support of Φ.

• Estimate on the first term [START_REF] Agoshkov | Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation[END_REF] Since E ǫ does not depend on v, we have:

E ǫ ∈ L ∞ t (L 3/2 x (L ∞ v (L 3/2 x ⊥ )))
In particular if we restrict to compact supports:

E ǫ ∈ L 3/2 t,x,v
The second point is that the differential operator applied in (1) involves only derivatives with respect to the x ⊥ variable and not in the parallel direction: this remark is fundamental for using the averaging lemma 3.1 (indeed, the case of a full derivative in x can not be handled).

Hölder's inequality simply implies that:

R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v)g ǫ Φ L γ x ⊥ ,v ⊥ ≤ E ǫ (t, x + R(-t/ǫ)v)Φ L 3/2 x ⊥ ,v ⊥ g ǫ L p x ⊥ ,v ⊥ (3.19) where 1 γ = 2 3 + 1 p . Hence: ∇ x .(R(t/ǫ)E ǫ (t, x+R(-t/ǫ)v)g ǫ Φ) W -1,γ x ⊥ ,v ⊥ ≤ E ǫ (t, x+R(-t/ǫ)v)Φ L 3/2 x ⊥ ,v ⊥ g ǫ L p x ⊥ ,v ⊥ (3.20)
Notice that the change of variables (x, v) → (x + R(s)v, v) has unit Jacobian for all s ∈ R, so that:

E ǫ (t, x + R(-t/ǫ)v)Φ L 3/2 x ⊥ ,v ⊥ = E ǫ (t, x) L 3/2 x ⊥ ,v ⊥ (3.21)
So finally we have, after integrating in t, x , v and thanks to Hölder's inequality:

∇ x . (R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v)g ǫ Φ) L γ t,x ,v (W -1,γ x ⊥ ,v ⊥ ) ≤ C E ǫ (t, x) L 3/2 t,x ,v (L 3/2 x ⊥ ,v ⊥ ) g ǫ L p t,x ,v (L p x ⊥ ,v ⊥ )
and C is a constant independent of ǫ.

Remark. The regularity of (1) with respect to v ⊥ is not optimal (since it involves no derivative in v ⊥ for g ǫ ). Nevertheless we are interested in the regularity of the whole right hand side, and we will see that the term (2) has this regularity in v ⊥ .

• Estimate on the second term (2)

By the same method one gets:

∇ v . (R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v)g ǫ Φ) L γ t,x (W -1,γ v (W -1,γ x ⊥ ,v ⊥ )) ≤ C E ǫ (t, x) L 3/2 t,x (L 3/2 x ⊥ ,v ) g ǫ Φ L p t,x (L p x ⊥ ,v )
Finally we see that the right hand side is uniformly bounded in:

L γ t,x ,loc (W -1,γ v ,loc (W -1,γ x ⊥ ,v ⊥ ,loc ))
• Regularity of the moments By lemma 3.1 , for all Ψ ∈ C ∞ c , the moment:

η ǫ (t, x, v ⊥ ) = g ǫ (t, x, v)Ψ(v )dv
is then uniformly bounded in the space W s,γ t,x ,loc (W -1,γ x ⊥ ,v ⊥ ,loc ) for any s > 0 with s < 1 2γ ′ .

We can now prove that the sequence of moments η ǫ is compact in a space of distributions which is the dual of some space where the sequence (E ǫ ) is uniformly bounded. Corollary 3.1. There exists θ ∈]0, 1[ and η ∈ W sθ,3 t,x ,loc (W -θ,3

x ⊥ ,v ⊥ ,loc ) such that for all ξ > 0, up to a subsequence:

η ǫ → η strongly in L 3 t,loc (L 3 x ,loc (W -θ-ξ,3 x ⊥ ,v ⊥ ,loc )) (3.22)
Proof. By assumption on the initial data, there exists q > 3 such that f 0 ∈ L q x,v ; thanks to the a priori L q estimate, we get g ǫ ∈ L ∞ t (L q x,v ). Define γ by:

1 γ = 2 3 + 1 q
The previous lemma shows that for some s > 0:

η ǫ ∈ W s,γ t,x loc (W -1,γ x ⊥ ,v ⊥ ,loc ) uniformly in ǫ Since g ǫ ∈ L q t,loc (L q x,v
) and Ψ has compact support, we get by Hölder's inequality:

η ǫ ∈ L q t,loc (L q x (L q x ⊥ ,v ⊥ ))
Since 1 γ > 2 3 > 1 3 and 1 q < 1 3 , there exists θ ∈]0, 1[ such that

1 3 = 1 -θ q + θ γ
By interpolation ( [START_REF] Bergh | Interpolation Spaces. An introduction[END_REF]) we deduce that:

η ǫ ∈ W sθ,3 t,x loc (W -θ,3 x ⊥ ,v ⊥ ,loc ))
This implies that:

η ǫ ∈ W sθ,3 t,loc (L 3 x (W -θ,3 x ⊥ ,v ⊥ ,loc )) uniformly in ǫ η ǫ ∈ L 3 t,loc (W sθ,3
x ,loc (W -θ,3

x ⊥ ,v ⊥ ,loc )) uniformly in ǫ

We then use the following refined interpolation result proved by Simon in [START_REF] Simon | Compact sets in L p (0, T ; B)[END_REF], which is, roughly speaking, an anisotropic adaptation of the classical Riesz-Fréchet-Kolmogorov criterion for compactness in L p : Theorem 3.2. Let 1 ≤ p ≤ ∞ and s > 0. Let T > 0 and X, B, Y be three Banach Spaces such that X ⊂ B ⊂ Y and with X compactly embedded in B. Let F be a bounded set of L p t ([0, T ], X) ∩ W s,p t ([0, T ], Y ). Then F is relatively compact in L p t ([0, T ], B). This entails, thanks to Sobolev's embeddings, that the sequence (η ǫ ) is strongly relatively compact in L 3 t,loc (L 3

x ,loc (W -θ-ξ,3

x ⊥ ,v ⊥ ,loc )), for all ξ > 0.

From now on, we consider ξ such that θ + ξ < 1, which is of course possible since θ < 1.

Remark. Following the remark in Step 2 and by uniqueness of the limit in the sense of distributions, we get:

η = GΨ(v )dv
Step 4: Passing to the weak limit

We will first need a technical lemma which is obtained directly from the 2-scale convergence of E ǫ .

Lemma 3.2. Up to a subsequence,

• R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v) 2-scale converges to 1 2π 2π 0 R(τ )E(t, τ, x + R(-τ )v)dτ ∈ L ∞ t (L ∞ 2π,τ (L 3/2 x (W 1,3/2 x ⊥ )) • R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v) 2-scale converges to 1 2π 2π 0 R(τ )E(t, τ, x + R(-τ )v)dτ ∈ L ∞ t (L ∞ 2π,τ (L 3/2 x (W 1,3/2 x ⊥ )) Proof. E ǫ is uniformly bounded in L ∞ t (L 3/2 x (W 1,3/2 x ⊥ )), so there exists E ∈ L ∞ t (L 3/2 x (W 1,3/2
x ⊥ ) such that E ǫ 2 scale converge to E.

We take Ψ(t, τ, x) a 2π-periodic w.r.t. τ test function and use the 2 scale convergence of E ǫ :

R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).Ψ(t, t/ǫ, x)dtdx = E ǫ (t, x). t R(t/ǫ)Ψ(t, t/ǫ, x -R(-t/ǫ)v)dtdx → 1 2π 2π 0 E(t, τ, x). t R(τ )Ψ(t, τ, x -R(-τ )v)dtdτ dx = 1 2π 2π 0 E(t, τ, x + R(-τ )v). t R(τ )Ψ(t, τ, x)dtdτ dx
The proof is the same for R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).

Now, we can write the weak formulation of the kinetic equation (3.6) against a smooth test function of the form Φ(t, x, v ⊥ )Ψ(v ) with compact support. If we can pass to the limit for such test functions, then by density it will be also the case for all test functions.

Noticing that div x v = 0 and that div

x (R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v)) + div v (R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v)) = 0
we get:

∂ t (Φ(t, x, v ⊥ )Ψ(v )) + v .∇ x (ΦΨ) + R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ x (ΦΨ) +R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ v (ΦΨ) g ǫ dtdx ⊥ dx dv ⊥ dv = -u 0 Φ(0, x, v ⊥ )Ψ(v )dxdv
We can easily take weak limits in the linear part ∂ t g ǫ + v .∇ x g ǫ .

Consider now the "non linear" term:

R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ x Φ(t, x, v ⊥ )g ǫ Ψ(v )dtdx ⊥ dx dv ⊥ dv = R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ x Φ(t, x, v ⊥ ) g ǫ Ψ(v )dv dtdx ⊥ dx dv ⊥
The convergence of this term can be established by the strong/weak convergence principle. Nevertheless, we have to carefully use this technique to get the result and we will explicitly evaluate the difference:

R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ x Φ(t, x, v ⊥ )η ǫ dtdx ⊥ dx dv ⊥ - 1 2π 2π 0 R(τ )E(t, τ, x + R(-τ )v)dτ.∇ x Φηdtdxdv ⊥ ≤ R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v) - 1 2π 2π 0 R(τ )E(t, τ, x + R(-τ )v)dτ .∇ x Φηdtdxdv ⊥ + R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ x Φ(t, x, v ⊥ ) (η ǫ -η) dtdx ⊥ dx dv ⊥
The first term of the right hand side converges to zero because of the 2-scale convergence of E ǫ (Lemma 3.2). We can control the second term by:

C E ǫ .∇ x Φ L 3/2 t (L 3/2 x (W θ+ξ,3/2 x ⊥ ,v ⊥ )) η ǫ -η L 3 t (L 3 x (W -θ-ξ,3 x ⊥ ,v ⊥ )) (3.23)
(these norms are actually taken on the compact support of Φ but we do not write it for the sake of simplicity) Using the fact that E ǫ is uniformly bounded in L )) (this is an easy consequence of Lemma 2.4) and Corollary 3.1,

η ǫ -η L 3 t ([0,T ],L 3 x (Kx ,W -θ-ξ,3 x ⊥ ,v ⊥ (Kx ⊥ ×Kv ⊥ ))) → 0, we can deduce that R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ x Φ(t, x, v ⊥ ) (η ǫ -η) dtdx ⊥ dx dv ⊥ → 0
The proof is of course the same for the other non-linear term:

R(t/ǫ)E ǫ (t, x + R(-t/ǫ)v).∇ v g ǫ
To conclude let us compute the asymptotic equation satisfied by the 2-scale limit of V ǫ denoted by V . We take Ψ(t, τ, x) a 2π-periodic w.r.t. τ test function. We write the weak formulation of the Poisson equation:

V ǫ ∇ x Ψ(t, t/ǫ, x)dtdx + ǫ 2 ∇ x V ǫ ∇ x Ψ(t, t/ǫ, x)dtdx + ∇ x ⊥ V ǫ ∇ x ⊥ Ψ(t, t/ǫ, x)dtdx = f ǫ (t, x, v)Ψ(t, t/ǫ, x)dtdvdx - f 0 dvdx Ψ(t, t/ǫ, x)dtdvdx
We then pass to the 2 scale limit:

1 2π 2π 0 V (t, τ, x)∇ x ⊥ Ψ(t, τ, x)dτ dtdx + 0 + 1 2π 2π 0 ∇ x ⊥ V (t, τ, x)∇ x ⊥ Ψ(t, τ ,
V -∆ ⊥ V = G(t, x + R(τ )v, R(τ )v)dv -f 0 dvdx
Moreover since E ǫ,⊥ = -∇ x ⊥ V ǫ and thanks to Remark 1 following Lemma 2.4, we easily get if we pass to the two-scale limit:

E = (-∇ x ⊥ V, 0)
4 Concluding comments

Comments on the result

Finally we can see as in [START_REF] Frénod | The Finite Larmor Radius Approximation[END_REF] (namely by performing the change of variables x = x cv ⊥ and looking at the new equations in the so-called gyro-variables (t, x c , v)) that the drift involving the electric field in the asymptotic "kinetic" equation corresponds to the electric drift that we mentioned in the introduction and which was expected to appear. We also notice that the Poisson equation we get in the asymptotic system is the same than the one used in the numerical simulations of tokamak plasmas (see for example the GYSELA code in [START_REF] Grandgirard | Global full-f gyrokinetic simulations of plasma turbulence[END_REF]). Nevertheless, physicists do not get it in the same formal way: they claim that it only expresses the quasineutrality of the plasma (there is no "real" Poisson equation involved) and the perpendicular laplacian happens to appear due from the socalled "polarization drift" ( [START_REF] Grandgirard | Global full-f gyrokinetic simulations of plasma turbulence[END_REF], see also [START_REF] Wesson | Tokamaks[END_REF] for a physical reference on the subject). It would be interesting to justify such a computation from a mathematical point of view.

At last, we wish to point out a really unpleasant feature of our model, which is that there is no parallel dynamics.

An alternative model

Let us give some comments on the gyrokinetic approximation of the system (4.2) which consists in considering a population of electrons in a fixed background of ions:

n i ǫ = f 0 dxdv (4.1)
Actually, quite surprisingly, this model engenders more interesting physical properties: in this case the parallel component of the electric field does not vanish but appears as a pressure in the end (which may bring difficulties both in the study of the asymptotic system and in numerical simulations).

       ∂ t f ǫ + v ⊥ ǫ .∇ x f ǫ + v .∇ x f ǫ + (E ǫ + v∧B ǫ ).∇ v f ǫ = 0 E ǫ = (-∇ x ⊥ V ǫ , -ǫ∇ x V ǫ ) -ǫ 2 ∆ x V ǫ -∆ x ⊥ V ǫ = f ǫ dv -f 0 dvdx f ǫ,t=0 = f ǫ,0 (4.2)
With the same computations as the present paper, we get:

f ǫ 2-scale converges to F (4.
3) E ǫ 2-scale converges to E (4.4)

In a formal sense, there exists a function G such that:

F (t, τ, x, v) = G(t, x + R(τ )v, R(τ )v) (4.5)
and (G, E) is solution to: The parallel component E has to be seen as a pressure (or the Lagrange multiplier) associated to the "incompressibility" constraint T 2 G(t, x+R(τ )v, R(τ )v)dvdx ⊥ = f 0 dvdx Let us just give a few words on the difficulties that arise with this model. The Poisson equation can be restated as:

∂ t G + v .∇ x G + 1 2π
-ǫ 2 ∆ x V ǫ -∆ x ⊥ V ǫ = f ǫ dv -f ǫ dvdx ⊥ + f ǫ dvdx ⊥ -f 0 dvdx (4.6)
so that thanks to the linearity of the Poisson equation we can study separately two equations. The first one states:

-ǫ 2 ∆ x V 1 ǫ -∆ x ⊥ V 1 ǫ = f ǫ dv -f ǫ dvdx ⊥ (4.7)
For this part of the electric potential we get the same estimates as in lemma 2.4. Indeed, f ǫ dv -f ǫ dvdx ⊥ dx ⊥ = 0 so that we can use elliptic estimates on the torus T 2 x ⊥ .

x)dτ dtdx = 1 2π 2π 0 Ff 0

 00 (t, τ, x, v)Ψ(t, τ, x)dtdvdx -, τ, x + R(τ )v, R(τ )v)Ψ(t, τ, x)dτ dvdxdvdx Ψ(t, τ, x)dτ dvdx from which we get the "Poisson" equation given in Theorem 1.4:

2π 0 R cos τ -sin τ 0 sin τ cos τ 0 0 0 1 

 01 (τ )E(t, τ, x + R(-τ )v)dτ .∇ x G )E(t, τ, x + R(-τ )v)dτ .∇ v G = 0 G |t=0 = f 0 E = (-∇ ⊥ V, E ), -∆ ⊥ V = G(t, x + R(τ )v, R(τ )v)dv -f 0 dvdxstill denoting by R and R the linear operators defined by:R(τ ) =  , R(τ ) = (R(-π/2) -R(-π/2 + τ ))

Acknowledements. The author would like to thank L. Saint-Raymond for having given him the idea to write this paper and for her careful reading and advice. The author is also indebted to M. Hauray for having pointed out to him the existence of plasma waves and for indicating Grenier's paper.

Consequently this electric potential does not give birth to any parallel dynamics, like in Theorem 1. [START_REF] Bergh | Interpolation Spaces. An introduction[END_REF].

The second one is:

This equation, associated to the one giving the electric field E 2 ǫ, = -ǫ∂ x V 2 , is similar to the one studied by Grenier in [START_REF] Grenier | Oscillations in quasineutral plasmas[END_REF], coupled to a Vlasov equation describing a quasineutral plasma. In this case it was shown that there exist plasma waves with both temporal and spatial oscillations with frequency 1 √ ǫ and magnitude of order 1 √ ǫ . Because of these waves, it is much more difficult to pass to the limit in order to get a kinetic equation. Grenier managed to prove the convergence only for distribution functions with special form and got in the end a Euler-like system with an electric field interpreted as a Lagrange multiplier. Hence, in our case, this part of the electric field may engender an non-zero E .

For these reasons, it seems much harder to expect to prove a rigorous result similar to Theorem 1.4 with such a model.

Prospects

A way to pass to the limit in this latest case would be to use a relative entropy method like in the papers of Brenier ([5]) and Golse and Saint-Raymond ( [START_REF] Golse | The Vlasov-Poisson system with strong magnetic field in quasineutral regime[END_REF]). This will be the object of a forthcoming work.

Another interesting issue would be to consider a "true" Boltzmann-Maxwell distribution for the electrons (not linearized like in this paper) and perform an asymptotic analysis, maybe also with a relative entropy method.