Maria Stitz

Cristian Chaparro

Zhigang Lu

V Janett

Janett Olzog

Christina E Weinberg

Jochen Blom

Alexander Goesmann

Christoph Grunau

Christoph G Grevelding
email: christoph.grevelding@vetmed.uni-giessen.dechristoph.grunau@univ-perp.fr

V Janett Olzog

Prof G Christoph

Prof Grevelding

Christoph Grunau

Justus Liebig

Satellite-like W elements: repetitive, transcribed, and putative mobile

Keywords: Schistosoma mansoni, genome evolution, W-element, mobile genetic element, non-56 coding RNA, ribozyme. 57 58

published or not. The documents may come

Introduction

69

Schistosomes are parasitic plathyhelminths and cause schistosomiasis (bilharzia). Listed as neglected 70 tropical disease (NTD) by the WHO, schistosomiasis ranks close to malaria in terms of parasite-induced 71 human morbidity and mortality (Hotez and Kamatha 2009;[START_REF] Colley | Human schistosomiasis[END_REF]. The life cycle of the 72 parasite is complex with a fresh-water snail of the genus Biomphalaria as intermediate host, producing 73 human-dwelling larvae (cercariae) that develop in the vertebrate host into adults, pair and produce 74 eggs that are excreted with the faeces and hatch snail-infecting miracidia when in contact with water.

75 Schistosomes are evolutionary unique as they are the only digenean parasites that have evolved 76 separate sexes [START_REF] Moné | Sexual biology of schistosomes[END_REF]. Heteromorphic sex chromosomes determine the sex of 77 schistosomes, with ZZ in the males and ZW in the females. Earlier studies from us and others has shown 78 that the females-specific region of the W chromosome is essentially composed of repetitive DNA 79 sequences (W-Elements, WE) arranged in large satellite blocks (Lepesant et al. 2012a;Lepesant et al. 80 2012b). WE sequences are abundant on the W chromosome, but may occasionally be transmitted to 81 autosomes, shown for the female W1 repeat sequence [START_REF] Grevelding | Genomic instability in Schistosoma mansoni[END_REF]. In most species with Y or W 82 sex chromosomes, it is found that: (i) repetitive sequences accumulate on these chromosomes, (ii) 83 large regions are heterochromatic, and (iii) these chromosomes deteriorate or are completely absent 84 in an extreme case. Among the accepted evolutionary theories is that the accumulation of repetitive 85 sequences on one of sex chromosomes has facilitated recombination suppression between the 86 heterochromosomes thus protecting sexually beneficial loci [START_REF] Ohno | Sex chromosomes and sex-linked genes[END_REF]. Another hypothesis is that 87 chromosome rearrangements, sequence accumulation, and amplification may have occurred near the 88 sex-determining locus as a result of suppression of recombination [START_REF] Charlesworth | The evolution of sex chromosomes[END_REF]Marais and N. 89 Galtier 2003). Heterochromatization of the W chromosome in schistosomes has long been known and 90 has even been used as a marker for sex identification at the cercaria stage, where male and female 91 phenotypes are indistinguishable (Grossman et al. 1980;Grossman et al. 1981;Liberatos & Short 92 1983). Our earlier results (Lepesant et al. 2012a) showed that the repetitive sequences located in the 93 heterochromatic region of the W chromosome carry a euchromatic signature in the miracidia stage 94 (presence of H3K9ac and H3K4me3), which is gradually lost during development into adults. In most 95 species, the structural change of the chromatin is a highly organised process, and the observed 96 heterochromatin/euchromatin cycle of W-specific repeats suggests functional importance. Full or 97 partial euchromatisation could not only allow transcription of functional WE but may also open space 98 for mobilisation of these elements. The WEs W1 and W2 were originally found in Puerto Rican isolates 99 of Schistosoma mansoni [START_REF] Webster | Isolation of a female-specific, highly repeated Schistosoma 1312 mansoni DNA probe and its use in an assay of cercarial sex[END_REF][START_REF] Drew | Female-specific sequences isolated from Schistosoma mansoni by 1017 representational difference analysis[END_REF][START_REF] Spotila | A cloned DNA probe identifies the sex of 1291 Schistosoma mansoni cercariae[END_REF]. Southern 100 blot analysis demonstrated their female-specific occurrence, and in situ hybridization localized W1 and 101 W2 in the heterochromatic region of the W-chromosome [START_REF] Hirai | Schistosoma mansoni: chromosomal localization of DNA repeat 1068 elements by in situ hybridization using biotinylated DNA probes[END_REF][START_REF] Spotila | A retroposon-like short repetitive DNA element in 1288 the genome of the human blood fluke, Schistosoma mansoni[END_REF].

102

According to their repetitive nature, both elements were classified as satellite-like DNA (satDNA).

103 Subsequent studies in the Liberian strain and further strains of S. mansoni demonstrated W1 and W2 104 occurrence in both sexes of nearly all investigated strains, including another isolate from Puerto Rico 105 [START_REF] Grevelding | The female-specific W1 sequence of the Puerto Rican strain of Schistosoma 1045 mansoni occurs in both genders of a Liberian strain[END_REF][START_REF] Quack | Schistosoma mansoni: the varying occurrence of 1259 repetitive elements in different strains shows sex-specific polymorphisms[END_REF]. In contrast to the polymorphic occurrence of W1 and W2, 106 another family of repetitive elements, called D9 [START_REF] Spotila | A retroposon-like short repetitive DNA element in 1288 the genome of the human blood fluke, Schistosoma mansoni[END_REF], repeatedly occurred in the 107 investigated schistosome strains [START_REF] Quack | Schistosoma mansoni: the varying occurrence of 1259 repetitive elements in different strains shows sex-specific polymorphisms[END_REF]. Molecular analyses of clonal populations 108 obtained from crossing experiments with characterized parental generations, including males with and 109 without W1 and W2, showed the variable emergence of both WE among 10 generations of male 110 progeny -even when the parental male exhibited no W1 and W2 [START_REF] Grevelding | Genomic instability in Schistosoma mansoni[END_REF]). These data 111 indicated a meiotic level of (illegitimate) recombination and suggested a Z-chromosomal or autosomal 112 existence of W1 and W2 in the male progeny of these crosses. However, diversity in W1/W2 113 abundance and presence occurred also among siblings of single offspring generations. This pointed to 114 an additional process generating variable copy numbers of WEs in the genome. Indeed, the analysis of 115 clonal cercariae, generated by monomiracidial snail infections, demonstrated differences in W1 and 116 W2 copy numbers within clonal cercariae obtained from the same snails but at different time points 117 post infection [START_REF] Grevelding | Genomic instability in Schistosoma mansoni[END_REF]. This was unexpected since clonal cercariae were considered to be 118 genetically identical (Jourdane and Theron 1980;[START_REF] Jones | Intra-specific variation in Schistosoma mansoni[END_REF]. Additionally, the genetic 119 heterogeneity found in individuals of the same clonal cercarial population led to the hypothesis that 120 mitotic recombination may contribute to variable WE copy numbers in males, because only asexual 121 but no sexual reproduction processes take place in the intermediate host [START_REF] Grevelding | Genomic instability in Schistosoma mansoni[END_REF]. Using an 122 in vitro technique that allowed the generation of defined clonal daughter sporocysts originating from 123 a single mother finally confirmed that W1 copy numbers can vary among daughter sporocysts 124 generated by one defined mother (Bayne and Grevelding 1993). These results clearly pointed to the 125 possibility of mitotic recombination or transposition in the intermediate host. The underlying 126 mechanisms of WE variability remained obscure.

127

A microarray study with RNA of female cercariae provided first evidence for W1 and W2 transcripts 128 [START_REF] Fitzpatrick | Use of genomic DNA as an indirect reference for identifying gender-1029 associated transcripts in morphologically identical, but chromosomally distinct, Schistosoma 1030 mansoni cercariae[END_REF]. The authors hypothesized that these transcripts may be associated to germ-129 line protection in female schistosomes such as interfering with retrotransposable element activity, as 130 previously proposed for repeat elements in Drosophila [START_REF] Pélisson | A novel repeat-associated small 1237 interfering RNA-mediated silencing pathway downregulates complementary sense gypsy 1238 transcripts in somatic cells of the Drosophila ovary[END_REF]. In 2012, a comprehensive 131 sequencing analysis to de-novo assemble S. mansoni repeat elements based on version 5 (V5) of the 132 genome [START_REF] Berriman | The genome of the blood fluke Schistosoma mansoni[END_REF]) predicted 36 WEFs (Lepesant et al. 2012b).

133

In addition to their peculiar evolutionary position as the only dioeciously living trematodes, another 134 unique feature of schistosome biology is the essentiality of a constant pairing contact for the sexual 135 maturation of the female. Pairing induces mitotic activity and differentiation processes in the female 136 that finally lead to the development of the female gonads, ovary and vitellarium (Popiel and Basch 137 1984;[START_REF] Hollander | Schistosoma mansoni: male stimulation and DNA synthesis by 1010 the female[END_REF][START_REF] Kunz | Schistosome male-female interaction: induction of germ-cell differentiation[END_REF][START_REF] Grevelding | Genomic instability in Schistosoma mansoni[END_REF]). This is a prerequisite for egg 138 production and closely associated with the pathology of schistosomiasis since eggs, which fail to reach 139 the gut lumen, migrate via the blood stream to liver and spleen. Here, these eggs lodge in the tissues 140 causing granuloma formation, inflammatory processes, and fibrosis [START_REF] Olveda | The chronic enteropathogenic disease schistosomiasis[END_REF]. The sexual 141 maturation status of a female is reversible. Upon separation, egg production stops and females de-142 differentiate to an immature status. Upon repairing, differentiation of the gonads and egg production 143 start again [START_REF] Clough | Morphology and reproductive organs and oogenesis in bisexual and unisexual 981 transplants of mature Schistosoma mansoni females[END_REF]. Although the underlying processes have not been completely understood, 144 recent transcriptomics approaches have highlighted a complex scenario of male-female interaction. A 145 comparative RNA-Seq analysis of paired and unpaired adult S. mansoni and their gonads showed the 146 occurrence of >7,000 gene transcripts in the gonads of both sexes, of which 243 (testes) and 3,600 147 (ovaries) were pairing-dependently transcribed. High numbers of differentially expressed genes in the 148 ovary were expected because of the pairing-induced sexual maturation of females [START_REF] Da | A comparative study of the reproductive system of mature, immature and 1023 "unisexual" female Schistosoma mansoni[END_REF]149 Shaw 1987;[START_REF] Kunz | Schistosome male-female interaction: induction of germ-cell differentiation[END_REF][START_REF] Neves | A new description of the reproductive system of Schistosoma mansoni 1220 (Trematoda: Schistosomatidae) analyzed by confocal laser scanning microscopy[END_REF]Beckman et al., 2010). Among others, evidence was 150 obtained for the participation of neuronal processes, guided by GPCRs (G protein-coupled receptors) 151 and neuropeptides, in male-associated processes of the male-female interaction [START_REF] Hahnel | Tissue-specific transcriptome analyses provide new insights into GPCR signalling 1056 in adult Schistosoma mansoni[END_REF]152 Lu et al. 2019). In addition, kinase signaling seem to dominate processes in females leading to gonad 153 differentiation and the maintenance of the sexual maturation status [START_REF] Grevelding | Kinases: molecular stage directors for schistosome 1053 development and differentiation[END_REF].

154

In this study, we challenge the classical view that repetitive DNA on the sex chromosomes is simply a 155 by-product of heterochromatization and provide further evidence for their functional importance. We 156 made use of genome sequencing updates in combination with recently obtained transcriptome data 157 sets to re-analyse WEFs, their structures, their chromosomal occurrence, their physical relationships, 158 and their developmentally regulated and strain-associated transcriptional activities. From the obtained 159 results we propose a hypothesis for their functional roles in schistosome biology and evolution. the Puerto Rican strain but not in schistosomula of the Guadeloupean strain. We found strain-309 dependent expression differences also between the Guadeloupean and Liberian strains. For example, 310 W11.2 expression in males of the Guadeloupean strain was higher than the respective levels of males 311 of the Liberian strain. Also with respect to schistosomula samples, we discovered differences between 312 strains, here Guadeloupean and Puerto Rican. This applied also to miracidia samples from the Liberian 313 and Puerto Rican strains, although here only one biological replicate was available for analysis.

314 Furthermore, differences in stage-dependent expression patterns occurred, within and among 315 strains. Within the Liberian strain, for example, W13.2 appeared to be expressed in miracidia and 316 paired females but not in sporocysts and males, respectively. In contrast, there was no W13.2 317 expression in females of the Guadeloupean strain. In the latter strain, no W13.2 transcripts occurred 318 in schistosomula in contrast to schistosomula of the Puerto Rican strain, which expressed W13.2. We 319 identified comparable differences for WEF expression comparing cercariae of both strains. 10).

364

According to the output of sequence similarities and structural predictions, some seemed more likely 365 than others (Supplementary Figure 11). This class of short ncRNAs is processed from stem-loop regions 366 of longer RNA transcripts and can influence post-transcriptional processes during gene expression 367 [START_REF] Bartel | Metazoan microRNAs[END_REF]. Transcripts of WEF possibly encoding miRNAs showed varying transcript amounts in life-368 cycle stages, strains, sex and tissue. For example, miRNA candidates mir-785 and mir-891 appeared to 369 form stable, miRNA-like hairpin structures (Supplementary Figure 11). Transcripts of mir-785 370 sequence-containing W12.2 occurred in a strain-influenced (lower expression levels in the 371 Guadeloupean strain compared to the other two strains; Figure 2 and Supplementary Figure 8) as well 372 as stage-restricted and sex/gonad-influenced manner (high expression levels only in miracidia, paired females and ovaries compared to males and testes of the Liberian strain; Supplementary Figure 9). For 374 mir-891 sequence-containing W8.2, we found sex-, gonad-, strain-, and stage-influenced patterns 375 (Figure 2). Interestingly, in the Guadeloupean strain a clear sex-biased expression level of W8.2 376 occurred. We detected W8.2 transcripts in female cercariae, female schistosomula and adult females, 377 whereas no W8.2 transcripts occurred in the male-sample counterparts of this strain (Supplementary 378 Figure 8). Among other possibilities, these observations suggest roles for W12.2 and W8. 12). The W2.2 expression level was higher in cercariae and 388 schistosomula of the Puerto Rican strain compared to the Guadeloupean strain (Supplementary 389 Figure 8). In the Liberian strain, W2.2 showed preferential expression in miracidia, paired females, and 390 ovaries (Supplementary Figure 9). Expression levels of WEFs containing snoRNA candidates TB11Cs2H1 391 (W26.2_W003) and GlsR19 (W36.2_W001) in adults of the Guadeloupean strain exceeded those from 392 adults of the Liberian strain (Figure 2). In the latter strain, W26.2_W003 and W36.2_W001 expression 393 dominated in miracidia compared to other samples (Supplementary Figure 9). Within the sequence of 394 W22.2, different snoRNA candidates were predicted, sR11, snoZ178, and SCARNA7 (Supplementary 395 Table 3 and Supplementary Figures 10 and11). W22.2 appeared expressed at higher levels in miracidia 396 and schistosomula of the Puerto Rican strain compared to the Guadeloupean strain. In the Liberian 397 strain, W22.2 expression is higher in miracidia than in other samples (Supplementary Figure 9).

398

In case of W5.2 and W11.2_ZW, StructRNAFinder predicted sequence parts for both miRNA and 399 snoRNA, which partly overlapped (Supplementary figs. 10 and 11). W5.2 and W11.2_ZW expression 400 levels appeared to be higher in the Puerto Rican strain compared to the Guadeloupean and Liberian 401 strains (Figure 2 and Supplementary Figure 8). However, compared to W11.2_ZW showing constant 402 expression among all samples of the Liberian strain, W5.2 expression was higher in miracidia and 403 sporocysts than all other stages and tissues of this strain (Supplementary Figure 9).

404

At this stage of the analysis, however, it is unclear whether these predictions of miRNA and snoRNA 405 correspond to biologically relevant ncRNAs. In addition, we found four WEF that contain sequences 406 reminiscent of the HHR class of ribozymes. 469 470 show sequence similarity to the highly conserved core sequences CUGANGA and GAAA (grey background), but 486 also deviations occurred (marked in yellow) [START_REF] Ruffner | Sequence requirements of the hammerhead RNA self-1270 cleavage reaction[END_REF][START_REF] Perreault | Identification of hammerhead ribozymes in all domains of life reveals novel 1241 structural variations[END_REF].

Smalpha ctaagtg----------------------------------------------------- 471 W36.2_335bp ctaagta-------------aatttcaact------------------------------ 472

487

Varying sequence similarity for HHRs occurred within these WEF (Figure 4 and Supplementary 488 Figure 13), which we also observed for W-chromosomal and autosomal copies of these WEF showing 489 incomplete or altered HHR consensus sequences with deletions, insertions, and point mutations. As 490 shown before, mutations in the CUGANGA and GAAA consensus motifs can lead to inactivity of HHRs 491 or diminished cleavage speeds [START_REF] Ruffner | Sequence requirements of the hammerhead RNA self-1270 cleavage reaction[END_REF].

492

Previous studies already confirmed the activity of the SMalpha-encoded type I HHR, represented by 493 W36.2_W001 [START_REF] Ferbeyre | Schistosome satellite DNA encodes active hammerhead 1026 ribozymes[END_REF][START_REF] Canny | Fast cleavage kinetics of a natural hammerhead ribozyme[END_REF]). Prior to experimental validation of activities of 494 predicted HHRs of other WEFs, we analysed the appropriate sequences for intact three-stem junctions 495 and perfect conservation in the CUGANGA and GAAA sequence motifs to exclude presumably inactive 496 HHRs (Supplementary Table 4) [START_REF] Ruffner | Sequence requirements of the hammerhead RNA self-1270 cleavage reaction[END_REF]). To investigate additional HHRs for their self-497 cleavage activity in vitro, we selected predicted candidates from different WEs (Figure 4). Because of 498 the high similarity between all candidates (Supplementary Figure 13), we selected only five 499 representatives. These candidates descend from WEF copies on autosomes and represent variants of 500 W25.2 and W36.2 (Figure 4C and Supplementary Table 4). Although the HHR core mainly comprised 501 the three-stem junction with its conserved CUGANGA and GAAA sequences (Figure 4A), additional 502 interactions outside the catalytic core increased ribozyme cleavage speed and structural stability 503 [START_REF] Khvorova | Sequence elements outside the hammerhead 1115 ribozyme catalytic core enable intracellular activity[END_REF][START_REF] Uhlenbeck | Less isn't always more[END_REF][START_REF] Martick | Tertiary contacts distant from the active site prime a ribozyme for 1203 catalysis[END_REF][START_REF] Perreault | Identification of hammerhead ribozymes in all domains of life reveals novel 1241 structural variations[END_REF]. Therefore, we 504 extended predicted HHR sequences by 11-13 nucleotides, which naturally occur at the 5'-and 3'-ends 505 of the motifs.

506

Our analysis confirmed the activity of HHR candidates 1-3, which cleave into the expected 5'-and 3'-507 subfragments (Figure 4D). Due to the imprecise run-off transcription of the T7 RNA polymerase, we 508 observed additional bands for the 3'-cleavage fragment [START_REF] Chamberlin | Characterization of T7-specific ribonucleic acid polymerase. 1. General 968 properties of the enzymatic reaction and the template specificity of the enzyme[END_REF]. Candidate 4 contains 509 a mutation in the catalytic core (CUGANAA), and candidate 5 has a deletion at the cleavage site. Both 510 alterations likely rendered these candidates inactive based on detailed studies on the effect of 511 mutations to the HHR core on cleavage [START_REF] Ruffner | Sequence requirements of the hammerhead RNA self-1270 cleavage reaction[END_REF]). In summary, we confirm all predictions in 512 this co-transcriptional cleavage assay.

583

For the HHR-containing WEF, species comparison showed a different picture. Looking again for best 584 hits, we detected orthologs of three of four WEFs (W25.2, W27.2, W36.2) in all examined schistosome 585 species, with S. mansoni showing the longest size variants. In nearly all cases, the integrity of the HHR 586 motifs was fully preserved. The exception was W26.2, of which only a small part without HHR motif 587 appeared to be conserved in S. japonicum (Figure 6). When we searched for the HHR sequence of 588 W26.2 in the S. japonicum genome by BLASTn, we found no hit (data not shown). 601 Discussion

602

The days of "junk DNA" are over. When the senior authors of this article studied genetics at their 603 respective universities, the common doctrine was that the non-protein coding part of eukaryotic 604 genomes consists of interspersed, "useless" sequences, often organized in repetitive elements such as 605 satDNA. The latter might have accumulated during evolution, e.g. as a consequence of gene 606 duplication events to separate and individualise gene function [START_REF] Britten | Repeated sequences in DNA. Hundreds of thousands of copies of DNA 942 sequences have been incorporated into the genomes of higher organisms[END_REF][START_REF] Comings | The structure and function of chromatin[END_REF]607 Ohno 1999). This view has fundamentally changed (Biscotti et al. 2015a) Arabidopsis [START_REF] Woodhouse | Transposed genes in Arabidopsis are often associated 1324 with flanking repeats[END_REF]. Whether this or similar mechanisms may apply to the WEF of S.

638

mansoni has still to be elucidated.

639

Based on evidence for transcriptional activity and dynamic occurrence, recent years have unravelled 640 novel roles for satDNA (Biscotti et al. 2015a). Among these is the establishment of heterochromatic

Figure 1 :

 1 Figure 1: Showcase for the new definition of WEFs in S. mansoni. A, Result of the dotplot analysis for WE W29 in

  320

Figure 2 .Figure 2 :

 22 Figure 2. WE Transcript Profiles Across Different S. mansoni Strains, Life Stages, Sexes, and

Figure 3 :

 3 Figure 3: Clustal-based alignment of the WEF W36.2, W25.2, W26.2 and W27.2 compared to the SMalpha family

Figure 4 :Figure 5 :

 45 Figure 4: HHR candidates as part of WEs were selected for functional analysis. A, Consensus sequence and

Figure 6 :

 6 Figure 6: Diagram of structural differences of WEF W25.2, W26.2, W27.2, W36.2 in S. mansoni, S. rodhaini,

  . and C.C. performed the bioinformatics analyses supported by advice and help from Z.L., J.B., and 896 A.G. J.O. and C.E.W. planned and performed the biochemical experiments with ribozymes and analysed 897 the data. C.G.G., C.G., and C.E.W. wrote the manuscript. C.G.G. and C.G. designed the study and 898 provided funding (FUGI). C.E.W. provided funds (DFG) for the ribozyme part.---------------------------------------------------W36.2_335bp ctaagta-------------aatttcaact---------------------------

160 161 Results 162 There are 19 W-Element Families in S. mansoni

	173	(Supplementary Figure 1). W1, W8, W10, W13, W17, W18, W20, W22, W31, W34, and W35 exclusively
	174	matched to W-scaffolds (Table 1). Some WEs, or partial versions thereof, also occurred on autosomes.
	175	BLAST analyses revealed that some WEs of a single WEF clustered in single chromosomal regions,
	176	whereas other WEFs occurred as separate clusters in different chromosomal regions. For WEF 26
	177	(W26.2), we detected clusters of different total lengths in two yet undefined regions of W, provisionally
	178	designated W003 and W010 in V7 (Table 1). Although WEs of all 36 examined WEFs aligned to V7,
	179	repeat units of W9, W15, W19, and W32 showed no clustering of multiple, aligned repeats. Instead,
	180	we found a wide distribution of mostly partial fragments of these WEs on sex chromosomes and
	181	autosomes (data not shown). Because these WEs presented no multiple repeat character, we excluded
	182	them from further analysis.
	183
	184
	163	To get a new overview of WEFs in S. mansoni, we performed a local BLAST (Basic Local Alignment
	164	Search Tool) search of the published WEF (Lepesant et al. 2012b), which were based on version 5 (V5)
	165	of	the	genome,	against	version 7 (V7;	PRJEA36577,	Puerto	Rican	strain;
	166	https://parasite.wormbase.org; Howe et al. 2017). Except for W32, we found all 36 examined WEF
	167	with a high percent identity on W-scaffolds, which cover DNA sequences, which are likely derived
	168	from/correspond to the female-specific sex chromosome W but have not been exactly assembled yet.
	169	We detected differences in DNA-sequence similarity among individual WEs of a single WEF. For
	170	instance, the ten most significant hits of W6 elements (monomer repeat length: 310 bp) showed no
	171	mismatch and correspond 100%, whereas the ten most significant hits of W23 elements (monomer
	172	repeat length: 125 bp) showed deletions and mismatches resulting in 76-100% identity

Table 1 . BLAST Results of WEFs in the Genome Version 7 of S. mansoni No. WE (V7) WE (V5) WE monomer lengths Number of copies Scaffold Start Stop Total length of the array

 1

		1	W1.2	W1; W23	475	270	SM_V7_W003	320,915	513,415	192,500
		2	W2.2	W2; W3	709-711	up to 34.1	SM_V7_W001	422,658	479,629	56,971
		3	W4.2	W4; W30	1,206	11.7	SM_V7_W004	1	14,111	14,110
		4	W5.2	W5	1,104	41.2	SM_V7_W018	1	45,515	45,514
		5	W6.2	W6; W18; W35	715-718	up to 45.7	SM_V7_W014	1	53,884	53,883
		6	W7.2	W7	980	29	SM_V7_W016	1	28,424	28,423
		7	W8.2	W8	538	60.6	SM_V7_W015	1	32,211	32,210
		8	W10.2	W10	671	59.2	SM_V7_W021	1	39,689	39,688
		9	W11.2_W002	W11; W14; W28	903	33.8	SM_V7_W002	712,429	755,874	43,445
			W11.2_ZW	W11; W14; W28	1,294	29.5	SM_V7_ZW 1,1804,837 11,863,496	58,659
		10	W12.2	W12	475-499	up to 46.1	SM_V7_W004	206,491	279,311	72,820
		11	W13.2	W13; W17; W20; W33	524-646	up to 23.8	SM_V7_W003	229,593	276,593	47,000
		12	W16.2	W16; W21	317	204.2	SM_V7_W007	57,292	122,025	64,733
		13	W22.2	W22	604	106.6	SM_V7_W008	1	64,369	64,368
		14	W24.2	W24	636	3.6	SM_V7_W020	1	2,259	2,258
		15	W25.2	W25	415-428	up to 115.4 SM_V7_W012	1	49,306	49,305
		16 W26.2_W003	W26	399-402	up to 146.7 SM_V7_W003	1	161,189	161,188
			W26.2_W010	W26	400	152.6	SM_V7_W010	1	61,065	61,064
		17	W27.2	W27; W29	403	101.3	SM_V7_W017	1	40,810	40,809
		18	W31.2	W31; W34	260	205.5	SM_V7_W004	41,351	114,609	73,258
		19 W36.2_W001	W36	333-335	173.4	SM_V7_W001	1	104,272	104,271
			W36.2_W005	W36	332	up to 450.5 SM_V7_W005	1	149,646	149,645
	185
	186	Tab. 1: Based on genome version V7, a number of 19 WEF (WE V7) has been newly defined (first column, No. 1 -
	187	19). Each WEF consists of one or more of those repeats that were previously found as individual WEs in version
	188	5 of the genome (WE V5; Lepesant et al. 2012a). In some cases (No. 9, 16 and 19), representing WEF (W11.2,
	189	W26.2 and W36.2, respectively) WEs were found on two different scaffolds each associated with the W
	190	chromosome and thus split into two subfamilies each. Furthermore, information of the length of WE monomers

and 97 bp, respectively

(Lepesant et al. 2012a)

, to be part of the same repeat 203 unit (Figure

1

). According to V7, the W27/W29 monomers are separated by spacer sequences of 134 204 by and 62 bp, respectively, which together with the W27/W29 monomers form a new monomer of a 205 total length of 403 bp (Supplementary Figure 2 H). This 403 bp monomer represents a single repeat 206 unit of the newly named W27.2 family. This designation relates to the former nomenclature used for

Table 2 . WEFs on Autosomes WE Length Presence on autosomes Minimal -maximal length Copy numbers on autosomes WE on autosomes

 2 the formerly described 36 WEF(Lepesant et al. 2012a), which resulted in newly 221 defined 19 WEFs (Table 1 and Supplementary Figure 2 A and B). These 19 WEFs vary in their copy 222 numbers from 3 (W24.2) to 450 (W36.2) times at one or several chromosomal locations of the V7 223 genome assembly (Table 1). These numbers may underestimate the actual copy-numbers of the WEF 224 in the genome since large satellite WEF-containing blocks are absent from the assembly. These findings suggested the presence of WE on Z and/or on autosomes. To find evidence for 228 one or both possibilities, we took WE monomers of the newly defined WEFs and WE-transcript 229 sequences detected in males for BLAST analyses against V7 in the NCBI database, which allowed Summary of the BLAST analyses of WEs against genome version V7 of S. mansoni in the NCBI database.

	225	Some WEs Are Also Present On Autosomes
	226	Former studies reported on the presence of WEs in male S. mansoni (Grevelding 1995; Quack et al.
	227	1998). W2.2	709-711	all	37 -705	68	parts
		W4.2	1,206	all	44 -390	1,101	parts
		W5.2	1,104	all	26 -742	3,454	parts
		W6.2	715-718	all	36 -161	5,382	parts
		W7.2	980	2,4,6,7	80 -156	5	parts
		W11.2	903-1,294	all	30 -514	18,610	parts
		W12.2	475-499	all	26 -102	172	parts
		W13.2	524-646	1,2,4,6	38 -131	19	parts
		W16.2	317	all	26 -314	203	parts
		W22.2	604	all	26 -232	1,897	parts
		W24.2	636	all	28 -450	2,302	parts
		W25.2	415-428	all	25 -428	39,648	parts and full-length (408)
		W26.2	399-402	all	25 -399	25,162	parts and full-length (321)
		W27.2	403	all	27 -403	55,303	parts and full-length (174)
		W36.2	333-335	all	25 -335	42,670	parts and full-length (858)
	235	Tab. 2:

2 (Table 1). 218 Therefore, with respect to the uninterrupted repetitive patterns of WEs within respective 219 chromosome regions as well as small distances or overlaps of different WEs within a genome locus, we 220 integrated some of 230 assigning hits to individual chromosomes. For 4 out of 19 WEFs (W1.2, W7.2, W8.2, W10.2), we found 231 no autosomal localization. Represented by either full-length or partial WE sequences, 15 WEFs were 232 distributed among all 7 autosomes (Table 2 and Supplementary Figure 3A). 233 234 236 Of the 19 newly defined WEFs, representatives of 15 WEF were found to be distributed among all 7 autosomes, 237 whereas W7.2 and W13.2 were found on 4 autosomes. WEs, their original length (bp), their copy numbers on 238 autosomes, and minimal and maximal lengths (bp) of the WE parts are listed. Next, the numbers of full-length 274 275 WE Transcripts

Occur in All Investigated Strains, Life stages, Sexes, and the Gonads of S. mansoni 276

  To get a first overview of WEF transcript occurrence in the different biological samples we performed 277 a sample-distance matrix analysis. By pairwise comparisons, this approach provided information about 278 WEF transcript amounts based on the summation of all reads of transcribed WEs -independent of the 279 composition of each WEF that contributed to the WE transcript pool of each sample. These samples 280 included miracidia, cercariae, sporocysts, female and male schistosomula, paired and unpaired males 281 and females, as well as testes and ovaries of paired and unpaired females and males, respectively, and

	282	also samples of other strains, depending on availability of sequencing data. We selected data sources
	283	with at least two biological replicates for the individual life stages, both sexes, and the gonads
	284	(Supplementary Table 1) and normalized the data using DESeq2 prior to further analysis. When
	285	available, we also included replicates.
	286	The result indicated remarkable differences in WE transcript amounts among the investigated samples.
	291	Rican strain (Cer1-3) and the Liberian strain (CerM1-CerM3; CerF1-CerF3). Generally, schistosomula
	292	samples (SomF/M) and samples from males and testes (sM1-sM3; bM1-sM3; sT1-sT3, bT1-bT3)
	293	seemed to be homogeneous with respect to transcript amounts. Bigger differences occurred between
	294	unpaired females (sF1-sF3), which revealed an overall higher WE transcript level compared to paired
	295	females (bF1-bF3). Finally, we found differences also among biological replicates as exemplified by
	296	samples from females and ovaries (bF1-bF3; bO1-bO3; sO1-sO2), and even among replicates as
	297	observed for schistosomula (SomM/SomF samples). In summary, this first overview indicated a high
	298	degree of variability in overall WE transcript amounts between life stages, strains, and sexes, but also
	299	between biological replicates.
	300
	301	WEF Expression Levels Differ Among and Within Strains, Life Stages, Sexes, Pairing-Status and
	302	Gonads of S. mansoni
	303
	304	Next, we performed differential expression analysis to investigate WEF expression profiles across
	305	strains, different life stages (from miracidia to adults), sex-, gonad-, and/or pairing-dependent
	306	expression. Again, we discovered remarkable differences (Figure 2 and Supplementary figs. 8 and 9).

287

Furthermore, we observed clustering of samples with similar or dissimilar levels of the total amount 288 of WE transcripts (Supplementary figs. 6 and 7). In the Liberian strain, miracidia (Mir_1, Mir_3, Mir_6) 289 showed high deviations of WE transcripts compared to other stages but these levels differed from 290 miracidia of the Puerto Rican strain

(Mir)

. We detected similar deviations for cercariae of the Puerto 307 Strain-dependent expression occurred, among others, for W7.2, with transcripts in schistosomula of 308

Table 2 ;

 2 The 338 color code indicates various levels of expression (from dark blue = no transcripts (0) to deep red = high transcript 339 level (25) in the form of log2-transformed normalized counts of all data sets used. Supplementary Figure

		available, biological and technical replicates were included. The first number behind a sample abbreviation
	330	indicates the number of the biological replicate. The second number indicates the technical replicate. For
	331	example, SomF1.1 indicates the first biological and first technical replicate of a female schistosomula, sample.
	332	SomF1.2 is the second technical replicate of this schistosomula sample. Samples without numbers had no
	333	replicate (Supplementary Table 1). Furthermore, biological symbols indicate female and male samples. In cases
	334

without symbol, the sample origin was mixed-sex. The horizontal line at the top of this figure shows a color code 335 for the different schistosome strains, which were the basis for generating RNA-Seq data (Supplementary Table

336

1). We used strains from Guadeloupe (beige), Liberia (purple), and Puerto Rico (grey). The dendrogram at the 337 outer left side indicates relationships between appropriate WEs/WEFs as labelled at the outer right side.

  , and our study is the first one 608 addressing this issue with structural, functional, and evolutionary aspects for the genome of a

	609	multicellular parasite.
	610	Using V7 of the S. mansoni genome we re-analysed WEFs, which previous studies described as
	611	organized in 36 families of repetitive, non-protein coding DNA elements occurring on the W-
	612	chromosome (Lepesant et al. 2012b). From our results, we deduce 19 WEFs based on the fact that
	613	many of the original 36 WEFs are fused together. These 19 WEFs exhibit a surprising diversity of
	614	features. As expected for repetitive DNA, there is high sequence similarity but also variation among
	615	the repeat units of each single WEF. Besides few partial WE sequences of some WEFs on the WZ
	616	chromosomes, we found for 15 out of 19 WEF copies also on autosomes. Here, copy numbers
	617	considerably varied as well as the lengths of WE units of the respective WEFs. The autosomal
	618	occurrence possibly indicates the potential of element mobility. Indeed, in some WEF sequences we
	619	found residues of features typical for mobile genetic elements such as retrotransposons or
	620	retroviruses. Although we found no evidence of complete ORFs coding for a transposase, in an
	621	exemplary case, we detected direct repeats and TSD, which are often, but not always, found at
	622	integration sites of mobile genetic elements (Jensen et al. 1994; Han et al. 2014). Repetitive elements
	623	such as satDNA have been discussed to originate from transposable elements (TEs). Both share
	624	sequence similarity and organization patterns, which suggests a mutual relationship between satDNA
	625	and TEs. This probably influenced satDNA evolution and its roles on genome architecture and function
	626	(Meštrović et al. 2015; Satović et al. 2016). The mechanism of satDNA formation from TEs is unclear
	627	yet. However, structural elements such as palindromes, direct/inverted repeats, and the ability of
	628	stem-loop formation may be involved as well as illegitimate recombination events and deletions based
	629	on double-strand breaks and excision. These factors might drive rearrangements of TEs and the
	630	production of sequence parts as templates for further amplification to form tandemly organized
	631	satDNA clusters. This may apply to at least some of the WEs described here (W2.2, W4.2, W5.2, W11.2,
	632	and W16.2), which showed partial sequence homology to known mobile genetic elements. SatDNA
	633	monomers might be recognized by transposase-driven mechanisms acting on short DNA sequence
	634	motifs of these DNA sequences, which are recognized by enzymes related to transposases (Meštrović
	635	et al. 2015). As alternative to transposon-like cut and paste mechanisms, reintegration of repeat
	636	elements may happen via circular DNA intermediates generated from tandem repeats, as shown in
	637

  nt) was gel-purified after in vitro transcription, dephosphorylated using the antarctic 877 phosphatase (NEB) and labelled at its 5'-end using the T4 polynucleotidyl kinase (NEB) and [γ 32 P]-ATP.878Following another gel-purification, we used the 5'-labeled RNA to create a size standard by alkaline 879 hydrolysis and RNase T1 digestion. For experimental validation and product visualisation, we separated used for analysis in this study are available at ENA (http://www.ebi.ac.uk/ena).886These data originated from published and yet unpublished studies covering adult and larval 887 schistosomes[START_REF] Grevelding | The female-specific W1 sequence of the Puerto Rican strain of Schistosoma 1045 mansoni occurs in both genders of a Liberian strain[END_REF][START_REF] Protasio | A systematically improved high quality genome and transcriptome of the 1255 human blood fluke Schistosoma mansoni[END_REF][START_REF] Wang | Functional genomic characterization of neoblast-like stem 1306 cells in larval Schistosoma mansoni[END_REF][START_REF] Lu | Schistosome sex matters: a deep view into gonad-specific and pairing-dependent 1185 transcriptomes reveals a complex gender interplay[END_REF][START_REF] Picard | Sex-biased transcriptome of Schistosoma mansoni: host-parasite interaction, 1244 genetic determinants and epigenetic regulators are associated with sexual differentiation[END_REF]; 888 Lu et al. 2017; https://www.ebi.ac.uk/ena). Sample types, strains, replicates, accession numbers and 889 references are listed in Supplementary Table 1. Further sequence information, as indicated in the text, 890 was obtained from WormBase ParaSite (version 15, October 2020; https://parasite.wormbase.org),

	747 782 852	chromatin stage, in which both euchromatic and heterochromatic states can be adopted. Indeed, this Materials and Methods Functional Prediction
	748 853	has been confirmed for S. mansoni by showing that euchromatic histone modifications around WEs For the prediction of sequences coding functional RNA, we used StructRNAfinder, an integrative tool
	749 854	dominate in the intermediate snail-host but decrease afterwards (Lepesant et al. 2012a). In allowing the identification, functional annotation, and taxonomic allocation of sequences to RNA
	750 855	P. falciparum, bistable chromatin has been discussed as a mechanism regulating variant gene families by secondary structure inference
	751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 W25_415_3 (154 880 expression patterns within clonal populations (Llorà-Batlle et al. 2019). In nematodes, mobile genetic elements contribute to genome plasticity in the absence of sexual reproduction (Castagnone-Sereno et al. 2014). This fits to the general view of the impact of mobile genetic elements on genome evolution and adaptation, which is important for organisms frequently facing new environments such as parasites (Schrader and Schmitz 2019), particularly those with complex life cycles. The evolution of repetitive DNA was also associated with reproductive isolation, founding new species, genome integrity, and karyotype evolution (Coghlan 2005; Biscotti et al. 2015a; Lower et al. 2018). Especially with respect to heterochromatic regions, remarkable diversity of karyotype patterns exist for different schistosome species leading to the definition of six clades correlating with the different geographical distribution as well as with the hypothesized Asian origin of schistosomes (Hirai et al. 2000; Lawton et al. 2011). Our comparison of WEFs in selected schistosome species of different clades provide a first hint for their potential contribution to karyotype variability, and thus speciation. This may include a varying repertoire of WEF-encoded regulatory RNAs, which differs between these species. Repetitive DNA discloses high sequence and copy number variability among and within species but also in closely related organisms, which points to rapid evolution (Lower et al. 2018). This may involve coevolution with regulatory RNAs as it was hypothesized for satellite repeats and long noncoding RNAs (Lee et al. 2019) as well as the dual relationship between Alu repeats (short interspersed nuclear elements of the human genome) and miRNAs. Duplication events involving Alu elements have favoured the expansion of miRNA clusters and their expression (Lehnert et al. 2009). From the data obtained in our study and against the background of recent literature, it is tempting to speculate that more of the WE "junk-DNA" than expected might be functional and relevant. WEs of all investigated WEFs exhibit a capricious incidence, , and they are transcribed in a stage-, sex-, pairing-, gonad, and strain-specific or -preferential pattern. From exemplary findings of features typical for the activity of mobile genetic elements, we hypothesize that W elements may have a mobile character. Together with previous findings of intra-clonal recombination events of WEs (Grevelding 1999; Bayne and Grevelding 2003), their presumptive role in sex chromosome emergence (Lepesant et al. 2012), their putative capacity to express regulatory RNAs, we propose that W elements might influence the biology of S. mansoni. Furthermore, based on the variable occurrence of WEFs in different schistosome strains, isolates, and even species, we hypothesize that the W elements represent one of the sources of heritable variability in the evolution of the family Schistosomatidae. (https://structrnafinder.integrativebioinformatics.me/run.html) (Arias-Carrasco et al. 2018). StructRNAfinder displays sequence consensus alignments for RNA families, according to Rfam database (RNA families, data base version 14.2, Kalvari et al. 2018; https://rfam.xfam.org), but also provides a taxonomic overview for each assigned functional RNA. As input, we generated FASTA files with complete RNAs from the Bowtie2 output by using the tool "bedtools MergeBED" (Galaxy Version 2.27.1) (Quinlan and Hall 2010) to merge overlapping and adjacent regions. We chose the option cmsearch and an E-value of 0.01. In the output options, we selected "report only best hit per sequence". Determination of Ribozyme Activity During in vitro Transcription We produced transcription templates containing the T7 promoter sequence by extension of partly complementary oligonucleotides (Supplementary Table 6) using the Phusion DNA Polymerase (ThermoScientific TM ). We purified templates by phenol/chloroform extraction and ethanol precipitation. For in vitro transcription, 1 µg purified template, 1x transcription buffer, 3% DMSO, 4 mM NTPs, 0.013 U thermostable inorganic pyrophosphatase (NEB), and 25 ng of T7 RNA polymerase (laboratory preparation) were combined in a 30 µl reaction and incubated at 37°C for 2 h. The 1x transcription buffer contained 80 mM HEPES-KOH (pH 7.5), 24 mM MgCl 2 , 2 mM Spermidine, and 40 mM DTT. Labelling of HHR candidates during transcription occurred by the addition of [α 32 P]-CTP. We stopped reactions by adding RNA loading-dye composed of 2.5 mM Tris-HCl (pH 7.6), 20% formamide, 0.06% bromophenol, and 0.06% xylene cyanol. A cleavage-deficient, elongated variant of the size standard and internally labelled ribozyme cleavage products by denaturing 20% 881 polyacrylamide gel electrophoresis (PAGE) and detected the bands with an Amersham Typhoon Imager 882 (GE Healthcare). 883 884 Data Availability 885 BioProject numbers: PRJEA36577 (S. mansoni), S. rodhaini (PRJEB526 -Republic of Burundi), All sequencing data 891 892 PRJNA520774 (HuSjv2, S. japonicum), and PRJNA78265 (S. haematobium).
	641 781	states at centromeres and telomeres, which is indispensable for preserving chromosome integrity and

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Acknowledgements

901

We thank all the members of the Grevelding laboratory, particularly Simone Häberlein, Thomas Quack 902 and Oliver Weth for fruitful discussions. The work was supported by the Wellcome Trust FUGI grant


Luigi C Berselli

Alex Kaltenbach

Roger Lewandowski

Michael R Ůž

I Čka

ON THE EXISTENCE OF WEAK SOLUTIONS FOR A FAMILY OF UNSTEADY ROTATIONAL SMAGORINSKY MODELS

Keywords: 2010 Mathematics Subject Classification. 35Q35, 76F02, 47H05, 47J35 Bochner pseudo-monotone operators, Rotational turbulence models, evolution equations. 1

In this paper we show that the rotational Smagorinsky model for turbulent flows, can be put, for a wide range of parameters in the setting of Bochner pseudo-monotone evolution equations. This allows to prove existence of weak solutions a) identifying a proper functional setting in weighted spaces and b) checking some easily verifiable assumptions, at fixed time. We also will discuss the critical role of the exponents present in the model (power of the distance function and power of the curl) for what concerns the application of the theory of pseudo-monotone operators.

Introduction

In this paper we introduce the unsteady general rotational Smagorinsky model for incompressible turbulence

∂ t v + ω × v + curl C α α |ω|ω + ∇q = f in (0, T ) × Ω, ω = curl v in (0, T ) × Ω, div v = 0 in (0, T ) × Ω, v = 0 on (0, T ) × ∂Ω, v(0) = v 0 in Ω, (1.1)
where Ω is a smooth bounded domain in R 3 , is the Prandtl mixing length, α > 0 is a given exponent, C α > 0 is a calibration constant, v is the mean velocity, ω is the mean vorticity, and q is the sum of the Bernoulli pressure of the fluid and certain potentials such as the turbulent kinetic energy and others. Here and in the sequel, for each smooth vector u : R 3 → R 3 we define as curl the vector

(curl u) i := 3 j,k=1 ijk ∂u j ∂x k ,
where ijk is the Levi-Civita totally anti-symmetric tensor.

Note that in the equations (1.1) the linear dissipative term -ν∆v is not present, since we are considering flows at very high Reynolds number, and then viscous effects are negligible compared to the Reynolds stresses. Note that the presence of the linear dissipative term, for any ν > 0 will allow for some simplifications of the proofs. Nevertheless, to obtain result and estimates independent of ν > 0, a treatment as the one we provide is requested.

According to standard assumptions (see (2.7) and (2.8) in Section 2.3), we will assume that behaves as the distance to the boundary. This means that (x) ≈ d(x) when x ∈ Ω and x is close to the boundary ∂Ω (see (2.9) and (2.10) in what follows). As it is common in turbulence modeling, we assume that the flow fields are stochastic processes, and the bar operator stands for the expectation in the Reynolds decomposition v = v+v , π = π +π , where π denotes the pressure and π the mean pressure (see Section 2.1 below, even if other choices are possible, as for instance denoting by the bar operator the long time-averaging).

The natural value of the parameter α is equal to 2 and this model is similar to the widely used Smagorinsky model, but the term div(C s 2 |Dv|Dv) is replaced here by curl(C 2 2 |ω|ω). The equivalence between both models can be understood for homogeneous isotropic turbulence, by the equality of the enstrophy |ω| 2 to the total mean deformation 2|Dv| 2 . The equivalence can be obtained by a straightforward generalization of [START_REF] Rebollo | Mathematical and numerical foundations of turbulence models and applications[END_REF]Lemma 4.7]. Then, according to this equality, in [9, Section 5.5.1] it is proved that the -5/3 Kolmogorov law yields to express the eddy viscosity as ν T = C 2 2 |ω|. The rotational structure of the eddy diffusion is a peculiarity of the model which is suitable for high-speed flows with thin attached boundary-layers. The mathematical treatment of rotational models is one of the main theoretical contribution of this paper.

The numerical performance of this model in the steady state case has been initially tested by Baldwin and Lomax [START_REF] Baldwin | Thin-layer approximation and algebraic model for separated turbulent flows[END_REF], so that this model is also known as the Baldwin-Lomax model. Numerical analysis foundations also in the statistical non-equilibrium setting can be found in [START_REF] Rong | Extension of a simplified Baldwin-Lomax model to nonequilibrium turbulence: model, analysis and algorithms[END_REF].

The analytical properties of a steady version of this model have been recently studied in [START_REF] Berselli | On the existence of weak solutions for the steady Baldwin-Lomax model and generalizations[END_REF] in the setting of weighted Sobolev spaces. Some unsteady versions, with the presence of a dispersive term -which allows for a more classical treatment-have been recently studied in [START_REF] Nguyen | Some results on turbulent models[END_REF][START_REF] Berselli | Rotational Forms of Large Eddy Simulation Turbulence Models: Modeling and Mathematical Theory[END_REF].

The steady version can be treated within the standard theory of monotone operators, plus a localization argument, while the unsteady one requires a more delicate argument to deal with the precise choice of spaces and formulation of the problem. As we will prove, a proper definition of the functional setting will make system (1.1) to fit into the framework of evolution problems with Bochner pseudo-monotone operators, for which the theory have been recently developed by two of the authors in [START_REF] Kaltenbach | Note on the existence theory for pseudo-monotone evolution problems[END_REF]. The theory developed in [START_REF] Kaltenbach | Note on the existence theory for pseudo-monotone evolution problems[END_REF] represents an extension and an adaption to unsteady problems of the classical theory of pseudo-monotone operators from Brézis [START_REF] Brezis | Les opérateurs monotones, Séminaire Choquet: 1965/66, Initiation à l' Analyse[END_REF], [START_REF] Brezis | Équations et inéquations non linéaires dans les espaces vectoriels en dualité[END_REF], already described in the classical monograph of Lions [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]. Our main result is the following, which covers all possible positive powers of the distance function which are strictly smaller than the critical value α = 2.

Theorem 1.1. Let us suppose that (x) = d(x, ∂Ω) and let α ∈ [0, 2), 0 < T < ∞, v 0 ∈ L 2 σ (Ω), and f ∈ L 3/2 (0, T ; (W 1,3 0 (Ω, d α ) * ).
Then, there exists a weak solution to the initial boundary value problem (1.1)

such that v ∈ C([0, T ]; L 2 σ (Ω)) ∩ L 3 (0, T ; W 1,3 0,σ (Ω, d α )), and for all t ∈ [0, T ] 1 2 v(t) 2 + ˆt 0 ˆΩ C α d α (x)|ω(s, x)| 3 dx ds = 1 2 v 0 2 + ˆt 0 f , v W 1,3 0 (Ω,d α )
ds. The limitation α < 2 seems to be intrinsic to the problem due to the fact that d α is not anymore a Muckenhoupt weight for α ≥ 2 (cf. Definition 4.4). Hence, for α ≥ 2 most of the analytical properties may fail, since we cannot ensure that the quantity from the energy estimates controls the (weighted) full gradient of the solution. For values of α larger or equal than 2, even the weak formulation, the density of smooth functions, and the meaning of the boundary conditions may fail; the solution of the problem, if possible, would pass through the introduction of a more general setting, of very weak solutions.

In the last section we will also consider the existence for a family of problems with different powers of the vorticity in the turbulent stress tensor, still with the distance function raised to any exponent smaller than the critical one, cf. Thm. 5.5.

Plan of the paper. In Section 2 we derive the Rotational Smagorinsky model from a classical turbulence modeling process, in Section 3 we define the notion of Bochner pseudo-monotone operators and we recall the main result for general evolutionary problems, in Section 4 we recall the main results on weighted spaces, which will be used to properly formulate the problem. Next in the final Section 5 we show how the hypotheses apply to problem (1.1), for relevant choices of the weight functions and discuss generalization and critical values of the parameters.

Modeling

v t + ω × v -ν∆v + ∇ π + |v| 2 2 = f in (0, T ) × Ω, ω = curl v in (0, T ) × Ω, div v = 0 in (0, T ) × Ω, v = 0 on (0, T ) × ∂Ω, v(0) = v 0 in Ω, (2.1)
where v = v(t, x, ω) is the velocity field, π = π(t, x, ω) the pressure, ω = curl v the vorticity, (t, x) ∈ R + × Ω, ω ∈ X(B, P ), where X(B, P ) is a given probability space on the space of initial data. For instance, if f = 0 (the argument can be adapted also to include a smooth enough external force) it holds that for each divergence-free element of v 0 ∈ H 1/2 (Ω) there exists a lower bound T = T ( v 0 1/2 ) > 0 for the life-span of the unique Fujita-Kato mild solution. Since the life-span can be estimated with the norm of the initial datum, by fixing X = B(0, R) ⊆ H 1/2 (Ω) ∩ {∇ • v = 0} for some R > 0, then the life-span is bounded from below by some T X > 0. This means that for each v 0 ∈ X, there exists a unique v ∈ C(0, T X ; H 1/2 (Ω)) solution of the NSE.

We then introduce P , which is a probability measure on the Borel sets of X. More specifically, P can be constructed as limit of averages of Dirac measures as in [START_REF] Rebollo | Mathematical and numerical foundations of turbulence models and applications[END_REF] or the renormalized Lebesgue measure constructed from the Borel sets of X. The final result does not depend on the choice of P . Let us denote the expectation with a bar, hence

v 0 = ˆX v 0 dP (v 0 ), and v(t, x) = ˆX v(t, x, v 0 ) dP (v 0 ) π(t, x) = ˆX π(t, x, v 0 ) dP (v 0 ).
More generally, for any field Ψ = ∂ t v, ω, ∇v, ∆v, |v| 2 2 ..., we can define the statistical mean as Ψ(t, x) = ˆX Ψ(t, x, v 0 ) dP (v 0 ), and consequently we can perform the usual decomposition of Ψ as Ψ = Ψ + Ψ , which is known as the Reynolds decomposition. The properties of the statistical averaging process imply (Reynolds rules) that for all Ψ, Θ ∈ X

∂ t Ψ = ∂ t Ψ, ∇Ψ = ∇Ψ, Ψ = 0, Ψ Θ = Ψ Θ,
hence, taking the expectation of the NSE (2.1) yields

v t + ω × v + ω × v -ν∆v + ∇ π + |v| 2 2 + |v | 2 2 = f , ω = curl v, div v = 0, v| ∂Ω = 0, v| t=0 = v 0 . (2.2)
The basic closure and modeling problems concern expressing ω × v in terms of averaged variables.

2.2. Rotational Reynolds stress. When taking the expectation of the NSE with the convective term written in the usual form, we get the term div(v ⊗ v ). The quantity σ (r) = v ⊗ v is called the Reynolds stress and the Boussinesq assumption consists in assuming that

σ (r) = -ν T Dv,
where ν T ≥ 0 is an eddy viscosity which remains to be determined and modeled in terms of v. If we want to use such a Boussinesq assumption, we must express the turbulent stress (which is a vector in the rotational formulation)

s := ω × v in terms of derivatives of mean quantities. This is similar to the approach used when modeling the more standard Reynolds stress tensor. We prove in what follows the following theorem Theorem 2.1. Assume that Ω is connected and of class C 1 . Then, there exists a vector a (R) = a (R) (t, x) and a scalar potential Φ = Φ(t, x) such that

v t + ω × v + curl a (R) -ν∆v + ∇(π + 1 2 |v| 2 + k -Φ) = f , div v = 0, v| ∂Ω = 0, v| t=0 = v 0 , (2.3)
where k = 1 2 |v | 2 is the turbulent kinetic energy. Proof. Let a (R) and Φ be given by:

a (R) (t, x) = 1 4π ˆΩ curl s(t, x ) |x -x | dx + 1 4π ˆ∂Ω s(t, x ) |x -x | × dσ(x ), Φ(t, x) = 1 4π ˆΩ div s(t, x ) |x -x | dx - 1 4π ˆ∂Ω s(t, x ) |x -x | • dσ(x ).
(2.4)

Therefore, by the Helmholtz-Hodge theorem, we have the relation

ω × v = curl a (R) -∇Φ. (2.5) Inserting (2.5) into (2.2) gives (2.3).
The vector a (R) , which is continuously and uniquely determined by formula (2.4), is called the rotational Reynolds stress tensor. From now on we write q instead of π

+ 1 2 |v| 2 + k -Φ. 2.3.
Closure assumption: Rotational Smagorinsky model. In order to finish the modeling of turbulent quantities, it remains to link a (R) to the mean vorticity ω. Notice that a (R) has the dimension of a squared velocity, while ω those of a frequency. Therefore, adapting the Boussinesq assumption to this case yields to assume

a (R) = ν T ω,
in which ν T ≥ 0 is a quantity with the dimensions of a viscosity. According to the -5/3 Kolmogorov law and following [9, Section 5.5.1], we can assume (for an homogeneous and isotropic flow, in the limit ν → 0)

ν T = ν T ( , |ω|),
where is the Prandtl mixing length. The dimensional analysis of the expression shows that a consistent expression is

ν T = C ω 2 |ω|, (2.6)
with C ω a dimensionless constant. This raises the question of the determination of . In the case of a flow over a plate, one finds in Obukhov [START_REF] Obuhov | Turbulence in an atmosphere with inhomogeneous temperature[END_REF] the following classical law: = (z) = κz, (2.7) where z ≥ 0 is the distance from the plate and κ the von Kármán constant. The Van Driest formula [START_REF] Van Driest | On turbulent flow near a wall[END_REF] defines by: (z) := κ z (1 -e -z/A );

(2.8)

here A depends on the oscillations of the plate and on the kinematic viscosity ν, while z ≥ 0 is again the distance from the plate.

According to these formula, we shall assume throughout the rest of the paper that the function : Ω → R + is of class C 2 and satisfies the two following properties:

a) (x) ≈ d(x, ∂Ω) for x close to ∂Ω; (2.9) b) ∀ K ⊂⊂ Ω, ∃ K > 0 s.t. (x) ≥ K > 0 ∀ x ∈ K, (2.10)
where d(x, ∂Ω) denotes the distance from the boundary. In practice, we could have directly assumed (x) = d(x), i.e.,

ν T = C ω d 2 |ω|. (2.11)
2.4. Generalised Rotational Smagorinsky models by dimensional analysis. The analysis of the previous section can be put also in a more general framework of Large Eddy Simulation (LES) models, looking also at possible modifications of the parameters present in the expression of the turbulent (rotational) stress vector. Let 0 > 0 be a typical length scale of the motion. For instance, in the case of a flow over a plate, one can take

0 = ν v * ,
where ν is the kinematic viscosity and v * is the so-called friction velocity (cf. [START_REF] Amrouche | Turbulent flows as generalized Kelvin-Voigt materials: Modeling and analysis[END_REF]). We consider (modulo introducing an appropriate non-dimensionalization of the equations) the following operator

curl 2-α 0 α |ω|ω (2.12)
with α ∈ [0, 2], which is degenerate at the boundary and for which the natural treatment is through scales of weighted Banach spaces.

We report some discussion about the relationships between the scaling of the weight and that of the power of the curl. In the framework of LES methods we show that even starting with

ν T = 2-α 0 α |ω| p-2
(2. [START_REF] Kaltenbach | Note on the existence theory for pseudo-monotone evolution problems[END_REF] this determines a link between powers α and p. Nevertheless, in the last section we will also point out the limiting behavior of the exponent p = 3 present in model ( 1 By extrapolation, motivated by experiments with central difference approximations to linear convection diffusion problems, the following alternate scaling has also been proposed (cf. [START_REF] Berselli | Mathematics of Large Eddy Simulation of turbulent flows[END_REF] and Layton [START_REF] Layton | A nonlinear, subgridscale model for incompressible viscous flow problems[END_REF]) ν T = (Cδ) p-1 |Dv| p-2 , and we consider here the rotational counterpart The justification of the presence of the critical value p -1 as power of the distance function can be done directly by dimensional arguments as in [START_REF] Berselli | On the existence of weak solutions for the steady Baldwin-Lomax model and generalizations[END_REF]. In fact, recall that both ∇v and ω have dimensions T -1 , where T is a time, and in (2.6) the turbulent viscosity

ν T = (Cδ) p-1 |ω| p-2 , 1 < p < ∞,
ν T = d 2 |ω| ∼ L 2 T -1 (
where L is a length) has the dimensions of a viscosity. This is the only way to identify (by using just a typical length and the vorticity) a quantity with the dimensions of a viscosity. Introducing as third parameter as the friction velocity v * ∼ LT -1 , one can consider more general combinations. The outcome is to find a turbulent eddy viscosity of the following form

ν T = v θ * d α |ω| p-2
, for some constants θ, α, p. It turns out (cf. [START_REF] Berselli | On the existence of weak solutions for the steady Baldwin-Lomax model and generalizations[END_REF]) that the dimensions of this quantity are ν T ∼ L θ+α T 2-θ-p , and to respect dimensions of the viscosity one has to fix θ = 3 -p and α = p -1.

A sound generalization of the rotational Smagorinsky model is then the one with rotational stress

S(v * , d, ω) = Cv 3-p * d p-1 |ω| p-2 ω
, and, after re-scaling, one can assume Cv 3-p * = 1. Note that, even for different values of p, the power of the distance is always the critical one (in terms of analytical properties of the weight functions), since d p-1 ∈ A p , cf. Lemma 4.5.

In the last section we will show that from the point of view of mathematical properties, the turbulent eddy viscosity

ν T = d p-1 |ω| p-2 ,
can be handled in terms of an existence theory by (pseudo)monotone operators only for p ≥ 3. Hence, the exponent p = 3 plays for the weighted rotational operators, the same role that the exponent p = 11/5 plays for the usual p-NSE with stress tensor S(Dv) = c|Dv| p-2 .

From now and so far no risk of confusion occurs, we do not write the bar anymore.

Evolution equations in an abstract setting

As already claimed in the introduction, a proper setting to the rotational Smagorinsky model is that of pseudo-monotone evolution problems so we briefly recall the abstract existence result we will use on the sequel.

For the convenience of the reader, we recall the following known definition.

Definition 3.1. Let X,Y be Banach spaces. An operator A :X → Y is called (i) bounded, if for all bounded M ⊆X, the image A(M )⊆Y is bounded.

(ii) coercive, if Y = X * and lim

x X →∞ Ax,x X x X = ∞. (iii) pseudo-monotone, if Y =X * and for any sequence (x n ) n∈N ⊆X from x n n→∞ x in X, lim sup n→∞ Ax n , x n -x X ≤ 0, it follows that Ax, x-y X ≤ lim inf n→∞ Ax n , x n -y X for all y ∈ X.
It is well-known that for each f ∈ X * , the steady problem Ax = f admits a solution if A is bounded, coercive and pseudo-monotone, see [START_REF] Brezis | Les opérateurs monotones, Séminaire Choquet: 1965/66, Initiation à l' Analyse[END_REF], [START_REF] Brezis | Équations et inéquations non linéaires dans les espaces vectoriels en dualité[END_REF]. A typical example of a pseudo-monotone operator is the sum of a hemicontinuous monotone and a compact operator. Recently, two of the authors in [START_REF] Kaltenbach | Note on the existence theory for pseudo-monotone evolution problems[END_REF] developed an abstract framework for evolution problems, by using the concepts of Bochner pseudo-monotone and Bochner coercive operators to generalize the ideas of [19, Sec. 2.5], [START_REF] Landes | A strongly nonlinear parabolic initial-boundary value problem[END_REF], [START_REF] Hirano | Nonlinear Volterra equations with positive kernels, Nonlinear and convex analysis[END_REF], [START_REF] Hirano | Nonlinear evolution equations with nonmonotonic perturbations[END_REF] and [START_REF] Shioji | Existence of periodic solutions for nonlinear evolution equations with pseudo-monotone operators[END_REF]. We want to access this theory for our concrete example. Therefore, for the remainder of this section, we assume that (V, H, id) is an evolution triple, i.e., V is a separable, reflexive Banach space, H a separable Hilbert space and V embeds densely into H. For I := (0, T ), T ∈ (0, ∞), and p ∈ (1, ∞), we set

X := L p (I, V ) and Y := L ∞ (I, H).
In this framework we have the following notion of a time derivative.

Definition 3.2. A function u ∈ X has a generalized time derivative if there exists a function w ∈ L p (I, V * ) such that

-ˆI (u(s), v) H ϕ (s) ds = ˆI w(s), v V ϕ(s) ds
for every v ∈ V and ϕ ∈ C ∞ 0 (I). Since such a function is unique, du dt := w is well-defined. By

W := W 1,p,p (I, V, V * ) := u ∈ X | ∃ du dt ∈ L p (I, V * )
, we denote the Bochner-Sobolev space with respect to the evolution triple (V, H, id).

In the context of evolutionary problems, the following generalized notions of pseudo-monotonicity and coercivity (cf. Definition 3.1) are particularly relevant and useful.

Definition 3.3 (Bochner pseudo-monotonicity). An operator

A :X ∩Y →X * is said to be Bochner pseudo-monotone if for a sequence (u n ) n∈N ⊆ X ∩Y from u n n→∞ u in X . u n * u in Y (n → ∞), u n (t) n→∞ u(t)
Au n , u n -u X ≤ 0, it follows that Au, u -v X ≤ lim inf n→∞ Au n , u n -v X for every v ∈ X .

Definition 3.4 (Bochner coercivity). An operator

A : X ∩ Y → X * is called: (i) Bochner coercive with respect to f ∈ X * and u 0 ∈ H if there is a constant M := M (f , u 0 , A) > 0 such that for every u ∈ X ∩ Y from 1 2 u(t) 2 H + Au -f , uχ [0,t] X ≤ 1 2 u 0 2 H for a.e. t ∈ I, it follows that u X ∩Y = u X + u Y ≤ M . (ii)
Bochner coercive if it is Bochner coercive with respect to f and u 0 , for every f ∈ X * and u 0 ∈ H.

The critical role of the above definitions is that they identify a vast class of problems for which existence can be established. In fact, if A : X ∩ Y → X * is bounded, Bochner pseudo-monotone, and Bochner coercive, then the corresponding evolution problem du dt + Au = f is solvable for any initial datum u 0 ∈ H. This result was recently obtained in [START_REF] Kaltenbach | Note on the existence theory for pseudo-monotone evolution problems[END_REF]Thm. 4.1]. This result is particularly relevant since the difficulty is then shifted to the verification of the properties of induced operators, which can be performed time-by-time in the known steady setting. We will not describe the full result, but we propose a particular, simplified setting enough to solve (1.1).

The existence result is mainly based on the following proposition giving sufficient conditions which have to be checked at any fixed time slice t ∈ I and which is a particular case of [START_REF] Kaltenbach | Note on the existence theory for pseudo-monotone evolution problems[END_REF]Prop. 3.13].

Proposition 3.5. Let A : V → V * be an operator. Assume that there exists a number p ∈ (1, ∞) and constants c 0 , c 1 > 0 such that 1 :

(C.1): For every v ∈ V there holds

Av V * ≤ c 0 v p-1 V . (C.2): A : V → V * is pseudo-monotone. (C.3): For every v ∈ V there holds Av, v V ≥ c 1 v p V .
Then, the induced operator

A : X ∩ Y → X * , for all u ∈ X ∩ Y and v ∈ X defined by Au, v X := ˆI A(u(t)), v(t) V dt,
is well-defined, bounded, Bochner pseudo-monotone, and Bochner coercive.

On the basis of Proposition 3.5, we immediately obtain the following existence result, which will be used to study the families of rotational models just checking that the conditions (C.1)-(C.3) are satisfied, after a proper choice of the functional setting. Theorem 3.6. Let A : V → V * be an operator satisfying (C.1)-(C.3). Then, for arbitrary u 0 ∈ H and f ∈ L p (I, V * ), there exists a solution u ∈ W of the evolution equation

ˆI du dt (t) + A(u(t)), v(t) V = ˆI f (t), v(t) V dt ∀v ∈ X , u c (0) = u 0 in H.
Here, the initial condition has to be understood in the sense of the unique continuous representation u c ∈ C 0 (I, H) of u ∈ W (cf. [START_REF] Zeidler | Nonlinear functional analysis and its applications[END_REF]Prop. 23.23]).

1 For a pseudo-monotone operator A : X → X * (local) boundedness implies demicontinuity, i.e., xn → x in X (n → ∞) implies Axn Ax in X * (n → ∞), hence we do not need here to make any further assumptions of demi-continuity.

Weighted spaces

Since (1.1) is a boundary value problem with the principal part given by a space dependent (and degenerate at the boundary) operator, a natural functional setting would be that of weighted Sobolev spaces. Apart from classical Lebesgue and Sobolev spaces, we will use their weighted counterparts. We follow the notation from the classical book of Kufner et al. [START_REF] Kufner | Weighted Sobolev spaces[END_REF].

A weight on R n is a locally integrable function satisfying almost everywhere 0 < (x) < ∞. The weighted space L p (Ω, ), 1 < p < ∞, is defined as follows

L p (Ω, ) := f : Ω → R n measurable ˆΩ |f (x)| p (x) dx < ∞ .
For p > 1 we have by using Hölder's inequality that [START_REF] Kufner | Weighted Sobolev spaces[END_REF]. In addition, L p (Ω, ) is a Banach space when equipped with the norm

-1/(p-1) ∈ L 1 loc (Ω) ⇒ L p (Ω, ) ⊂ L 1 loc (Ω) ⊂ D (Ω), allowing to work in the standard setting of distributions. It turns out that C ∞ 0 (Ω) is dense in L p (Ω, ) if the weight satisfies -1 p-1 ∈ L 1 loc (R n ), see
f p, := ˆΩ |f (x)| p (x) dx
W k,p 0 (Ω, ) := {φ ∈ C ∞ 0 (Ω)} . k,p, .
In our application the weight (x) will be a power of the distance d(x) ≥ 0 of the point x ∈ Ω from the boundary ∂Ω. Consequently, we specialize to this setting and give specific notions regarding these so-called power-type weights, see Kufner [START_REF] Kufner | Weighted Sobolev spaces[END_REF]. First, it turns out that W k,p (Ω, d α ) is a separable Banach space provided α ∈ R, k ∈ N and 1 ≤ p < ∞. In this special setting, since d(x) ≥ C K > 0 for each compact K ⊂⊂ Ω, several results are stronger or more precise due to the inclusion L p (Ω, d α ) ⊂ L p loc (Ω), valid for all α ∈ R. We recall the following classical result about the distance function (cf. [START_REF] Kufner | Weighted Sobolev spaces[END_REF]). Lemma 4.1. Let Ω be a domain of class C 0,1 , which means that in a small enough neighborhood Ω P , for P ∈ ∂Ω, the boundary ∂Ω∩Ω P can be expressed (after a rigid rotation) as x 3 = a(x 1 , x 2 ) for Lipschitz continuous a. Then, there exist constants 0 < c 0 , c 1 ∈ R such that

c 0 d(x) ≤ |a(x ) -x 3 | ≤ c 1 d(x) ∀ x = (x , x 3 ) ∈ Ω P .
One of the most relevant properties of the distance function is that the following embedding holds true

L p (Ω, d α ) ⊂ L 1 (Ω) if α < p -1. ( 4
α p p = α p -1 < 1.
In the same way we have also that

∀ α ∈ [0, p -1[ L p (Ω, d α ) ⊂ L q (Ω) ∀ q ∈ 1, p 1 + α . (4.2)
As in [14, Prop. 9.10] it can be shown that: In this case functions from W 1,p 0 (Ω, d α ) are zero on ∂Ω in the sense that they can be approximated by smooth functions with compact support. In the sequel we will use certain Hardy-Sobolev inequalities. Note that inequalities of this kind, when d is replaced by |x| = d(x, 0) are known as Caffarelli-Kohn-Nirenberg inequalities [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF].

Lemma 4.3. Let Ω ⊆ R n be a bounded Lipschitz domain. For p ∈ [1, n), α = p -1 and q ∈ [p, np n-p there exists a constant c > 0 such that for all f ∈ W 1,p 0 (Ω, d α ) there holds ˆΩ d q p (n-p+α)-n |f | q dx 1 q ≤ c ˆΩ d α |∇f | p dx 1 p . (4.3)
Proof. This follows from the definition of the space W 1,p 0 (Ω, d α ), [18, Theorem 2.1] and the classical (p, α) Hardy inequality

ˆΩ d α-p |f | p dx 1 p ≤ c ˆΩ d α |∇f | p dx 1 p , (4.4)
which is valid for all p ∈ (1, ∞) and α = p -1, for functions in W 1,p 0 (Ω, d α ) (cf. [START_REF] Nečas | Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle[END_REF], [START_REF] Kufner | Function Spaces[END_REF]Theorem 8.10.14]).

In addition to (4.1) and its role in Hardy-type inequalities, the critical nature of the power α = p -1 also occurs in the notion of Muckenhoupt weights and their relation with the maximal function.

Definition 4.4. We say that a weight ∈ L 1 loc (R 3 ) belongs to the Muckenhoupt class A p , for 1 < p < ∞, if there exists C such that sup

Q⊂R n Q (x) dx Q (x) 1/(1-p) dx p-1 ≤ C,
where Q denotes a cube in R 3 .

The powers of the distance function belong to the class A p according to the following well-known result for general domains (say it is enough that ∂Ω is a n -1-dimensional closed set, see [START_REF] Dyda | Muckenhoupt Ap-properties of distance functions and applications to Hardy-Sobolev-type inequalities[END_REF]). Here and in the sequel the boundary will be at least locally Lipschitz to have the outward unit vector properly defined. α is a Muckenhoupt weight of class

A p if and only if -1 < α < p -1.
4.1. Solenoidal spaces. A standard approach in fluid mechanics, is to incorporate the divergence-free constraint directly in the function spaces. These spaces are built upon completing the space of solenoidal smooth vector fields with compact support, denoted as φ ∈ C ∞ 0,σ (Ω). For α ∈ R define

L p σ (Ω, d α ) := φ ∈ C ∞ 0,σ (Ω) . p,d α , W 1,p 0,σ (Ω, d α ) := φ ∈ C ∞ 0,σ (Ω) . 1,p,d α
.

For α = 0 they reduce to the classical spaces L p σ (Ω) and W 1,p 0,σ (Ω). Next, we will extensively use the following extension of classical inequalities linking curl/divergence and full gradient estimates (cf. [START_REF] Berselli | On the existence of weak solutions for the steady Baldwin-Lomax model and generalizations[END_REF]). Lemma 4.6. Let 1 < p < ∞ and assume that the weight belongs to the class A p . Then, there exists a constant C, depending on the domain Ω and on the weight ∈ A p , such that

∇u p, ≤ C( div u p, + curl u p, ) ∀ u ∈ W 1,p 0 (Ω, ).
In particular, we will use the latter result in the following special form Corollary 4.7. For -1 < α < p -1 there exists a constant

C = C(Ω, α, p) such that ˆΩ d α |∇v| p dx ≤ C ˆΩ d α | curl v| p dx ∀ v ∈ W 1,p 0,σ (Ω, d α ). (4.5)
5. Application to the rotational turbulence models: the proof of Theorem 1.1

In this section we verify that the initial boundary value problem (1.1), after a proper selection of parameters, and definition of both the operators and functional spaces, can be put in the framework of the abstract Theorem 3.6. This will be enough to give a proof of the main result of this paper, that is the existence of weak solutions in Theorem 1.1.

In our setting the choice of the natural spaces is determined by the problem itself which yields, by the a priori estimate obtained by testing with the velocity v, that the integral

ˆT 0 ˆΩ d α | curl v| 3 dxdt
is finite. Hence, for almost all t ∈ [0, T ] the integral ´Ω d α | curl v| 3 dx will be finite, determines the choice for the Banach space V . In order to identify the evolution triple to be used for the proper formulation, we need to clarify the relationship with the L 2 (Ω) norm. We have the following result which immediately derives from the basic results on weighted spaces of the previous section. (5.1)

Proof. For α < 2, combining (4.3) with q = 3p 3-p+α and (4.5), it follows for every p ∈ (α + 1, 3)

ˆΩ |u| 3p 3-p+α dx ≤ c ˆΩ d α |∇u| p dx q p ≤ c ˆΩ d α | curl u| p dx q p
, for all v ∈ C ∞ 0,σ (Ω). Since 2 ≤ 3p 3-p+α the assertion follows from Hölder's inequality as Ω is bounded. Lemma 5.1 shows that one can work with the following evolution triple for all α ∈ [0, 2)

(V, H, id) := W 1,3 0,σ (Ω, d α ), L 2 σ (Ω), id .
Proof. The proof is based on the estimation of the space integral, by using appropriate weighted version of classical Sobolev spaces tools. We have in fact, for all smooth functions with compact support the following inequality (obtained multiplying and dividing a.e. x ∈ Ω by the positive function

d α/3 ) ˆΩ(curl v × u) • w dx ≤ ˆΩ d -α/6 |u| d α/3 | curl v| d -α/6 |w| dx ≤ ˆΩ d -α/2 |u| 3 dx 1/3 ˆΩ d α | curl v| 3 dx 1/3 ˆΩ d -α/2 |w| 3 dx 1/3
.

It remains to show that all u ∈ V also belong to the weighted space L 3 (Ω, d -α 2 ), with a continuous embedding. From (4.3) it follows for all p ∈ [1, 3), α ∈ [0, 2) and u ∈ W 1,p 0,σ (Ω,

d α ) that ˆΩ d -α 2 |u| p(6-α) 2(3-p+α) dx 1 q ≤ c ˆΩ d α |∇u| p dx 1 p , with q := p(6 -α) 2(3 -p + α) < p * .
One easily checks that for all α ∈ [0, 2) there exists a p ∈ (1 + α, 3) such that 3 < q < p * . Since Ω is bounded we deduce from this V → L 3 (Ω, d -α 2 ) by using Hölder's inequality.

Once the integral ´Ω(u × curl v) • w dx is well-defined for u, v, w ∈ V , it immediately follows that Bu, u V = 0 for all u ∈ V , since a.e. in Ω it holds

(v × curl v) • v = 0.
This is enough for what concerns the growth and coercivity. We need now to show compactness for B in order to prove pseudo-monotonicity. Lemma 5.4 (Compactness of B). Let α ∈ [0, 2). Then, the weak convergence u n u in V implies (up to a sub-sequence) that

Bu n → Bu in V * ,
i.e., the operator B is compact.

Proof. By the boundedness of the weakly converging sequence (u n ) n∈N ⊆ V and by (4.2) we get that

u n W 1,r (Ω) ≤ C ∀ r ∈ 1, 3 1 + α .
Hence, by the usual (unweighted) compact Sobolev embedding W 1,r (Ω) → → L r(Ω), valid for all r < ( 3 1+α ) * = 3 α we get also that (up to a sub-sequence)

u n → u a.e. and in L r(Ω).

By using the definition of B, the properties of the curl (with summation over repeated indices), and integration by parts, we have that for all u, v ∈

V Bu, v = ˆΩ jkl jlm u k v i ∂u m ∂x l dx = ˆΩ(δ kl δ im -δ km δ il )u k v i ∂u m ∂x l dx = -ˆΩ u k u i ∂v i ∂x k dx = -ˆΩ(u ⊗ u) : ∇v dx.
Hence, we have

Bu n , v -Bu, v = -ˆΩ(u n ⊗ u n ) : ∇v -(u ⊗ u) : ∇v dx = - ˆΩ (u n -u) ⊗ u n : ∇v + u ⊗ (u n -u) : ∇v dx.
By Hölder inequality we get, as in the proof of Lemma 5.3,

ˆΩ (u n -u) ⊗ u n : ∇v dx ≤ ˆΩ d -α/2 |u n -u| 3 dx 1/3 ˆΩ d α |∇v| 3 dx 1/3 ˆΩ d -α/2 |u n | 3 dx 1/3 ≤ ˆΩ d -α/2 |u n -u| 3 dx 1/3 v V u n V .
We now observe that the last two terms are uniformly bounded, while

d -α/2 |u n -u| 3 → 0 a.e. x ∈ Ω.
Consequently, to show that the integral vanishes it is enough to prove that for some q > 3 there holds

u n -u L q (Ω,d -α/2 ) ≤ C
uniformly in n ∈ N, which permits to apply the Vitali theorem in the weighted space L 3 (Ω, d -α/2 ). However, this was already obtained in the proof of Lemma 5.3. The other term in the decomposition of Bu n , v -Bu, v can be treated in the same way.

Proof of Theorem 1.1. In the previous lemmas we have proved that A = S + B is continuous and pseudo-monotone since it is the sum of a monotone continuous and a compact one. Collecting the estimates we have that in particular that the boundedness and coercivity are as follows This shows that the induced operator A is Bochner pseudo-monotone and coercive, hence all the hypotheses of the abstract existence Theorem 3.6 are satisfied. This proves the main result of this paper, that is the existence of weak solution in Theorem 1.1.

Av V * ≤ c 0 v 2 V , Av, v V ≥ v 3 V , since Bv, v V = 0.
5.1. The case p > 3. In this section we show that most of the results of the previous section can be extended (even with easier proofs) to the system with the following operator

S p v, w V := ˆΩ d α | curl v| p-2 curl v • curl w dx with p > 3, 0 ≤ α < p -1,
while the use of the tools typical of pseudo-monotone operators fails for p < 3. We can then prove the following result Theorem 5.5.

Let p > 3, α ∈ [0, p -1), 0 < T < ∞, v 0 ∈ L 2 σ (Ω), and f ∈ L p (0, T ; (W 1,p 0 (Ω, d α ) * ).
Then, there exists a weak solution to the initial boundary value problem

∂ t v + ω × v + curl d α |ω| p-2 ω + ∇q = f in (0, T ) × Ω, ω = curl v in (0, T ) × Ω, div v = 0 in (0, T ) × Ω, v = 0 on (0, T ) × ∂Ω, v(0) = v 0 in Ω, such that v ∈ C([0, T ]; L 2 σ (Ω)) ∩ L p (0, T ; W 1,p 0,σ (Ω, d α )) and for all t ∈ [0, T ] 1 2 v(t) 2 + ˆt 0 ˆΩ C α d α (x)|ω(s, x)| p dx ds = 1 2 v 0 2 + ˆt 0 f (s), v(s) W 1,p 0 (Ω,d α ) ds.
The proof of this result is again just a verification that the hypotheses of the abstract theorem are satisfied, but we will highlight the critical role of the parameters.

First note that for p > 3 and 0 ≤ α < p -1 the inclusion W The properties of the operator S p are practically the same as those of the operator S, hence one can directly show

S p v V * ≤ v p-1 V ∀ v ∈ V, S p v, v V = v p V ∀ v ∈ V.
On the other hand, the properties of B are to be checked. The operator is the same as before, but the functional setting is different.

To show the boundedness of B : V → V * for 0 ≤ α < p -1, we proceed as in Lemma 5.

This shows that

Bu, w ≤ C(Ω, α, p) u 2 V w V , and the compactness of B follows with the same arguments used in the previous section (almost everywhere convergence and Vitali theorem). , and then estimate the first and third integral with (4.3) for q = 2p < p * , which holds for p > 9 5 . Hence, to apply (4.3) the precise exponent will be q = p p-1 3p-3-α 3-p+α , and since q ≥ 2p this implies that we have to request for α ≤ 5p -9 3 .

Since we would like to treat cases with α smaller but "arbitrarily close" to p -1, the inequality p -1 ≤ 5p -9 3 , should be correct. On the other hand the latter can be satisfied only for p ≥ 3. Since we are out of the range of permitted p this shows that the estimate can not be used. Being the inequalities Hardy-Sobolev inequalities sharp, this proves that operator B is not bounded for 9 5 < p < 3, when α is close to p -1, hence the basic assumptions to use the pseudo-monotone methods are not satisfied. The existence of weak solutions, if possible, should be obtained with different methods and possibly considering different weak formulations of the problem.

2. 1 .

 1 Reynolds decomposition. Let us consider the Navier-Stokes equations (NSE in the sequel) written with the convective term in the rotational formulation:

  .1), when p = α + 1. If one thinks of a flow as composed of eddies of different sizes in different places, then in a region of large eddies the changes of velocity and its curl are both O(1) of the typical distance. In a region of smaller eddies the velocity changes over a distance of O(eddy length scale), so the local deformation is O(1/eddy length scale), cf. [4, § 3.3.2]. Hence, the rotational Smagorinsky model introduces a turbulent viscosity ν T = (Cδ) 2 |ω|, where δ is the (local) smallest resolved scale, such that ν T = O(δ 2 ) in regions where |ω| = O(1), O(δ) in the smallest resolved scale where |ω| = O(δ -1 ).

  which resembles general power laws for non-Newtonian fluids. The above choice of ν T satisfies ν T = O(δ p ) in regions where |ω| = O(1), O(δ) in the smallest resolved scale where |ω| = O(δ -1 ),

.

  Next, we define weighted Sobolev spacesW k,p (Ω, ) := f : Ω → R n D α f ∈ L p (Ω,) for all α s.t. |α| ≤ k , equipped with the norm f k,p, := |α|≤k D α f p p, 1/p , and, as usual, we define W k,p 0 (Ω, ) as follows

Lemma 4 . 2 . 1 p

 421 The quantity ´Ω d α |∇f | p dx is an equivalent norm in W 1,p 0 (Ω, d α ), provided that 0 ≤ α < p -1.

Lemma 4 . 5 .

 45 The function (x) = d(x)

Lemma 5 . 1 . 2 ≤

 512 Let u ∈ C ∞ 0,σ (Ω) and α ∈ [0, 2). Then, there exists C = C(α, Ω) such that ˆΩ |u| 2 dx 1/C ˆΩ d α | curl u| 3 dx 1/3 .

  Hence all hypotheses from (C.1) to (C.3) are satisfied.

2 p (α-p) |u| 2 d 2 p≤

 22 1,p 0 (Ω, d α ) ⊂ L 2 (Ω) holds true. Directly by Hölder's inequality with δ = p/2, δ = p/(p -2), and Hardy inequality (4.4) we get ˆΩ |u| 2 dx = ˆΩ d (p-α) dx ≤ ˆΩ d α-p |u| p dx C(p, α) ˆΩ d α |∇u| p dx 2 p , since p -α > 0. This shows that one can work with the evolution triple (V, H, id) := W 1,p 0,σ (Ω, d α ), L 2 σ (Ω), id with p > 3, 0 ≤ α < p -1.

1 .

 1 3 and we getˆΩ(curl v × u) • w dx ≤ ˆΩ d -α/2p |u| d α/p | curl v| d -α/2p |w| dx ≤ ˆΩ d -αp /p |u| 2p dx 1 2p ˆΩ d α |∇v| p dx 1 p ˆΩ d -αp /p |w| 2p dx 1 2p.Next, observe that 2p = 2p/(p -1) < p is satisfied for p > 3. Consequently, in this case we can directly apply Hölder inequality with exponents δ = (p -1)/2 and δ = (p -1)/(p -3) to bound the first and third integrals as follows:ˆΩ d -α/(p-1) |u| 2p dx = ˆΩ d (α-p)2/(p-1) |u| 2p d (2p-3α)/(p-1) dx ≤ ˆΩ d α-p |u| p dx 2 p-1 ˆΩ d (2p-3α)/(p-3) dx p-3 p-The first term from the right-hand side is bounded with ´Ω d α |∇u| p dx 2 p-1 by using (4.4), while the second is finite if 2p -3α p -3 > -1 ⇐⇒ α < p -1.

Remark 5 . 6 .

 56 The case 1 < p < 3 does not fit with the theory for the reasons we now explain. The argument with Hardy inequality as in the previous lemma requires p > 3. If we try to apply the same argument used for p = 3 with Hardy-Sobolev inequality (4.3), we can writeˆΩ(curl v × u) • w dx ≤ ˆΩ d -αp /p |u| 2p dx 1 2p ˆΩ d α | curl v| p dx 1 p ˆΩ d -αp /p |w| 2p dx 1 2p

  .1) It follows directly from Hölder's inequality ˆΩ |f | dx = ˆΩ d α/p |f |d -α/p dx ≤ ˆΩ d α |f | p dx

	1/p	ˆΩ d -αp /p dx	1/p

, and using Lemma 4.1 the latter integral is finite if and only if

Acknowledgments

Luigi C. Berselli was partially supported by a grant of the group GNAMPA of INdAM and by the University of Pisa within the grant PRA 2018 52 UNIPI: "Energy and regularity: New techniques for classical PDE problems."

and as functional setting for (1.1) we use the following spaces and operators, where 0 ≤ α < 2

and define the operator

The induced operator S : X ∩ Y → X * inherits the properties of the operator S (cf. [START_REF] Zeidler | Nonlinear functional analysis and its applications[END_REF]Chapter 30]). Note that S is a strictly monotone, bounded, coercive, and continuous operator. These properties are practically the same known for the p-Laplace operator. In fact, from the definition, one obtains directly the following two inequalities:

The monotonicity of S derives from the following lemma (cf. [3, Lemma 3.3]).

Lemma 5.2. For smooth enough vector field ω i (it is actually enough that

for any (not necessarily the distance) bounded function such that d : Ω → R + for a.e. x ∈ Ω.

The proof of the above lemma is based on the observation that it can be proved that

Then weighted integrability of the functions this used to prove that the integral is finite.

To treat the operator B, and the induced one B : X ∩ Y → X * , we need to properly adapt the estimates on the convective term in weighted spaces and this is mainly based on the previously Hardy-type inequalities (4.3).

Lemma 5.3 (Boundedness of B). For all

V v V and Bu, u V = 0, for all u, v ∈ V .