S.G. Chefranov 1 ) and A.S. Chefranov 2 ) Summary

Transcription

1 Exac Te-Dependen Solon o he Three-Densonal Eler- Helholz and Reann-Hopf Eqaons for Vorex Flow of a Copressble Med and one of he Mllenn Prze Probles S.G. Chefranov and.s. Chefranov, Obhov Inse of ospherc Physcs of he Rssan cadey of Scences, Moscow, Rssa schefranov@al.r a.chef@b.r Sary For he frs e he exac solon n nbonded space s represened for he wo and hreedensonal Eler-Helholz (EH vorex eqaon n he case of a nonzero-dvergence velocy feld for an deal copressble ed flows. Ths solon corresponds o he obaned n Eler varables wo and hree-densonal Reann-Hopf (RH eqaon solon. necessary and sffcen condon of he onse of a snglary n he evolon of ensrophy n fne e for hs solon s forlaed. The exac closed descrpon of he evolon of ensrophy and all oher one-, wo-, and l pon oens of he vorex feld s now possble. Besdes, he proble of closre, ha s he an proble of he rblence heory, s no arsng here a all. The sooh connaon of he obaned solon o he EH an RH eqaons for, de o nrodcon of a farly large hoogeneos frcon or an arbrary sall effecve vscosy, s saed. new analyc, sooh for all e, solon o he wo and hree-densonal Naver-Soes (NS eqaon s obaned. Ths solon concdes wh he above-enoned sooh solon o he EH and RH eqaons, whch aes no accon he vscosy effec of a copressble ed and also a nown lnear relaon beween he flcaons of pressre and he velocy feld dvergence n syses far fro he eqlbr. Ths gves he posve answer o he generalzaon of he nown Mllenn Prze Proble ( on he case of he copressble Naver-Soes eqaon. Indeed, earler only a negave answer for he proble of he exsence of sooh solons for any fne e has been a pror consdered for he copressble case. Keywords: hydrodynacs, copressbly, vscosy, rblence, vorex waves PCS nber(s:

2 Conen Inrodcon. The Cachy proble for he copressble Naver-Soes (NS eqaon. Energy and enropy balance eqaons. New solon o he NS eqaon 4. Exac solon o he Eler Helholz (EH and Reann-Hopf (RH eqaons 5. Ensrophy balance eqaon and he hoogeneos frcon 6. To exsence of he dvergence-free solons o he NS eqaon Dscsson and Conclsons ppendx Exac solon of n-d RH eqaon (n=,, ppendx B Exac vorex solon of -D EH eqaon Lerare Inrodcon. The ndersandng of any naral and echnologcal processes s relaed o exsence of he basc and appled probles of rblence whch rean nsolved already drng ore han hndred years by vre of he absence of analyc e-dependen sooh vorcal solons o he Naver-Soes (NS eqaon. The developen of he sascal approach o s solon gave any neresng resls b also led o a new and so far nsolvable proble of closre n descrpon of varos oens of he vorex feld whose approxae solon has been proposed by Kologorov, Hesenberg and oher []. In fac, so far only a few exac solons are nown n hydrodynacs. However, none of hese solons s e-depended and defned n nbonded space (or n space wh perodc bondary condons [ - ]. There are only wea e-dependen solons whch descrbe, for exaple, dynacs and neracon of snglar vorcal obecs n a wo- and hree-densonal deal ncopressble ed [,4,5]. For hree-densonal flows of an deal ed here s a concep of he possbly of exsence of e-dependen solons o he Eler-Helholz (EH eqaon only on a bonded e nerval (see [,,5,6] and he lerare ced heren. In he case of he ncopressble ed he lengh of hs e nerval s deerned only by he hreedensonal effec of exenson of he vorex flaens whch can lead o explosve nbonded growh of he ensrophy (negral of he vorcy sqare over he space n a fne e [,, 5, 6]. On he oher hand, he exac seady-sae reges of flows of a vscos ncopressble ed n he for of he Brgers and Sllvan vorces are well nown []. For he he effec of exenson of he vorex flaens poenally dangeros for he developen of snglary s exacly copensaed by he vscosy effec. However, for hese solons here exss no convergen energy negral over he enre nbonded space.. The qeson of exsence of sooh e-dependen nonzero-dvergence and dvergencefree solons o he hree-densonal NS eqaon n nbonded space (or n space wh perodc bondary condons and on an nbonded e nerval reans open drng alos wo hndred years (sarng fro [, 7 - ]. The porance of hs proble s deerned by no only prely aheacal b also praccal neres n connecon wh he basc and appled proble of predcably of hydro-eeorologcal and oher felds developed n sng he ehods of nercal solon of he NS eqaon [8, 9].

3 In hs connecon, n he proble of exsence of a sooh e-dependen vorcal solon o he hree-densonal NS eqaon on an nbonded e nerval was saed by he Clay Maheacs Inse as one of seven basc Mllenn Prze Probles [7]. However, n [7] was proposed o consder he solon of hs proble no for he coplee NS eqaon [] b only for an eqaon obaned fro he laer nder he asspon on he dvergence-free velocy feld for he ncopressble fld. Evdenly, n he choce of sch a forlaon a pror was assed ha for flows wh nonzero dvergence of he velocy feld he coplee Naver- Soes eqaon canno delberaely have sooh solons on he ndefne e nerval. In fac, n [] was wren abo hs: The Mllenn Prze Proble relaes o ncopressble flows snce s well nown ha he behavor of copressble flows s abonable. In hs connecon he exaple for a shoc wave, developed n a copressble ed when a body oves hrogh a a velocy hgher han he sonc speed of he ed, s gven []. However, hs does no exclde he possbly of exsence of sooh solons wh nonzero dvergence of velocy o he coplee NS eqaon de o he effec of vscosy.. Therefore, he forlaon of he proble gven n [7] can be well generalzed o nclde he case of flows wh nonzero dvergence of velocy for he copressble ed. Ths s he sbec of he presen sdy. In fac, a new analyc e-dependen vorex solon o he coplee -D NS eqaon s obaned here on he bass of exac vorex solon for he -D Reann-Hopf (RH eqaon and s odfcaons whch aes no accon he effecve vscosy or frcon [-6] (see also ppendx. Ths solon reans sooh on any arbrary large e nervals precsely owng o fneness of he vscosy forces. In hs case he solon o he NS eqaon can be conned n he Sobolev space H q ( R for any q and, where s n e of he developen of a snglary (collapse for he correspondng exac solon o he RH eqaon a zero vscosy. For he Sobolev space H q ( R he nor s defned n he for [7]: ( ( d x H q R q In [7] he heore of exsence of a solon o he -D Eler eqaon (ha s NS eqaon wh zero vscosy local n e was forlaed for he dvergence-free deal ncopressble fld flow. In accordance wh hs heore, a sooh solon o he Eler eqaon exss f he nal velocy feld belongs o he Sobolev space H q ( R when q wh N cons for a ceran N. Then here exss a e ( N, whch depends only H N, sch ha for all / (I. on, he Eler eqaon has a solon belongng o he class q q C, ; H C, ; H, where he nor s defned n (I.. Then, for he exac solon o he Eler-Helholz (EH and RH eqaons consdered n he presen sdy (see ppendx and ppendx B, n he case of deal copressble fld flow s possble o conne hs solon for es only n he Sobolev space H ( R and already s possble o conne hs solon n he Sobolev space H ( R a e, when hreshold vale q nsead of he q n [7]. The obaned analyc solon o he NS eqaon corresponds o a fne dvergence of he velocy feld. Ths ndcaes ha a pror asspon on he absence of sooh solons o he copressble -D NS eqaon s nvald.

4 4 Or way of ang he vscosy no accon s a parclar exaple of slaon of rblence when a rando velocy feld s nrodced nsead of he rando force [8]. However, n [8] only he large-scale rando velocy feld was consdered and he drf par of hs velocy dependen only on e was elnaed. veragng over he rando velocy dependen only on e s ensres he slaon of he effecve vscosy n he presen paper. In hs case s of porance ha hs way of slaon of he vscosy effec does no change he srcre of he vscosy force F v whch eners no he NS eqaon and, for exaple, for he ncopressble ed has he for F v []. In fac, s nown [8] ha he exsence of a solon o he NS eqaon rns o o be proved f a er proporonal o he hgher dervave (of he flow velocy of he for 5, (see [9, ] s added o he sal vscosy force F. Ths er changes he 4 srcre of he vscosy force of he nal NS eqaon. I s also shown ha he elnaon of snglary of he solons o he EH and RH eqaons always aes place n nrodcng a farly large hoogeneos (or exernal frcon wh coeffcen whch s sasfy he condon (5. and corresponds o he change n he NS eqaon. The new solon o he hree-densonal NS eqaon s obaned provdng he zero oal balance of he noral sresses cased by he pressre and he vscosy of flow wh nonzero dvergence of velocy for a copressble ed. s shown n Secon, hs corresponds o he sffcen condon of posve defneness of he negral enropy growh rae. Ths aes possble o redce solon of he NS eqaon o solon of a hree-densonal analog of he Brgers eqaon and hen also o solon of he hree-densonal RH eqaon and s generalzaon o nclde he case of ang no accon he vscosy forces (exernal frcon and he above-enoned effecve frcon relaed o he rando velocy feld. We also noe ha n he general case he vorcal solons o he hree-densonal RH eqaon concde wh he solons o he hree-densonal EH eqaon for descrbng vorex flows of an deal copressble ed wh nonzero dvergence of he velocy feld [9, -6] (see also ppendx and ppendx B. In fac, all real eda are copressble and generally her flows s be descrbed by precsely solons o he copressble NS eqaon. On he oher hand, for a convenonally ncopressble ed he flows wh nonzero dvergence of velocy can correspond o he presence of dsrbed sorces and socs whose slaon s sccessflly sed n nonrelavsc and relavsc hydrodynacs [-4]. 4. We noe ha n [5] an exac solon o he hree-densonal RH eqaon, whch descrbes he explosve e evolon of he arx of he frs dervaves of he velocy feld only n he Lagrangan varables, was obaned. Ths does no ae possble o oban exac solons o he hree-densonal EH eqaon for he vorex feld on s bass, as ade n [, ] and here (see ppendx B n he Eler represenaon of he exac solon of RH eqaon (see ppendx. he sae e, n he presen sdy s shown ha he exac solon o he -D RH eqaon for he velocy feld obaned here (see forla (.5 below and (. n ppendx gves he expresson (. n he Lagrangan represenaon for he evolon of he arx of he frs dervaves of he velocy. Ths forla exacly concdes wh a forla gven n [5] (see forla ( n [5]. In he wo- and hree-densonal cases new analyc solons for he evolon of he nenses of vorex and of he helcy of he Lagrangan lqd parcles are also obaned (see (4.4 (4.6. In [6] a srcrally slar for (4.7 of he solon o he EH eqaon (see

5 5 forla ( n [6] was consdered on he bass of sng a cobnaon of he Eleran and Lagrangan descrpons n represenaon of vorex lnes. However, on he bass of heory [6] does no possble explcly o descrbe he e evolon of he vorcy and s hgher oens, b hs s possble de o or solon (4.5. In he presen sdy we obaned a new necessary and sffcen creron of pleenaon of he explosve snglary (collapse n fne e (see (.7 and (.8 for he nvscd solon o he RH and EH eqaons n he one-, wo-, and hree-densonal cases. he sae e, n [5] he negral creron s gven n he for (.9 (see forla (8 n [5]. Ths creron deernes only he sffcen condon for pleenaon of collapse of he solon. For exaple, when he nal velocy feld s dvergence-free he collapse s possble only n accordance wh he necessary and sffcen creron (.8 b already canno be esablshed fro he creron of [5]. Moreover, fro consderaon of he explosve rege carred o n [5] for he solon o he hree-densonal RH eqaon a conclson s ade ha hs solon canno be conned n nfne e n he Sobolev space H ( R. Ths dffers fro he aboveenoned resl obaned n he presen sdy. In he wo-densonal case here s an exac correspondence beween he creron (.7 and a slar creron gven n [7] (see forla (9 n [7] n connecon wh he solon of he proble of propagaon a flae fron (generaed by a self-ssaned exoheral checal reacon on he bass of he splfed verson of he Svashsy eqaon [8]: f U s f f (I. In Eq. (I. he fncon x f x, x, deernes he flae fron whch represens he ( nerface beween a cobsble aer ( x and he cobson prodcs ( x, U s, are consan posve qanes whch characerze he fron velocy and he cobson nensy, respecvely. For eqaon (I. concdes wh he Halon-Jacob eqaon for a free non-relavsc parcle. In he wo-densonal case (ore exacly, n s odfcaon wh accon for he hoogeneos frcon wh he coeffcen when, he exac solon (.5 of he RH eqaon gves also he exac solon of Eq. (I.. In hs case he solon (.5 descrbes a poenal flow wh velocy U s f a zero vorcy. 5. n poran resl of he presen sdy consss n obanng a closed descrpon of he e evolon of he ensrophy and any hgher oens of he vorex feld, as well as he velocy feld n he -D and -D cases. Ths s reached on he bass of he correspondng analyc solons o he EH, RH, and NS eqaons n he case of zero vscosy and n he case of ang he hoogeneos frcon and he effecve vscosy no accon. s a resl, he proble of closre n he heory of rblence s solved here exacly and no approxaely, as sal []. In he presen sdy hs becae possble only owng o a relavely sple dependence on he nal condons for he exac solons obaned for he EH and RH eqaons for he velocy feld (.5 and he vorex feld ((4. and (4.. Ths s absen n he well-nown exac solon o he Brgers eqaon obaned sng he nonlnear Cole-Hopf ransfor. In parclar, owng o hs acheveen, on he bass of he exac solons (4. of EH eqaon he followng esaes for he negrals of he vorcy feld can be obaned n he hree-densonal case n he neghborhood of he oen of snglary of he solon when :

6 6 ( d x O( ; ( ( ( d x O, when,,,.... ( Hence here drecly follows he neqaly ( / ( O ;, (I. whch ndcaes srong nerency of he vorex feld n he neghborhood of snglary. We noe ha sally he neqaly s assed o be really flflled n he case of ( ( srong vorex nerency [8]; however, earler, cold no be obaned fro he exac solon of he proble of closre n heory of rblence, as ade n obanng he esae (I.. 6. In he fnal par of he presen sdy he possbly of exsence of no only he dvergen b also sooh dvergence-free solons o he NS eqaon s dscssed on he bass of an analyss of he exac closed solon o he ensrophy balance eqaon (5.6 and he varaon rae of he negral nec energy n (6. - (6.4.. The Cachy proble for he copressble Naver-Soes (NS eqaon. The redcng of he NS eqaon o odfcaon of he Reann-Hopf (RH eqaon In he general case he eqaons of oon of a copressble vscos ed can be represened n he followng for []: ( p ( ( ; (. x x x x x ( (. x Fro he for of he second er on he rgh-hand sde of Eq. (. follows ha for a vscos copressble flow he noral sresses are deerned by no only he pressre b also by he dvergence of he velocy feld. In (. and (. s he velocy of he ed, he saon convenon s assed o be appled o he recrrng sbscrps varyng fro o n (here, n s he densonaly of space and, n wha follows, we wll consder he cases of n,,, and p,,, are he pressre, he densy, and he consan vscosy and dlaaonal (second vscosy coeffcens, respecvely []. The syse of eqaons (. and (. havng for eqaons for fve nnowns s no closed. To ge s closre, provdng correcness of he Cachy proble n he nbonded space, one ore eqaon s needed. Usally, as sch an eqaon, a sae eqaon s sed ha relaes he pressre and densy [9, ], or he pressre, densy, and eperare (or enropy []. In he laer case, he syse (., (. s o be copleened wh an eqaon for he eperare (or enropy [, ]. However, for he non-eqlbr flow syse wh fne velocy, loos probleac n general defnng of he adeqae for of any sae eqaon who dependence of pressre on velocy or s dervaves. Moreover, for relavely sall flow veloces (copared o he speed of sond n he gven ed, nsead of he sae eqaon, for he syse of eqaons (., (. closre, he ncopressble approxaon s sed when he velocy feld s assed dvergen-free and pressre s srongly depended on velocy. Ths approxaon s also refleced n he Mllenn proble defnon where only dvergen-free NS eqaon solons shall be consdered [7].

7 7 Insead of hese approxae closre ehods for he syse (., (., n he second secon, fro he sffcen condon of he posve-defneness of he enropy ncrease rae, a lnear relaonshp s derved beween he pressre and he dvergence of velocy for vscos copressble ed. Correspondng closng he syse (., (., he ffh eqaon s as follows: p ( dv (. Under (., he second er n he rgh-hand sde of he eqaon (. rns o zero dencally. Ths corresponds o he zero oal balance of he noral sresses and n he l he expresson (. has a for ypcal for syses far fro he eqlbr (see (8.4, (8.6 n []. The represenaon of hs ype for he pressre s also sed n [] nsead of he ed eqaon of sae. In he resl, he syse (.- (. s already closed and for he Cachy proble can correcly defned n he nbonded space wh any denson nber n. Eqaon (. nder (. becoes eqvalen o he n-densonal Brgers eqaon l (.4 xl Syse of eqaons (., (.4 s already closed b raher coplcaed for beng solved n qadrare or even n explc analycal for for respecve Cachy proble. I s de o he presence of he dependence on he densy n he rgh-hand sde of (.4 characerzng effec of he volerc vscosy force. Conseqenly, nsead of he exac acconng for he vscosy forces n he eqaon (.4, he approxae effec of he exernal frcon force wh consan coeffcen s sed. Ths accordng wh (. allows geng he Cachy proble solon for he -densonal NS eqaon analycally on he base of exac vorex solon of -D Reann-Hopf (RH eqaon (see ppendx, obaned fro (.4 when rgh-hand sde of (.4 s replaced by:, (.5 noher possbly, realzed n Secon, o oban analycal solon of NS eqaon (. when (. aes place s he odelng of effecve vscosy wh help of nrodcng of he rando Gassan dela-correlaed n e velocy feld V ( (by sbson V ( n (.4 nsead of rgh-hand sde er n (.4. I s poran ha he las represenaon sed for he vscosy force does no lead o he nrodcng of he ore rapd dapng of he hgh-freqency haroncs (n he l of large wave nbers copared o ha realzed n he sandard varan of he - densonal NS eqaon (.e. wh he Laplacan n he frs b no hgher order. I aers for he possbly of bacng of he nference of he exsence a all es of sooh analycal solon for he -D copressble NS eqaon obaned heren. Ths, f n (.4 o ae he sbson (.5 hen fro (.4, we shall ge odfcaon of he RH eqaon as follows l (.6 xl n expresson for he exernal frcon force n he rgh-hand sde of (.6 also corresponds o he approxae acconng for he vscosy force (descrbed by he rgh-hand sde of (.4 acally sed n any nercal solon of he NS eqaon when wh necessy cng s ade for large

8 8 wave nbers or correspondng sall scales ha corresponds o he sbson (.5 where n (.5 we have,, n. In he hree-densonal case (n= afer applyng he crl operaor o boh sdes of Eq. (.6, we oban he Eler Helholz (EH vorex eqaon n he odfcaon wh hoogeneos frcon force (for zero frcon wh -see he eqaon EH (B. n ppendx B: dv. (.7 x x For zero dsspaon eqaon EH (.7 (when can be also obaned drec fro correspondng o (. Eler eqaon n he case when he relaon ro( p s ae place for non-zero dvergence of velocy. Ths s poran o oban he exac solon of (.7 on he base of he exac solon of (.6 (see ppendx and ppendx B becase gves analycal vorex solon of NS eqaon (.. In parclar, s shown ha for he case wh zero exernal frcon coeffcen he sooh exac solon of (.6 and (.7 obaned can exs only on a bonded e nerval (he vale of wll be deerned fro he solon o Eq. (.7. In wha follows we wll fnd a hreshold vale of he exernal frcon coeffcen (see (5. so ha for he greaer vale of he coeffcen h he solon of NS eqaon (. (whch s redced o (.6 or (.7 on he base of (. and (.5 becoes reglar for any e. The ore effecve reglarzaon for any e we oban also n he case, when he rando velocy feld V ( wh arbrary sall aplde s nrodced nsead of exernal frcon when.. The saeen of he Cachy proble for he redced NS syse s n [7], b for he copressble ed we gve here he nex saeen of he Cachy proble for syse (. and (.6 (or (.7, when eqaon (.6 (or (.7 s obaned fro NS eqaon (. on he base of (. and (.5. The eqaons (., (.6 and (.7 descrbe he vorex oon of copressble ed n n R (n=, or. These eqaons are o be solved for an nnown velocy, vorcy n n ( x, R ; ( x, R and densy ( x n, R, defned for poson x R and e. We n pay aenon here only o he copressble fld fllng all of R. In addon o eqaons (. and (.6, (.7 we consder he nal condons ( x, ( x; ( x, ( x (.8 ( x, ( x ro Here ( x n - s a gven, C vecor feld on R and ( x -s gven C scalar feld on R. For physcally reasonable solons of (., (.6 and (.7 wh condons (.8, we wan o ae sre x, ( does no grow large as x. Hence we shall resrc aenon o nal ( x condons ha sasfy ( x x ( x or C K ( x x K (.9 n on R or R -respecvely for any non negave and K. We accep a solon of (. and (.6 and (.7 as physcally reasonable only f sasfes

9 and bonded energy 9 ( x,, ( x,, ( x, C ( R n, (. n R n d x ( x, C, (. The solon for he feld of pressre p s deerned fro (.. In Secon, we gve an exac solon of he defned above Cachy proble for he eqaon (.6, and n Secon 4 also an exac solon of he Cachy proble for he eqaon (.7 s obaned (for, and also for. In Secon 5, a solon s gven of he Cachy proble for he eqaon (. ha follows fro he solon of Cachy proble for he eqaon (.6 ( when b acconng for he effecve vscosy relaed wh nrodcon of he rando Gassan dela-correlaed n e velocy feld V (.. Energy and enropy balance eqaon. Usally, n consderaon of he syse of for eqaons (., (. for fve nnown fncons, a relaon beween he densy and he pressre (eqaon of sae of he ed s appended n order o eqalze he nbers of he eqaons and he nnown fncons. Insead of hs ehod, we shall now derve a slar eqaon, closng he syse (., (., for dvergen flows of a copressble ed whch wll replace he condon of he dvergence-free velocy feld for ncopressble fld flows. For hs prpose we shall oban he energy and enropy balance eqaons whch follow boh fro (., (. and fro sal herodynac relaons []. In he case of a sngle-coponen ed hese relaons ae he for [] (see (4., (5.6, and (5.7 в []: p Ts (. dp sdt d (. p d Tds d (. In (. - (. T s he eperare;, s, and are he nernal energy, he enropy, and herodynac poenal or he Gbbs free energy (of n ass of he ed, respecvely []. The eqaon (. drecly follows fro Eq. (4. n [9] and s n exac accordance wh Eqs. (. and (. for any fncon. For he sngle-coponen ed consdered, provdng nvarably of he parcle nber, we wll asse ha n (. and (. d or cons. In wha follows, we shall se Eq. (. n he for (see also p. 7 n []: s p T (.4. Fro Eqs. (., (. we can oban he followng balance eqaon for he negral nec energy E d n x : de n n d x( d x p ( dv dv (.5 d x For he dvergence-free ncopressble fld flow he forla (.5 concdes exacly wh forla (6. n [ ] and serves as a generalzaon of he laer o he case of copressble vscos

10 fld flow. To derve Eq. (.5 s sffcen o lply (. scalarly by he vecor, lply Eq. (. by he scalar, add he expressons obaned, and negrae he resl over he enre space. We noe ha n he case of an deal (nvscd ed fro (.5 follows ha he negral nec energy s an nvaran only for he dvergence-free flows, whle for he dvergen flows he nvaran s only he oal negral energy ( E h d x assed o be conserved also for he vscos ed [].. On he bass of Eqs. (., (., (. and (.4 we shall now derve he oal energy balance eqaon for a vscos copressble ed and he correspondng enropy balance eqaon. s dsnc fro he dervaon gven n [], we shall drecly se Eq. (., wren wh allowance for he above-enoned eqaly cons. s a resl, ang (. no accon, fro (.4 we oban: ( T ( s (.6 In Eq. (.6 s convenen o represen he second er on he rgh-hand sde wh allowance for (. n he for dv(. In hs case fro (., (., and (.6 we can oban he followng oal energy balance eqaon: B ( ( ( p dv T s x x ( ( ( (, T (.7 B ( p dv dv x ( s n [], ang no accon he reqreen of vanshng he dervave of he negral oal d energy wh respec o e E h, we oban he followng enropy balance eqaon: d B ( s, (.8 T where he expresson B s gven n (.7. The energy and enropy balance eqaons (.7 and (.8 do no concde wh he eqaons gven n [] n forlas (49. and (49.4, respecvely. However, fro he balance eqaon (.7 we can exacly oban hese eqaons (49. and (49.4, as well as he negral enropy balance eqaon (49.6 gven n []. For hs prpose, nsead of Eq. (.6, n (.7 s necessary o se he p s eqvalen represenaon ( ( T (sed n [] regardless of (. and he asspon on he eqaly cons. More essenally, n addon o above n order o oban he concdence beween (.7 and (49.7 n [], s necessary he pressre graden n (.7, p p s n accordance wh [], o express n he for ( T followng fro he x x x herodynac relaon (. (by addng he er dp o boh sdes of (.. Sch a herodynac represenaon for he pressre graden enerng no (.7 (and no (. corresponds o he

11 ordnary eanng of he pressre whch copleely descrbes he noral sresses for copressble and ncopressble eda only a zero vscosy. I does no correspond o ha new eanng of he pressre whch arses naely n he case of descrpon of dynacs of a vscos copressble ed n (. by vre of appearance of addonal noral sresses proporonal o dvergence of he velocy feld (abo hs see also p. 75 n []. In wha follows, hs saeen on no qe adeqae represenaon of he pressre graden (n forlas (.7 and (. on he bass of sng he herodynac relaon (. wll be confred by eans of he fndaenal relaon (., obaned below, beween he e varaon rae of he negral enropy and he negral nec energy. In fac, he relaon (. drecly follows fro (.5 and he negral enropy balance eqaon wren s n he for (.9 on he bass of (.8. On he oher hand, hs relaon (. canno delberaely be obaned fro (.5 and he negral enropy balance eqaon n he for gven n [] (see (49.6 n []. 4. Fro he enropy balance eqaon (.8 here follows he balance eqaon for he negral enropy S d xs n he for (for he sae of splcy, as n (.7 and (.8, we o he ers descrbng he flxes cased by he eperare graden: d S d x d x dv p d T x T dv ( ( (.9 s noed n he prevos e, he balance eqaon (.9 dffers sgnfcanly fro he negral enropy balance eqaon gven n [] (see forla (49.6 n []. In he case of consan eperare T T n (.9, fro (.9 and (.5 here drecly follows he exac flfllen of he fndaenal relaon ds de T (. d d beween he echancal energy varaon rae and he negral enropy growh rae [] (p.4 n []. de The expresson for he represened n he forla (79. n [] canno be drecly derved d fro (. and (., as ade for Eq. (.5, b only s nrodced on he bass of he relaon (. sng he negral enropy balance eqaon (49.6 gven n []. I s precsely he forla (.5 for de he qany s a generalzaon of he forla (6. n [] o case of dvergen flows of he d copressble ed, b no he forla (79., as saed n []. 5. Ths, fro (.5 and (.9 follows ha he negave defneness of he dsspaon rae of he negral nec energy and he correspondng posve defneness of he growh rae of he negral enropy are possble for dvergen flows of a copressble ed only nder he condon of vanshng he second er on he rgh-hand sdes of (.5 and (.9 when he relaon (. holds. Fro Eq. (. follows ha n (.5 he decrease rae of he negral nec energy of dvergen flows s now deerned only by vscos dsspaon, as for dvergence-free flows (see (6. n []. When Eq. (. s flflled, he posvely defned growh rae of he negral enropy n (.9 rns o o be apprecably less han he growh rae of he negral enropy gven by forla (49.6 n []. In fac, n (49.6 here s a er proporonal o he second vscosy coeffcen, whle n (.9 here s no sch a er when he condon (. s flflled. Provdng he flfllen of (., hs decrease n he enropy growh rae n (.9 corresponds o he relave decrease n he nec energy dsspaon rae n (.5, as copared wh he expresson (79. n []. In hs case here s a correspondence wh he Prgozhn prncple of n enropy prodcon (see [9]. Ths, for dvergen flows of a copressble ed on he bass of he reqreen of posve defneness of he growh rae of he negral enropy n (.9 and negave defneness of he

12 growh rae of he negral energy n (.5 we oban he addonal eqaon (. and hs eqaon closes he syse (., (.. Therefore, for dvergen flows of he copressble ed eqaon (. s replace he dvergence-free condon sally sed o close he syse (., (. n he case of he ncopressble-ed approxaon.. New solon o he NS eqaon. Relaon (. deernes he exac al copensaon of he noral pressre sresses and he noral vscos sresses of copressble dvergen flow. s a resl of hs copensaon, he second er on he rgh-hand sde of Eq. (. vanshes. In hs case he eqaon (. concdes exacly wh he n-densonal generalzaon of he Brgers eqaon (.4. The syse (., (.4 s already closed and descrbes he evolon of he densy and he velocy feld of he ed n s neral oon n he presence of dapng relaed o only he acon of shear vscos sresses correspondng o he nonzero rgh-hand sde of Eq. (.4. he sae e, f n (.4 he vscosy coeffcen s eqal o zero, hen fro (.4 we can oban he n-densonal RH eqaon for whch he exac vorcal solon was obaned n [, ] (see also ppendx. In wha follows, we shall consder hs solon and generalze by ang he exernal frcon and he effecve vole vscosy no accon on he base of eqaon (.6. We noe ha, as dsnc fro consderaon of he vorcal solons carred o n he presen sdy and [-6], only he vorex-free solon o he -D Brgers eqaon was nvesgaed earler (when cons n (.4. Ths solon corresponds only o he poenal flow and can be obaned sng a odfcaon of he nonlnear Cole-Hopf ransfor [4, 5]. Wh he a o nrodce effecve vole vscosy (n addon o exernal frcon n (.6 le he change V ( be pleened n (.6, where V ( s a rando Gassan delacorrelaed-n-e velocy feld for whch he relaons hold V ( V ( ( (. V ( In (. s he Kronecer dela, s Drac-Heavsde dela-fncon, and he coeffcen characerzes he acon of he vscosy forces. In he general case he coeffcen can be a fncon of e when descrbng he effecve rblen vscosy b also can concde wh he consan neac vscosy coeffcen when he rando velocy feld consdered corresponds o oleclar flcaons. We wll resrc or aenon o consderaon of he case of consan coeffcen n (.. s was noed n Inrodcon, hs change n (.6 relaed o nrodcon of a rando velocy feld corresponds o he ehod [8] of obanng he sochasc NS eqaon no de o he se of a rando force b by addng a rando velocy o he velocy feld whch eners no he sal deernsc NS eqaon. s dsnc fro [8], n he presen sdy we consder he saon n whch sch a rando velocy feld depends only on e (n [8] sch a velocy feld s called by he drf par of a large-scale rando feld and ang hs feld no accon s eqvalen o nrodcon of a bl vscosy force whch concdes n he srcre wh he sal frcon force n he NS eqaon. Fro Eq. (.6 averaged wh allowance for (. we can oban he eqaon, (. x

13 where he broen braces denoe he operaon of averagng over he rando Gassan velocy feld V (. In dervng Eq. (. fro (.6, (., n addon we se he followng relaon whch s a conseqence of he Frs - Novov forla [6-8]: V x (. Eqaon (. for he case of zero exernal frcon wh n (. can also correspond o Eq. (.4 f he eqales and x x are flflled (see [9] and f n (.4 we prelnarly perfor he change n(. Sch an ncoplng of correlaons s possble n he case of exac separaon of e scales relaed o large-scale neral oons and oons wh he characersc scale of vscos dsspaon [9, 6].. Raher han o solve approxaely (e.g., see [9] he closre proble o fnd he average velocy feld by consderng drecly Eq. (., we shall se he nal eqaon fro whch s he eqaon (. follows exacly. Ths nal eqaon (.6 for he case aes he for of he n-densonal RH eqaon [9, ]: ( V ( (.4 x Indeed, f we ae average n (.4 on V ( he eqaon (. s exacly obaned f (. s sed. s shown n ppendx, n he case of an arbrary densonaly of he space (n=,,, ec. eqaon (.4 has he followng exac solon (see also [-6]: n x, d ( ( x B( ( de ˆ, (.5 where ( B ( d V (, ˆ, de  - s he deernan of he arx Â, and n n n ( s an arbrary sooh nal velocy feld. The solon (.5 sasfes Eq. (.4 only a sch x es for whch he deernan of he arx  s posve for any vales of he spaal coordnaes,.e., de ˆ. Therefore, everywhere n wha follows we shall no se he sybol of absole vale n wrng de Â, f he oppose s no enoned. Only n he case of he poenal nal velocy feld he solon (.5 s poenal for all sccessve nsans of e, correspondng o zero vorex feld. On he conrary, n he case of nonzero nal vorex feld he solon also deernes he evolon of velocy wh nonzero vorex feld (see nex secon. In wha follows, we shall resrc or aenon o consderaon of only he vorex solons n (.5. However, we noe ha n [7] s precsely he poenal solon o he wo-densonal RH eqaon (.4 (when B n (.4 was obaned n he Lagrangan represenaon whch also exacly follows fro (.5 for n=, as was already noed n Inrodcon n connecon wh he possbly of descrpon of he solon of he Svashnsy eqaon (I. sng he poenal solon n (.5. In he one-densonal case (n= n (.5 we have ˆ d de and he solon (.5 d concdes exacly wh he solons obaned n [6, 7]. The solon (.5 can be obaned f we se he negral represenaon for he plc solon of Eq. (.4 n he for

14 4 ( x, ( x B( ( x, wh he se of he Drac dela-fncon (see ppendx or [, ]. fer averagng over he rando feld B ( (wh he Gassan probably densy, fro (.5 we can oban he exac solon of Eq. (. (for n he for: n x d ˆ ( ( ( de exp n (.6 ( 4 s dsnc fro (.5, he average solon (.6 of eqaon (. s already arbrary sooh on any nbonded e nerval and no only provdng he posveness of he deernan of he arx Â. The solon (.6 s also he solon of NS eqaon (. when he condons (. and (.5 are ae place n l.. If we neglec he vscosy forces when B ( n (.5, he sooh solon (.5 s defned, as was already noed, only nder he condon de ˆ [, ] (see ppendx. Ths condon corresponds o a bonded e nerval, where he n lng e of exsence of he solon can be deerned fro he solon o he followng nh-order algebrac eqaon (and sccessve nzaon of he expresson obaned, whch depends on he spaal coordnaes, wh respec o hese coordnaes: d( x de ˆ(, n dx de ˆ( dv de ˆ U, n (.7 de ˆ( (de ˆ de ˆ de ˆ dv de ˆ U U U U, n where ˆ n deu s he deernan of he arx U n, and x ˆ de U s he deernan of a slar arx n he wo-densonal case x x x x for he varables ( x, x. In hs case de U ˆ ˆ,deU are he deernans of he arces n he wo-densonal case for he varables ( x, x and ( x, x, respecvely. We noe ha n he wo-densonal case eqaon (.7 exacly concdes wh he collapse condon obaned n [7] n connecon wh he proble of propagaon of a flae fron nvesgaed on he bass of he Svashnsy eqaon (I.. In hs case for exac concdence s U s (exp( necessary o replace b( n (.7. In he one-densonal case, when n=, fro Eq. (.7 we can oban he n e of appearance of he snglary. In parclar, for he nal dsrbon d( x ax dx x L e L ( x aexp(, a follows ha obaned for he vale x xax. In hs L a case he snglary self can be pleened only for posve vales of he coordnae x when eqaon (.7 has a posve solon for e.

15 5 Ths eans ha snglary (collapse of he sooh solon can never occr when he nal velocy feld s nonzero only for negave vales of he spaal coordnae x. Slarly, we can also deerne he vorex wave brs e for n>. For (.7 n he wodensonal case (when he nal velocy feld s dvergence-free for he nal srea fncon n x x he for ( x, x a L L exp(, a we oban ha he n e of exsence of L L e L L he sooh solon s eqal o. a In he exaple consdered hs n e of exsence of he sooh solon s pleened x x for he spaal varables correspondng o pons on he ellpse. L L In accordance wh (.7 he necessary condon of pleenaon of he snglary s he condon of exsence of a real posve solon o a qadrac (when n= or cbc (when n= eqaon for he e varable. For exaple, n he case of wo-densonal flow wh he nal dvergence-free velocy feld dv, n accordance wh (.7, he necessary and sffcen condon of pleenaon of he snglary (collapse of he solon n fne e s he condon de U (.8 x x For he exaple consdered above fro (.8 here follows he neqaly. When hs L L neqaly s sasfed, for n= here exss a real posve solon o he qadrac eqaon n (.7 e L L for whch he n collapse e gven above s obaned. a On he conrary, f he nal velocy feld s defned n he for of a fne fncon whch s x x nonzero only n he doan, hen he neqaly (.8 s volaed and he developen L L of snglary n a fne e rns o already o be possble and he solon reans sooh n nbonded e even regardless of he vscosy effecs. The condon of exsence of a real posve solon of Eq. (.7 (e.g., see (.8 s he necessary and sffcen condon of pleenaon of he snglary (collapse of he solon, as dsnc fro he sffcen b no necessary negral creron whch was proposed n [5] (see forla (8 n [5] and has he for: di d xdv Uˆ ( de ; I d xde Uˆ (.9 d In fac, n accordance wh hs creron proposed n [5], he collapse of he solon s no possble n he case of he nal dvergence-free velocy feld,.e., when dv. However, n hs case he volaon of creron (.9 does no exclde he possbly of he collapse of he solon by vre of he fac ha he creron (.9 does no deerne he necessary condon of pleenaon of he collapse. cally, n he exaple consdered above (n deernaon of he n e of pleenaon of he collapse e L L for wo-densonal copressble a

16 6 flow he nal condon corresponded s o he nal velocy feld wh dv n (.7 when n=. 4. On he bass of he solon (.5, sng (.7 and he Lagrangan varables a (where x x (, a a ( a, we can represen he expresson for he arx of he frs dervaves of he velocy Uˆ n he for: x ˆ ˆ U (, ( a U a ( a, (. In hs case he expresson (. exacly concdes wh he forla ( gven n [5] for he Lagrangan e evolon of he arx of he frs dervaves of he velocy whch s sasfy he hree-densonal RH eqaon (.4 (when B ( n (.4. In parclar, n he one-densonal case when n=, n he Lagrangan represenaon fro (.5 and (.7 we oban a parclar case of he forla (.: d ( a ( x, ( da x x( a,, (. x d ( a da where a s he coordnae of a fld parcle a he nal e. The solon (. also concdes wh he forla (4 n [5] and descrbes he caasrophc process of collapse of a sple wave n a fne e whose esae s gven above on he bass of he solon o Eq. (.7 n he case n wh he se of he Eler varables. 4. Exac solon o he EH and RH eqaons The velocy feld (.5 corresponds o he exac solon for he vorex feld whch has he followng for n he wo- and hree-densonal cases, respecvely (see ppendx B or [, ]: ( x, d ( ( x B( ( (4. ( ( x, d ( ( ( x B( ( (4. where n (4. ro and n (4. s he nal vorcy dsrbon n he wo-densonal case. The solon (4., (.5 corresponds o he followng exac expresson for he helcy: o H d ( ( ( x B( ( (4. The represenaons for he hree-densonal vorex and velocy felds (4. and (.5 exacly sasfy (see ppendx B he hree-densonal EH and RH eqaons (.7 and (.4 for he case of zero exernal frcon wh n (.7 and when n (.7 we also ae he change B(, as n (.4. fer averagng carred o n (4. - (4. over he rando Gassan feld B ( and ang (. no accon, we oban expressons n whch s necessary o sbse an exponenal fncon wh a noralzng facor, as ha n (.6, n place of he dela fncon n he negrands n (4. - (4.. Only afer hs averagng he exsence of no only he average vorex feld and helcy heselves s ensred on any e nerval, b also he exsence of he correspondng hgher dervaves and hgher oens. In parclar, hs s vald for he ensrophy (negral of he vorcy

17 7 sqare over he enre space and he hgher oens of he vorex feld for whch explc analyc expressons wll be eleenary obaned n he nex secon who solvng any closre proble.. In he Lagrangan varables, n he case B ( he expressons correspondng o he Eleran vorex and helcy felds (4., (4., and (4. can be represened n he for: ( a ( a,, (4.4 de ˆ( a, ( a ( ( a ( a a ( a,, (4.5 de ˆ( a, ( a ( ( a ( a ( a ( a H( a,, (4.6 de ˆ( a, ( where ˆ a de de(. a Fro (4.4 (4.6 follows ha n he wo- and hree-densonal cases for a Lagrangan fld parcle he snglary of boh he vorex and he helcy aes place as when de ˆ( a, and he vale of fne e of exsence of he correspondng sooh felds can be deerned fro he Lagrangan analog of he condon (.. In hs case fro (4.5 and (4.6 follows ha he hree-densonal effec of exenson of he vorex lnes leads only o a weaer power-law b no he explosve ncrease n he vales of he vorex and helcy, as dsnc fro he caasrophc process of collapse of vorex waves n fne e s for he dvergen flow of he copressble ed. We noe ha n [6] (see he forla ( n [6] he represenaon of he solon o he EH eqaon (.7 (when n (.7 was obaned n he for: ( a R ( a, (4.7 J a x In (4.7 n J de s he Jacoban of he ransforaon o he Lagrangan varables a. In hs a case ( a s a new Cachy nvaran (concdng wh he nal vorcy characerzed by zero ( a dr dvergence, whle x R ( a, and Vn ( R,, where V n s he velocy coponen a d noral o he vorcy vecor so ha dvv n for [6]. s dsnc fro (4.4 and (4.5, he expresson (4.7 does no gve any explc represenaon for he solon o he EH eqaon snce n (4.7 no defne dependence s gven for he Jacoban J and he vecor R. he sae e, here s a srcral correspondence beween (4.7 and (4.4, (4.. In he case of he neral oon of Lagrangan fld parcles, for he Jacoban n (4.7 we can already se he explc represenaon J de ˆ, where de  can be deerned fro ( Ensrophy balance eqaon and he hoogeneos frcon

18 8. Fro (4., (4. we can oban he exac closed descrpon of he ensrophy of wo- and hreedensonal flows of an deal copressble ed n he for [, ] (see also ppendx B: d x ( x, d ( /de ˆ (5. / de ˆ (5. d x ( x, d ( In obanng he expressons (5. and (5. here was no need for solvng he closre proble sally exsng n heory of rblence. In he presen sdy hs proble can be go rond owng o he relavely sple represenaon of he exac solon o he nonlnear EH eqaon for descrbng vorex flow. The expressons (5. and (5. end o nfny n fne e, whch can be deerned fro he solon of he algebrac eqaon (.7 and he sccessve nzaon of hs solon n he spaal coordnaes. Usng he exac solon o he EH eqaons n he for of (4. and (4., we can oban he closed descrpon of he e evolon of no only he ensrophy, as ade n (5. and (5., b also for any hgher oens of he vorex feld. For exaple, n he wo-densonal case, ang no accon fro (4. and (.5 (.7 (see ppendx we oban ( ( ( d x d ; ( ;,,,... de ˆ d x d de ˆ In Inrodcon we gave he esae (I. for he relaon beween dfferen oens of he hreedensonal vorex feld whch was obaned on he bass of he expressons of he sae ype fro (4. and (.5- (.7. The esae de ˆ O( pleened n he l as s also sed n obanng (I.. In hs case he n collapse e can be deerned fro (.7.. We wll now ae no accon he exernal frcon. For hs prpose s necessary o consder he case wh n Eq. (.7. In hs case we can also oban he exac solon fro he expressons (.5, (4., and (4. changng n he he e varable by he varable exp( (see (. n ppendx and [, ]. The new e varable now vares whn he fne ls fro (when o (as. Ths leads o he fac ha n he case of flfllen of he neqaly, (5. for gven nal condons he qany de ˆ n he denonaor of he expressons (5. and (5. so ha he denonaor canno vansh a any nsan of e, snce he necessary and sffcen condon of pleenaon of he snglary (.7 wll be no sasfed. The change ( s also be carred o n he condon (.7. Provdng (5., he solon o he hree-densonal EH eqaon s sooh on an nbonded nerval of e. The correspondng analyc dvergen vorcal solon o he hree-densonal NS eqaon also reans sooh for any f he condon (5. s sasfed. In NS eqaon s necessary o ae he relaons (. and (.5 no accon.

19 9 We noe ha nder he foral concdence of he paraeers (see he Svashnsy eqaon (I. n Inrodcon he eqaly ( b( aes place provdng he pleenaon of snglary (.7 when n= and n accordance wh he solon of he Svashnsy eqaon n [7].. For nvscd (deal ncopressble fld flows wh he dvergence-free velocy feld he explosve growh of he ensrophy s characersc of he hree-densonal flows only, whle for he wo-densonal flows he ensrophy s an nvaran. noher saon aes place for dvergen flows of he copressble ed consdered n he presen sdy. In fac, for dvergen flows of an nvscd ed n he wo- and hree-densonal cases we have he ensrophy balance eqaons whch follow fro he EH eqaon (see (B. n ppendx B, or when n (.7: d d d dv, d d d dv d (5.4 Fro (5.4 we can see ha n he hree-densonal case he e evolon of he ensrophy s deerned by no only he effec of exenson of he vorex flaens (he frs er on he rghhand sde b also by he second er cased by he fneness of dvergence of he velocy feld. For wo-densonal flow he e evolon of he ensrophy s pleened only for nonzero dvergence of he flow velocy feld. For he solon (.5 he dvergence of he velocy feld aes he for [, ]: de ˆ n d ( x B( ( (5.5 x The negral of he rgh-hand sde of (5.5 over he enre nbonded space s eqal o zero by vre of flfllen of denes (.4 (.6 (see ppendx and he condon of vanshng he nal velocy feld a nfny. s a resl, he eqaly d n xdv holds for he solon nder consderaon. Ths eqaly ensres flfllen of he conservaon law for he oal fld ass and he exac al negral copensaon of he nenses of dsrbed sorces and sns. For he hree-densonal case n (5.4, sng (.5, (.4- (.6, (4., and (5.5, we can oban exac expressons for he frs and second ers on he rgh-hand sde of (5.4 whch descrbe he conrbon of he effec of exenson of he vorex flaens and of nonzero dvergence of he velocy feld, respecvely, o he ensrophy growh rae. We can readly verfy ha he sae expressons for he wo ers enoned-above can be also obaned by drecly dfferenang he expresson for he ensrophy n (5.. s a resl, we oban he eqaly d ˆ d ˆ de ( / de d ( / de ˆ (5.6 d In (5.6 he frs and second ers on he rgh-hand sde correspond exacly o he frs and second ers on he rgh-hand sde of (5.4, respecvely. Fro (5.6 follows ha n he nvscd case boh ers end o nfny as, when de ˆ n accordance wh (.7. The orgn of boh frs ers on he rgh-hand sdes of (5.6 and (5.4 relaes o he effec of exenson of he vorex flaens. Ther negrands are proporonal o O (. Evdenly, hs er n (5.6 aes de ˆ a relavely lesser conrbon o he explosve growh rae of he ensrophy as copared wh he second er n (5.6 whose negrand s proporonal o O ( and whch exss only n he case de ˆ of dvergen flows wh nonzero dvergence of he flow velocy feld.

20 Snce, as noed above, ang he vscosy no accon (n parclar, n ang no accon he exernal frcon when he condon (5. s flflled leads o reglarzaon of even dvergen solons o he NS eqaon, we can expec ha s also possble for he solons wh nonzero dvergence. For he laer a slar reglarzaon wll also appear possble by vre of he relavely weaer (n he sense noed above effec of exenson of he vorex flaens as copared wh he process of wave collapse n dvergen flow. Ths qeson wll be also dscssed n he nex secon. 4. fer averagng over a rando feld B ( wh he Gassan probably densy, fro (. and (5.5 and ang (. no accon, we oban he followng represenaon for he pressre ˆ n de ( x ( p ( d exp n (5.7 ( 4 The expresson for he densy correspondng o Eqs. (. and solon (.5 aes he for [, ]: d n ( x B( ( (5.8 ( fer averagng n (5.8 and ang (. no accon, we oban he followng expresson for he densy of he ed whch s sooh for any es: n ( x ( d ( exp n (5.9 ( 4 fer replacng, n (5.9, we oban an expresson for he wo-densonal vorex feld snce he expressons (5.8 and (4. have he sae srcres. 6. To exsence of he dvergence-free solons o he NS eqaon The fac self of he analyc represenaon of he sooh dvergen solon o he NS eqaon (. and eqaon (. obaned above n he for (.6, (5.7, (5.9 for an arbrary sooh nal condons, proves ha he proble of exsence and nqeness s solved for hs eqaon. I s of porance ha n hs case s he rando Gassan dela-correlaed-n-e velocy feld was nrodced for slang he vscosy effec. Ths leads o he effecve vscosy force whose srcre exacly corresponds o he srcre of he vscosy force n he NS eqaon, as dsnc fro he hgher dervaves han he Laplacan n he defnon of he vscosy force consdered n [9,]. We wll carry o a coparave analyss of he negral qanes for dvergen and dvergencefree flows whch characerze he e evolon of he negral nec energy whose fneness n [7] s he basc creron of exsence of he solon o he NS eqaon. For hs prpose, we wll consder he negral nec energy balance eqaon (.5 nder he condon (.. In hs case fro (.5 we can oban he expresson de F ; d (6. F d x( x The for of he balance eqaon (6. exacly concdes wh he for of he negral nec energy balance eqaon for dvergence-free ncopressble fld flow gven n [] (see forla (6.. For dvergen flow he fnconal F n (6. s conneced wh he ensrophy d x( ro by he followng relaon

21 F D; D d x( dv (6. The rgh-hand sde of (6. s delberaely greaer han he vale of he fnconal F F for he solon wh zero dvergence of he velocy feld, when D n (6.. For he exac solon obaned, he expresson for he ensrophy on he rgh-hand sde of (6. has he for (5. and fro (5.5 and (.5 - (.7 we obaned he followng expresson for he negral of he dvergence sqare ˆ de D d / de ˆ (6. Fro a coparson of (6. and (5. follows ha n he neghborhood of he solon as (see (.7 he vales of he frs and second ers on he rgh-hand sde of (6. are of he sae order of agnde. In addon, for he fnconal F n (6. we can oban he followng pper esae sng he Cachy-Schwarz-Bnyaovs neqaly: F d x d x d x( d x d x ( roro ( graddv (6.4 In accordance wh (6. - (6.4 he dvergen flows have, all oher facors beng he sae, he delberaely larger vale of he fnconal F as copared wh he dvergence-free flows for whch here s no second er n he sqare braces on he rgh-hand sde of (6.4. Fro or consderaon he conclson on exsence of sooh dvergence-free solons o he NS eqaon ay follow fro he proved fac of exsence of sooh dvergen solons o he NS eqaon on nbonded e nerval when ang he effecve vscosy (or he exernal frcon no accon provdng (5. Dscsson and Conclsons Ths, n (.6, (5.7, and (5.9 we represen he analyc solon o he NS eqaon (. and he conny eqaon (. for dvergen flows wh nonzero dvergence of he velocy feld (5.5. In he hree-densonal case fneness of he energy negral E d follows explcly fro (.6. Ths sasfes he an reqreen n he forlaon of he proble of exsence of he solon o he NS eqaon [7]. long wh hs, he reqreen, saed n [7] for arbrary soohness of he solon on any e nervals when descrbng he velocy and pressre felds, s also flflled. We noe ha for he exac solon (.5 obaned he energy negral E d x d de ˆ also reans fne who averagng (for exaple, n he case B ( for any fne nsan of e, alhogh n he l as he energy also ends o nfny n accordance wh he power law O ( (see (.7. In hs case he solon (.5 can be conned for any fne e n he Sobolev space H ( R. Ths eans ha n he case of an deal ed he flow energy s sasfy he cla lad n [7]. However, n hs case he ensrophy negral n (5. has already he explosve nbonded growh (n a fne e, deerned fro (.7, when O( n he case of a sngle real posve roo of Eq. (.7. Ths eans ha he obaned exac solon o he EH eqaon n he for (.5

22 and (4. canno already be conned n he Sobolev space H ( R for e n he nor (I.. Tang he vscosy no accon aes possble o avod he snglar behavor of he ensrophy and he hgher oens of he vorex feld. Ths eans ha he solons o he EH and NS eqaons can be conned for any n he Sobolev space H q ( R already for any q. In forlaon [7] of he proble of exsence of he solon o he hree-densonal NS eqaon was proposed o resrc consderaon only o he case of solons wh he dvergencefree velocy feld. However, n [7] was noed he porance of consderaon of precsely hreedensonal flows for whch he effec of exenson of vorex flaens n fne e can lead o consran of exsence of he solon o he NS eqaon only n sall. The conclson obaned concernng exsence of sooh dvergen solons o he hreedensonal NS eqaon de o ang even a low vscosy no accon ndcaes also he possbly of posve solon of he proble of exsence of sooh dvergence-free solons on an nbonded e nerval. In fac, as esablshed n (5.6, he effec of exenson of vorex flaens aes a qe saller conrbon o he pleenaon of snglary of he solon as copared wh he collapse of he vorex wave n dvergen copressble flow. Boh neqaly (6.4 and eqaly (6. for he varaon rae of he negral nec energy also ndcae hs possbly. We s noe ha he exac solon o he EH and RH eqaons obaned n [-6] and here gves he closed descrpon of he e evolon of he ensrophy and any oher oens of he vorex, velocy, pressre, and densy felds. The possbly of a closed sascal descrpon of he reges of rblence who pressre (slaed by eans of he nonlnear hree-densonal RH eqaon (.4 was frs noed n 99 n []. We noe ha he general heorec-feld approach o heory of rblence who pressre was developed n 995 n [4] by Polyaov for he hreedensonal RH eqaon wh a rando force of he ype of whe nose (dela-correlaed-n-e, where he relaon beween he volaon of Gallean nvarance and nerence was esablshed. However, a parclar solon of he closre proble was obaned only n he one-densonal case n he for (see forla (4 n [4] of he explc expresson for he probably dsrbon w (, y of he velocy dfference a pons locaed a a dsance y fro each oher. In he presen sdy he approach whch aes possble o ae exacly no accon he pressre s developed. Owng o hs, he analyc solon o he coplee NS eqaon for flow of a vscos copressble ed s obaned. In hs case he an proble of heory of rblence [] s acally solved. The solon obaned can gve he exac represenaon for he al characersc fnconal of he velocy and densy felds of he ed (n hs case he pressre feld can be nqely deerned fro (.. Earler, he solon of he an proble of heory of rblence was consdered o be nachevable for he copressble fld and n [] n hs connecon was wren: Unfornaely, hs general proble s oo dffcl and a presen we canno see even an approach o s coplee solon (see p. 77 n []. Usng he obaned exac solons of he EH, RH, and NS eqaons he rblen reges can be also slaed on he bass of he randozaon ehod for negrable hydrodynac probles proposed by Novov [4] and developed n [9]. For hs prpose s necessary o nrodce he probably easre on an enseble of pleenaon of he nal condons whch n hs case s be consdered as rando fncons. The possbly of exsence of he solon o he Naver-Soes eqaon esablshed n he presen sdy s based on a new e-dependen analyc solon of hs eqaon, earler consdered o be possble [, 7]. In hs case s revealed ha for exsence of he solon on an nbonded e nerval s necessary o ae precsely he vscosy forces no accon. On he oher hand, he qeson of sably of he solon obaned s be consdered on he bass of he exsng resls