1
To:
Ma in R B idson FRS
P esiden , Clay Ma hema ics Ins i u e
F om
Solomon I. Khmelnik
ORCID: h ps://o cid.o g/0000-0002-1493-6630
The open le e o he Clay Ma hema ics Ins i u e (CMI)
«On he six h p oblem o millennium»
In 2010, I ha e sen my me hod o esol ing o his p oblem o he Clay Ma hema ics Ins i u e
(CMI). I ecei ed he ollowing eply:
«CMI does no ecei e no commen on p oposed solu ions o hese p oblems. The
p oposed solu ion mus be e e eed and published in a leading ma hema ics jou nal
o a pe iod o a leas wo yea s be o e CMI will conside i .»
I am no p o essional ma hema ician, do no wo k a any uni e si y (cu en ly I am 86, my sho
CV can be ound in my book on he Na ie -S okes equa ion) and know he “wish” o leading
ma hema ics jou nals o publish pape s by au ho s like me. The e o e, I unde s ood ha I canno
ind such leading ma hema ics jou nal, ha can publish and mo eo e p esen my wo k on his own
ini ia i e o he CMI. The e o e, I did no y. Howe e , I con inued my wo k in his di ec ion and
ha e published my esul s bu no in leading ma hema ics jou nals.
Bu in 2021, D . A.A. Zakha enko ( he Edi o o he Canadian Jou nal o Pu e and Applied
Sciences) became in e es ed in my wo ks and he d ew Ch is ophe 's a en ion o hem (Ch is ophe
O’Neill BA Fine A s Media, Uni e si y College Dublin, I eland.)
Ch is ophe O'Neill has w i en his le e o he CMI, see Appendix 1. The esponse om he CMI
was ecei ed in h ee hou s, almos ins an ly, see Appendix 2. This mean ha he CMI was
amilia wi h my wo k and belie ed ha i did no esol e he speci ied p oblem. So, I wan o
espond o he esponse om he CMI in he ollowing open le e :
The absolu e ad an age o he CMI is he o mula ion o he Millennium p oblems and he o e
o Millennium p izes o hei solu ion om i s no in ini e budge . Bu he e y o mula ion o
he p oblems should no be he p e oga i e o he CMI: hese p oblems can be adjus ed by scien is s
o ou ime and he scien is s o u u e gene a ions ha will wo k du ing his millennium.
A e such a s a emen , I can be suspec ed o pulling ou he Millennium p ize wi h my ee h.
The e o e, i s o all, I say ha I did no ul ill he condi ion o ecei ing he Millennium p ize
in he wo ding ha is compiled in he CMI [1], namely:
Exis ence and smoo hness o he Na ie -S okes equa ion. (1)
I ha e no o mal igh o apply o he Millennium p ize. Howe e , I do no ag ee wi h o m (1)
i sel and wi h he wo ding o he e usal, which is gi en in he men ioned CMI answe , bu I
belie e ha I ha e ac ually esol ed he six h p oblem.
Bu , i s o all, i is necessa y o say abou he o mula ion o p oblem (1). I wan o ci e he
opinion o Olga Ladyzhenskaya, who is an indispu able au ho i y on he p oblems o di e en ial
equa ions, hyd odynamics, and he Na ie -S okes equa ions, see he wo k [4] in Appendix 3.
Appendix 3 con ains a ansla ion o an exce p om pape [3] o Appendix 3, whe e
Ladyzhenskaya, in pa icula , w i es he ollowing.
2
The esea che does no need any p esc ip ions in ad ance conce ning in ini e o any
o he smoo hness o solu ions. One hing mus be demanded, namely he uniqueness
heo em should ake place in he selec ed class o gene alized solu ions.
Thus, acco ding o Ladyzhenskaya, he ollowing p oblem is impo an :
Exis ence and uniqueness o he solu ion o he Na ie -S okes equa ion. (2)
Howe e , elsewhe e on he CMI websi e, he p oblem is o mula ed by ollowing way ([2] in
Appendix 3):
Na ie –S okes Equa ion
This is he equa ion which go e ns he low o luids such as wa e and ai . Howe e ,
he e is no p oo o he mos basic ques ions one can ask: do solu ions exis , and a e
hey unique? Why ask o a p oo ? Because a p oo gi es no only ce i ude, bu also
unde s anding.
I almos coincides wi h o mula ion (2) bu signi ican ly di e en om o iginal o mula ion (1).
I esol ed he p oblem in o mula ion (2), no only o incomp essible luid bu also o
comp essible luid, as well as (wi h addi ional assump ions) o u bulen solu ions (whe e i is no
necessa y o speak abou he smoo hness o solu ions).
As o he esul , i is ob ious. A unc ional wi h a single op imal poin is ound and i is shown
ha he necessa y and su icien condi ion o he op imum o his unc ional coincides wi h he
Na ie -S okes equa ions. I someone unde akes o p o e ha he p oblem is no esol ed, he mus
p o e he allacy o his conclusion. The las edi ion o my book wi h my solu ion is in [5] in
Appendix 3. As o he answe om he CMI i sel , i su p ised me no wi h a denial bu wi h an
explana ion o his denial. Fi s , my ask was o p o e exis ence and uniqueness, no smoo hness.
Uniqueness is mo e impo an han smoo hness, and mo e di icul o p o e. Secondly, he answe
assumes ha I only ha e heu is ics and nume ical me hods. Appa en ly, my me hod is called a
heu is ic because he p oo does no mo e along a well- odden ack. Howe e , i was ob ained
no by nume ical bu by an analy ical me hod. Nume ous p og ams illus a e no nume ical
calcula ion me hods, bu analy ical esul s o mula ed in he MATLAB language. In addi ion, hey
can be used o possible applica ions.
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Appendices
Appendix 1
The le e by Ch is ophe O'Neill o he CMI
F om: Ch is ophe O'Neill <[email p o ec ed]>
Subjec : Millennium P ize: Na ie -S okes Equa ion p oblem, D . Solomon I. Khmelnik
Da e: 4 Augus 2021 a 15:18:26 BST
To: admin@clayma h.o g
To Naomi K ake ,
I ha e been commissioned by he edi o o he Canadian Jou nal o Pu e and Applied Sciences,
D . A.A. Zakha enko, o p esen a solu ion o he Na ie -S okes Equa ion p oblem. The solu ion
3
was de ised by D . Solomon Khmelnik and is laid ou in his book; ‘Na ie -S okes equa ions: On
he exis ence and he sea ch me hod o global solu ions’, which accompanies his le e . The book
consis s o 10 Chap e s and a o al o 261 pages. I was i s published in 2011 and is cu en ly in
i s Six h Edi ion. The publishe in ques ion is Ma hema ics in Compu e Co p. (MiC), which
supplies he ollowing con ac in o ma ion:
BOX 15302, Bene-Ayish, Is ael, 0060860
email: [email p o ec ed]
This o ganiza ion is in connec ion wi h he DNA Jou nal
[h p://dna.izda els wo.com/homeng.h m], who has a lis o employees; E kin, Vale y A.;
G insh ein, Ma k M.; Samokh alo , Vladimi N.; Vilshansky, Alexande N. Cu en ly and D .
Khmelnik himsel [1]. The jou nal is a ailable in bo h Russian and English. As pe he ‘Millenium
P ize Desc ip ion and Rules’, i will be le o he disc e ion o Clay Ma hema ics Ins i u e o
disce n whe he o no ei he o hese publishing en i ies cons i u e a “Quali ying Ou le ” [2].
D . Khmelnik’s solu ion has ecei ed inc easing a en ion om he global ma hema ical
communi y in he las numbe o yea s. Recen ly, a ques ion pos ed on he popula in e ne o um
Resea ch Ga e ga ne ed o e a 100 page iews in a ma e o days [3]. Mo eo e , his book has
ga ne ed 8 ci a ions, om o e 12 au ho s wo king in a wide ange o physics ields [4]. O hese
8 ci a ions, 6 ha e been in he las 3 yea s, and hal in he las yea alone. This indica es ha D .
Khmelnik’s wo k is ga he ing a ou among he global scien i ic communi y.
The eason behind his sudden up ick in ci a ions may be a ibu ed o he addi ion o “compu e
algo i hms” o la e edi ions o his book. Indeed, a ull qua e o he mos ecen ci a ions appea
in he Jou nal o Applied and Compu a ional Mechanics [h ps://jacm.scu.ac.i /], which is a
ecognized by and egis e ed wi h he DOAJ (Di ec o y o Open Access Jou nals) [5][6]. This
sugges s ha he me hod is inding a ou wi h physicis s and ma hema icians who a e wo king in
Compu e Science i s and o emos .
One o hese pape s desc ibes how D . Khmelnik’s solu ion o he Na ie -S okes equa ion can be
used in conjunc ion wi h he equa ion o con ec i e hea ans e o p o ide “ he ma hema ical
model o he p ocesses o hea and mass ans e a PCM phase ans o ma ions” [6]. The inclusion
o his equa ion and he compu a ional simula ions accompanying i s ongly sugges he eal-
wo ld applica ions o Khmelnik’s me hod. Acco ding o one sou ce, he Na ie -S okes equa ions
a e desc ibed as “ he ex emum condi ions o some unc ional” wi h Khmelnik’s me hod o
inding a solu ion o hem being desc ibed as “ he g adien mo ion o he ex emum o his
unc ional”[7].
Since ely,
Ch is ophe O’Neill BA Fine A s Media,
Uni e si y College Dublin, I eland
Re e ences:
[1] h p://www.hmel.i i-as.o g/index_e.h ml
[2] h ps://www.clayma h.o g/si es/de aul / iles/millennium_p ize_ ules_0.pd
[3] h ps://www. esea chga e.ne /pos /The_me hod_o _ esol ing_o _ he_Na ie -
S okes_equa ions_by_D _Khmelnik# iew=60 125060b20020db459865
[4] h ps://www.seman icschola .o g/pape /Na ie -S okes-equa ions%3A-On- he-exis ence-and-
he-Khmelnik/a1becbda4 ce352 19 048354dacd0cc1171bc30?so = ele ance
4
[5] Go obe s, V., T okhaniak, V., Bohdan, Y., & An ypo , I. (2020). Nume ical Modeling o Hea
T ans e and Hyd odynamics in Compac Shi ed A angemen Small Diame e Tube
Bundles. Applied and Compu a ional Mechanics.
[6] An ypo , I., Go obe s, V., & T okhaniak, V. (2020). Expe imen al and Nume ical In es iga ion
o Hea and Mass T ans e P ocesses o De e mining he Op imal Design o an Accumula o
wi h Phase T ans o ma ions. Applied and Compu a ional Mechanics.
[7] Fedosin, S. G. (2014). Fou -dimensional equa ion o mo ion o iscous comp essible and
cha ged luid wi h ega d o he accele a ion ield, p essu e ield and dissipa ion ield.
In e na ional Jou nal o The modynamics, 18, 13-24.
------------------------------------------------------------------------------------------
Appendix 2
The answe -le e om he CMI
On 4 Aug 2021, a 18:47, Ma in B idson <p esiden @clayma h.o g> w o e:
Dea M O'Neill
I am eplying o you le e on behal o CMI. No hing ha I say should be in e p e ed in any way
as indica ing ha CMI will de ia e om he published ules o he millennium p izes, which a e
a ailable on ou websi e.
The Na ie S okes p oblem, as desc ibed in he o icial p oblem desc ip ion, asks o igo ous
ma hema ical p oo o he exis ence and smoo hness o solu ions o he Na ie S okes equa ions
on R^3, he 3- o us R^3/Z^3, o else a p oo o b eakdown o solu ions as desc ibed in (C) o (D)
o he p oblem s a emen . *Rigo ous ma hema ical p oo * is key. No hing sho o his is ele an
in he con ex o he millennium p oblem, al hough heu is ics and nume ics, o example, migh
be o in e es o many s udying luid lows in science and enginee ing.
The con en s o D . Khmelnik's book do no al e he ac ha he millennium p oblem conce ning
exis ence and smoo hness o he Na ie S okes equa ions emains open.
You s since ely,
Ma in B idson
Ma in R B idson FRS
P esiden , Clay Ma hema ics Ins i u e
h p://www.clayma h.o g
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Appendix 3.
O.A. Ladyzhenskaya. Six h p oblem o he millennium: Na ie –S okes equa ions, exis ence
and smoo hness. [3]
A he middle o 2000 he si e o he Clay Ma hema ics Ins i u e (h p://clayma h.o g/) has
published one a icle called “Millennium P ize P oblems”. This a icle iden i ies se en p oblems
called he p oblems o he new millennium. Each o he p oblems has i s own subsec ion. The six h
subsec ion in [1] was en i led “Na ie -S okes, exis ence and egula i y” (now his i le was
changed), whe e i is conce ned only on incomp essible luids. The au ho o he a icle (Cha les
L. Fe e man, he P ince on Uni e si y, USA) The au ho con ines himsel o he wo simples
p oblems o he Na ie -S okes equa ions: he Cauchy p oblem and he p oblem wi h bounda y
condi ions pe iodic in spa ial a iables. In hese simples p oblems, he luid ills he en i e space.
The e a e no obs acles (bodies) on he way o he luid. These p oblems e lec some o he
5
di icul ies ha exis in he ini ial bounda y alue p oblems o luids ha ill only pa o he space
Rn, n = 2 o 3.
Cha les Fe e man conside s physically meaning ul only hose solu ions ha a e in ini ely smoo h
unc ions and, in he case o he Cauchy p oblem, dec easing as e han any deg ee o x. Fo bo h
asks, he ollowing wo ques ions a e posed:
1) has i a solu ion o he speci ied ype o all he ime momen s Є R1+ = [0, ∞) o
2) is he e a ini e poin in ime 1, a which he equi ed smoo hness is iola ed? In his case,
he gi en pa ame e s o he p oblem ( he ini ial eloci y ield and ex e nal o ces) a e
assumed o be in ini ely smoo h and apidly dec easing a x o (x, ) app oaching an
in ini y.
Wi h my almos i y yea s o expe ience in he s udy o Na ie -S okes equa ions and mo e han
hal a cen u y o expe ience in he s udy o bounda y alue and ini ial bounda y alue p oblems
o linea and nonlinea pa ial di e en ial equa ions o a ious ypes, I would o mula e he main
ques ion ega ding he Na ie -S okes equa ions in a comple ely di e en way, namely:
P oblem 1. Can o canno he Na ie –S okes equa ions oge he wi h ini ial and bounda y
condi ions gi e de e minis ic desc ip ion o he dynamics o an incomp essible luid?
In his a icle, I wan o b ing o a wide ange o ma hema icians wha i is enough o do o he
Na ie -S okes equa ions o gi e an answe o P oblem 1. Impa ien eade o hose who like o
ead wo ks (books) in e e se o de can ge acquain ed wi h my conclusions in he "Conclusion"
sec ion a he end o Pa ag aph 4.
When sol ing P oblem 1, he choice o he phase space and he class o gene alized solu ions
should be le o he esea che , no p esc ibe o he esea che in ad ance an in ini e o any o he
smoo hness o solu ions. One hing mus be demanded: he uniqueness heo em should ake place
in he selec ed class o gene alized solu ions. I is ad isable o s a he s udy o any ini ial
bounda y alue p oblem (as well as he Cauchy p oblem) by inding uniqueness classes.
Re e ences:
1. Cha les L. Fe e man. Exis ence & smoo hness o he Na ie -S okes equa ion,
h p://www.clayma h.o g/si es/de aul / iles/na ie s okes.pd
2. Na ie -S okes equa ion, h p://www.clayma h.o g/millennium-p oblems
3. O.A. Ladyzhenskaya. Six h p oblem o he millennium: Na ie –S okes equa ions, exis ence
and smoo hness. Russian Ma hema ical Su eys, 2003, olume 58, issue 2(350),
h p://www.ma hne . u/links/5 7064e54 b50225e7443 91783364c8/ m610.pd
4. Olga Ladyzhenskaya, h ps://en.wikipedia.o g/wiki/Olga_Ladyzhenskaya
5. Khmelnik Solomon. Na ie -S okes equa ions. On he exis ence and he sea ch me hod o
global solu ions. Ve sion 3, pp. 1–129. "MiC" - Ma hema ics in Compu e Co p., Is ael,
h ps://doi.o g/10.5281/zenodo.5037579
