Optimal control of fluid-structure interaction systems : the case of a rigid solid
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Optimal control of fluid-structure interaction systems : the case of a rigid solid Marwan Moubachir — Jean-Paul Zolésio. N° 4611 Octobre 2002. ISSN 0249-6399. ISRN INRIA/RR--4611--FR+ENG. THÈME 4. apport de recherche.
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(55) lmZ\^ x egv^X|lvegnsp/nw ]<nwlmeonWp e ¤ Y&Z ^ h\e x egujsumuzh\]<^ x lmnqO^jq§fegum|%nWh uAeopO|%ns]<r\m^{umuveo\ig^Cp\^{&lvnsp egjsp Oh\e x ¤ * lmu^§Wnsightlvegnsp`eku x ^{um|%megO^ x fy eolmu§s^ignt|%eoly u jwp x eolmur v^Xuvuvh\m^ p ¤²Y&Z ^<|nsh\r io^ (u, p) uvjwlveku´ ^XulvZ\^²|igjWuvuveg|{jwiC¦Rj¨§feg^ ¢ cflvns©W^{u ^Xªh jFlmeonWp uSqveolzlm^peopp\nWp ¢ |%nspOuz^{v§FjFlmeo§W^+ ¡nWv] f 0. s 0. t. . f. 0. 1. f. s t. 2. . ∂t u + D u · u − ν∆u + ∇p = 0, div(u) = 0, u = u∞ , u(t = 0) = u0 ,. _. Qf (V ) Qf (V ) Σf∞ Ωf0. . qZ ^m^ ν uzlmjsp x uS ¡ns&lmZ\^©feop ^] jFlmeg|N§fegum|%nuzeolyjsp x u eku&lmZ\^` ßjwv´ ^i x §s^ignt|%eoly ´ ^{i x ¤ R^p |^s½tlmZ\^`r\mnw^{|%lv^ x h e x ignWj x u ZOj¨§s^lvZ\^` ¡nWioignFqeop b^}tr\v^Xuvuvegnsp F ∞. . f. Ff = −. qZ ^m^. ¤. σ(u, p) = −p I +ν(D u + ∗ D u) (D u)ij = ∂j ui = ui,j. Z. σ(u, p) · n Γst. !. J. · e2. uzlmjsp x u ¡ns/lvZ ^ h e x ulmv^Xuvulv^{p uznWegp uve x ^ Ω ½RqeolvZ f t. Û¡Ý'Þ*ÛÜá.
(56)
(57) ! " $# % & (')*,+ .- / 0. V. ®^<|%ns]<r\ig^%lm^
(58) lmZ\^<qZ\nsig^<uzytuzlv^] qeolvZ¬©feop\^{] jFlvek|<|%nWpWlmeopfh\eoly£|%nWp x eolvegnsp u`jFlNlvZ ^ h e xf¢ uzlvmh |lmh\v^ egplv^{z ßjW|%^ Γ nWp Σ (V ) u = V = d˙ e , Yn uzh ]b] jsveg¯^W½t^`s^l&lvZ\^` ¡nWioignFqeop <|nsh\r io^ x uvytulm^] s t. . s. 2. . ¶ (ßsS¼ ns. ' _5 4 . ∂t u + D u · u − ν∆u + ∇p = 0, div(u) = 0, u = u ∞, u = d˙ e2 , ! Z ¨ md+kd = − σ(u, p) · n · e2 , Γst h i u, d, d˙ (t = 0) = [u0 , d0 , d1 ] ,. Qf (V ) Qf (V ) Σf∞ Σs (V ) (0, T ) Ωf0 × R2. lvZ ^` ¡nsigionFqegp\b]begp\eg]<eo¯XjFlvegnspr\mns io^{]/½. u∞ ∈ U c. min j(u∞ ) u∞ ∈ U c h i h i j(u∞ ) = Ju∞ ( u, p, d, d˙ (u∞ )) u, p, d, d˙ (u∞ ). qZ ^m^ qe«lmZ jsp x J ekuqj<v^Xjwi: ¡h\p |%lvegnsp jsi*nw lvZ\^` ¡nWioignFqegp\b ¡nsm]. . V . W. egu"jNS^{js©
(59) uznWioh\lvegnsp<nw Ar\mns\ig^]. V . u∞. Z i γ h i α Th |d − d1g |2 + |d˙ − d2g |2 + ku∞ k2Uc Ju∞ ( u, p, d, d˙ ) = 2 0 2. j x ]<e«lu`jFl`io^Xjsuzl`j/uznWioh\lvegnsp |nsp x e«lmeonWp u½. X. qZ\ek|Z¬umjFlmeguz´ ^Xu+lmZ\^b ¡nsigionFqegp\p ^{|%^Xuvumjwmy´ muzl ¢ ns x ^{Rnsr\lveg]<jsioeoly X ∇j(u ) = (σ(ϕ , π ) · n)| +γu =0 qeolvZ (ϕ, π, b , b ) jwm^uznWiohtlmeonWp uSnw 'lvZ\^` ¡nWioignFqegp\bj x nsegplRuzytuzlv^{]/½ u∗∞. ∗ ∞. 1. u∗ ∞. u∗ ∞. Σf∞. ∗ ∞. 2. −∂t ϕ − D ϕ · u + (∗ Du) · ϕ − ν∆ϕ + ∇q = 0, div(ϕ) = 0, ϕ = b 2 e2 , ϕ = 0, ϕ(T ) = 0,. ÞÞ È ]_^`/aa. Qf (V ) Qf (V ) Σs (V ) Σf∞ ΩfT. s. . X. . −b˙1 + k b2 = α(d − d1g ), (0, T ) b1 (T ) = 0,. . . −∂t λ − ∇Γ λ · V = f, Σs (d˙ e2 ) λ(T ) = 0, ΓT (d˙ e2 ). . {.
(60) W. -
(61) ; . qeolvZ. i h ˙ 2 (D ϕ · e2 ) · e2 f = −d˙ b˙2 + ν (D ϕ · n) · (D u · n) − |d| Z. ¶
(62) t. Γst (V. ). jwp x. Z h i λ n = −b1 − m b˙2 − α(d˙ − d2g ) e2 +. {_. . σ(φ, π) · n Γst (V. ). 1 . A
(63) _ Y 0 Y-$ (λ, b ) I. " +Z 0% 0 9 +* _ Y ! Y- "A )1 _ -0 _ ' Z+ E
(64) (ϕ, π, b2 ). . {. ! Z 1 2 m ¨b2 + k b2 = α(d − dg ) − α(d˙ − d˙g ) + ∂t σ(φ, π) · n · e2 Γst (V ) Z h i ˙ 2 (D ϕ · e2 ) · e2 − ν (D ϕ · n) · (D u · n) · n | d| + Γst (V ) h i b2 , b˙ 2 (T ) = [0, 0] ,. {_. . /J. (0, T ). q# % m \¬
(65) #q /#q .#0 S %#q *) !'#+ ®^+jwm^&eoplm^m^{uzlv^ x egpbm^{|{jwigioegp\Nsigns jsit^%}tekulm^p |^Sm^{uvh\iolmu' ¡nW"^Xjw©uvnsightlvegnspOu'lvn`lvZ ^qeop\eolvekjwi OnWh\p xf¢ jsvy£§Fjsioh\^<r\mns io^{] V¤p\^ uzZ nsh\i x ^}trO^X|l
(66) uvns]<^m^{uzlvmek|lvegnspOunWp¥lmZ\^ ^%}tekulm^p |^blveg]b^< ¡nW`lvZ\^ uvnsightlmeonWp uquzegp |^ x ^r:^p x egp\ nWplmZ\^ x jFlj\½flmZ\^uznWioe x ]<j¨y§¨jsp\ekuzZ/nshtluze x ^`lmZ\^|%nWpWlmvnWiA§snsigh\]<^ Ω ¤ Y&Z eguqr vnW\io^{] &jsu&m^{|^plvigy²egp§W^{uzlvegWjwlv^ x fyuv^§W^jwi*jwh\lvZ\nWmu Åw ؽ J ؽ gs ؽ g/ J ؽ g9 X U½ w_ U¤¬^ m^% ¡^{Nlvn o/ W °jwp x g9 X C ¡nWj/|ns]<r\ig^%lm^bm^§feg^¤ * p¬nWh\|jsuv^s½!S^<nsp\igy Z j¨§W^
(67) nWp\^ x ^Wv^{^ns ° ¡m^^ ¢ x ns] ¡nslvZ\^`uvnsige x ]bnslvegnspjsp x S^jsv^ x ^{jsioegp\qe«lmZ²lSn x eg]<^p uveonWp jwiA¦Rj¨§feg^ ¢ cflvns©W^{uC^Xªh jFlmeonWp u¤ ¦R^§W^vlvZ\^{io^Xuvu{½A^ x ^{jsiCqe«lmZ¬p nsp ¢ Z\ns]<nWs^p ^nshOu Heomeg|Z io^lN:nsh\p x jwmy |%nWp x eolvegnsp u`jFlNlvZ ^ ßjsz´O^i x Oh\e x :nsh\p x jwmys¤ O 8 ¶fº ¶
(68) 1 * _ ' $ Z+ 0_ # . . . . . . . . Ωs0 Ω0. . . . . . . " ' $ . . C2. a = dist(Γf∞ , Ωs0 ) > 0. . . . . u0 ∈ (L2 (Ωf0 ))2 div(u0 ) = 0, u0 = d 1 e 2 ,. u∞ ∈ (H m (Σf∞ ))2 f Ω0s Γ0. m>. 3 4. {. . _ _" R % % " R _ I' !0 s _" Y $ E , T 0 R= T 0(u ! 0, a, (- Ω 0,EΩ 0 )0 % A" Y-$ T ∈ (0, T0) d ∈ W 1,∞ (0, T ; R) ∩ C 0 ([0, T ]; R) ˙ ∈ L2 (0, T ; Vd(.) ) ∩ L∞ (0, T ; Hd(.) ) (u, d). Û¡Ý'Þ*ÛÜá.
(69)
(70) ! " $# % & (')*,+ .- / 0 . X. Hd(t) ≡ (v, `) ∈ (L2 (Ω0 ))2 × R,. div(v) = 0, v · n = 0,. A. Vd(t) ≡ (v, `) ∈ (H 1 (Ω0 ))2 × R,. div(v) = 0, v = 0,. A* _ ' $ Z+. 0 # − +. Z Z. T 0 T 0. "Z Z. Ωft. Ωft. u · ∂t v + md˙ `˙ − k d `. . . Γf∞ ,. v|Ωst = ` · e2 , supp(v) ⊂ Ωft. o. Γf∞ ,. v|Ωst = ` · e2 , supp(v) ⊂ Ωft. o. #. [(D u · u) · v + ν D u · · D v] = md1 `(0) +. Z. Ωf0. u0 · v(0). %V . ˙ )=0 v(T ) = `(T ∀ (v, `) ∈ C ([0, T ]; Vd(.) ) 7 Z+F" , ' " \ Y M "$ _ "
(71) + $ 0 A . A % #. ¶
(72) t . . 1. . ' u∞ . 1. 1. u∞ · τ ∈ L2 (0, T ; (H 2 (Γf∞ ))2 ) ∩ H 4 (0, T ; (L2 (Γf∞ ))2 ) 3. 1. u∞ · n ∈ L2 (0, T ; (H 2 (Γf∞ ))2 ) ∩ H 4 (0, T ; (H −1 (Γf∞ ))2 ). #& #1/'
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(75) uznWioh\lvegnsp<nw Ar\mns\ig^] V jsp x J ekuqj<v^Xjwi: ¡h\p |%lvegnsp jsi*nw lvZ\^` ¡nWioignFqegp\b ¡nsm] Z h h i i γ 9 X α J ( u, p, d, d˙ ) = |d − d | + |d˙ − d | + ku k . . . . . . . . ∞. ∞. ∞. u∞. ∞. c. . ∞. u∞. T. u∞. ÞÞ È ]_^`/aa. 2. 0. 1 2 g. 2 2 g. 2. 2 ∞ Uc.
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(79) lmZ\^qh\p\©fp\nFqp x ns] jwegp Ω qZ\ek|Z x ^{rO^{p x u"jsiguvnNnsplvZ\^+|nsplvmnsif§Fjwmegjs\ig^ u ¤'Y&Z egu r:nsegplqqeoigi*:^uznWio§W^ x yh uzegp\blvZ ^R *2] jwr T egplvmn x h |%^ x r v^{§egnshOuzigys¤ . f t. ∞. t. .
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(112) lvZ\^<ns x ^{Rnw °lvZ\^ uvnsige x WnF§s^{vp\egp\²^Xªh jFlmeonWp fy x ^%´ p eop\ lmZ\^signsOjwi!uznWioe x §¨jsvekjw io^W½ w d d = d˜ = d d˙ ig^{j x egp\blvnblvZ\^`´Omuzlqns x ^qnW x egp jsvy x e :^{v^{plvekjwi*^XªWhOjFlvegnsp½ Z V ˙ M d˜ + K d˜ = F 1 2. ÞÞ È ]_^`/aa.
(113) ¨. -
(114) ; . qeolvZ M=. 1 0 0 m. . 0 k. F =. . K=. jsp x. . −1 0. 0 Ff. . . p/lmZ\^|jsuv^`nw 'lvZ\^|nsh\r io^ x h e xf¢ uvnsige x uzytuzlv^{]/½\lvZ\^`ignWj x u x ^r:^p x nWplmZ\^ h\e x uljFlm^N§Fjsvekjw\ig^s½ *. Z. Ff (u, p) = −. σ(u, p) · n. !. · e2. ®^ egpWlmvn x hO|%^<lvZ ^uljFlm^²jwp x ]
(115) h i«lmeor\igeg^
(116) uvr js|^{u`egpns x ^`lmn x ^%´Op\^²jr\mnsr:^uvnsige x S^{jw©¬uljFlv^ nWrO^{mjwlvnW{½ . . W X = d ∈ C ([0, T ]) . . X X = d ∈ C ([0, T ]) . s Y = b ∈ C ([0, T ]) . \ Y = b ∈ C ([0, T ]) jsp x lvZ\^ignWj x uvr js|^s½ s 1. 1. s 2. 2. s 1. 1. s 2. 2. Γst. 1. 1. 1. 1. . L = F ∈ C 1 ([0, T ]). Y&Z eguqjsioignF+uSh u&lvn x ^´ p\^j<uvnsige x uzlmjFlm^`nsr:^jFlvnW{½. es : X1s × X2s × L −→ (Y1s × Y2s )∗. qZ nWuv^`jW|lmeonWpeku x ^%´Op\^ x fy lvZ ^` ¡nsigionFqegp\<e x ^plveolys½ hes (d1 , d2 , F ), (b1 , b2 )i = +. Z. Z. T 0. T 0. h. h. i −d1 b˙1 − d2 b1 − d01 b1 (0) + d1 (T )b1 (T ). Z i −m d2 b˙2 + k d1 b2 − m d02 b2 (0) + m d2 (T ) b2 (T ) −. T. F b2 0. Û¡Ý'Þ*ÛÜá.
(117) X.
(118) ! " $# % & (')*,+ .- / 0 . . 838!. . +. < +%. * p/lmZ\eguRuz^X|lvegnsp½\^uvh\]<] jwmeo¯{^`lvZ\^ x e A^m^pl+nsrtlmeonWp uSlvZOjFlRS^`Z j¨§s^|Z\nuz^{p ¡nW&lvZ\^
(119) jwWmjsp\sekjwp ¡jw]<^{nWv© jwp x/x ^%´ p\^`lmZ\^`§Fjwmegjwlvegnsp jsiAuzlmjwlv^nWrO^{mjwlvnsq|nsp uzlvjwegpl{¤ * pnW x ^{Slvn x ^{jsi!qe«lmZvegse xx eguvr\ikjs|%^{]<^pl+§s^X|lvnWS´ ^i x u{½t^egplvmn x h |%^NlvZ ^` ¡nsigionFqegp\<uzr jW|%^Xu +eoWe x/x ekuzr\ikjs|^]<^pl+uvr js|^{u eop Ω H = φ ∈ (H (Ω )) ; ∇φ = 0, eop Ω , φ = 0, nsp Γ H = φ ∈ (H (Ω )) ; ∇φ = 0, ¶
(120)
(121) 7 % & + _. A
(122) " . ϕ∈H Ω C∈R . . 1. def. 1 0. def. 1. 0. 1. 0. 1. 2. s 0. 2. s 0. . . s 0. f ∞. . 2. . . ¥p lvZ\^ uv^{ªh\^{iH½!S^qegigi'p ^^ x lvn x ^%´Op\^<r\v^X|%ekuz^buljFlm^bjsp x ]h\iolvegr\ioeg^uvr js|^{ueop®ns x ^{+lvn^{p x nF nWh\qr\mns\ig^] qe«lmZj< jwsjwp sekjwp ¡h\pO|lvegnspOjwiA ¡jw]<^Snsm©A¤ ¼ nsigionFqegp\lvZ\^^}teguzlv^{p |%^`m^{uvh\i«lRuzlmjwlv^ x r\m^§fegnsh uvigys½f^egplvmn x h |%^NlmZ\^ h\e x uljFlv^uvr js|^ ϕ|Ωs0 = C. *. . n. def. Xf. =. def. Zf. u ∈ H 2 (0, T ; (H 2 (Ωft ))2 ∩ H1 ) . p ∈ H 1 (0, T ; (H 1 (D))2 ). o. ®^jwikuvnp\^{^ x lv^Xulq ¡h p |lmeonWpuvr jW|%^{ulmZ jFl+qegioiO^h uv^% ¡h i!lvn x ^´ p\^ jssjwp\W^R]h\i«lmeor ioeg^u =. Yf. def. Vf. def. Wsf. def. =. =. . o v ∈ L2 (0, T ; (H 2 (Ωft ))2 ∩ H01 ) . q ∈ H 1 (0, T ; (H 1 (D))2 ) . (v, b2 ) ∈ Y f × Y2s , v|Γst = b2 · e2. n. ®^ x ^´ p\^`lvZ ^ O h\e x S^{js©²uljFlm^NnWrO^{mjwlvnW{½. efu∞ : X f × Z f × U s −→ (Y f × V f )∗. qZ nWuv^`jW|lmeonWpeku x ^%´Op\^ x fy hefu∞ (u, p, us ), (v, q)i = +. Z. T 0. −. ÞÞ È ]_^`/aa. Z. Γf∞. Z. T 0. =. . Z. T 0. Z. Ωft. [−u · ∂t v + (D u · u) · v − νu · ∆v + u · ∇ q − p div v]. u∞ · (σ(v, q) · n) + Z. Z. T 0. v · (σ(u, p) · n) + Γst. Z. Z. [us · (σ(v, q) · n) − (u · v) hus , ni] Γst. u(T ) · v(T ) − ΩT. Z. u0 · v(t = 0) Ω0. ∀ (v, q) ∈ Y f × V f.
(123) {_. -
(124) ; . . . . +%3. 7 !. . 1. +. <. +'. h °]<^{|Z jsp\ek|jwiOuzytuzlv^] |nsp uveguzl°ns juznWioe x r jszljsp x j Oh\e x r jwvl{¤CY&Z\^Xuz^Ruvh\ uvytulm^] u"ZOj¨§s^+:^^{p m ^r v^Xuz^{pWlm^ x lvZOjwp\©tuqlmnj²uvnsige x jwp x j h\e x uljFlm^
(125) nsr:^jFlmnsX¤ * lekuRp\nF r:nWumuzeg\ig^lvn/|%nWh\r\ig^lvZ\^Xuz^ lSn<nsr:^jFlmnsu°egpns x ^Slvn h\eoi x jwpj xf¢ Z\nt|`|%nWh\r\ig^ x uzytuzlv^] nsr:^jFlvnW{¤ Y&Z ^] jFnW&rOnWeopl&Z\^m^s½tekulvnbp\nwlmeg|^lmZ jFl&lmZ\^ h\e x ignWj x F jsr\r:^{jwuSeoplvZ ^ h e x uljFlm^NnWrO^{mjwlvnW jsp x lvZ\^{p<|{jwp<:^+|%nsh r\io^ x qeolvZblmZ\^+uznWioe x r jwvllvZ jsp\©tulvnNlvZ ^qeop\r htl"ionj x F ns OlmZ\^+uvnsige x nsr:^jFlmnsX¤ Ynjs|Z\eg^§W^SlvZ\eku"|%nWh\r\igeop ½wS^&p\^^ x lmn x ^{|e x ^qqZ\^%lmZ\^"nWCp\nwllmZ\^ h\e x jwp x lmZ\^+uznWioe x ]
(126) h\iolvegr\igeo^{mu ] jwlm|Z²jFl°lvZ\ ^ h\e xf¢ uvnsige x eoplv^{z ßjW|%^W¤ * *p\nsl{½^RZ j¨§s^&lmn
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(128) qeolvZ lmZ\^ h\e x |nsp uzlvjwegpl"lv^{p uznW°jFl lmZ\^ h e xf¢ uvnsige x :nsh\p x jwmys½WqZ jFl°] j¨yO^+p nwl|nspf§s^{p\eo^{pl x h\^qlmnv^{sh\ikjwmeolyv^Xªh\eom^]<^{pWlX¤ R^p |^s½ S^|Z nnuz^
(129) lvnnWv©qeolvZ¬|nsplvegpfh\nshOu+lm^{uzl ¡h\pO|lvegnspOunWp ¤`Y&Z eguN]<^{jsp u+lmZ jFl`S^buvZ jwigi|Z\nfnWuv^ lmZ\ ^ h\e x jwp x uv^{|%nWp x uvnsige x ]
(130) h\iolvegr\igeo^{ uzr jW|%^Xulmn¬:^lmZ\Γ^uzr jW|%^ W ¤®^ x ^%´ p ^lmZ\^/|nsh\r io^ x uvytulm^] S^{js©uzlmjFlm^`nsr:^jFlvnWqjsuS ¡nsigignF+u½ f. s t. f s. eu∞ : Y f × Z f × X1s × X2s −→ (Wsf × V f × Y1s )∗. qZ nWuv^`jW|lmeonWpeku x ^%´Op\^ x fy lvZ ^` ¡nsigionFqegp\<e x ^plveolys½. heu∞ (u, p, d1 , d2 ), (v, q, b1 , b2 )i = hes (d1 , d2 , Ff ), (b1 , b2 )i + hefu∞ (u, p, d2 e2 ), (v, q, d1 · e2 )i Z TZ = [−u · ∂t v + (D u · u) · v − νu · ∆v + u · ∇ q − p div v] Ωft. 0. +. +. . Z. Z. T. 0. Z. Γf∞. u∞ · (σ(v, q) · n) +. u(T ) · v(T ) − ΩT. + Z. Z. T 0. h. Z. T. 0. .-. Γst. [(d2 e2 ) · (σ(v, q) · n) − (u · v) hd2 e2 , ni]. i Z ˙ −d1 b1 − d2 b1 +. u0 · v(t = 0) −. Ω0. . Z. T 0. d01 b1 (0). h. i. + d1 (T ) b1 (T ) − md02 b2 (0) + m d2 (T ) b2 (T ) ∀ (v, b2 , q, b1 ) ∈ Wsf × V f × Y1s. . . −m d2 b˙2 + k d1 b2. #+':# 1. p2lmZ\ekuuv^{|lmeonWp*½&^£egplvmn x h |%^lmZ\^¥ikjwWmjsp\sekjwp ¡h\p |%lvegnsp jsijsumuvnf|egjwlv^ x qeolvZ r\mns io^{] V jwp x r vnW\io^{] /W W L (u, p, d , d ; v, q, b , b ) = J (u, p, d , d ) − he (u, p, d , d ), (v, q, b , b )i qeolvZ *. . def. u∞. 1. 2. 1. 2. α Ju∞ (u, p, d1 , d2 ) = 2. 1. u∞. "Z. 2. u∞. T. |d1 − 0. d1g |2. + |d2 −. d2g |2. #. 1. 2. +. γ ku∞ k2Uc 2. 1. 2. Û¡Ý'Þ*ÛÜá.
(131) %J.
(132) ! " $# % & (')*,+ .- / 0. uzegp\blvZ egu& ¡h p |lmeonWp jwiU½tlvZ ^nsrtlmeo] jwi*|nsplvmnsiAr\mns io^{] / WS|jsp/:^r\htl+egp/lvZ\^N ¡nWioignFqeop ¡nWv] min. min. max. . Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ). s Sy hOuzegp\lvZ ^ K£eop ¢ K jw}¥ ¡jw]<^Snsm©A½!^²j¨§Wnse x lmZ\^|%nW]<r\htlmjwlvegnsp®nw SlvZ\^uljFlv^ x ^{veg§FjFlmeo§W^bqeolvZ m^{uvr:^{|llmn u ¤ ¼ eoul ¢ nW x ^nsrtlmeo] jsioeoly
(133) |%nWp x eolvegnsp uqegigit ¡h\mp\eguvZblvZ\^qWmj x eg^plCnw lmZ\^qnsmegsegp jwit|%nul ¡h p |lmeonWp/hOuzegp\blvZ\^uvnsightlmeonWpns jsp/j x nsegpl&r vnW\io^{]¤ ®^NnWh\i x ioeg©s^lvn<jwr r\ioy ]<egp ¢ ]<jw} x e:^{v^{pWlmegjs\egioeoly<v^Xuzh\iolmulmnr\mns io^{] s¤CY&Z\^N] jwegpekuvuvh\^Negu lmn<r\vnF§W^RlmZ jFlqlmZ\^]<eop ¢ ] jF}/uzh\ r\vnW\ig^] w _ min max L (u, p, d , d ; v, q, b , b ) j x ]<e«luqjFl+ig^{jWulqnsp ^`umj x x io^`r:nsegplq ¡ns u ∈ U ¤ » F\;» 5 ¶ 4 6 [1 ¶ » ' ¶ ®^qjsumuzh ]b^SlmZ jFlClmZ\^q|%nsp x e«lmeonWp u'lvnjsr\r\igylvZ\^ K eop ¢ K jw}r\vegp |eor\ig^ Ojwm^ ¡h\ioio´ igio^ x uzn`S^&|{jwpbfy ¢ rO jsumulvZ\^ x ^meg§¨jwlvegnsp qeolvZ²m^{uvr:^{|l°lvnlmZ\^|nsplvmnsi §¨jsvekjw io^ u lvmnsh\WZ<lvZ\^R]begp ¢ ] jF}²uzh \r\mns\ig^] w_ ¤ * l+io^Xj x uSlmn<lvZ\^` ¡nWioignFqegp\m^{uvh\iol O 8 ¶fº ¶
(134) 7 I A _&! $ (' u ∈ U (u , p , d˜ , ϕ , π , ˜b ) u∞ ∈ U c. (u,p,d1 ,d2 )∈X f ×Z f ×X1s ×X2s. (v,b2 ,q,b1 )∈Wsf ×V f ×Y1s. . ∞. (u,p,d1 ,d2 )∈X f ×Z f ×X1s ×X2s. 1. u∞. (v,b2 ,q,b1 )∈Wsf ×V f ×Y1s. 2. 1. 2. c. ∞. . ∞. . E" -$ A " #. . _ + ∞ " ! c (' _ %u A' uE u ∞. ∞. ∞. j. u _ u ∞. . ∞. u∞. u∞ ∈ Uc. + -0 ' $ +. º*Aº @ Ruvegp\lvZ ^nsm^] ¡mns] W ؽS^Rfyfr jWuvu"lvZ\^ x ^meo§Fjwlvegnsp qeolvZm^{uvr:^{|llvn ]<egp ¢ ]<jw}uvh\\r\mns\ig^] w_ . . egp uze x ^lvZ\^. YJ. ∇j(u∞ ) = (σ(ϕu∞ , πu∞ ) · n)|Σf∞ + γ u∞. u∞. D Lu (uu∞ , pu∞ , d1u∞ , d2u∞ ; φu∞ , πu∞ , b1u∞ , b2u∞ ), δu∞ i Du∞ ∞ D Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ), δu∞ i h = min max (u,p,d1 ,d2 )∈X f ×Z f ×X1s ×X2s (v,b2 ,q,b1 )∈Wsf ×V f ×Y1s Du∞ Z TZ Z TZ u∞ · δu∞ = (ν ∂n φu∞ − πu∞ n) · δu∞ + γ 0. < j (u∞ ), δu∞ >= h. 0. . Γf∞. 0. Γf∞. ' %:) / * p2lmZ\ekuuv^{|%lvegnsp*½&S^¥jwm^£eoplm^m^{uzlv^ x egp ^{uzlmjs\igeguvZ\egp\lvZ ^£´ ulnW x ^²nWrtlveg] jwige«ly2|%nWp x eolvegnsp ¡ns r vnW\io^{] w_ ½C:^%lzlm^<©fp\nFqpjsu jwmh um|Z ¢ `h\Z\p ¢ YhO|©s^`nWrtlveg] jwige«ly|%nWp x eolvegnsp u{¤ Y&Z\egubuzlv^r egu
(135). . ÞÞ È ]_^`/aa.
(136) X. -
(137) ; . |vhO|%ekjwiU½W:^{|{jwh uv^ReolSig^{j x u°lvnlvZ\^ ¡nsm]
(138) h\ikjFlmeonWp²nw !lmZ\^`j x nWeopl&r\vnW\ig^] uvjwlveku´O^ x fyblvZ\^` jssjwp\W^ ]h\iolvegr\ioeg^u (ϕ , π , b , b ) ¤"®^Nm^{|{jwigiAlvZ\^^}tr\v^Xuvuvegnspns lmZ\^ jssjwp\Wegjsp*½ s L (u, p, d , d ; v, q, b , b ) = J (u, p, d , d ) − he (u, p, d , d ), (v, q, b , b )i Y&Z ^ Y?uzytuzlv^{] qeoigi*Z j¨§W^NlvZ\^N ¡nsigignFqeop\ uzlvmh |lmh\m^ u∞. 1 u∞. u∞. 2 u∞. def. 1. u∞. 2. 1. 2. 1. u∞. 2. 1. u∞. 2. 1. 2. . clmjwlv^`°ªh jwlvegnsp u 0, x nsegplq°ªh jFlmeonWp u. ∂(v,q,b1 ,b2 ) Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) · (δv, δq, δb1 , δb2 ) = 0, ∀ (δv, δb2 , δq, δb1 ) ∈ Wsf × V f × Y1s → ∂(u,p,d1 ,d2 ) Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) · (δu, δp, δd1 , δd2 ) = ∀ (δu, δp, δd1 , δd2 ) ∈ X f × Z f × X1s × X2s → . 46. N 1 1. . ¶
(139)
(140). &$> $% . . /. ß»2\» º 5 4 ² · { ¶
(141). ( ;5. . <. . & . < (. . +2 . $> >&7 >:# . . . u∞ ∈ Uc (p, b1 , b2 , v, b2 , q, b1 ) ∈ Z f × X1s × X2s × Wsf × V f × Y1s ) " -$ F" _ A Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) u∈ Yf. . h∂u Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ), δui = Z TZ [−δu · ∂t v + [D δu · u + D u · δu] · v − νδu · ∆v + δu · ∇ q] − 0. Ωft. +. Z. T. Z. (δu · v) hd2 e2 , ni −. Z. δu(T ) · v(T ). ∀δu ∈ X f. ?p ns x ^lvn nstljwegp³j uzlvmnsp\ ¡nsm]
(142) h\ikjFlmeonWp nw lmZ\^ \h e x j x nsegpWl/r\mns\ig^]½+S^¬r:^v ¡nsm] uznW]<^ egplv^{sjFlvegnspfy²r jwvlmu ¶
(143)
(144) 6 *. 0. Γst. ΩT. *. . Z. (D δu · u) · v. = −. l+ig^{j x uSlvnblmZ\^` ¡nsigionFqegp\<e x ^plveoly Ωft. Z . Ωft. [D v · u + div(u) v] · δu +. Z. Γf∞ ∪Γst. (δu · v) hu, ni. h∂uˆ Lu∞ (u, p, d1 , d2 ; ϕ, π, b1 , b2 ), δui = Z Z − [−∂t ϕ + (∗ Du) · ϕ − (Dϕ) · u − div(u) ϕ − ν∆ϕ + ∇π] · δu −. ¶
(145)
(146). Qf. N. . . . ΩfT. ϕ(T ) · δu(T ). f s s f f s u∞ ∈ Uc (u, )b 1 , b2",v, b2 ,-q,$ b 1) ∈F"X _× X 1 × X2 ×f W As ×V × Y1 Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) p∈ Z Z TZ δp div ϕ, ∀δp ∈ Z f u, p, d1 , d2 ; ϕ, q, b1 , b2 ), δpi = h∂p Lu∞ (ˆ 0. . Ωft. Û¡Ý'Þ*ÛÜá.
(147) V.
(148) ! " $# % & (')*,+ .- / 0. Y&Z\ekuqig^{j x uSlvnblvZ ^` ¡nsigionFqegp\ \h e x j x nWeopl+ulmvnWp\ ¡nsm]
(149) h\ikjFlmeonWp*½. . . .+% 8!. −∂t ϕ − D ϕ · u + (∗ Du) · ϕ − ν∆ϕ + ∇q = 0, div(ϕ) = 0, ϕ = b 2 · e2 , ϕ = 0, ϕ(T ) = 0,. +,,.
(150) !. Qf Qf Σs Σf∞ ΩfT. ZV . 1. ®^m^{|{jwigi"lmZ jFlblvZ\^/R ] jwreku\h\egi«l<jsu
(151) lvZ\^& nF j£§s^X|lmns´O^i x V lmZ jFl<] jFlm|Z ^{ulmZ\^uvnsige x §W^ignt|%eoly®jwl
(152) lvZ &^ h e xf¢ uvnsige x egplv^{z ßjW|%^jsp x ekujsv\eolvjwmy¥egp uve x ^²lvZ ^& h e x x ns] jwegp*½eU¤ ^/h uveop\¥lvZ\^ m^ x h |%^ x n x ^{q]bn x ^{iH½ V (x, t) = d2 · e2 , V (x, t) = Ext(d2 · e2 ), V (x, t) · n = 0,. . x ∈ Ωst x ∈ Ωft x ∈ Γf∞. W. R^p |^s½WlvZ\^R *] jsr x ^r:^p x u"nsp d lmZ\vnWh\sZ V ¤ ¼ h\vlvZ\^{v]<nsm^s½wlvZ\^ x ^meo§FjFlmeo§W^qqe«lmZ²m^{uvrO^X|lClvn ] j¨y:^
(153) uveo]<r\ig^uzegp |%^eol x nf^Xuqp\nwleopf§snWio§W^ x ^meo§Fjwlveg§s^qeolvZ£m^{uvrO^X|lqlvn<lmZ\^W^ns]<^lvmys¤Y&Z ^p*½ d S^`Z j¨§s^lvZ ^` ¡nsigionFqegp\<v^Xuzh i«lX½ ¶
(154)
(155) [ E u ∈ U (u, p, d , d , v, b , q, b ) ∈ X × Z × X × X × W × V × Y . 2. 1. _I+ " + . ∞. 1. c. 2. Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ). . f. &2 1" -$. . f. s. s. F 1
(156) 2 . f s. d1 ∈ X1s. s. f. A 1 . h∂d1 Lu∞ (u, p, d1 , d2 ; ϕ, π, b1 , b2 ), δd1 i = Z Th i α (d1 − d1g ) δd1 − k δd1 b2 + δd1 b˙ 1 − δd1 (T ) b1 (T ). ¼ vnW] qZ\ek|ZS^ x ^ x h |%^NlmZ\^` ¡nsigionFqegp\ j x nsegpWl H +½ 0. Y&Z ^ x ^meo§FjFlmeo§W^&nw lmZ\^+ jssjwp\WegjspqeolvZ<v^Xuzr:^{|%llvnNlvZ\^+uvnsige x §W^ignf|e«ly§Fjsvekjw\ig^ d egpf§snWio§W^{u'uzZ jsrO^ x ^meo§FjFlmeo§W^{uSnw x nW]<jseop/egpWlm^Wmjsigu{¤ Y&Z eguCr:nsegpl"Z jsu"O^{^p egp§W^{uzlvegWjwlv^ x r\m^§fegnsh uvioy
(157) y :nsi%{uvegnNegp X ؽ Å W :jwp x egp o{_ U½ o¨ ؤY&Z ^p<^ p ^^ x lmn²egpWlmvn x hO|%^`lvZ ^
(158) |%nWp |%^{rtlRnw "Ymjsp uv§s^uv^ ¼ eg^i x jsumuznt|egjwlv^ x lvnj rO^{zlmh\m jFlmeonWp/ns lvZ\^
(159) uvnsige x §W^ignt|%eolys¤ ®^egpWlmvn x hO|%^jbrO^{zlmh\vOjFlvegnsp nF W jsumuvnf|egjwlv^ x lmnlmZ\^r:^vlvh\m jFlmeonWp δd e ¤ ¼ nsq^}\jw]<r\ig^s½ . X. −b˙1 + k b2 = α(d1 − d1g ), (0, T ) b1 (T ) = 0,. 2. . . # . 2 2. W = Ext(δd2 · e2 ). ÞÞ È ]_^`/aa. .
(160) /W. -
(161) ; . Y&Z egu nF?W^p\^{mjwlv^Xu&p\^ \h e x jwp x uznWioe xx ns] jwegp u&lmZ\mnsh\WZlmZ\^
(162) * ] jwr½ T (V + ρW ) ½ qeolvZ ¤C®^uz^l ρ≥0 t. Ωf,ρ t. def. =. Tt (V + ρW )(Ωf0 ). Ωs,ρ t. def. Tt (V + ρW )(Ωs0 ). =. ¼ ns&lmZ\^uvjs©s^Nnw uzeg]<r\ioek|%eolys½ ^uv^%l def. Ttρ = Tt (V + ρW ). Y&Z ^NnWt^{|%lveg§s^`nw 'lvZ eguqrOjwjwsjwr Z²eguSlmn |%ns]<r\h\lv^`lvZ ^` ¡nsigionFqegp\ x ^meo§Fjwlveg§s^ . .
(163)
(164) d Lu∞ (ˆ u, p, d1 , d2 + ρδd2 ; vˆ, q, b1 , b2 )
(165)
(166) dρ ρ=0. F 4 D ¶ ¨ ¶
(167) ? 4 » D ¶ w º ¶ (ß» cteop |^S^Snsh\i x igeo©W^²lmn x e:^{v^{plvekjFlv^lvZ\^ jssjwp\Wegjsp¥ ¡h p |lmeonWp jwiSqe«lmZ m^{uvrO^X|llvn ρ jFlbrOnWeopl ½teolSeku&|%nWp§W^p\eg^pllvnbnWv©<qeolvZ ¡h\p |lmeonWpjsiom^{j x y x ^%´ p ^ x egp Ω = Ω jsuS^Nr\mnt|%^^ x ρ lmnb=lvZ\0^igeo]<eol ρ → 0 ¤ YnblvZ\ekuq^{p x ½\S^`eoplvmn x h |^`jbp\^{?] jsr/jWu&eop Å X _W T = T ◦T : Ω −→ Ω Ω −→ Ω Y&Z egu&] jwrekuqjs|%lvh jsioigy lvZ ^ nF³ns 'j<§s^X|lmnsS´ ^i x ¡nsigionFqegp\ ¡ns ρ ∈ (0, ρ ) ½ O 8 ¶fº ¶
(168) < _ " R _ ' R " $ E % ' $ # T Z ∂T _ Z = Z (ρ, .) = ◦ (T ) N 7 1. O. f,ρ=0 def t. f t. . t def ρ. ρ t. f t s t. −1 t. f,ρ t s,ρ t. 0. . ρ t. t def ρ. . . t ρ. t ρ. t. . t −1 ρ. ∂ρ. & _ (' ' $ +M 0 E \ 0%
(169) Q#. . Ttρ (Zρt ) : Ω −→ Ω. . x 7−→. x(ρ, x) ≡ Ttρ (Zρt )(x). d x(ρ) = (ρ, x(ρ)), dρ x(ρ = 0) = x,. ρ≥0. . _. . Ω. Û¡Ý'Þ*ÛÜá.
(170) /X.
(171) ! " $# % & (')*,+ .- / 0. cteop |^s½*S^<nsp\igy¥|nsp uve x ^ x ^meg§¨jwlveg§s^XuNjwlrOnWeopl ½*S^<uv^%l ¡mns] oX qZ\ek|Z]<egsZl+:^h uz^ ¡h\i! ¡nW&lvZ\^uv^{ªh\^{ρiH½ = 0 O 8 ¶fº ¶
(172) 6 _ * AZ+ . . def. t Zt = Zρ=0. ¤<¬^<m^{|jsioiCj/m^{uvh\i«l. . [0, ρ0 ] −→ C 0 ([0, T ]; W k−1,∞ (Ω)) ρ 7−→ Tt (V + ρW ). . ) ! ! 0 " -$* A def. t. S (ρ, .) = ∂ρ [Tt (V + ρW )] =. º º ((ßt · S. 1. Z. t. [D(V + ρW ) ◦ Tτ (V + ρW ) · S τ (ρ, .). _W. 0. . +W ◦ Tτ (V + ρW )dτ ]. _) (' ' $ + 0E" -$ . def. St (.) = S t (ρ = 0, .) ∂t St − (D V ◦ Tt ) · St = W ◦ Tt , Ω0 × (0, T ) St=0 = 0, Ω0. t. s_ _ m^ikjFlv^ x. . 0 A -. . ¡h\p x jw]<^plmjsim^{uvh\iol ¡h vp\ekuvZ lvZ ^ x yfp js]<eg|{jwiuzytuzlv^] umjFlmeguz´ ^ x ylmZ\^b§s^{|%lvnW+´ ^{i x Z lmnblvZ\^§W^{|lmnsS´ ^{i x u (V, W ) ½ O 8 ¶fº ¶
(173) N 1 6
(174) _ %
(175) $ _ ' _ ' $ + Z . . . . . . . t. . def. _. ZJ. ∂t Zt + [Zt , V ] = W, Ω0 × (0, T ) Zt=0 = 0, Ω0. [Zt , V ] = DZt · V − DV · Zt. E% ' _ R-" Y ' __ . (Zt , V ). . ' °¶ » ¶ 5 D t/5 D ¶ * p2lmZ\^¬uv^{ªh\^{iH½R^ qegigiRp\^{^ x W^p\^{mjsi+v^Xuzh i«lu|%nWp |%^{vp eop\uvZ jsrO^ x ^meo§Fjwlveg§s^{u²nw egpWlm^Wmjsi+nF§s^{ x ns] jwegp u&ns&:nsh\p x jwmeo^Xu¤®^qeoigi*h uv^NlvZ\^` ¡jw]<^{nWv© x ^{§s^{ionWrO^ x egp cfns©WnsignF+uz©fe ¢ :nWiXuzegn Å J ؤ ¶
(176)
(177) !
(178) Z Z Z
(179) _ d
(180) G(ρ = 0)hZ , ni dΓ ∂ G(ρ) dΩ + = G(ρ) dΩ
(181)
(182) dρ B ¶
(183)
(184) F !
(185) Z Z h
(186) i _V d
(187) = φ(ρ) dΓ
(188) φ + Hφ hZ , ni dΓ N 7 7. . 8 ?. . Ωf,ρ t. Ωft. ρ=0. t. ρ. Γst. 0. dρ. . . 0. φΓ. ÞÞ È ]_^`/aa. Γs,ρ t.
(189). ρ=0. Γst. Γ. A ' _)
(190) Z+Z! _R % R '. t. φ(ρ, .) ∈ L1 (Γst ).
(191) s. -
(192) ; . ®^m^{|jsioi!|igjWuvuveg|{jwi x ^´ p\eolvegnsp uqns uvZ jsrO^ x ^{veg§FjFlveg§s^N ¡h\pO|lvegnspOu /¶ 4 5¡/ 5 º 4 1 I φ(ρ, x) ∈ C ((0, ρ ; C (Γ )) _ * . 0. ' $ +M E " #. 0. 1. s,ρ t. . . % + % -0 _.
(193)
(194) d t
(195) ˙ φ(ρ, .) ◦ Tρ
(196) φ= dρ ρ=0. _F _)
(197) Z+Z! __R %* ('. φ. + -0* _ ' $ Z+M A " . 0 def φΓ = φ˙ − ∇Γ φ · Zt. ¶
(198) t 6 ' . φ. "Y ' %
(199) $. φ˜. YA %. Ω. . . _ _ . 0 def 0 φΓ = φ˜ |Γst + ∂n φ˜ hZt , ni. . . φ˜ |Γst 0.
(200)
(201)
(202)
(203) d ˜
(204) φ(ρ, .)
(205)
(206)
(207) = dρ ρ=0
(208). . def. ¡nWioignFqegp\
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