Диссертация (1149369), страница 7
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Îïåðàòîð ΠT îáëàäàåò ñëåäóþùèìè ñâîéñòâàìè:1. ΠT óíèòàðåí;2. äëÿ îãðàíè÷åííûõ ñêàëÿðíûõ ôóíêöèé φ âûïîëíåíî ñîîòíîøåíèå ΠT φ = φ ΠT ;3. îòîáðàæåíèå ΠT ñîõðàíÿåò ãëàäêîñòü: ΠT [LνT ∩C ∞ (ΩT ; R3 )] = FνT ∩C ∞ (ΘT ; R3 ).Äîêàçàòåëüñòâî.Âõîäÿùèå â ïðàâóþ ÷àñòü îïðåäåëåíèÿκ ôóíêöèè ÿâëÿþòñÿ ïîëî-u, v ∈ LνT èìååì()∫J(3.1.3)(u, v)LνT = u · v dx =u(x(γ, τ )) · v(x(γ, τ )) c(γ, τ ) dΓ dτ =J0TΩT)(Θ )()∫ ()ccJ(3.1.9)(3.1.19) ( T=πu (γ, τ ) ·πv (γ, τ ) c(γ, τ ) dΓ dτ =Π u , ΠT v F T ,νc0c0J0æèòåëüíûìè è ãëàäêèìè íàΘT .∫Äëÿ ëþáûõΘTòàêèì îáðàçîìΠTåñòü èçîìåòðèÿ.
Íåòðóäíî ïîêàçàòü, ÷òî RanΠT = FνT .Ñâîéñòâî(2) ÿâëÿåòñÿ ñëåäñòâèåì îïðåäåëåíèé è (3.1.10); ñâîéñòâî (3) ëåãêî ñëåäóåò èç äèôôåîìîðôíîñòè îòîáðàæåíèÿ3.23.2.1i.Ëåììà äîêàçàíà.ÈçîáðàæåíèÿÏðîåêòèðîâàíèå â ïðîñòðàíñòâå ïîòåíöèàëüíûõ ïîëåé. ïðîñòðàíñòâå ïîòåíöèàëüíûõ ïîëåé (1.2.4)G = {h ∈ H | h = ∇φ, φ ∈ H 1 (Ω)}âûäåëèì öåïî÷êó ïîäïðîñòðàíñòâ}{G ξ := h ∈ G supp h ⊂ Ωξ ,06ξ6T;îòìåòèì íåêîòîðûå ñâîéñòâà èõ ýëåìåíòîâ [33].Ïðåäëîæåíèå 3.2.1. Ïóñòü T < T reg è ξ ∈ (0, T ) ôèêñèðîâàíî;1. ñëåä h|Γξ−0 ïîëÿ h ∈ G ξ , ãëàäêîãî â Ωξ , åñòü ïîëå, íîðìàëüíîå ê Γξ ;442.
ëþáîå ãëàäêîå íîðìàëüíîå ïîëå íà Γξ åñòü ñëåä ïîëÿ èç G ξ , ãëàäêîãî â Ωξ .QξÎáîçíà÷èì ÷åðåçñòâî{Qξ }ïðîåêòîð âíàG ξ (T < T reg ).Ìîæíî ïîêàçàòü, ÷òî ñåìåé-íåïðåðûâíî:0 6 ξ 6 T;s-lim Qτ = Qξ ,τ →ξÏóñòüGTT < T reg .Q0 = OG T ;Îïèøåì ïðåäñòàâëåíèå äëÿh = hθ + hν = ∇φ = (∇φ)θ +∂φν∂ν∈ GT ,QT = IG T .Qξ : G T → G ξ .ξ ∈ (0, T )ôèêñèðóåìΩξ ,∆r = 0â(∇r)θ = hθ = (∇φ)θíàÂûáåðåì ãëàäêîå ïîëåè ðàññìîòðèì çàäà÷ó(3.2.1)∫Γξ ,r dΓξ = 0 ,(3.2.2)Γξ∂r=0∂νíàΓ.(3.2.3)Ïåðâîå èç óñëîâèé â (3.2.2) ðàâíîñèëüíî ñîîòíîøåíèþr = φ + constíàΓξ ;(3.2.4)ïî âòîðîìó îäíîçíà÷íî íàõîäèòñÿ ïîñòîÿííàÿ â (3.2.4).
Èç ýòîãî ñëåäóåò, ÷òî çàäà÷à(3.2.1)(3.2.3) ðàçðåøèìà åäèíñòâåííûì îáðàçîì; åå ðåøåíèåΩξr = rξ (x)åñòü ãëàäêàÿ âôóíêöèÿ.×åðåçΥTîáîçíà÷èì ìíîæåñòâî ðåøåíèé ñåìåéñòâà çàäà÷ (3.2.1)(3.2.3) ïðè âñå-âîçìîæíûõ ãëàäêèõh ∈ GT:{Υ := rξ ∈ C ∞ (Ωξ ) rξ Tåãî çàìûêàíèå âL2 (ΩT )}ðåøåíèå (3.2.1)(3.2.3),îáîçíà÷èì0<ξ6T ;(3.2.5)ΥT .Ëåììà 3.2.1.
Äëÿ ëþáîãî ãëàäêîãî ïîëÿ h ∈ G T cïðàâåäëèâî ïðåäñòàâëåíèåξQ h =h − ∇rξâ Ωξ ,0â Ω \Ω ,â êîòîðîì rξ ðåøåíèå çàäà÷è (3.2.1)(3.2.3)T(3.2.6)ξ45Äîêàçàòåëüñòâî.Ïóñòüξh =Îáîçíà÷èìhξ⊥ := h − hξ ,h − ∇rξâΩξ ,0âΩT \Ωξ .òàê ÷òîh = hξ + hξ⊥è îòìåòèì ñëåäóþùèå ñâîéñòâà(3.2.7)hξ :1.hξ = h − ∇rξ = ∇φ − ∇rξ = ∇(φ − rξ )2.(3.2.2)hξθ Γξ−0 = (h − ∇rξ )θ Γξ−0 = hθ Γξ−0 − (∇rξ )θ Γξ−0 = 0 .Ñëåäñòâèåì (1) è (2) ÿâëÿåòñÿ âêëþ÷åíèå(hξ⊥èìååì:∇r · w dx =Ωξ∫(∇r ) ψ dΓ −ξ ν(∇r ) ψ dΓ +Γ∇rξ · ∇ψ dx =Ωξ∫ξ ν=êîòîðîå ìîæíî ïðåäñòàâèòü â âèäå∫ξ, w)H =∫Òàêèì îáðàçîì,∫Ωξ ;hξ ∈ G ξ .w ∈ G ξ ∩ C ∞ (ΩT ; R3 ),Äàëåå, äëÿ ëþáîãîw = ∇ψ : ψ Γξ = const,âξΓξ∆rξ ψ dx(3.2.1)−(3.2.3)=0.Ωξ(hξ⊥ , w)H = 0 è, ïî ïëîòíîñòè ãëàäêèõ w â G ξ , ïîëó÷àåì hξ⊥ ∈ G T ⊖ G ξ ,ò.å. â (3.2.7) ñëàãàåìûå îðòîãîíàëüíû.
Ëåììà äîêàçàíà.Îòìåòèì âàæíûé ôàêò: ïîëåQξ hΓξ−0åñòü ïîëå,íîðìàëüíîåQξ h, âîîáùå ãîâîðÿ, ðàçðûâíî íà Γξ , ïðè÷¼ì ðàçðûâêΓξ : (3.2.2)(3.2.6)Qξ hΓξ−0 = [hθ + hν − (∇rξ )θ − (∇rξ )ν ]Γξ = [hν − (∇rξ )ν ]Γξ .3.2.2(3.2.8)Îïåðàòîð ÊàëüäåðîíàÔèêñèðóåìξ : 0 < ξ 6 T < T regDom Λξ = C∞ (Γξ ),èäåéñòâóþùèé ïî ïðàâèëó:ââåä¼ì∂q ,Λ g=∂ν Γξ−0ξîïåðàòîðΛξ : L2 (Γξ ) → L2 (Γξ ),46ãäåqåñòü ðåøåíèå çàäà÷è∆q = 0ñ çàäàííîé ãëàäêîé ôóíêöèåéÝòî èçâåñòíûégâΩξ ,q = gíàΓξ ,∂q= 0∂νíàΓíàΓξ .îïåðàòîð Êàëüäåðîíà(èëè îòîáðàæåíèå Dirichlet-to-Neumann); îò-ìåòèì íåêîòîðûå èç åãî ñâîéñòâ:1. îïåðàòîðΛξíåîãðàíè÷åí è ñèììåòðè÷åí:∫∫ξξφΛξ ψ dΓξ ,Λ φ ψ dΓ =Γξ2. îïåðàòîðΛξ(3.2.9)Γξïîëîæèòåëåí:(Λξ g, g)L2 (Γξ ) > 0 , g ̸= 0è, ñëåäîâàòåëüíî, èíúåêòèâåí;3. îïåðàòîð4.ΛξñΛξñîõðàíÿåò ãëàäêîñòü:Λξ C ∞ (Γξ ) = C ∞ (Γξ ), ξ > 0 ;åñòü ýëëèïòè÷åñêèé ïñåâäîäèôôåðåíöèàëüíûé îïåðàòîð (ÏÄÎ) 1-ãî ïîðÿäêàãëàâíûì ñèìâîëîì |k|g, ãäå()1/2;|k|g := g αβ (γ 1 , γ 2 , ξ)kα kβçäåñük1 , k 2 ïåðåìåííûå, äâîéñòâåííûå ê ïåðåìåííûì[35, 36].3.2.3ÎïåðàòîðÔèêñèðóåìT < T regΛè îòìåòèì ïðåäñòàâëåíèåΩT =∪06ξ6TΓξ .(3.2.10)γ 1 , γ 2 ; γ = (γ 1 , γ 2 ) ∈ Γ47L2 (ΩT ) îïðåäåëèì îïåðàòîð Λ, Dom Λ = C∞ (ΩT ), ïðîñòðàíñòâå ñêàëÿðíûõ ôóíêöèéäåéñòâóþùèéïîñëîéíî(â ñîîòâåòñòâèè ñ ïðåäñòàâëåíèåì) ïî ïðàâèëó:(Λφ)Γξ := Λξ [φΓξ ] , 0 < ξ 6 T .Îòìåòèì íåêîòîðûå èç åãî ñâîéñòâ:1.
îïåðàòîðΛíåîãðàíè÷åí è èíúåêòèâåí; îí íåëîêàëåí, ò.å. íå ñîõðàíÿåò íî-ΩT .ñèòåëü ôóíêöèè âξ ′′ξ′supp Λφ ∈ Ω \Ω òî æå âðåìÿ, âêëþ÷åíèå′âëå÷åò(0 < ξ ′ 6 ξ ′′ 6 T ) ;2. èñïîëüçóÿ ãëàäêèé õàðàêòåð çàâèñèìîñòèãëàäêîñòè:′′supp φ ∈ Ωξ \ΩξΛξîòξ , ìîæíî óñòàíîâèòü ñîõðàíåíèåΛC ∞ (ΩT ) ⊂ C ∞ (ΩT ).Ëåììà 3.2.2. Ñïðàâåäëèâî ñîîòíîøåíèåΛ∗ = c−1 Λ c .Äîêàçàòåëüñòâî.Äëÿ ëþáûõ ãëàäêèõφ, ψèìååì∫∫T(Λφ, ψ)L2 (ΩT ) =Λφ ψ dx ==∫∫0=τφ Λ (cψ) dΓ =Γτ(3.2.9)=Γτ∫dτ0cΛτφ ψ dΓτdτ∫Tτdτ∫0ΩT∫T(3.2.11)1cφ Λτ (cψ) dΓτ =cΓτ11φ Λc ψ dx = (φ, Λc ψ)L2 (ΩT ) = (φ, Λ∗ ψ)L2 (ΩT ) .ccΩTËåììà äîêàçàíà.3.2.4Ïîïåðå÷íûé ãðàäèåíò è ïîïåðå÷íàÿ äèâåðãåíöèÿ îïèñàíèè ââîäèìûõ íèæå îïåðàòîðîâ èñïîëüçóþòñÿ ïîëóãåîäåçè÷åñêèå êîîðäèíàòû(ìû ïîëàãàåìT < T reg ).Íàïîìíèì, ÷òîLTθåñòü ïðîñòðàíñòâî ïîïåðå÷íûõ âåêòîðûõïîëåé (3.1.13).
Çàïèøåì âûðàæåíèå ãðàäèåíòà è äèâåðãåíöèè â ïîëóãåîäåçè÷åñêèõêîîðäèíàòàõ:[((∇φ)(x) =gαβ)() ]∂φ00 ∂φrα + gr0 (γ, τ ) ;∂γ β∂τ(3.2.12)48[]1 ∂1 ∂α0(div y)(x) =(cJy ) +(cJy ) (γ, τ ) ,cJ ∂γ αcJ ∂τãäå{g αβ } ìàòðèöà, îáðàòíàÿ ê{gαβ }, g 00 =• ïîïåðå÷íûé ãðàäèåíò ∇θ : L2 (ΩT ) → LTθ ,ïðàâèëó:gαβv = v α rαÎïðåäåëèì:ΩTôóíêöèè ïî) ]∂φrα (γ, τ ) ;∂γ βT• ïîïåðå÷íóþ äèâåðãåíöèþ divθ : LTθ → L2 (Ω ),ïîëÿy = y α rα + y 0 r0 .äåéñòâóþùèé íà ãëàäêèå â[((∇θ φ)(x) =1;c2(3.2.13)äåéñòâóþùóþ íà ãëàäêèå ïîïåðå÷íûåïî ïðàâèëó:[Îòìåòèì èõ ïîñëîéíûé(divθ v)Γξ = 0.]1 ∂α(divθ v)(x) =(cJv ) (γ, τ ) .cJ ∂γ αõàðàêòåð: ðàâåíñòâà φ ξ = 0, v ξ = 0ΓΓâëåêóò(∇θ φ)Γξ = 0,Ñïðàâåäëèâî ñîîòíîøåíèå(∇θ φ, v)LTθ = − (φ, divθ v)L2 (ΩT ) ,(3.2.14)êîòîðîå ëåãêî âûâîäèòñÿ ïîñëîéíûì èíòåãðèðîâàíèåì ïî ÷àñòÿì.Íèæå íàì ïîíàäîáÿòñÿ îïåðàòîðû ïîïåðå÷íîãî ãðàäèåíòà è ïîïåðå÷íîé äèâåðãåíöèè ñ íåñêîëüêî áîëåå óçêèìè îáëàñòÿìè îïðåäåëåíèÿ, ñâÿçàííûìè ñ ñåìåéñòâîì çàäà÷(3.2.1)(3.2.3).Ñóçèì îáëàñòü îïðåäåëåíèÿ∇θíà ìíîæåñòâî ãëàäêèõ ðåøåíèéΥT(3.2.5); çà ïî-ëó÷èâøèìñÿ îïåðàòîðîì ñîõðàíèì ïðåæíåå îáîçíà÷åíèå.
 ñîîòâåòñòâèè ñ (3.2.1)5(3.2.3) ,∇θ rξ := hθÝòîò îïåðàòîð äåéñòâóåò èçîáðàçîì,∇θ : ΥT → HθT ,Dom∇θËåãêî âèäåòü, ÷òî Ker∇θðàâíîñèëüíîhθ = 0 ;ΥTíà0 < ξ 6 T.Γξ ,6â ïîäïðîñòðàíñòâî= ΥT ,= {0}.Ran∇θÄåéñòâèòåëüíî, ïî (3.2.15) ðàâåíñòâîÒàêèì∇θ rξ = 0â ñèëó ýòîãî, äëÿ ðåøåíèÿ çàäà÷è (3.2.1)(3.2.3) èìååìïîëüçóåìñÿ òåì, ÷òî ∇θ := [∇(·)]θPθT : HT → LθT îïðåäåë¼í ôîðìóëîé (3.1.15)6 ïðîåêòîðHθT := PθT G T ⊂ LθT .= HθT ∩ C ∞ (ΩT ; R3 ).Ωξ .5 ìû(3.2.15)rξ ≡ 0â49Ñêàçàííîå ïîçâîëÿåò ñâÿçàòü ñ ñåìåéñòâîì çàäà÷ (3.2.1)(3.2.3) îáðàòíûé îïåðàòîðTT∇−1θ : Hθ → Υ ,äåéñòâóþùèé íà ïîïåðå÷íûå êîìïîíåíòû ãëàäêèõ ïîëåéh ∈ GTïîïðàâèëó:ξ∇−1θ hθ := rêàê íåòðóäíî ïðîâåðèòü,Ïîñêîëüêó Dom∇θLθT ),òî ñóùåñòâóþò∇θ ∇−1=θ= ΥT0<ξ6T;idθ (idθ òîæäåñòâåííûé îïåðàòîð â∇∗θ : HθT → ΥT(∇∗θΓξ ,HθT ).L2 (ΩT )) è Ran∇θ = HθT (çàìûêàíèå â( −1 )∗∇θ: ΥT → HθT , ïðè÷¼ì âûïîëíåíî ðàâåí-(çàìûêàíèå âñòâî [41]:Îïåðàòîðíàè∇−1θ)∗= (∇∗θ )−1 .(3.2.16)îïðåäåëÿåòñÿ ïåðâûì èç ðàâåíñòâ (ñ ïðîèçâîëüíûìè ãëàäêèìèrξ ∈ ΥT ; hθ ∈ HθT ):(rξ , ∇∗θ hθ)ΥT()= (∇θ rξ , hθ )HθT = − rξ , divθ hθ ΥT ;èç âòîðîãî ðàâåíñòâà, àíàëîãè÷íîãî (3.2.14), ñëåäóåòåòñÿ ñóæåííàÿ íà ãëàäêèå ïîëÿ èçHθT ⊂ LθT∇∗θ = −divθ , ãäå ïîä divθïîíèìà-ïîïåðå÷íàÿ äèâåðãåíöèÿ (çà îïåðàòîðîìñîõðàíÿåì ïðåæíåå îáîçíà÷åíèå).Ó÷èòûâàÿ (3.2.16), çàêëþ÷àåì ñóùåñòâîâàíèå îáðàòíîãî îïåðàòîðàHθT ,äåéñòâóþùåãî (ïîñëîéíî) íà ôóíêöèè èç3.2.5DomN T = G T ∩ C ∞ (ΩT ; R3 )Òàêèì îáðàçîì, îáðàç{Qξ }îïðåäåëÿåò îïåðàòîðN T : GT →ïî ïðàâèëóN T h = Qξ hΓξ−0ΓξΥT ⊂ L2 (ΩT ).N T -ïðåîáðàçîâàíèåÂâåä¼ííîå â ï.
3.2.1 ñåìåéñòâî ïðîåêòîðîâLνT ;divθ−1 : ΥT →íàΓξ , 0 < ξ 6 T < T reg .N T h ñîñòàâëÿåòñÿ èç ðàçðûâîâ, ïîÿâëÿþùèõñÿ íà ïîâåðõíîñòÿõïðè ïðîåêòèðîâàíèèhíàGξ .h = hθ + hν èìååì ïîñëîéíîå ïðåäñòàâëåíèå( ξ )∂r TξN h = hν − (∇r )ν = hν −ν íà Γξ , 0 < ξ 6 T < T reg .∂ν ΓξÑîãëàñíî (3.2.8), äëÿ(3.2.17)50Èñïîëüçóÿ îïåðàòîðûΛè∇−1θ ,ïðåäñòàâëåíèå (3.2.17) ìîæíî çàïèñàòü â ôîðìå()N T h = hν − Λ∇−1hν.θθ(3.2.18)Ïðåäëîæåíèå 3.2.2.
Îïåðàòîð N T èçîìåòð÷åí è ðàñøèðÿåòñÿ ïî íåïðåðûâíîñòèäî óíèòàðíîãî îïåðàòîðà èç G T íà LνT 7 .Ýòîò ôàêò óñòàíàâëèâàåòñÿ íà òîì æå ïóòè, ÷òî è â [33].Íàïîìíèì ðàçëîæåíèå ÂåéëÿHT = J T ⊕ G T ;PGTïóñòüåñòü ïðîåêòîð âHT = H[ΩT ]íà(3.2.19)GT .Ëåììà 3.2.3. Ñîïðÿæåííûé îïåðàòîð (N T )∗ : LTν → G T îïðåäåë¼í íà ãëàäêèõ ïðî-äîëüíûõ ïîëÿõ v ∈ LTν è äîïóñêàåò ïðåäñòàâëåíèå([])(N T )∗ v = PGT v + divθ−1 c−1 Λc v ν ,(3.2.20)ãäå v ν = v · ν .Äîêàçàòåëüñòâî.(N T h, v)LνTÄëÿ ãëàäêèõ(3.2.18)=h = hθ + hν ∈ G T(hν − (Λ∇−1θ hθ )ν, v)LνT(3.2.16)=èv ∈ LνTèìååì(hν , v)LνT − (hθ , (∇∗θ )−1 Λ∗ v ν )LθT =(3.2.14), (3.2.11)(hν , v)LνT + (hθ , divθ−1 [c−1 Λc v ν ])LθT =(())= (hθ + hν , v + divθ−1 [c−1 Λc v ν ])HT = h, PGT v + divθ−1 [c−1 Λc v ν ] G T .=Ëåììà äîêàçàíà.3.2.6Àêóñòè÷åñêàÿ ïîäñèñòåìàÂåðíåìñÿ ê àêóñòè÷åñêîé ïîäñèñòåìåïóñòüφgαTp .ÎïåðàòîðαpT . Âûáèðåì ïðîèçâîëüíîå óïðàâëåíèå g ∈ MTp ;åñòü ðåøåíèå ñèñòåìû (2.2.9)(2.2.11).
Ïðèìåíÿÿ îïåðàòîðâèäåòü, ÷òî∇φg∇ê (2.2.9), ëåãêîóäîâëåòâîðÿåò óðàâíåíèþ(∇φg )tt = [∇κ div − rotµrot]∇φg7 çà∇κ divðàñøèðåíèåì ñîõðàíÿåì îáîçíà÷åíèå N TâQT(3.2.21)51è óñëîâèÿì∇φg |t=0 = (∇φg )t |t=0 = 0Òàêèì îáðàçîì,∇φgþùåå óïðàâëåíèþåñòü ðåøåíèå ñèñòåìû8′f ′ = uf ΣT = ∇φg ΣTâΩ.(3.2.22)(2.2.1)(2.2.3):∇φαTg= uf′, ñîîòâåòñòâó- ∂φggT∇φ · νR g = ∂ν = p .= (∇φg )θ∇θ g∇θ g(3.2.23) ñîîòâåòñòâèè ñ (2.2.7), çíà÷åíèå îïåðàòîðà ðåàêöèè íà ýòîì óïðàâëåíèè áóäåò TggRgκdiv∇φκ∆φφgttg(2.2.9)T ′T p = tt .R f =R===∇θ gµ rot ∇φg × ν000RT , íàõîäèì, ÷òî νgtt RpT g = [RT ]−1 .0Îòñþäà, ó÷èòûâàÿ ïî (2.2.8) îáðàòèìîñòüÈç (3.2.23) è åäèíñòâåííîñòè ðåøåíèÿ ñèñòåìû∇φαTäëÿ ëþáîãîg ∈ MTp(3.2.24)ñëåäóåò, ÷òîufïðåäñòàâèìî â âèäåòîãäà è òîëüêî òîãäà, êîãäà êîìïîíåíòû óïðàâëåíèÿ ñâÿçàíû ñîîòíîøåíèåìfθ =∇θ [RpT ]−1 f ν 9 .Ó÷èòûâàÿ, ÷òî ïî (3.2.23) íîðìàëüíóþ è êàñàòåëüíóþ êîìïîíåíòû óïðàâëåíèÿf′íåëüçÿ çàäàâàòü íåçàâèñèìî, ïðè÷¼ì çàäàíèå íîðìàëüíîé êîìïîíåíòû îäíîçíà÷íîîïðåäåëÿåò êàñàòåëüíóþ, åñòåñòâåííî ðàññìîòðåòü ñëåäóþùóþ ïîñòàíîâêó íà÷àëüíîêðàåâîé çàäà÷è.
Îáîçíà÷èìh = ∇φgè äîïîëíèì óðàâíåíèå (3.2.21) ñ íà÷àëüíûìèäàííûìè (3.2.21) óñëîâèåì íà íîðìàëüíóþ êîìïîíåíòó íà ãðàíèöåν íîðìàëü êh = hf (x, t)htt = ∇κdiv hâQT ,(3.2.25)h|t=0 = ht |t=0 = 0âΩ,(3.2.26)hν = fíàΓ, hν = (h · ν)ν , f ∈ FνT ⊂ F Tìîæíî ðàññìàòðèâàòü êàêΣT ;(3.2.27) óïðàâëåíèå.
Ðåøåíèå ýòîé çàäà÷èG T -çíà÷íóþôóíêöèþ, çàâèñÿùóþ îò âðåìå-íè. Cîîòâåòñòâóþùóþ äèíàìè÷åñêóþ ñèñòåìó ìû íàçûâàåì âåêòîðíîé àêóñòè÷åñêîé(ïîä)ñèñòåìîé è îáîçíà÷àåì8 îïåðàòîð9 ìîæíîαTp .ðåàêöèè ñêàëÿðíîé àêóñòè÷åñêîé ñèñòåìû îïðåäåë¼í ôîðìóëîé (2.2.12)ïîêàçàòü, ÷òî îïåðàòîð RpT îáðàòèì52Íà ïîëÿõ êëàññàH2 (Ω)ââåä¼ì îïåðàòîðL := ∇κ div,(3.2.28)îïðåäåëÿþùèé ýâîëþöèþ ñèñòåìû (3.2.25)(3.2.27).Ëåììà 3.2.4.  ïîäîáëàñòè ΩT , ïîêðûâàåìîé ñèñòåìîé ïîëóãåîäåçè÷åñêèõ êîîðäè-íàò, äëÿ ãëàäêîãî ïîëÿ y = y 0 r0 + yθ èìååò ìåñòî ïðåäñòàâëåíèå 10[∂]1 ∂ c0cJy+cJdivyθ θ c2 ∂τ J ∂τ(Ly)0Ly == .(Ly)θ[]∂∇θ Jc ∂τcJy 0 + cJdivθ yθÄîêàçàòåëüñòâî.(3.2.29)Èñïîëüçóÿ âûðàæåíèÿ (3.2.12) è (3.2.13) ãðàäèåíòà è äèâåðãåíöèèκ = c2 , g 00 = c12 , çàïèøåì()1 ∂00 ∂φ20∇φ = gr0 + ∇θ φ;κ divy = ccJy + divθ yθ ;∂τcJ ∂τ()∂cJy 0 + c2 divθ yθ )c ∂1 ∂( Jc ∂τ(3.2.28)02r0 + ∇θcJy + c divθ yθ .Ly = ∇κ div y = 2c∂τJ ∂τâ ïîëóãåîäåçè÷åñêèõ êîîðäèíàòàõ è ó÷èòûâàÿ, ÷òîËåììà äîêàçàíà.3.2.7ÎïåðàòîðN T (∇κ div)(N T )∗h ∈ G T , v ∈ LνT −1100h1−Λ∇θ h cTT N h=N=,hθ00hθ ñîîòâåòñòâèè ñ (3.2.18) è (3.2.20), äëÿ ëþáûõ ãëàäêèõ 00vv(N T )∗ v = (N T )∗ = PGT −1 −1 2 0divθ [c Λc v ]0(ìû ó÷ëè, ÷òîÏóñòüwν=r0|r0 |= 1c r0èèìååì(3.2.30)(3.2.31)v ν = v 0 |r0 | = cv 0 ).åñòü ïðîèçâîëüíîå ãëàäêîå ïîëå âΩT .Ïî ðàçëîæåíèþ Âåéëÿ (3.2.19),èìååìw = PGT w + PJT w,10 çäåñüè äàëåå èñïîëüçóåòñÿ Ñîãëàøåíèå 3.1.2 î ìàòðè÷íîé çàïèñè(3.2.32)53ãäåPGT ïðîåêòîð âHTñîõðàíÿåò ãëàäêîñòü:íàG T , PJT ïðîåêòîð âHTíàJ T ; îòìåòèì, ÷òî ïðîåêòîð PGTPGT C ∞ (ΩT ; R3 ) ⊂ G T ∩ C ∞ (ΩT ; R3 )(ñì.
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