Е.В. Радкевич - Лекции по урматфизу (1120444), страница 13
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Äîêàæåì ñïðàâåäëèâîñòü(104) äëÿ ëþáîé v(x, t) = vk (x)θ(t), ò.å. äëÿ ëþáîé v ∈ M. Ïîòîì äîêàæåì âñþäó ïëîòíîñòü M â1ee (QT ).HÔèêñèðóåì m. Äëÿ k ≤ m óìíîæèì (0.2) íà θ(t) è ïðîèíòåãðèðóåì ïî (0, T ). ÒîãäàZZ[(k∇ wm ∇ vk + awm vk )θ − ∂t wm vk θ0 ]dxdt =f vk θ dxdtQTQTÈç ñëàáîé ñõîäèìîñòè ñëåäóåò, ÷òîZZ[(k∇ u∇ vk + auvk )θ − ∂t u vk θ0 ]dxdt =QTf vk θ dxdtQT1ee (QT ).
Äëÿ ýòîãî äîñòàòî÷íî óñòàíîâèòü, ÷òî ìíîæåñòâîÒåïåðü äîêàæåì, ÷òî vk θ ïëîòíû â H1ee (QT ), ìîæíî àïïðîêñèìèðîâàòü ôóíêöèÿìè èç∀ η(x, t) ∈ C 2 (QT ), η|ST ∪ DT = 0, ïëîòíîå â HM.Ïî íåðàâåíñòâó Ôðèäðèõñà èíòåãðàë ÄèðèõëåZ(((∂t f )2 + |∇ f |2 )dxdt)1/2QTee 1îïðåäåëÿåò ýêâèâàëåíòíóþ íîðìó â H(QT ). M ëèíåéíàÿ êîìáèíàöèÿ vk∗ θ , Vk∗ îðòîíîðìèðîâàííîãî1R◦áàçèñà â H (Q), ãäå ñêàëÿðíîå ïðîèçâåäåíèå (f, g) ◦ 1= Q ∇ f ∇ g dx. Èç vj ïîëó÷àåì îðòîíîðìèðîâàííóþH (Q)ñèñòåìó ìåòîäîì Ãðàìà-Øìèäòà.e1 = h1 /kh1 k, e2 = (h2 − (h2 , e1 )e1 )/kh2 − (h2 , e1 )e1 k, .
. . ,em = (hm − (hm , e1 )e1 ) − · · · − (hm , e1 )em−1 )/khm − (hm , e1 )e1 ) − · · · − (hm , e1 )em−1 k◦1Äëÿ ëþáîé ôóíêöèè η(x, t) ∈ C 2 (QT ), η|ST = 0, äëÿ ëþáîãî t ∈ [0, T ] èìååì η, ∂t η ∈ H (Q)◦1ïîýòîìó èõ ìîæíî ðàçëîæèòü â ñõîäÿùèéñÿ â ìåòðèêå H (Q) ðÿä Ôóðüåη(x, t) =∞Xηk (t)vk∗ (x),j=1ãäå ηk (t) =RQ∂t η(x, t) =∞Xηk0 (t)vk∗ (x)j=1∇η(x, t)∇ vk∗ (x)dx, ïðè ýòîìZ∞X0 22(ηk + (ηk ) ) =(|∇η(x, t)|2 + |∇∂t η|2 )dx.Qj=1Îáîçíà÷èì ÷åðåç ηN (x, t) =◦1PNj=1ηk (t)vk∗ (x). Îòñþäà äëÿ ëþáîãî N ≥ 1 ïðè âñåõ t ∈ [0, T ] ôóíêöèè∂t η − ∂t ηN ∈ H (Dt ).
Ïîýòîìó íà îñíîâàíèè íåðàâåíñòâà Ñòåêëîâàk∂t η − ∂t ηN kL2 (Dt ) ≤ Ck∂t η − ∂t ηN k ◦ 1H (Dt )ãäå ïîñòîÿííàÿ C > 0 çàâèñèò òîëüêî îò îáëàñòè Q. Ñëåäîâàòåëüíî, äëÿ ëþáîãî N ≥ 1 ïðè âñåõt ∈ [0, T ]k∂t η − ∂t ηN k2L2 (Dt ) + kη − ηN k2◦ 1H (Dt )=∞X≤ C 2 k∂t η − ∂t ηN k ◦ 1H (Dt )(ηk2 + (ηk0 )2 ) → 0, N → ∞j=N +1+ kη − ηN k ◦ 1H (Dt )=Ïîýòîìó íà îñíîâàíèè òåîðåìû ËåâèZkη − ηN k2e 1e (QT )H=0T(k∂t η − ∂t ηN k2L2 (Dt ) + kη − ηN k2◦ 1H (Dt ))dt → 0, N → ∞ ñèëó åäèíñòâåííîñòè îáîáùåííîãî ðåøåíèÿ âñÿ ïîñëåäîâàòåëüíîñòü wm ñëàáî â H 1 (QT ) ñõîäèòñÿê u.Òåîðåìà ËåâèÒåîðåìà 0.10 Ëþáàÿ ìîíîòîííàÿ ï.â.
ïîñëåäîâàòåëüíîñòüfk (x), èíòåãðèðóåìàÿ ïî Ëåáåãó â QRñ îãðàíè÷åííîé ïîñëåäîâàòåëüíîñòüþ èíòåãðàëîâfk (x) dx ≤ M , ï.â. ñõîäèòñÿ ê íåêîòîðîéèíòåíðèðóåìîé ïî Ëåáåãó ôóíêöèè f (x), òàê ÷òîZZlim (L)fN dx = (L)f dxN →∞QQÏðèëîæåíèå ê Ëåêöèè 12.Çàäà÷èËåêöèÿ XII. Ñìåøàííàÿ çàäà÷à äëÿ ñèììåòðèçóåìûõ ñèñòåìïåðâîãî ïîðÿäêà. Ìû èññëåäóåì ñìåøàííóþ çàäà÷ó äëÿ ñèñòåìû ïåðâîãî ïîðÿäêà ñ ïîñòîÿííûìèêîýôôèöèåíòàìè âèäàLu = ∂t u +nXAj ∂x u = f(105)j=1Aj − êîìïëåêñíûå ïîñòîÿííûå ìàòðèöû ïîðÿäêà N × N .1. Ïåðâîãî îãðàíè÷åíèåñèñòåìà (121)íåõàðàêòåðèñòè÷åñêàÿ, ò.å.
ìàòðèöà An -íåâûðîæäåííàÿ.2. Ñèììåòðèçóåìîñòü ïî Ôðèäðèõñó:Îïðåäåëåíèå 0.7 Ñèñòåìà (121) íàçûâàåòñÿ ñèììåòðèçóåìîé ïî Ôðèäðèõñó, åñëè ñóùåñòâóåòñèììåòðè÷íàÿ, ïîëîæèòåëüíî îïðåäåëåííàÿ ìàòðèöà S , òàêàÿ ÷òî ìàòðèöû SAj , j = 1, . . . , n,ñèììåòðè÷íû.Íàøà öåëüèññëåäîâàòü óñëîâèÿ òàê íàçûâàåìîé L2 − êîððåêòíîñòè ñìåøàííîé çàäà÷è äëÿ ñèñòåìû(121) ñ ãðàíè÷íûì óñëîâèåì:Γ u|xn =0 = g(106)n+1â îáëàñòè R+= {(t, xe, xn ), (t, xe) ∈ Rn , xn > 0} äëÿ ñèììåòðèçóåìîé ñèñòåìû (121). Ñèñòåìà (106)èìååò 1 ≤ k ≤ N óðàâíåíèé.Îïðåäåëåíèå 0.8 Ñìåøàííàÿ çàäà÷à (L, Γ) (121), (106) íàçûâàåòñÿ ñòðîãî L2 − êîððåêòíîé,n+1åñëè ñóùåñòâóåò ïîñòîÿííàÿ C > 0, ÷òî äëÿ ëþáûõ γ > 0, f ∈ eγ t L2 (R+), g ∈ eγ t L2 (Rn )n+1ñóùåñòâóåò è åäèíñòâåííî ñëàáîå ðåøåíèå u ∈ eγ t L2 (R+ ) ýòîé çàäà÷è òàêîå ÷òîZ ∞Z ∞−2γ t2γeku(xn , t, ·)kL2 (Rn ) dt +e−2γ t ku(0, t, ·)k2L2 (Rn−1 ) dt ≤(107)+−∞−∞ZZ∞∞ih1e−2γ t kf (xn , t, ·)k2L2 (R+e−2γ t kg(t, ·)k2L2 (Rn−1 ) dt≤ Cn ) dt +γ −∞−∞Çäåñü èñïîëüçîâàíî îïðåäåëåíèå ñòðîãîé L2 êîððåêòíîñòè, â ñèëó êîòîðîãî èç íåðàâåíñòâà (107)ñëåäóåò L2 îöåíêà ñëåäà u|xn =0 .Ðàâíîìåðíîå óñëîâèå Ëîïàòèíñêîãî.
 îáðàçàõ Ôóðüå (x0 , t) → (η, τ ) èç ñìåøàííîé çàäà÷è(121), (106) ïîëó÷èì ðåçîëüâåíòíîå óðàâíåíèådeue − G(Λ) ue = A−1n f,dxnΓ(Λ)eu(0) = ge,n−1³´XG(Λ) = −A−1(γ+iτ)E+iηj Aj , Λ = (τ, η, γ).n(108)(109)j=1Èç ãèïåðáîëè÷íîñòè ñèñòåìû (121) ñëåäóåò, ÷òîËåììà 0.4 Äëÿ γ > 0 ìàòðèöà G(Λ) íå èìååò ÷èñòî ìíèìûõ ñîáñòâåííûõ çíà÷åíèé.Äåéñòâèòåëüíî, åñëè G(Λ)R = iµ R, òî äëÿ ξ = (η, −µ) èìååìdet(−iµ An + (iτ + γ)E + in−1Xηj Aj ) = iN det((τ − iγ)E +j=1n−1Xη j Aj − µ A n ) = 0j=1÷òî ïðè óñëîâèè γ > 0, ïðîòèâîðå÷èò óñëîâèþ ãèïåðáîëè÷íîñòè.Ðàññìîòðèì óñòîé÷èâîå ïîäïðîñòðàíñòâî E− (Λ) ìàòðèöû G(Λ). Èç ëåììû (0.4) ñëåäóåò, ÷òî åãîðàçìåðíîñòüdim E− (Λ) = n− = const, ∀Λ ∈ P,ãäå P = {(τ, η, γ) : γ > 0, (τ, η) ∈ Rn }.
Åñëè âûáðàòü Λ = (0, 0, 1), η = τ = 0, γ = 1, ïîëó÷èì ÷òîn− ðàçìåðíîñòü íåóñòîé÷èâîãî ïðîñòðàíñòâï ìàòðèöû −(An )−1 .Îïðåäåëåíèå 0.9 Áóäåì ãîâîðèòü, ÷òî äëÿ çàäà÷è (L, Γ):du(xn ) − G(Λ) u(xn ) = f (xn ), xn ∈ R+ ,dxnΓ(Λ)u(0) = g ∈ Cn ,(110)(111)âûïîëíåíî ðàâíîìåðíîå óñëîâèå Ëîïàòèíñêîãî, åñëè1. k = rank Γ(Λ) = dim E− (Λ), ∀Λ ∈ P2. Ñïðàâåäëèâî íåðàâåíñòâî|v| ≤ C|Γ(Λ) v|, ∀ v ∈ E− (Λ)(112)Äîêàæåì ñëåäóþùþþ òåîðåìóÒåîðåìà 0.11 Äëÿ íåõàðàêòåðèñòè÷åñêîé ñèñòåìû (121), ñèììåòðèçóåìîé ïî Ôðèäðèõñó, ñìåøàííàÿçàäà÷à (L, Γ) ñòðîãî L2 êîððåêòíà òîãäà è òîëüêî òîãäà, êîãäà Γ óäîâëåòâîðÿåò ðàâíîìåðíîìóóñëîâèþ Ëîïàòèíñêîãî.Äîêàçàòåëüñòâî ñîñòîèò èç äâóõ øàãîâ.
Íà ïåðâîì øàãå ìû äîêàæåì, ÷òî äëÿ íåõàðàêòåðèñòè÷åñêîéñèììåòðèçóåìîé ïî Ôðèäðèõñó ñèñòåìû (121) èç ñòðîãîé L2 êîððåêòíîñòè íåêîòîðîãî ãðàíè÷íîãîóñëîâèÿ Γ∗ ñëåäóåò ñòðîãàÿ L2 êîððåêòíîñòü ëþáîãî ãðàíè÷íîãî óñëîâèÿ (106), óäîâëåòâîðÿþùåãîðàâíîìåðíîìó óñëîâèþ Ëîïàòèíñêîãî. Íà âòîðîì øàãå ìû äîêàæåì ñòðîãóþ L2 êîððåêòíîñòü òàêíàçûâàåìîãî ìàõñèìàëüíî äèññèïàòèâíîãî óñëîâèÿ Ôðèäðèõñà Γmax .Äîêàçàòåëüñòâî ñïðàâåäëèâî äëÿ áîëåå øèðîêîãî ñëó÷àÿ, êîãäà P− ñâÿçíîê îòêðûòîå ìíîæåñòâî.Ðàññìîòðèì ðåçîëüâåíòíîå óðàâíåíèå (108).Óñëîâèå 0.1 Îñíîâíîå óñëîâèå: ìàòðèöà G(Λ) íå èìååò ÷èñòî ìíèìûõ ñîáñòâåííûõ çíà÷åíèéäëÿ ëþáîãî Λ ∈ P .
Òîãäà dim E0 (Λ) = const äëÿ ∀Λ ∈ P .Îïðåäåëåíèå 0.10 Äëÿ íåêîòîðîãî α = α(Λ) > 0 çàäà÷à L(Λ), Γ(Λ) äëÿ ðåçîëüâåíòíîå óðàâíåíèåðàâíîìåðíî óñòîé÷èâî, åñëè äëÿ ëþáîãî u ∈ H 1 (R+ ), Λ ∈ P, ñïðàâåäëèâà àïðèîðíàÿ îöåíêàα(Λ)kuk2 + |u(0)|2 ≤ Ch 1ikL(Λ)uk2 + |Γ(Λ) u(0)|2α(Λ)(113)Îïðåäåëåíèå 0.11 Ðåçîëüâåíòíîå óðàâíåíèå (108) óäîâëåòâîðÿåò ðàâíîìåðíîìó óñëîâèþ Ëîïàòèíñêîãî,åñëè äëÿ ∀Λ ∈ P :1. k = rank Γ(Λ) = dim E− (Λ),2.
Ñïðàâåäëèâî íåðàâåíñòâî(114)|v| ≤ C|Γ(Λ) v|, ∀ v ∈ E− (Λ).Ëåììà 0.5 Óñëîâèå 2. â ðàâíîìåðíîì óñëîâèè Ëîïàòèíñêîãî íåîáõîäèìî è äîñòàòî÷íî, ÷òîáûL2 (R+ ) ðåøåíèå óðàâíåíèÿL(Λ) u = 0, t > 0,(115)|u(0)|2 ≤ C |Γ(Λ) u(0)|2(116)óäîâëåòâîðÿëî îöåíêå ñëåäàñ ïîñòîÿííîé C > 0, íåçàâèñèìîé îò Λ ∈ P .Äîêàçàòåëüñòâî. L2 ðåøåíèå u(xn ) îäíîðîäíîãî óðàâíåíèÿ (115) îïðåäåëÿåòñÿ îäíîçíà÷íîu(xn ) = exnG(Λ)u0î äîëæíî óáûâàòü ïðè xn → ∞.
Ñëåäîâàòåëüíîu0 ∈ E− (Λ)Óñëîâèå 0.2 Ïîòðåáóåì ñóùåñòâîâàíèå íåêîòîðîãî ãðàíè÷íîãî óñëîâèÿ Γ∗ , äëÿ êîòîðîãî çàäà÷àdeu(xn ) − G(Λ) u(xn ) = A−1n f,dxnΓ∗ (Λ) u(0) = ge,äëÿ ëþáûõ f ∈ L2 (R+ ), g ∈óäîâëåòâîðÿþøåå îöåíêåC k , k = rank Γ èìååò åäèíñòâåííîå ðåøåíèå u ∈α(Λ)kuk2 + |u(0)|2 ≤ C(117)L2 (R+ ),ih 1kL(Λ)uk2 + |Γ(Λ) u(0)|2 ,α(Λ)ïîñòîÿííàÿ C > 0 íå çàâèñèò îò f, f, Λ ∈ P .Íèæå ìû ïîêàæåì, ÷òî ìàñèìàëüíî äèññèïàòèâíîå óñëîâèå Ôðèäðèõñà óäîâëåòâîðÿåò óñëîâèþ (0.2).Ïðåäëîæåíèå 0.6 Ïóñòü äëÿ ðåçîëüâåíòíîãî óðàâíåíèå (117) âûïîëíåíî óñëîâèå (0.2) ïðè íåêîòîðîìâûáîðå ïàðàìåòðà α(Λ).
Òîãäà äëÿ ëþáîãî ãðàíè÷íîãî óñëîâèÿΓ(Λ) u(0) = g ∈ Cn ,óäîâëåòâîðÿþùåãî ðàâíîìåðíîìó óñëîâèþ Ëîïàòèíñêîãî, çàäà÷à (110)ñòðîãî óñòîé÷èâî ñ òåìæå ïàðàìåòðîì α(Λ).Äîêàçàòåëüñòâî. Ïóñòü u(xn ) ∈ H 1 (R+ )-åñòü ðåøåíèå çàäà÷è (110). Ââåäåì âñïîìîãàòåëüíóþçàäà÷ó äëÿ ãðàíè÷íîãî óñëîâèÿ Γ∗ èç (0.2 ):dw − G(Λ)w = f, xn > 0,dxnΓ∗ (Λ) w(0) = g,äëÿ êîòîðîé â ñèëó óñëîâèÿ (0.2 ) ñóùåñòâóåò åäèíñòâåííîå L ∗ 2− ðåøåíèå òàêîå, ÷òîα(Λ)kwk2L2 (R+ ) + |w(0)|2 ≤ C∗1kf k2L2 (R+ ) .α(Λ)Òåïåðü ðàññìîòðèì êîððåêòîð v = u − w ∈ L2 (R+ ).
Òîãäàdv − G(Λ)v = 0, xn > 0,dxnΓ(Λ) v(0) = Γ(Λ)(u(0) − w(0)) = g − Γ(Λ)w(0).Èç ðàâíîìåðíîãî óñëîâèÿ Ëîïàòèíñêîãî (114)|v(0)|2 ≤ C |Γ(Λ)v(0)|2 ≤ C[|g|2 + |Γ(Λ)w(0)|2 ] ≤ 2C[|g|2 + C1 C∗1kf k2L2 (R+ ) ]α(Λ)Ñ äðóãîé ñòîðîíû v(xn ) ÿâëÿåòñÿ ðåøåíèåì ðåçîëüâåíòíîãî óðàâíåíèÿ ñî ñïåöèàëüíûì ãðàíè÷íûìóñëîâèåì Γ∗dv − G(Λ)v = 0, xn > 0,dxnΓ∗ (Λ) v(0) = Γ∗ (Λ)(u(0) − w(0)).Îòñþäàα(Λ)kvk2L2 (R+ ) + |v(0)|2 ≤ C∗ |Γ∗ (Λ) v(0)|2 ≤ 2c2 C∗ [|g|2 + C1 C∗1kf k2L2 (R+ ) ],α(Λ)ãäå C1 − ìàòðè÷íàÿ íîðìà Γ(Λ), C2 − ìàòðè÷íàÿ íîðìà Γ∗ (Λ).
Çäåñü ìû èñïîëüçóåì ðàâíîìåðíóþîãðàíè÷åííîñòü ýòèõ íîðì â P . Ñóììèðóÿ ýòè îöåíêè, ïîëó÷èìα(Λ) kuk2L2 (R+ ) + |u(0)|2 ≤ 2α(Λ) kvk2L2 (R+ ) + 2|v(0)|2 + 2α(Λ) kwk2L2 (R+ ) + 2|w(0)|2 ≤≤ 4C∗ C2 [|g|2 + C1 C∗111kf k2L2 (R+ ) ] + 2C∗kf k2L2 (R+ ) ≤ c0 [kf k2L2 (R+ ) + |g|2 ]α(Λ)α(Λ)α(Λ)Ýòî çàâåðøàåò äîêàçàòåëüñòâî Ïðåäëîæåíèÿ (0.6). Òåïåðü ïåðåéäåì êî âòîðîìó øàãó â äîêàçàòåëüñòâåÒåîðåìû (0.11)ïîñòðîèì ìàêñèìàëüíî äèññèïàòèâíîå ãðàíè÷íîå óñëîâèå Ôðèäðèõñà, óäîâëåòâîðÿþùååóòâåðæäåíèÿì óñëîâèÿ (0.2).Ìàêñèìàëüíî äèññèïàòèâíîå ãðàíè÷íîå óñëîâèå Ôðèäðèõñà.
Ïîñòðîèì ãðàíè÷íîå óñëîâèå(106) c ìàòðèöåé ΓP h , òàê íàçûâàåìûå ìàêñèìàëüíî äèññèïàòèâíîå ãðàíè÷íîå óñëîâèÿ Ôðèäðèõñà ,n+1äëÿ êîòîðîãî ñìåøàííîé çàäà÷à (L, ΓP h ) ñòðîãî L2 − êîððåêòíà â îáëàñòè R+= {(t, xe, xn ), (t, xe) ∈ Rn , xn >0} äëÿ ñèììåòðèçóåìîé ñèñòåìû (121).n+1×åðåç L2γ (R+), γ ≥ 0, îáîçíà÷èì ãèëüáåðòîâî âåñîâîå ïðîñòðàíñòâî ôóíêöèé u ∈ Lloc , òàêèõ ÷òîn+1v = exp(−γ t) u ∈ L2 (R+).
ÒîãäàLγ v = ∂t v + γ v +nXAj ∂x v = F, F = exp(−γ t)f.(118)j=1n+1Äëÿ v ∈ H 1 (R+), èíòåãðèðóÿ ïî ÷àñòÿì, ïîëó÷èìZ∞[(∂t v, Sv)dt + γ(v, Sv) +−∞nXj=1(SAj ∂x v, v)]dt =(119)Z∞=γ−∞n) −(v, Sv)L2 (R+12Z∞−∞(SAn v|xn =0 , v|xn =0 )L2 (Rn−1 ) dtÇäåñü ìû âîñïîëüçîâàëèñü ñèììåòðèåé ìàòðèö SAj , j = 1, . . .
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