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On the existence of weak solutions of the Navier-Stokes equations Wei, David Yuen 1970

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On the Existence of Weak Solutions of the Navier-Stokes Equations by David Yuen Wei B . S c , Univers i ty of B r i t i s h Columbia, 1967 A THESIS SUMBITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF. MASTER OF SCIENCE In the Department of NMATHEMATICS We accept th is thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1970 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Depa rtment The University of British Columbia Vancouver 8, Canada i i . A b s t r a c t The e x i s t e n c e o f a weak s o l u t i o n u ( x , t ) , i n t h e -sense o f J . Le'ray ( [ 7 ] ) , i s e s t a b l i s h e d f o r t h e i n i t i a l - b o u n d a r y v a l u e p r o b l e m f o r t h e N a v i e r - S t o k e s e q u a t i o n s : ' 9u n j± - vAu + I u k D k u + g r a d x p = f k = l • d i v u = 0 . The s o l u t i o n i s r e q u i r e d t o s a t i s f y t h e i n i t i a l c o n d i t i o n u ( x , 0) = u Q ( x ) f o r x e 0, and t h e boundary c o n d i t i o n u ( x , t ) = 0 on 3P. x [ 0 , T ] , where i s an open bounded domain i n 3R n, w i t h 2 <_ n <_ 4. G a l e r k i n ' s method i s employed t o f i n d a weak s o l u t i o n u as t h e l i m i t o f a p p r o x i m a t e s o l u t i o n s { u m } . The co n v e r g e n c e o f t h e i s g u a r a n t e e d by some compact embedding theorems, w h i c h depend on a p r i o r i e s t i m a t e s f o r t h e t u m } and t h e i r f r a c t i o n a l t i m e d e r i v a t i v e s o f o r d e r Y , 0 < Y < T 1 . i i i . Table of Contents Chapter 1. Introduction 1 1.1 C l a s s i c a l Navier-Stokes Equations 1 1.2 Notation 2 1.3. Formulation of Solutions 3 1.4 Prel iminary Theory . 5 1.5 Existence Theorem 9 Chapter 2. 2.1 Existence of an approximate so lu t ion u of the Navier-Stokes 10 r r m ' . ' equations. . 2.2 A p r i o r i estimates for and i t s f r ac t iona l der iva t ives . 15 2.3 Convergence of the approximate solut ions {i^} to a weak so lu t ion 22 u of the Navier-Stokes Equations. Bibliography 28 Acknowledgements The author would l i k e . to thank Dr. A .T . Bui for h is guidance an'd assistance during the w r i t i n g of th i s thes is . The author i s also deeply indebted to Dr. J . Heywood for his careful reading, h is c r i t i c i sms and his corrections of this- thes is . The f i n a n c i a l support of the Univers i ty of B r i t i s h Columbia and the National Research Council of Canada i s gra tefu l ly acknowledged. A spec ia l thanks to Mrs. Y . S . Chia Choo for her prompt and excel lent typing of th is thes i s . - 1 -Chapter I Introduction t 1.1 C l a s s i c a l Navier-Stokes Equations. In the study of hydrodynamics, one i s int e r e s t e d i n determining the subsequent v e l o c i t y , within a f i x e d domain M n, of a viscous incompressible f l u i d , which has been i n i t i a l l y set into motion and i s governed by the Navier-Stokes equations. To be more pr e c i s e , one seeks the v e l o c i t y u = u(x,* t) and the pressure p = p(x, t) which, f o r x e ft and t > 0, s a t i s f y : ' 3 n j± - vAu + : I u kD ku + grad x p = f • k=l div u = 0 and ob£y the i n i t i a l and boundary'conditions -* u(x, 0) = u Q ( x ) , u(x, t) = 0 on 8ft x [0, T]. Here f i s a given force density; v > 0 i s a f i x e d constant; and A = A 2 2 - 2 -1.2 Notation i) Let ft be a bounded open subset of H n, with 2 <_ n <_ 4. Ii) H 1^) = { u : u e L 2(fi), D.u = e L2(fi). } . Here u is real valued and the derivatives, are taken in the distribution sense. H 1^) is a Hilbert space with : n - P l l « l l H l ( f l ) - < l l < ^ ) + J J I V l | 2 L W and (f, g ) L 2 ( n ) f(x)g(x)dx for f, g £ L 2(fi). i i i ) H*(fl) is the closure of V(Q) in H 1^), where P(fi) = c£(fi). iv) , H(fi) = H = (L2(J2))n (taken as the nth product space). Thus f • e H i f f = (f1, '", f ) where. f i e L2(fl)., i = 1, n. For n f, g e H we get (f, g) R = . ( f . , 8 ^ 2 ^ . v) V will be the closure in ,the product space (H^fi)) 1 1 of the subspace of functions i|> = 0|> , i|/n),/where ^ e P(fi) and div = 0. : For u, v e V, we set n n (u, v) = y y (Dn.u., D.v .) T 2, i=l j=l J 1 J L < a n d II u N v . = (u> u )v " n |-u^ D^ v^ w^ dx is defined on vi). The trilinear form . b(u,v,w) = £ i,k=l L 4 x v x L 4, where Lk = (L4(fi))n . - 3 -v i i ) HY(a,b; V ; H ) = { u : u e L 2 ( a , b ; V ) ; | T | Y | u ( T ) | e L 2 ( a , b ; H ) } w h e r e a ) b ) 0 < Y < 1 ; L ( a , b ; V) = { u : u ( x , t ) e V f o r e a c h t e ( a , b ) a n d rb J c) ||.u | | v d t < oo } ; L ( a , b ; H) = { u - : u ( x , t ) e H f o r e a c h t e ( a , b ) a n d | | u | | 2 H d t < co } ; d) e ) , f ) A ' ' 2 u ( x ) i s t h e L F o u r i e r t r a n s f o r m o f u w i t h r e s p e c t t o t ; . we s a y t h a t u e L ( a , b ; H) h a s a f r a c t i o n a l t i m e d e r i v a t i v e D^u £ L 2 ( a , b ; H) i f . a n d , o n l y i f | T | Y | | U ( T ) | | „ e L 2 ( a , b ) ; t . . . . . . . n fb rb u | , H Y ( a , b ; V ; H) . p (J a . u il v 4t + | T | 2 Y | | u ( T ) | P d T ) H a n d H^(a,b; V ; H) i s a H i l b e r t s p a c e . 1 . 3 F o r m u l a t i o n o f S o l u t i o n s . /' I n t h e r e s e a r c h o f J . L e r a y d u r i n g 1 9 3 3 - 3 4 ( [ 7 ] ) t h e c o n c e p t o f a w e a k s o l u t i o n o f t h e i n i t i a l b o u n d a r y v a l u e p r o b l e m f o r - t h e N a v i e r - S t o k e s e q u a t i o n s (1) w a s f o r m e d f o r Q ] R n , w i t h n = 2 , 3 . I n 1 9 5 1 E . H o p f ( [ 5 ] ) e x t e n d e d t h i s f o r m u l a t i o n t o a n y d i m e n s i o n n > 1 . F o r U Q ( X ) b e l o n g i n g — 2 t o t h e c l o s u r e V o f V i n H , a n d f g i v e n i n L ( 0 , T ; H ) , we s a y u i s a w e a k s o l u t i o n o f ( 1 ) i f u e L ( 0 , T ; V) a n d - 4 -(2) { v ( u ( t ) , <j>(t))v + b (u ( t ) , u ( t ) , <j>(t)) - (u ( t ) , <J> ( t ) ) H }dt ( f ( t ) , <j>(t))Hdt- + ( u 0 , <K0))H for a l l T > 0 and a l l <j> e ( C Q ( [0 , co) x fl)) n wi th d iv <}> = 0. In 1957, A. Kice lev and 0. Ladyzenskaya ([13]) proved uniqueness w i th in the class of weak solutions such that II u | | and || u 11 . are i V t H uniformly bounded on f i n i t e time in te rva l s for dimensions n = 2 or 3. In the case of small data they proved existence wi th in th i s c l a s s . J . S e r r i n ([16]) extended the resul ts of uniqueness and, for .small data, existence, wi th in th i s class to dimension n = 4. For the two dimensional problem, Ladyzenskaya has demonstrated the existence of such solut ions for a r b i t r a r i l y large data ([14]) . I t i s not yet known i f a so lu t ion i n th is class exis t s for dimensions larger than 4, or i n dimensions 3 and 4 for large data. \ This thesis w i l l be an expos i t ion 'of part bf the paper ([9]) by J . L . Lions . Lions showed the existence/of a weak so lu t ion u of (1) belonging to L (0,co;V) and having i n addi t ion a f r ac t iona l time der iva t ive 2 1 D^u e L (0,co; H) for any Y , 0 < Y < . Here we consider only the case of a bounded domain Q. , i n R n , wi th 2 <_ n <^  4. The rest of the chapter contains some prel iminary lemmas and a statement of the existence theorem. The second chapter w i l l concern i t s e l f with a proof of the existence theorem. -5 -1.4 Preliminary Theory. Proposition 1 : Let ft be a bounded, open and connected subset of K.n, with n <_ 4. Then there exists a natural continuous injection mapping from V into (L 4(ft)) n. The proposition follows as a special case of the Sobolev embedding theorem ([10]). The natural injection mapping of H*(ft) into L^Cft) i s continuous. Corollary 1. : Let u, v, w e V, then b(u, v, w) is continuous on V x V. x V. Proof : |b(u, v, w) | <_ £ i,j=l IuiI IDivi''wiIdx'» f o r a ny u> v> w e V* ft . From Proposition 1, |b(u, v, w)| < C|| u|| L,|| v|| v||w|| L , < C||u|| v|| v|| v||w|| Q.E.D. Proposition 2 : Let u(x, t) E L 2(0,co; V) and <)> e C"([0, oo); V) , then b(u, u, <f>) £ L 1(0, co) Proof : By Corollary 1 of Proposition 1, we have |b(u, u, •)'( < C||u<t)||2v||*l|v. -6 -Thus b(u, u, <j>) e L 1 ( 0 , oo). Q.E.D. Proposi t ion 3 : Suppose u , v , w e V, then b(u, v , w) + b(u, w, v) =-0. Proof : For u , v , w E V(Q) and divergence free, we can integrate by parts : . ' n r ,dx »(u, v , w) - I [ u .D ± v w < i , j = l J f i 2 2 u .v .D-w^x] . Q  J J D.u.v.w.dx fi 1 1 2 2 Since v . , w. e V(Q) and d iv u = 0 ; 3 3 b(u, v , w) = - b(u, w, v) \ Hence for any u, v , w e V the desired resu l t follows by passing to the l i m i t using Corol lary 1 of P r o p o s i t i o n ! . Proposi t ion 4 : For 0 < Y < ^ , there ex is t s 3 > -j and C1 e II such , , , 2 Y C l ( 1 + H> that | x | <_ — for a l l x e C. i + Proof : Since 0 < 2y < j we have ' | x | 2 Y - 7 -Hence, 2(1+ | x | ) > | x | 2 Y + | x | + l > . | t | ' 2 Y + IT | x | 2 Y ( l + I x l 1 - ^ ) , I |'2Y „ 2(1+ | x | ) and so | x | <_ — ±2 i + i - , - 2 v /Thus we'may l e t " Cj = 2 and 3 = 1 - 2Y . ' "»r>*:'' ; Q . E ; D r ; " ' Proposi t ion 5 : V Has a countable basis consis t ing of elements w = ( W p , . - . , w n ) , where e P(fi), i = 1, •••» n ; and d iv w = 0 . Proof : Let S = {w : w = fr , w n ) ; w i £ fl(ft), i = 1, • • •,. n ; d iy w = 0 } • -V • Since (C(fi)) n i s separable, S i s separble, ([3], Th. 1.6.12). Hence S has a countable bas i s . { vr. , • • • , w^ •:• }' , ([15]) . ' / ' Since V i s the closure S of S i n ( H 1 ^ ) ) 1 1 , for any element v e V, there exis t s a sequence ^ Sp^ S such that s^ -r—> v as p —> co , i n (H 1 ( f i ) ) n . . . Hence the Proposi t ion fo l lows. Q.E.D. Proposi t ion 6 : A weak so lu t ion-of the in i t ia l -boundary value problem •sa t i s f ies the Navier-Stokes equations (1) i n the d i s t r i b u t i o n sense. Conversely, 2 a d i s t r i b u t i o n so lu t ion u e L (0,oo ; V) of the Navier-Stokes equations (1) . v - 8 -i n Si x ( 0 , oo) s a t i s f i e s the i n t eg ra l i den t i t y (2) for a l l (j) e C Q ( ( 0 , OD ) ; V ) . Proof : Suppose u e L 2 ( 0 , co; V) s a t i s f i e s ( 1 ) . Let cb e (C£(sl x ( 0 , . c o ) ) ) n and *div <J> = 0 , then (Au, 4>)Hdt = Jo fc(t), <|>(t))„dt and fct, *)Hdt = - j o u (t) , <J>t(t))Rdt - (u Q , * C 0 ) ) H By Proposi t ion 2, b(u, u , <J>) e L ( 0 , oo ) and hence .oo n J 0 k=l k k H b(u( t ) , u ( t ) , <KO)dt Since <}> e (Oft x ( 0 , co))) and d iv <j> = 0 we have (grad^ p, *) R dt = (J o ±=Vn o x i n I ( P * i l 9 f t 0 1=1 3cb . p r - p i x ) d t ft d x i r<*> n r 9<j). 0 i=l Jfl d x i 0 0 r ' ' ' ri 9<J>. . p I g-pixdt = 0 o'ft i = i d x i Conversely, l e t u sa t i s fy ( 2 ) . Define a d i s t r i b u t i o n T i n P X ( f t x ( 0 , c o ) ) n by r T:=|^> v A u + .J ^ u - f . •'" k=l By the preceding arguments, (T, <j>)„dt = 0 for a l l $ e (C°°(ft x (0, c o ) ) ) n wi th d iv <j> = 0 . ri o Hence T = grad p , ( [6] , Th . l ) Q.E.D. 1.5. Existence Theorem. Let Q be an open, bounded and connected subset of IR , with 2 — 2 <_ n <_ 4. For any f e L (0, oo ; H) , v > 0, and u Q (x) e V, there exis t s a function u e L (0, co ; V) which s a t i s f i e s (5) { v ( u ( t ) , <J.(t))y + b (u ( t ) , u ( t ) , <J>(t)) - (u( t ) , * t ( t ) ) H }dt ^ ( f ( t ) , <(.(t))Hdt + (u Q + <j>(0))H for a l l , (j) e C^([0, co) ; V) wi th $ t e L 2(0, co ; H) . Moreover DYu e L 2 ( 0 , co ; H) , - 1 0 -Chapter I I The proof of the existence theorem w i l l be given i n a series of . lemmas which w i l l be broken in to three sect ions. In the f i r s t sect ion the existence of approximate solut ions i s proven. In the second sect ion a p r i o r i estimates for the and the i r f r ac t iona l time der ivat ives are es tabl ished. F i n a l l y we use these estimates along with some compact embedding theorems to show the convergence of the to a weak so lu t ion u of the in i t ia l -boundary value problem for the Navier-Stokes equations. Proof of the Existence Theorem 2 .1 . Existence of an approximate so lu t ion XL^ of the Navier-Stokes  equations. . s Let w 1 , •••> wg.«-»^ be a basis for V as defined i n Proposi t ion 5. Then Wj = ( w ^ , * • •, w j n ) > w j k e V(Q),. with k = 1, n; and div Wj = 0, j = 1, • • • . Since u Q e y V , there exis t s r ea l such that m / a.w. —-> u i n H as m—> co . x=l Lemma 1 n Let u m = I § ( t )w. (x) . Then the i n i t i a l value problem j=l jm J . (6) (um' VH + V(um' wj> V + b(um» V wj> " <f' w j V J = 1>"-»m m - -11 -NOTE : . - u at has a l o c a l so lu t ion u (t) e L (0, T : V) m ' m Proof' ; To prove the lemma we s h a l l apply the Caratheodory Existence Theorem ( [2] , p. 43). Rewriting (6), for j =.1, m, '"'•fe(V'wi^-(f» W 1 > H _v(um* w i V b ( ^ V w i } (7) • .. dF (V w m ) H = ( f > wm>H _ v ( u m> Wm>V + b ( V V V ' ' n Since , u m = £ g i m ( t ) w i ( x ) we have i = l ' (8) dt H t <wi» wm> H <wm» wm>H - v mm J (f, » j ) E ' ( w i ' wi >v '•' ( w m' W l } V - v • ( w l ' wm>V ( wm» wm>V -- 12 -+ b(u , u , w„) m' m' m' J We w i l l now l e t (8) be wr i t t en i n the matrix-vector form dt -AX(t) = F( t ) •+ vB-X(t) + P(X(t) ) with A = (Wj, w j ) H ......... (w m, « 1 ) H . <wm» ( w m ' wm>H J B = (w x , w x ) y . . . . . . . . (wlf wm), (wm, w 1) y ( wm» wm>V J - 13 -F(t) = x ( t ) = 8mm^ > J f b(um> V ] P(X(t)) = I b(u , u , w ) m m i=l 1=1 m '• . I , gim ( t )Sjm ( t ) b ( wi> Wj> V ' i , j = l J J . 1 m 1 1 , •8 i n(t)8 J B l(t)b(w 1, w wm) J i,j=l - 14 -Since {wn} i s a basis for V C H, ( w n ) ^ s l i n e a r l y independent i n . H, hence determinant A i s the Gram-Schmidt determinant. Therefore A i s nonsingular and A ^ e x i s t s . So (6) may be wr i t ten as (6') ^ - X ( t ) = A _ 1 . F ( t ) - v A - 1 . B . X ( t ) + A _ 1 . P ( X ( t ) ) dt 1 1 - 1 M = A X F ( t ) - vA .B-X(t ) + A • I g±m-C±-X(t), i = l where : = b ( w ± , w m , w x) b(w., w, , wm) b(w-,' w , wm) i m m ' Let the r ight 'hand side of (6') be denoted by G(X( t ) ) . For f ixed X ( t ) , G(X(t)) i s ce r t a in ly measurable in ; t,. as (f,••w.-)„ i s measurable i n t .'• '"'X'-' '." " 1 11 i • ; by hypothesis. For f ixed t , the f i r s t two terms of G(X(t)) are l i n e a r m V i n X( t ) and since P(X(t)) = 2 S i m C^X(t ) i s a continuous function of i = l X ( t ) , G(X(t)) i s continuous i n X ( t ) . Let R be the rectangle 0 < t < T , with T m > 0, and ° — — m m ' m X(t) - X(0) | = | g i m ( t ) - a . . J <_ a, wi th a > 0. Then using the matrix i = l lm norm, | | « | |= absolute value of the largest component of (•)> - 15 -|G(X( t ) ) | < HA"1!! | F ( t ) | + vIlA" 1!! ||B|| | X | + HA"1!! | P ( X ( t ) ) | where 6 X = 11 A-111 ; 32=||B|| ; ^ = max | g i m ( t ) | ; and | = max | | c j l ' "^m 1 Hence on 0 < t <_ T m , • (3 j |F(.t) | + vg 8 p + £p 2 ) i s Lebesgue integrable . Thus by Ca ra theodory theo rem, there exis t s a so lu t ion , defined on (0, T ) . . . Q.E.D. 2 .2. A p r i o r i estimates for u and i t s f r ac t iona l time der iva t ives . — c •  m The fol lowing two lemmas contain a p r i o r i estimates for the { u m ^ and the i r f r ac t iona l time der iva t ives . Lemma 2 : The solut ions {um). of the system (6) are uniformly bounded  i n the norms of L 2 ( 0 , co ; V) and L°°(0, T; H), wi th f i n i t e T. Proof : Mul t i p ly both sides of '(6) by g-m(t) and sum with respect to j = 1, • • • m . Since um(t) = I g- (t)w (^x), (6) i s replaced by i = l = ( f , \ W ) H • (By Proposi t ion 3, bCu^, u^, um) = 0) - 16 -We now show the are uniformly bounded i n L M ( 0 , T; H) , for a l l f i n i t e T. From (9) . ' i l E i i v t ) i i 2 H < V C » H • Thus i k W l l H f e l l V O l l H i N ' N H l l y ^ l l H ' m Since J an-w. —> u i n H we have II u (0)|L, < C- I! u l| T T and hence L x i o 1 1 m • " H — 11 o" H i = l in tegra t ing from 0 to t , t _< T m < co , f t | | f ( a ) | | da 0 Since f e L 2 ( 0 , co ; H) we have sup , | | u (t) | | <_ J < co . te[0, T m ] m Thus we may appeal to the global existence theorem ([17], p.122) for ordinary d i f f e r e n t i a l equations to conclude that u m ex is t s for a l l t e (0,* CD ) Hence sup | | u (t) || „ <_ J < oo , for a l l f i n i t e T. te[0, T] m We show the {um) are uniformly bounded i n L (0, oo; V ) . From the d e f i n i t i o n of (•, • )„ t l 4 v*). v » H - -1 i= i J <dt~ "mi ' U m i ) L 2 ( f t ) d t m i = l Jft 2 dt ' mi' Integration of (9) from 0 to t y i e lds Thus (10) rt ( f (a ) , u j a ^ d a •0 ft IKCcOllv da < J o | | f (a) | | H | |u m (a) | | H da + - | Using the inequal i ty (10') 1 f f.*8 1. "2 (~ + ^ 2 S 2 ) > w i t h n > 0 we get n j j | £ f o ) l l H l | u 1 I , ( < ' ) l l H d o ±k f0h H f < « > l l 2 H d ° + i | t n 2 i l V > l l l H * By the Poincare" inequal i ty ( [1] , p .73) , - 18 -(10") l l % l l H < d | | u j | v , d > 0 V Takxng n = — we o b t a i n •d 1 2 f t 2 . . . rt n 2||V^H 2 H d a^f J II V*>Hvda Hence i t follows that (11) uJOJI^dt < j \ < CO Q.E.D. '/...•In order to consider the f r ac t iona l time der ivat ive of u^ we u m ( t ) , t > 0 .extend i t s 'domain of d e f i n i t i o n by se t t ing ^ ( t ) = 0 , t < 0 or equivalent ly ^ ( t ) = u ^ O H C t ) where H(t) i s the Heaviside function. Then {^(t) £ L 2 ( -co , co ; V) . Lemma 3 : For f i x e d Y, 0 < Y < , u m ( t ) £ H^(-oo , co ; V; H) and the {{im,(t)} are u n i f o r m l y bounded i n H^Ctco , oo ; V; H) . Proof : We w i l l show that for k <_ m, ^ ( t ) s a t i s f i e s : ( 1 3 ) . dt" ( { i > n ( t ) » W k } H " V ( { i m ( t ) ' w k > V + ' b ( S t a < t > » ^ ( t ) > wk> (f, w k ) H + (%(0), wK)<5 , - 1 9 -where f = f-H(t) and <S i s Dirac measure on H . This follows from J- ( ^ ( t ) , w k ) H = ^ (u m ( t )H( t ) , w k ) R where = H ( t ) I t - (um(t>» wk>H + (um(t)> W k ) H IF H ( t ) > J£ CV C >» w k ^ = _ V ( u m ( t ) > wkV " b ( um(t), ^ ( t ) , w k) + (f , w k ) H and 4r H(t) = 6 . dt Since b(u, v , w) i s continuous on " V x V x V (Corollary 1, Proposi t ion 1) , the Riesz Representation theorem gives : b ( u m , u^, w^) = (h m ( t ) , w k ) v , where (14) II V ^ H v ^ C 2 l l V ^ H v- •:• Taking the Fourier transform of (13) wi th respect to time we obtain (15) ^ ( ^ ( x ) , w k ) R + V C G ^ T ) , w k ) v + ( h M(T), w k ) v = (f(T)', w k ) R + (1^(0), w k ) H . Mul t i p ly ing (15) by. S i m ( f ) and sumning with respect to i = 1, m, we get . v - 20 -(16) - ( f ( T ) , G m ( T ) ) H + (U m ( 0 ) , {^(T)) V Taking the imaginary part of (16) and applying the Holder inequal i ty we have (17) | T | | | u m ( T ) | | 2 H < H V O l l v l | u m ( T ) | | v + | | f ( T ) | | H l | u m ( T ) | | H + II V°>I.IHIIVT' H From (17) we w i l l show f | T | 2 Y | | ^ ( T ) | | 2 d T < J 3 < CO . J —oo Indeed, s ince- h ( T ) = - i x t e h^CtOdt, we have J -o b m ( t ) 11 v d t 1 C 2 vt)irvdt (by (14)) i C 3 (by lemma 2) Hence (17) y i e lds a s ) . IT I I I V T ) ! ! ^ < c j | u M ( T ) | | V + | | f ( T ) | | H | | u M ( T : H ' 2 1 -where = (C 3 + d - | | u Q | | ) For 0 < y < — , we have by Proposi t ion 5, T|*IIV T>IJH< c i ( 1 + l T l ) ( 1 + I ^ ^ I I ^ H H 3 . - 1 . - V 1 + iTrr 1 ! !^)!! 2 H + C J T K 1 + I T I * ) - 1 ! ! ^ ) ^ Now subs t i tu t ing (18) in to the previous inequal i ty and in tegra t ing from -co to co we have : I x l ^ H ^ l ^ d x < C l C | f a + I t l 3 ) - 1 ! ! ^ ) ! ! ^ + c j a + Irl^ iifcon IIG^CT: «* —oo + c, ^)H 2H< 1 + l T| P>' l d T Let I , , I 2 , I 3 be the three in tegra ls on the r igh t hand side of the previous inequa l i ty . Since 11 u m l I L 2 > 0 0 . V ) 1 J i ,00 hiW ( 1 + | T | B ) " 2 d T ) * ( H^OIP dT) J - c o - J —00 <_ M 2 < oo - 22 -We have i 2 i v . a + I x i V^lfllJI^coll^ T < C ( — o o J — OO By Plancherels f theorem and the Poicard inequal i ty (10") I 2 £ M 3 . Also e by (10") and the Plancherel theorem Ig <_ . Thus M 2 Y I I^ C O | | 2 H d T < J , Q.E.D. 2.3. Convergence of the approximate solut ions {u^} to a weak so lu t ion u of the Navier-Stokes Equations. In the l a s t three lemmas of th i s chapter we w i l l show that there exis ts a subsequence {"mp) of which converges, i n a sense to be spec i f i ed , to a weak so lu t ion u of the Navier-Stokes equations. Lemma 4 : . There exis ts a subsequence { u mp} ojE {i^} , which for  s i m p l i c i t y we denote by ^Up} , such that : (19) u„ -> u weakly i n L (0, co ; V) , (20) U p > u i n the weak star topology of L°°(0,T; H) , for a l l f i n i t e -T, (21) • u p > u strongly i n L 2 ( 0 , T ; H) for a l l f i n i t e T. - 23 -Proof : The unit sphere i n a r e f l ex ive Banach space i s weakly compact. Since i s uniformly bounded i n L 2 ( 0 , oo ; V) and L°°(0,T; H) , there exis ts a subsequence ^ up^ sa t i s fy ing (19) and (20), ( [3] , Th. V . 4 . 2 ) . The natural i n j ec t ion mapping of V in to H i s compact since • 0, i s bounded. Thus by the compactness theorem of Lions and Hormander, (Lions [10], Prop. 4 .2 ) , the natural i n j ec t i on mapping : H y (0 ,T ; V; H) > L 2 ( 0 , T ; H) i s compact. By Lemma_ 3, II "mil ^ (Q J . y . JJ) — ^3> hence there exis t s a subsequence ^ u pJ °f ^um^ such that U p > u strongly i n L 2 ( 0 , T ; H), Q . E . D . Since U p ( t ) i s a so lu t ion of the system (6), the fol lowing equation holds : . ' (22) B {v(up(t) , * ( t ) ) v + b ( U p ( t ) , u p ( t ) , <Kt)) + (up'(t), <J»(t))H}dt ( f ( t ) , $ ( t ) ) H dt , 0 H for a l l <j> of the form <J>(t) ='• 1 ^ - ( t ) w . ( x ) , where u < p and *j"Ct) e CQ([0, oo)) j = l 3 J Integration by parts y i e lds : - 24 -(23) Jo { v ( u p ( t ) , < K t ) ) v + b(up( t ) , u p ( t ) , (f)(t)) - ( u p ( t ) , (j)'(t))H}dt ( f ( t ) , * ( t ) ) R dt + (u p (0 ) , <j)(0))H . Lemma 5 : The l i m i t u s a t i s f i e s the equation : (24) fB 0 fB {v(u( t ) , * ( t ) ) v + b (u ( t ) , u ( t ) , Ht)) - (u ( t ) , <(.'(t))H}dt ( f ( t ) , * ( t ) ) H d t + (u(0), (J)(0))H , for a l l <j> of the form (25) <J> = I V i C O w , (x) ; iK e cj([0, oo)) j=l J J Proof : Let p > oo , . then from (19) and (21) , we have B rB v(Up(t) , <f)(t))vdt > 0 v ( u ( t ) , K t ) ) v d t ; I] < V •*(t)) H dt (u, <*>' ( t ) ) R d t From the i n i t i a l conditions for the system (6), (up(0), <j>(0))H > (uQ> $(0))^ since / a.w. —> u„ i n H as - p —> oo . . - i i 0 i = l I t remains to show that rB b ( u p ( t ) , u ( t ) , <|)Ct))dt > b(u ( t ) , u ( t ) , * ( t ) ) d t From the d e f i n i t i o n of b(u, v , w) we have to show (26) rB r rBf u p k (x , t )D k u p i (x , t ) ( J ) 1 dxd t > u k(x,t)D ku 1(x,t)<(> 3.dxdt 0 £l for p —> co ; k, 1 = 1, • • • , n . Now = < u Pk " V ^ p i * ! + ( D k u p l " D k u ± > u k * i This implies o-'n ( u p k D k U p l * l " U k D k u i * i d x d t l [ ( u p k " V ^ P i * * + ( D k u p l " D k u i ) u k ^ ] d x d t | ^ l'lUpk -•ukH L 2 ([0,B)xC2) II D k^l^iH L 2([0,B)x.Q) + I oh ( D k u p l " D k u i ) u k + i d x d t l The expression on the r ight tends to zero since U p k > u k s trongly i n L 2 ( [ 0 , B ) x fi), D k u p l > D k u i weakly i n L 2 ( [ 0 , B ) x Q), and ^ e CQCIO, oo) x f l ) ... - 26 -Lemma 6 The equation (27) {v(u( t ) , * ( t ) ) y + b (u( t ) , u ( t ) , <J.(t)) - (u( t ) , <j.'(t))H>dt ( f ( t ) , <Kt)) Rdt + (u Q , * (0 ) ) H holds for a l l functions <j> such that (28) 4 e L z ( 0 , oo j V) , 4»* e L 2 ( 0 , co ; H) , with (j>(t) having compact support i n t >_ 0 . Proof : By Lemma 5, (27) holds for a l l function <j> of the form (25). Let the lefthand side of (27) be denoted by L(<j>). For any <j> sa t i s fy ing (28), there exis t s a sequence of functions <j> k s a t i s fy ing (25) with support contained i n [0 ,B) , such that <j>k—> <J> i n L 2 ( 0 , B ; V) and <f>k —> 4>' i n L 2 ( 0 , B ; H) as k —> co , ([10]). Moreover, rB | L ( * k ) - L(<j>) | < v | | u(t) | | | | <frk - <j>| dt JQ rB + C i i u ( t ) i r v i u k - *n v dt ' u + j f |u(t) | | H |U k - * | | H d t - 27 -< v | | u | i L 2 ( 0 j B . V ) C j IUk-<Hi2vd t ) % B II * k- *H2vdt)' + C i H u i l L 2 ( 0 , B ; V ) ( . H | U I I l 2 ( 0 , B ; H ) ^ ! J l + i - ^ l l ^ t ) < <»IMIL2(0,B; V) + C l l N l L 2 ( 0 , B ; V) rB + II UIIL 2 (0 ,B; H ) ' K J l l * k - * l l 2 v + U*k < e Clear ly j \±> v k f ( f , O d t + ( u 0 , * k ( 0 » ^ ° converges to ( f , 4>)dt + ( u 0 , *(0)) as k —> co - 28 -Bibliography [I] Agmon, S. , " Lectures on e l l i p t i c boundary value problems, " D. Van Nostrand Company, Inc . , Princeton, (1965). [2] Coddington E. and N. Levinson, " Theory of ordinary d i f f e r e n t i a l equations, " McGraw-Hil l , (1955). [3] Dunford, N. and J .T . Schwartz, " Linear operators, Part I , " Interscience Publishers Inc, New York (1963). [4] Hormander, L . , " Linear P a r t i a l D i f f e r e n t i a l Operators, " A c a d . . ."Press .Inc, New York, (1963).- •  [5] Hopf, E . , " Uber die anfangswertaufgabe fl ir die hydrodynamischen grundgleichungen, " Math. Nachr. 4, (1951), pg 213-231. [6] Ladyzenskaya, 0 : A . , " The mathematical theory of viscous incompressible flow, " Gordon and Breach, New York, (1963). [7] Leray, J . , " Etudes des di'verses equations i n t £ g r a l e s non-lineare et de quelques problemes que pose 1'hydrodynamique, " J . Math, pures. appl . 12, (1933), pg 1-82. " Essai sur les mouvements plaus d'un l i qu ide visqueux que l imi t en t des parois , " J . Math, pures. appl 13, (1934), pg 331-418 " Sur les mouvements d'un l iqu ide visqueux emplissant l ' espace, " Acta Math. 63, (1934), pg 193-248. [8] Lions , J . L . , " Sur l ' ex i s t ence des solut ions des equations de Navier-Stokes, " C R . Acad. S c i . , Par is 248, (1959), pg 2847-2850. [9] " Quelques resul ta ts d'existence dans des equations aux der iv£es p a r t i e l l e s non- l inea i res , " B u l l . Soc. Math. , France, (1959), pg 245-261. [10] , " Equations d i f f e r e n t i e l l e s operat ionelles et problemes aux l i m i t e s , " Springer^-Verlag, B e r l i n , (1961). [II] Riesz , F. and S. Nagy, " Functional ana lys i s , " Blackie and Sons, (1956). [12] Schwar tz , 'L . , " Theorie des d i s t r i bu t i ons , T. 2, " P a r i s , Hermann, (1950). . • - 29 -[13] K ice l ev . A and 0. Ladyzenskaya, " On the existence and uniqueness of the so lu t ion of the nonstationary problem for a viscous incompressible f l u i d s , " Izv. Akad Nauk SSSR, Ser. Math. 21, pg 655-680, (1957). [14] Ladyzenskaya, O .A. , " Solut ion " i n the large " of the boundary- -value problem for the Navier-Stokes equations 'for the case of two space var iab les , " Dokl . Akad. Nauk SSSR; 123, pg 427-499, (1958). [15] Epstein, B . , " P a r t i a l d i f f e r e n t i a l equations : an in t roduct ion , " McGraw-Hill , (1962), pg. 98. [16] Se r r in , J . , " The i n i t i a l value problem for the Navier-Stokes equations, Nonlinear Problems, Univ. of Wise. Press, (1963). [17] Birkhoff , G. and G. Rota, " Ordinary D i f f e r e n t i a l equations, " Ginn and Company, (1962). 

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