On Square Summability and Uniqueness Questions Concerning - -• Nonstationary Stokes Flow i n an Exterior Domain by CHUNKING MA B.Sc, Hong Kong Baptist College, 1969 M.A. , Un i v e r s i t y of New Brunswick, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mathematics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1975 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree l y ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is fo r f i nanc ia l gain sha l l not be allowed without my writ ten permission. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 i i Supervisor: Dr. John G. Heywood ABSTRACT In t h i s thesis we investigate the square summability, uniqueness, and convergence to steady state of solutions of the nonstationary Stokes equations i n an exterior domain. A cl a s s of generalized solutions (which w i l l be c a l l e d class H q s o l u t i o n s ) , whose members are required a p r i o r i to have f i n i t e D i r i c h l e t i n t e g r a l s but not necessarily 2 to have f i n i t e L norms, has been introduced by J.G. Heywood for the purpose of studying the convergence of nonstationary solutions to stationary ones as time t -> °°. In our present work, we prove that, i n the case of an exterior domain Q, of R n(n > 2) , such solutions are necessa r i l y square-^summable i f both the i n i t i a l data and the force are square-summable. 2 We give a p a r t i a l r e s u l t for J2 i n R . Furthermore, we prove 3 that i f Q CZ R the unique class H sol u t i o n i s i d e n t i c a l o 2 with the unique f i n i t e energy s o l u t i o n ( i . e . L (Q)) of various classes when the data permits existence of both types of s o l u t i o n s . This has enabled us to show that the f i n i t e energy solutions of a p a r t i c u l a r nonstationary Stokes problem converge to solutions of steady state as t + », We have also succeeded i n extending the d e f i n i t i o n of class H solutions o i i i to nonstationary- Stokes problems with general nonhomogerieous boundary values i n such, a way that the uniqueness theorem for such, s o l u t i o n s i s preserved. i v TABLE OF CONTENTS Page Section 1 : Introduction 1 Section 2 : Preliminaries 8 Section 3 : Square Summability of Class H q Solutions . 18 Section 4 : Uniqueness of Solution Classes 33 Section 5 : The Case of Nonhomogeneous Boundary Values 42 References ' 53 ACKNOWLEDGEMENT I am greatly indebted to Dr. J.G. Heywood for suggesting t h i s topic and his generous assistance during the preparation of t h i s t h e s i s . My thanks i s extended to Dr. R. Adams and Dr. J. Fournier for t h e i r h e l p f u l comments on the d r a f t of t h i s work. The f i n a n c i a l support of the National Research Council of Canada and the Uni v e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. I dedicate t h i s work to my wife for her encouragement throughout my study and for her excellent typing of t h i s t h e s i s . 1. Introduction This thesis deals with square summability questions associated with a cl a s s of solutions of the i n i t i a l boundary value problem for thennonstationary Stokes equations. If u(x,t) i s the v e l o c i t y f i e l d of a f l u i d , i f p(x,t) i s the pres-sure, and i f f ( x , t ) i s a given external force density, then the equations for nonstationary Stokes flow i n a space-time region fix (0,°°) are Au = - Vp + f 3t v (1) V«u = 6 Here fi i s a domain i n R n(n >^ 2). Solutions of (1) are sought to s a t i s f y prescribed i n i t i a l and boundary conditionsre pie general i n i t i a l boundary value problem (Section 5) can be reduced to that with homogeneous boundary values: (2) u(x,0) = a(x) , x e fi (3) u(x,t) = 0 , (x,t) e 3fi x (o,») (4) u(x,t) 0 as |x| ->• 0 0 i f Q i s unbounded. -2-We w i l l be p a r t i c u l a r l y interested i n the case of an e x t e r i o r domain ft i n R n with n > 2. Because equations (1) - (4) admit a formal energy i d e n t i t y , i t i s natural to a n t i c i p a t e that solutions 2 of (1) - (4) w i l l belong to L (ft) f o r every t < T. i f the i n i t i a l 2 2 data a e L (ft) and i f the prescribed force f e L (ft x (o,T)). Our main objective i s to prove t h i s square summability f o r a class of solutions of (1) - (4) which are defined without being required, 2 a p r i o r i , to have f i n i t e L norms i n ft. In order to study the convergence of nonstationary solutions to stationary ones as t °° i n the case of an e x t e r i o r s p a t i a l domain, Heywood [2] has recently introduced a cla s s of solutions which are required, a p r i o r i , to have f i n i t e D i r i c h l e t 2 i n t e g r a l s but not nec e s s a r i l y to have f i n i t e L norms. Solutions of t h i s class are characterized c h i e f l y by membership i n a c e r t a i n function space H q , and we follow Heywood i n c a l l i n g them class H solutions. For some classes of i n i t i a l data and forces o which need not be square-summable, he proved the existence and uniqueness of a cla s s H q s o l u t i o n of problem (1) - (4) without appealing to energy estimates. This c l a s s of solutions proved us e f u l i n t r e a t i n g problems of convergence to steady state p r i n c i p a l l y because solutions of the e x t e r i o r stationary Stokes problem possess f i n i t e D i r i c h l e t i n t e g r a l s but not always f i n i t e norms. - 3 -Since class H solutions of ( 1 ) - ( 4 ) were studied i n o [ 2 ] without use of any energy estimates, i t i s not c l e a r whether class H q solutions w i l l be square-summable i f the i n i t i a l data and the force are square-summable. In f a c t , Heywood has given a negative answer to t h i s question i n the case of a two-dimensional e x t e r i o r domain. He has shown i n [ 3 ] that there are choices of square-summable data for which the unique class H q s o l u t i o n tends, a f t e r a f i n i t e time, continuously to a nonzero l i m i t at i n f i n i t y and i s thus not square-summable. The case of an ex-t e r i o r domain i n R n with n >>2, nevertheless, contrasts sharply with the case of n = 2 , because when n > 2 the c l a s s H s o l u t i o n o tends to zero at i n f i n i t y i n a generalized senseuwhichuprecludes the p o s s i b i l i t y of i t s tending continuously to a nonzero l i m i t at i n f i n i t y . It i s therefore reasonable to expect that the class 2 H q s o l u t i o n of problem ( 1 ) - ( 4 ) w i l l possess a f i n i t e L norm i f the i n i t i a l data and the force do. In t h i s t h e s i s , we w i l l prove t h i s conjecture i n the case of an e x t e r i o r domain i n R n with n > 2 . For a r b i t r a r y domains i n R n with n > 2 , the problem i s s t i l l open. In the case of an e x t e r i o r domain i n 2 R , there remains a question of what conditions on the forces w i l l ensure that class H q solutions are square-summable. It i s important to determine such conditions for the forces i n order to c l a r i f y the ph y s i c a l s i g n i f i c a n c e , i n the case of a two-dimensional e x t e r i o r domain, of a theorem of Heywood [ 3 ] which states the a t t a i n a b i l i t y of stationary solutions as l i m i t s of nonstationary solutions. We do give a p a r t i a l r e s u l t f o r the case of a two-dimensional e x t e r i o r domain; we show that i f the force i s square-summable then the time d e r i v a t i v e of the class H q s o l u t i o n tends to a l i m i t at i n f i n i t y i n a generalized sense f or each f i x e d time. Our method of proving that the class H q s o l u t i o n of (1) - (4) i s square-summable, i f the data i s , involves proving that u f c can be expressed i n the form u f c = Vq + g, where g i s square-summable i n ft x (0,T), and where q i s harmonic i n the space v a r i a b l e s . By expanding q i n a neighborhood of i n f i n i t y as a s e r i e s i n s p h e r i c a l harmonics, and by using L 2 estimates for g and Vu , we show that Vcq behaves l i k e — as |x| °°, z i , . i n - l l x l and t h i s implies that Vq i s square-summable over Q, f o r every t e (Q,T). The fact that Vq, and thus u , i s square-summable over fix (0,T) i s then shown by obtaining a uniform estimate for || Vq(« ,t) | l L2(j3) i n 0<t<T. The proof of t h i s estimate i s based i n part on an i n e q u a l i t y of Payne and Weinberger [8]. From the fac t that u e L2(£2 x (0,T)), our r e s u l t that u i s square-summable i n x (0,T) i s e a s i l y proved. D e t a i l s are contained i n Section 3. Under the hypotheses of L data, i t i s known that -5-problem (1) - (4) possesses a unique f i n i t e energy ( i . e . L (fi) ) s o l u t i o n i n various classes. Solutions of a class introduced by Ladyzhenskaya [6], which we c a l l class J solutions, belong to the function space J defined as the completion i n L 2 norm ||•|| of the set of a l l smooth solenoidal R n-valued functions with compact supports. She has studied the class J solutions i n d e t a i l only i n the case of a bounded s p a t i a l domain. We have extended the d e f i n i t i o n of class J solutions to cases of unbounded s p a t i a l domains. Another class of f i n i t e energy solutions studied i n [6] i s characterized c h i e f l y by membership i n the space defined as the completion i n norm ^ n (||•|| 2 + | | V ' | | 2 ) 2 of the set of a l l smooth solenoidal'Ri-valued functions with compact supports. I t i s natural to ask whether the class H s o l u t i o n i s i d e n t i c a l with the solutions of these f i n i t e o energy classes. For an e x t e r i o r domain i n R 2 the answer i s negative because, as mentioned above, there are forces for which the c l a s s H o s o l u t i o n f a i l s to be square-summable. For an e x t e r i o r domain i n R°, however, our demonstration that the c l a s s H q s o l u t i o n i s square-summable, combined with a recent c h a r a c t e r i z a t i o n of the space by Heywood [5], proves that the class H q s o l u t i o n and these several f i n i t e energy solutions are n e c e s s a r i l y a l l i d e n t i c a l . The proof i s i n Section 4. It remains unknown, for a r b i t r a r y unbounded domain i n R 3, whether these s o l u t i o n classes are a l l i d e n t i c a l f o r smooth L 2 data. -6 -Heywood has shown i n [2] that solutions of the e x t e r i o r stationary problem occur as l i m i t s of nonstationary class solutions. In p a r t i c u l a r , he established the convergence to steady state of the class H q s o l u t i o n of the problem which models the following ph y s i c a l experiment. An object i s i n i t i a l l y at rest i n a three dimensional space f i l l e d with a Stokesian f l u i d . It i s then smoothly accelerated u n t i l a given v e l o c i t y i s attained, a f t e r which i t i s kept i n motion with the same v e l o c i t y . Because of the uniqueness of s o l u t i o n classes of (1) - (4), we conclude i n Section 5 that the f i n i t e energy s o l u t i o n of t h i s nonstationary flow problem converges as t -> »' to the s o l u t i o n of the e x t e r i o r stationary problem. In order to define the class H q s o l u t i o n f or problems with general nonhomogeneous boundary values i n such a way as to preserve the uniqueness theorem, i t i s necessary to define a class of admissible extensions of the i n i t i a l and boundary i ; values i n t o the space-time region, and to prove that the diffe r e n c e betweenaany two such extensions belongs to the space H . The study of c l a s s H solutions i n [2] was l i m i t e d to o J o problems with constant prescribed boundary values; the method :'..< i n [2] of defining extensions of constant boundary values does not extend to more general boundary values. Our r e s u l t that the cla s s H s o l u t i o n i s i d e n t i c a l with the c l a s s J i s o l u t i o n has o 1 enabled us to define a reasonable class of such extensions for -7-much more general boundary values, at le a s t for the case of an exterior domain i n R 3. -8-2. Preliminaries By an ex t e r i o r domain fi i n R n (n >_ 2) , we mean an open set which contains a complete neighborhood of i n f i n i t y {xeR11: | x| >^ R > 0}. Throughout t h i s t h e s i s , we assume fi to be an exterior domain i n R n with n > 2 unless otherwise stated. The space-time region fi x (e,T) i s denoted by Q , and simply E , 1 by i f e = 0. The closure, boundary, and complement of a set — c S are denoted by S, 9S, and S respectively. The i n t e r i o r of a sphere of radius r centered at the o r i g i n 0 i s denoted by S ; i t s boundary i s denoted by SS^ and i t s exterior by E^. A l l functions i n th i s thesis are either R or R n -valued. We use l e t t e r s h, k, p, q, a, 3 , etc., to denote R -valued functions and l e t t e r s u, v, w, <f>, IJ J , £ , n , f, g, etc., to denote R n - valued functions. We w i l l use the same symbol to denote a function space of either R - valued or R n - valued functions. The d i s t i n c t i o n w i l l be clear from context. Thus, L (fi) denotes the space of a l l R - valued or R - valued functions which are square-summable i n fi, X L 2 (fi) denotes the space of a l l loc R - valued or R n - valued functions which are square-summable i n 00 compact subsets of fi, and C n(fi) denotes the space of a l l smooth -9-R - valued or Rn - valued functions with compact supports in Q. Spaces L Z(Q T), L^ o c(Q 0 O)> a n d c Q ( f i x [0,T]), etc., over space-time regions are defined similarly. We employ the usual notation of vector analysis and in addition the following notation: (u,v) = J^u-v dx = J J - u ^ dx, ||u|| = (u,u)z; ; i=l n n 8u. 3v., ^ ( v u , w ) = / o V u : V v d x = !&l J s F 1 ^ ^ ' I W u | | = ( v u , v u ) 2 i=lij-l j j (u,v) 1 = (u,v) + (Vu,Vv) , J j u j I x = (u,u) 2 ; (u,v) = / (u,v) dt , I M L = (u,u) ; (u.v)^ = il (u.vjj dt , I M | = ( ( u , u ) 1 ) Q J t ) 1 . £>•>• e>L' We w i l l also use extensively the following definitions of R n - valued function spaces: { <|>(x) : <(> e C~(fi) and V«<j> = 0 } ; Completion of D(fi) in the norm ||•|| ; Completion of D(f2) i n the norm | |v»| | ; Completion of in the norm | | • | | • D(fi) J(fi) J (ft) = o Let D(Q T) = { <f>(x,t) : <f> e CQ(ft x [0,T]) and V'<|> = 0 }. The -10-spaces J ( Q T ) , J 0 ( Q T ) , and ^ ( 0 ^ ) are defined to be the completions of D(Q ) i n the norms ||'||n » Il v*l1 n ' a n d I I'Mi n r e s p e c t i v e l y . Below, we state without proofs three well-known lemmas which w i l l be used throughout the l a t t e r sections. We r e f e r the reader to [6,pp.16-31] for proofs of Lemmas 1 and 2. Lemma 3 follows from Lemma 2. Lemma 1. Let fi be any domain i n R n with n > 2. Then the orthogonal complement of J(fi) i n L 2 ( f i ) i s G(fi) = { <j>(x) : <j) E L 2 ( f i ) and <j) = Vp for some p E L 2 q c (fi) }. That i s , every function u i n L 2(fi) can be expressed uniquely as u = v + Vp, where v e J( f i ) , Vp;e G(fi), and J v V p dx = 0. Lemma 2. Suppose fi i s a domain i n R n with n > 2. Then fo r any <J) E J q (fi) , <Kx): d x i fcr) L <v*>2 dx • 2 - 4i-2 ..x-y . where y i s an a r b i t r a r y point i n Rn. If fi C R 2 i s a domain such that fiC contains a disk { x : |x| <_ r ^ }, then every <j> e J_(fi) s a t i s f i e s o k2 fi | x | 2 l o g 2 | x / r o | " c (V(f))2 dx fi -11-Lemma 3. Suppose fi i s eit h e r a domain i n R with n > 2 or a two-dimensional domain whose complement contains a disk {x : |x| _< If fi' i s a bounded subset of fi there e x i s t s a constant C , such that j , §2 dx <^ C 2, | |vcf)| | 2 holds f o r a l l If JWJ !W! <f) e J Q ( f i ) . The following two lemmas w i l l be needed i n Section 3. Lemma 5 i s due to Payne and Weinberger [8]. We include i t s proof here because i t plays a fundamental r o l e i n our work. Lemma 4. Let fi be a-domain .asiin -Lemma 3".' Suppose eith e r u e J(fi) or u e J Q ( f i ) . Then u-Vh dx = 0 holds f o r a l l OO . . h e C Q ( f i ) . Proof. Suppose u e J(fi). I t i s evident from Lemma 1 that J U'Vh dx = 0 f o r a l l h e C^(fi) because Vh e G(fi) whenever 00 . v h e C D ( f i ) . Suppose u e J D ( f i ) . The i n t e g r a l u«Vh dx with 00 -h e C o(fi) e x i s t s since u e ^ | o c ( ^ ) by Lemma 3. For any e < 0, there e x i s t s <f> e D(fi) such that | |vu - Vd> | | < e. Thus | |V'u| | = | |v-u - V - dp [ | < C| |Vu - V<)>| | < Ce.-. f o r some C > 0 and th i s implies that V-u = 0 i n fi since e i s a r b i t r a r y . Hence we have, upon an int e g r a t i o n by parts, u«Vh dx = -/^(V'u)h dx = 0 CO for any h e G Q ( f i ) . Lemma 5. Let E be the ext e r i o r of a sphere S of -12 -radlus R centered at 0 In R n(n > 2). Suppose that <j> is_continuously d i f f e r e n t i a b l e i n E and that V cf> e.L 2(E ). Then there i s a constant function d> such that o ^ / 9 S I* " dCT <• /E l V * | 2 d x ' R R where da denotes the area of a surface element. PrOof. We consider the s p h e r i c a l coordinates x = (r,0) where r = |x| and 9 stands for the angular coordinates ( 0 ^ , , . . . , 9 ]_) Let V <|> denote the vector obtained by m u l t i p l y i n g the angular component of V<j> by r . Then (V(j))2 = f-rM + —(VQ<}>)2,. For 3r^ _2 ° x o i l l u s t r a t i o n , l e t us consider the case of n = 3. We introduce s p h e r i c a l coordinates x^ = r cos 0^ s i n Q^, x 2 = r s ^ n s ^ - n ®2' ^ = r cos Let a, b, c be a right-handed orthogonal system f unit vectors at x = (r, 0^, Q^) i n the d i r e c t i o n s of the coordinates r, 0^, 0.-^. Then the gradient of $ may be expressed as and (V Q<j>) 2 = (||-) + L_j|£_J) # Now i n t e g r a t i n g (V<}>)2 over 2 s i n 2 &2 1 -13-the annular domain S_ = {x e R n : R < Ixl < p}, we obtain, in R,p 1 1 spherical coordinates, the expression R,p 8S1 ^ R / S N I 2 dx = / ,.T © 2 r ^ d r da + R / vn 3 / (Vfi<j>)2 da dr, where 3S^ is the surface of the unit sphere S^ . By an inequality of Wirtinger ( a sketch of proof is given in [l,pp. 273-274]) and by the Schwartz inequality, we have Us |V6*|2 da > (n-l) f / U|2 da - ^ ( , / * do) ^ d 1 H <• 9S1 "n 3S1. ±2> where UJ is the area of the spherical surface 8S_. Using the Schwartz n 1 inequality again, we have f giL)2 r n - l d r > {fM.dl i ^ J — i 8r R R fp dr ' R r -14-Thus, / |v<j)|2 dx >_ SR ,P as, K 3r J / i n - l (5) (n-l) rp R •R n-3 R r da + / <f>2 da - ~{j <j> da) 2"| 3S, n 8S, dr. Since both i n t e g r a l s on the r i g h t of (5) are nonnegative, each i s bounded uniformly i n p by / | Vc() | 2 dx. Now the f i r s t i n t e g r a l R i s > (n-2) R n" 2 / 9 S «>(p) - K R ) ) 2 da and so (<f>(p) - M R ) ) 2 da must converge to zero as R «>. Thus <|>, as a .function of 6 -• on 3S^, converges i n the L 2 norm over 3S^ as R ». It follows from the Schwartz in e q u a l i t y that B = li m / <j> (R) da e x i s t s . By adding R-*=° 1 B B to <j>, we can make the l i m i t of (<f>(R) ) da as R 0 0 to CO ob, CO n 1 n g be zero. Denoting — by <j> and replacing (j> i n (5) by $ = <j) - <f> , n we obtain / |v<}>|2 dx > (n-2) R n 2 / (<j,(p) - <j>(R))2 da + (6) R,p 3S„ (n - l ) / r 1 1 " 3 (/ | (j) | 2 da — —• * da) ) dr. R 3S, n 3S, Since the second i n t e g r a l on the ri g h t of (6) converges as p -> °°, and since we have shown that i t s integrand converges as r -> °°, -15-i t s Integrand must tend to zero as r ->• °°. This implies that l i m / (((>(r) - <j> ) 2 do = 0 since lim./._ (<|)(r) - <j> ) da = 0. obi O db~ O r-H» 1 r-»co 1 Neglecting the second term on the r i g h t of (6) and l e t t i n g p •+ 0 0, we obtain our desired r e s u l t . Next, we investigate the space H (fi), fi being any domain o i n R n with nr>_ 2. For reference, see Heywood [2]. We denote by K Q(fi) the set of a l l u e J (fi) such that (Vu,Vc|>) = (-f,(j)) f o r some f e D(fi) and a l l (j) e J Q ( f i ) . The map A' : K (fi) •> J(fi) i s defined by s e t t i n g A u = f. K d ( f i ) c i s givenf.theeinnerup.ro"ductni(Vuy.fSv^d+bjJ-Au.jdAy^cand. associated norm :(| |Su:|.s|,2;h|u|iAui|4.2').)2 • It i s known that A i s closa b l e . The space H (fi) i s defined to be the completion of K (fi) i n the norm ( | [ v « I I 2 + I I A' I I 2 ) ' 2 and i t can be i d e n t i f i e d as a subset of J Q ( f i ) . The map A can be extended to the completion H (fi) . The spaces K (Q _) and H (Q m) are de f d h e d l s i m i l a r l y by s u b s t i t u t i n g o e,T o e,T Q for fi, ( • , ' ) n for (•,•), and .|.| • I L for | | • | | i n the E ' 4e,T ge,T corresponding d e f i n i t i o n s of the spaces over fi. The next two lemmas may also be found i n [2] and they w i l l be referred to i n our laterrsections.? 0 -16-Lemma 6. The equation (ViJj,V<j>) = (^ ,-A<|>) holds under eit h e r one of the following conditions: ( i ) <f> e K Q(ft) and e J Q ( f t ) , or ( i i ) <j) e H Q(ft) and $ e J Q(ft) fl L 2 ( f t ) . Lemma 7. I f u e H q ( f t ) , then u has second order derivatives Ux.x. E L i o c ^ ^ a n d A u = A u + V p f o r s o m e p e L l o c ^ W l t h V p e L l o c ^ F i n a l l y i n t h i s section, we state a r e s u l t f o r harmonic functions which i s important i n the work of Section 3. We re f e r the reader to Poincar£ [10], du P l e s s i s [9], and Hochstadt [12] for proofs. Lemma 8. Suppose q i s harmonic i n an open set containing the closure of an annular domain S = {x e R n : p < Ixl < R}. If p ,R n > 3, q has a se r i e s expansion i n S i n sph e r i c a l harmonics of P »R the form. °° N(n,k) „ 0> Q W - I I ( c k m r + c k „, ') Y (5) . k=0 m=l ' H e r e ? = jxT' Ck,m a n d Ck,m a r e instants, N ^ k ) ^ ^ 2 - k+n-3 k-1 for k > 1 and N(n,0) = 1, and Y i s a spher i c a l harmonic — k,m -17-function of order k. The spherical harmonics satisfy the orthonormality conditions where 6, . denotes the Kr.onec'kercidelta which.lisc.tequal t o - l . k,J - - -i f k = j and equal to 0 i f k ^ j . If n = 2, q has an expansion in S in spherical harmonics P >K of the form oo ^ (7)' q(r,6) = q Q + q Qlogr + I [ ( ^ r ^ + a k r k ) cos k9 + k=l ( b k r ~ k + b f cr k) sin k9] where a, ,b, ,a, ,b, ,q ,and q are a l l constants. -18-3. Square Summability of Class H Solutions Let ft be a domain i n R n with n >_ 2. Suppose f e L 2 (Q ) and a e J (ft). By a class H so l u t i o n of loc <» o v o equations (1) - (4), we mean a function u which s a t i s f i e s the conditions: (8) u e H (Q_) Iclf .^pvl5] (and u c J- (0 0 T) for a l l 0 < e < T < 0 0 , (9) | |Vu(t) - Va| | ->• 0 as t -»• 0 +, and (10) u e L 2 (Q ) and there e x i s t s a sc a l a r function x.x. l o c » i J p e L 2 (Q ) with Vp E L 2 (Q ) so that u^ - Au = -Vp + f holds loc 0 0 l o c 0 0 t almost everywhere (a.e.) i n Q^. The existence and uniqueness of class H q solutions of (1) - (4) has been studied by Heywood [2]. I f the i n i t i a l data a e H Q(ft) and i f the force f e J (Q™) for a l l T > 0, the clas s H s o l u t i o n o T o u s a t i s f i e s the a p r i o r i i n e q u a l i t y / | | Vu^ _ | | 2 dt <_ | | Aa | | + JQ I|vf|| 2 dt for a l l T > 0; see Theorem 2 of [2]. On the other hand, i f a = 0 and i f f, f f c e L 2 ( Q T ) f or a l l T > 0 with f(x,0) = 0, u s a t i s f i e s the a p r i o r i i n e q u a l i t y | | V u t ( t ) | | 2 <_ ||f || 2 dx; see Theorem 3 of [2]. In view of these a p r i o r i i n e q u a l i t i e s , we -19-may suppose that u f c e J q (Q^ ,) f o r a l l T > 0 i f we assume that a e H Q(ft) and f = f 1 + f 2 where ^ e J Q ( Q T ) for a l l T > 0 and f 2 , f e L 2(Q ) f o r a l l T > 0. In f a c t , t h i s i s the case because, as demonstrated i n the proof of Theorem 3 of [2], u can be obtained as a sum u^ +-1*2 + u^, where ( i ) u^ i s a s o l u t i o n on [0,°°) subject to the force f ^ , and equal to a(x) at t = 0, ( i i ) U 2 i s a s o l u t i o n on [-1,°°), equal to zero at t = -1, and subject to the force f extended to be defined on [-1,°°) i n such a way that f 2 ( * , - l ) = 0 and f 2 , f 2 t e L 2 ( ^ x [-1,T]) for a l l T > -1, and ( i i i ) u 3 i s a s o l u t i o n on [0-,°°) subject to zero force, and equal to -t^CxjO) at t = 0. Now suppose that f e L 2(Q ) for a l l T > 0 and that there e x i s t s a class H q s o l u t i o n u of (1) - (4). According to Lemmas'-l and 7, the functions f and Au i n (10) may be decomposed as f = F + Vp 1, Au = Au + Vp 2 where F e J(Q ),"Au e J(Q T>, Vp^ e L 2 ( Q T ) f o r a l l T > 0, and where p^, P2 and VP2 belong to L? (Q ). The equation u - Au = -Vp + f i n (10) thus becomes loc 0 0 t (11) Vq = u t - g , where Vq = V(p 1 + p 2 - p) and g = Au + F e J(Q T) for a l l T > 0. Our main e f f o r t i s devoted to showing that Vq e L2(Q^,) f o r a l l T > 0 i f ft i s an e x t e r i o r domain i n R n with n > 2, from which -20-i t follows that u t and hence u belong to L2(Q,j,). To accomplish t h i s , we begin by showing that, f o r f i x e d t , the term q i n (11) i s harmonic i n fi and that Vq behaves l i k e — - — r - at i n f i n i t y as i x r 1 a function of the s p a t i a l v a r i a b l e s x. Lemma 9. Suppose fi i s ei t h e r a domain i n R n with n > 2 or a two-dimensional domain whose complement contains a disk {x : |x| <_ r Q } . Then the function q(x,t) i n equation (11) i s harmonic i n the s p a t i a l v a r i a b l e s x i n fi for almost a l l t > 0. If fi i s an exterior domain i n R n and i f n > 2, then,' f or each 1 tv- 0, Vq = 0 in-1 Nx i n a neighborhood of i n f i n i t y ; further Vq e L 2(fi) for each f i x e d t > 0. Proof. By Lemma 4, we have / ufc'Vh dx = 0 and g'Vh dx = 0 for a l l h e C^(fi) and almost alh't > • 0 since'ii (t) e'J( and g('t)'eJ(fi)'for-almost a l l t\> 0. . Bence for almost a l l t > 0, (12) / Vq'Vh dx = 0 00 holds f or a l l h e C Q ( f i ) . Given any bounded domain fi such that fi" i s compact and fi"cfi, we l e t fi' be a subdomain of fi such that fi' i s compact and fi"c fi'cz fi'cr fi. Let t, be an R - valued 00 function i n C (fi') such that c = 1 on fi". Put the function o h = (£ 2Aqp)p i n (12) where the subscript p denotes an averaging convolution q^(x) = / q(x-py)k(y) dy with kernel k e C Q(|xj < 1) s a t i s f y i n g / k(x) dx = 1. The support of the function h i s -21-contained i n ft' i f p i s small enough. Using the well-known 9 i d e n t i t i e s -—(<b ) = 3x. p 1 we get from (12) 34> 3x. l P and = ((f),^ ) for convolution, (13) / n ( ? A q p ) 2 dx = 0 through integrations by parts. But since A(?q ) = £Aq +2v?'Vq +(Ac)q , P P P P we have, i n v i r t u e of (13), the i n e q u a l i t y (14) l|A ( ? S )H a „ i C c l | v , p | | B . + C j | | q p | | B . where C* and C1^ are constants depending only on £. From the fa c t that q and Vq are l o c a l l y square-summable, the r i g h t side of (14) w i l l be bounded by some constant C depending only on £ and ft'. Thus, ||q ^ \ | < | | ( W ) _x_ || < | | A ( W ) | | , < C. Taking the l i m i t as p + 0 y i e l d s q e L 2 ( f t " ) . Therefore, X . x, 1 J q e L 2 (ft). Furthermore, i t i s c l e a r that f Vq «Vh dx = x.x. l o c Jft ^x. /CO Vq^h dx = 0 for a l l h e C (£>) since the derivatives of \li x. o X oo oo functions h e C Q(ft) also belong to C Q(ft). By what we have j u s t shown, q has l o c a l l y square-summable t h i r d d e r i v a t i v e s . An induction argument shows that u has l o c a l l y square-summable 00 d e r i v a t i v e s of a l l orders. Therefore, q e C (ft) by a w e l l -known theorem of Sobolev (see for example [7]). F i n a l l y , we -22-note that / (Aq)h dx = -/ Vq'Vh dx = 0 for a l l h E C™(fi); thus Aq = 0. Suppose now that fi i s an exterior domain in Rn with n > 2. Without loss of generality, we may assume that q is harmonic in fi for every time t > 0. Let E , R > 1, be such that E ^ d fi. K K According to Lemma 8, for a fixed t > 0, q as a function of the space variables has an expansion in E of the form, K , - 0 0 N(n,k) * n i i \ (15) I ( c k m r + C k , m r ) Yk,m < 5 ) k=0 m=l where E = - i — r and Y, is ap.spherical harmonic of order k. This |x| k,m series may be differentiated term by term because i t and i t s differentiated series are uniformly convergent in every compact subset! of E . Thus, differentiating (15) with respect to r gives °° N(n,k) , u n !*•- I I (kCv r ^ 1 - (n-2+kK r- ( n- 1 + k )) Y. (O . 8r , L n L, K k,m v k,m J k,m w k=0 m=l ' By the orthonormality of Y, (£) over the surface of the unit k,m sphere, we have R< x dx = oo N(n,k) 1 1 ' k=0 m=l ,*1 R / on\^. - ( n - l + k ) i 2 n-1 _ (n-2+k)C. r ') r dr k,m ^ where R.. > R. Now since e J (fi), Lemma 2 implies that 1 t o v ' ^ fi o r^ dx i s f i n i t e . This together with the fac t that J — dx < 0 0 implies E„ o R r z © 2 r Vq r that - 1 — 3 J — dx < 0 0. In p a r t i c u l a r , the i n t e g r a l J r dx i s R r' R r^ f i n i t e . It then follows that the c o e f f i c i e n t s C, must vanish for k,m k >_ 1. Thus the expansion for q becomes (16) °° N(n,k) , ° k=0 m=l K , m k,m where we denote CQ ^ by q Q. BecauseGfchewseriesv.in: (16),xisrunif ormly convergentoon the unit sphere'[and because:,'i ;r' V= , i x . >. 1 .-of or 2—n x e E i f * i s not hard 'to-see' fromd(16) that —q, T qiE.=^0(rj.-• ) . and thus v-~ r Vq = .0(r"!" n ) i n E sinee^.-.q \?is/harmonie. sHencel^Vq. e'>L'2 .(&,«>. On the other hand, since Vq = u f c - g and since u e J Q ( f i ) and g e J ( f i ) , - 2 4 -i t follows from Lemma 3 that Vq e L 2 i n ft OS*. Consequently, Vq e L 2(ft) and our lemma i s proved. I f ft i s an ex t e r i o r domain i n R n with n > 2, we w i l l replace q i n (11) by q - q^ arid w i l l denote q - q^ by q f o r s i m p l i c i t y . In order to show that Vq i s square-summable over the space-time region Q f o r a l l 0< e < T < ° ° , w e w i l l obtain an estimate for ||Vq(',t)|| uniformly i n [e,T]. More p r e c i s e l y , we w i l l prove Lemma 10. Suppose ft i s an exterior domain i n R n with n > 2. Then there i s a constant C > 0 such that, f o r a l l 0 < e < T < oo, (17) ||Vq|| 1 C ( ||Vut|| + IU1L ) • If u t e J Q ( Q T ) for a l l T > 0, then Vq e L 2 ( Q T ) for a l l T > 0. •Proof. Let S n = { x e R n : R1 < I x I < R } be an R^, &2 ^ a r b i t r a r y fixed annular domain such that S a ft. We have 2 shown i n Lemma 9 that q i s harmonic i n ft and thus Vq i s uniformly continuous i n S D . C l e a r l y , l ( r ) = / |Vq(x)| 2 da i s continuous R1' R2 3S • r i n [R^jR^] as a function of r and i s nonnegative. By the w e l l -known Mean Value Theorem there ex i s t s some R^ e [Rn ,R0] such that -25-(18) I(R o) as. R vq| 2d^ = ^ A - / 2 I(r) dr = - ± — / V R1 S. V R1 R. Vq 2dx. R1' R2 Observe that for any R > R (19) / Vq'Vq dx = / qVq-v da + / qVq'V da; R ,R o as R as R v being the unit outward normal vector. This identity i s obtained through an integration by parts and by using the fact that Aq = 0. The second integral on the right of (19) tends to zero as R -> °° since q behaves like ^ as r ^ °°. The f i r s t integral on the r right of (19) can beeestimatedgbysusingetsheySchwagtz-inequality andq(18) :> a. d' (18) i (20) |/ qVq>v da | < as. . o 1 U2-RJ v. / k l 2 da *iif | Vq | 2 da R as / k l 2 da 4 / I Vq I 2 dx R R-^ In the last integral on the right of (20), we use (11), Minkowski's inequality and Lemma 3 step by step to get (21) / |Vq|2 d x K < C i / J V u J 2 dx ^ + / . n l g | dx R1' R2 -26-where i s some constant depending only on and R^ . Applying Lemma 5 and inequality (21) to the integrals on the right of (20), one ^ obtains j (22) |/ qvq-vdaj 1 3S R V R1 R n-2 /|v q|2dx%C. (/|VuJ 2dx)^ + Vo J I x o z (/!g|2dx)-1 K||Vq||(||Vut|| + ||g||) where K is a suitable constant independent of time t. Thus, letting R -»- °° in (19) and applying inequality (22) gives (23) / |Vq|2 d x < K||Vq||(||Vu || + ||g||) ER o On the other hand, i t is evident from (11) and Lemma 3 that (24) / |Vq|2 dx < L||Vq||(||Vu.|| + ||g||) R O where L is some constant independent of t. We can now combine (23) with (24) to obtain ||Vq||fi < C(||Vufc|| + ||g|| ), where C = K + L. Our assertion follows. -27-Now i f u e J (Qm) for a l l T > 0, then Vq e L2(Q_) t o T . 1 for a l l T > 0. It follows from (11) that u e L 2(Q T) for a l l T > 0. This enables us to show that u possesses a f i n i t e L 2 norm over Q^, for a l l T > 0 and that u assumes the i n i t i a l data a in a L 2 sense by the same arguments as those in Lemma 3 of Heywood [4]. We give the proof here. Theorem 1. Suppose that ft is an exterior domain in R N with n > 2 and that u is the class H q solution of equations (1) - (4) i n which the i n i t i a l data a e H (ft) fl L 2 (ft) and the force f = f±+ f 2 where f± e J q ( Q t ) fl L 2 (QT) and f 2 , f 2 t e L 2 (QT) for a l l T > 0. Then u is square-summable over the space-time region Q and lim ||u(',t) - a|| = 0. €->0 Proof. Under the above hypotheses on the prescribed data, u e J(Q_,) for a l l T > 0 (see the remark at the beginning t o 1 of this section). We see readily from Lemma 10 that u e L 2 (Q^ ,) for a l l T > 0. It i s enough to show ||u(*,t)-aj | 2 <_ t||ut|I£2(Q ) for a l l 0 < t < T. According to the definition of class H solutions 0 o and Lemma 3, we have u e L 2(ft' x [0,T)) and lim ||u(*,t)-a|| 0, = 0 t-K) for every bounded subset ft' of ft. Hence for almost every x e ft there holds u(x,t) - a(x) = u ( X , T ) dx. Now, using the inequality ab < | ( a 2 + b 2 ) , -28-(u(x,t) - a ( x ) ) 2 = u t(x,x) u t(x,a) da dr < \ fi Jl [ u 2 ( x , T ) + u 2 ( x > a ) ] , d a v j d T = II II U 2 ( X , T ) da dx; thus ||u(-,t) - a|| 2 < t | | u t | | 2 2 ( Q y The above demonstration shows that the class H s o l u t i o n o of equations (1) - (4) i n an e x t e r i o r domain ficzRn(n > 2) does i n fact s a t i s f y the boundary condition at i n f i n i t y i n a L 2 generalized sense. I f ficrR2, however, Heywood [3] showed that the boundary condition at i n f i n i t y i s not s a t i s f i e d by the unique class H Q s o l u t i o n f o r some forces. The rest of t h i s section i s devoted to proving that the behavior at i n f i n i t y of the cl a s s H q s o l u t i o n i n the case of a two-dimensional e x t e r i o r domain i s r e s t r i c t e d i n such a way that i t s time d e r i v a t i v e n e c e s s a r i l y tends to a d e f i n i t e l i m i t as |x| •+ 0 0 i n a L 2 generalized sense for each fixed time. Lemma 11. Let fi be an exterior domain i n R n, n > 2. Suppose that w e C ^ f i ) , V.w = 0 and that there e x i s t s a sequence of functions {(j)^ } i n D(fi) such that | | ^ - w | | , 0 as k -»- » for every bounded subset fi' of fi. Then f o r any s p h e r i c a l surface 3S of radius p enclosing fiC,, (v>0S: hrise a / c i r c l e d if=n/= 2)I,.'we have P P 3 £ r / w v da = 0, where v i s the unit outward normal and da denotes 8S P -29-the area of a surface element (if n = 2, da denotes the length of a line element). Proof. Let the annular domain {x e Rn : p < |x| < p + 1} be denoted by A. If n = 2, we introduce polar coordinates x^ = r cos 0, x^ = r sin 0. The radial and angular components of w, wr and w»,are related to the''Cartesian components by w, = w cos 0 - w sin 6, 0' 1 r FI w„ = w sin 0 + w.cos0 and i t follows that Iwl 2 = |w I 2 + w„ For the case of n > 2, we introduce spherical coordinates (r,9 1,...,0 1) 1 n-l and we can check easily that |w|2 21 | w r| 2« Thus, by hypothesis, one can find a sequence {$ } in D(fi) such that !!(<}>,) - w | | A 0 as k •+ °°; therefore lim /»(<)>,) dx = /. w dx by the Schwartz - A iC. TT A IT k-*» inequality; Furthermore, for each k = 1, 2, — , <j>k i s divergence free and is compactly supported, thus the integral j (<J> ) da = 0 3S _/ k r p+a by the divergence theorem, where S is a sphere (a disk,, i f n = 2) centered at the origin of radius p + a, a > 0. Hence j (<Jv) dx = A K. TC j j (<$> ) dada = 0 and i t follows that / w dx = 0. But u d o , K r A r p+a w e C 1(fi) and V'W = 0, one has, again by the divergence theorem, j w da = / w da for any a > 0. Thus, /. w dx = 3S r 9S A r A r p p+a f i w da da = /" „ w da and our result follows readily from J0 ' 3S , r J 3S r 1 p+a p the above arguments. -30-Lemma 12. Let ft be a two-dimensional exterior domain whose complement ft contains S for some r > 0 and let t > 0 be r o o fixed. Then, in a neighborhood of i n f i n i t y , the function q(',t) in equation (11) can be expressed as a series in spherical harmonics which has the form -k ( 2 5) q = q Q + (a 1cose+b 1sine)r + £ (a^oske+b^inke) r k=l where q Q, a^, b^, a^, b^, k = 1, 2, are a l l constants. Proof. With u a class H q solution of (1) - ( 4 ) , i t has been shown in Lemma 9 that, for fixed t > 0, the function q(«,t) in equation (11) is harmonic in ft. Thus, in a neighborhood of i n f i n i t y , say E cn ft, q has a series expansion of the form P q(r,e) = q Q + q ologr + £ [(a f cr k+a f cr k)coske+(b kr k+b kr k)sinke] k 1 which may be differentiated term by term. We w i l l now show that the coefficients a^ and b^ vanish for a l l k > 1. It is clear that (IS2 rS .«5 r 2 l o g 2 ( r / r ) ^ dx =2d n r 3 l o g 2 ( r / r ) dr i-R ^ (-k a^r ^ + k a^r"* J") k=l J -k-1 , , ~ k-1,2 r l o g 2 ( r / r Q ) dr + -31-k=l fR (-k b k r k 1 + k b k r k ^ r l o g 2 ( r / r Q ) dr Now slncei:p^>; r;i ^ i t / ' i s ea'syptb see that,the integral.. ( — ) 2 — — dx i s f i n i t e i n v i r t u e of [ Vq | 2 (^ -) j.... J E r 2 l o g 2 ( r / r ) d r p 6 o' Vq = u f c - g, and Lemma 2. Thus, a k and b k must be zero f o r a l l k > 1 and the expansion f o r q i s reduced to q(r,9) = q Q + q Q l o g r + ( a ^ o s e + b ^ i n B ^ + I (a kcosfo+b ksinke) k=l where the c o e f f i c i e n t q Q i s given by -|— 7 ^ p d e - h -TT da 9v 9S P oo It remains to show that -qi = 0. Evidently Vq e C (fi) and V'Vq = 0. We w i l l show that there e x i s t s a sequence ii>^} i n D(fi) such that | |<j>. - Vq | | n, 0 f o r every bounded subset fi' of fi. Since u e J (fi) and g e J( f i ) , there are sequences {?v^ and {n } i n D(fi) such that | |V£, .- Vu | | -> 0 and | |n - g | L -»- 0 as k •> ». Because of Lemma 3, | | £, - u | | , 0 as k «>; thus | j — rj, — Vq j | T 0 as k -> °° since Vq = u - g. Hence the functions <j>k = £ k - 1j,> k = 1> 2, form the desired sequence. In view of Lemma 11, we have thus q< = ~ J " Vq«vda = •_L-/. _ da = 0. o 2TT 9S 2TT 9S 3r P P This completes the proof. -32-From Lemma 12, we see that Vq = WQ+V( £ (a^cosk0+b^sink6)r k) k=l where Wq = (a^,b^). Since the second term on the r i g h t behaves l i k e as r -> °°, the i n t e g r a l / j Vq - w j 2 dx i s convergent. On r 2 x|>p the other hand, i t i s c l e a r that / |vq - w | 2 dx i s f i n i t e . n'fKx: |x|<p} ° Thus, / | Vq - w | 2 dx < 0 0 which i n turn implies that J" Ju - w 12dx < °°. o \i O O u t O We have proved Theorem 2. Let Q, be an e x t e r i o r domain i n R 2 such that c 0, contains S for some r > 0. Suppose u i s a cla s s H s o l u t i o n r o r r o o of (1) - (4) with a e J Q(Q) and f e L 2(Q T) for a l l T >-0. Then for each "t: >" 0, ^ there„exists a-constant yector.^w :• :depending on; t such that V/ O k ( u t - W o ) 2 d x - K ~--33-4. Uniqueness of Solution Classes If ft is an exterior domain in R N with n > 2, i t was shown in Section 3 that the unique class H q solution of the nonstationary Stokes problem (1) - (4) is square-summable, i f the prescribed data are, despite the fact that the class H q theory is developed without using energy estimates. In this section, we show further that, i f ft is an exterior domain in R 3, the class H o solution is identical with-.the solutions of various f i n i t e energy classes. We f i r s t consider a class of f i n i t e energy solutions treated by Ladyzhenskaya in [6,pp.81-104]. To define such a class of solutions for (1) - (4) in the case of an unbounded spatial domain ft C3-RN (n _> 2) , i t i s necessary to introduce an auxiliary linear operator & relat-edctoitheffdlfowihg-Mriearjstationary problem:; is-^ra:;<• ed to the rjxmowing Muear s i ^ i o r . . ; • (26) Au - u (27) V.u (28) u = (29) u -y Vp + f 0 0 0 in ft in ft on 9ft as |x| -> a> -34-Given f e L 2 ( f t ) ; we say that u i s a generalized s o l u t i o n of equations (26) - (29) i f and only i f u e J-^ft) and / (Vu:V<|>-hi*<|>+f'<|>)dx = holds for a l l <f> e J^(ft). I t i s not hard to see that f*(j> dx defines a bounded l i n e a r f u n c t i o n a l on all«.$ e J (ft) . An a p p l i c a t i o n of the Riesz representation theorem (see for example Taylor [11,p.245]) proves the existence of a unique generalized s o l u t i o n for (26) - (29). Now, we define the operator A^ as follows. Let V denote the set of a l l generalized solutions of (26) - (29) corresponding to f's which belong to J(ft). Define a map A : V + J(ft) by s e t t i n g A^u = f where u e V i s the s o l u t i o n of (26) - (29) corresponding to f e J(ft). It i s c l e a r that A^ i s well defined, one-one, and 'Pc:J^(ft). Moreover, A^ i s a closed operator on V. In f a c t , i f {u^} i s a sequence i n V such that u. tends to u i n J,(ft) and that A,u. tends J 1 1 J to f i n L 2(ft) as j -> °°, then f o r each j = 1, 2, ... the equation / (Vu. :VV<f> + u.-cJO dx + / A u. • <j> dx = 0 holds for a l l 4> e J (ft) . ft J J S' 1 j 1 By taking the l i m i t s as j 0 0, one obtains (Vu : V<J) + u*<|> + f •((>) dx =' for a l l c|) e J-^(ft). This implies that u e V and that A^u = f. Set P = {• ?(x,t) : 5(x,t) = <j>(x) + T|»(X,T) dx, where <)>, ijj(',t) eft f o r every t and where i|i, A^,and Vip depend continuously on t as elements i n L 2(ft) } . I t i s not hard to see that i s a subset of J 1 ( Q T ) for a l l T > 0. Given a e L 2(ft) and f e L 2 ( Q T ) f o r a l l T > 0; we c a l l u a c l a s s J s o l u t i o n of equations (1) - (4) i f , -35-for a l l T > 0, u e J(Q ) and v = ue t s a t i s f i e s the i d e n t i t y (30) / v>((f> + A cf>) dx dt + / fe t-<S> dx dt + L • (x,0>a(x) dx = 0 T T fo r a l l cf> e with <j>(',T) = 0. This i d e n t i t y can be obtained formally by s u b s t i t u t i n g v = ue t i n equation (1), mu l t i p l y i n g the r e s u l t i n g equation by <))(x,t) e with <K'»T) = 0, i n t e g r a t i n g over Q^, and carrying out integrations by parts with respect to t and x. Whether the cla s s J solutions are unique i s not r e a d i l y seen from the above d e f i n i t i o n . The uniqueness proof below follows the same idea as that f o r the case of a bounded domain presented i n [6]. Proposition 1. Let a e L 2(fi) and f e L 2 ( Q T ) for a l l T > 0. Equations (1) - (4) have at most one class J s o l u t i o n . Proof. Suppose that u^^ and u 2 are two cla s s J solutions of (1) - (4). Let w = u^e t - u 2e t . It follows from the d e f i n i t i o n of class J solutions that w e J(Q T) for a l l T > 0 and that (31) /Q $K<l>t + A.,^ ) dx dt = 0 for a l l <f> e V± with <$>(',T) = 0. We assert that i f g(x,t) e V and i f g(x,x) dx e V for every t, then -36-(32) A 1 g(x,x) dx = A L G ( X , T ) dx . Indeed for any n e D(fi), / QA 1/^g(x,x)dx«n(x)dx = -/f2(v/^g(x,x)dx:Vn(x)+/^g(x,x)dx'n(x))dx = /^/^A 1g(x,x)dx'Ti(x)dx , and our assertion follows. Now set \b = f/ t w(x,x) dx) . For each 1 w o t > 0, <jj(x,t) e V since ^ w(x,x) dx e J(fi). So does A_"*"w(x,x) dx ' o o 1 belong to V. It is then seen from equation (32) that / o ^ w(x,x) dx = ^ jQ w(x,x) dx = ^(x,t) Hence ^ = A^w and A^iJ^) = w = ( A ^ ^ i ) t . Substituting <j) (x,t)=/^(x, x)dx into equation (31), we get / Q A1iJ;t.(iJH-/^A1^(x,x)dx)dxdt = /Q A 1^ t-#xdt+/ Q A1i|;t./jA1^(x,x)dxdtdx = 0 . Applying the definition of A^ to the f i r s t integral on the right and integrating by parts with respect to t in the second integral on the right, one'obtaihs's - 3 7 -0 = -/ (V^tVi/rf-^.^dxdt + / n'(A 1ip./jA 1-i ( d T)|^dx-/ n/^(A 1^) 2dtdx = -I(||vMx,T)|| 2-: + ||*(x,T)|| 2) - / J l l v U J d t . I t follows that A ij)(x,t) = 0 f o r a l l t e (0,T) and f o r a l l x e fi; t h i s In turn Implies that w = 0. Our proof i s complete. Let fi be an e x t e r i o r domain i n R 3. Suppose that the i n i t i a l data a belongs to H o ( f i ) f l L 2 ( f i ) and that the prescribed force f equals f± + f 2 where f± e J Q(Q T) D L 2(Q t) and e L2(QT> for a l l T > 0. We w i l l show that the class H q s o l u t i o n of (1) - ( 4 ) , whose existence and uniqueness are guaranteed by our hypotheses, i s i d e n t i c a l with the class J s o l u t i o n of (1) - ( 4 ) . Instead of proving d i r e c t l y that these solutions are i d e n t i c a l , i t i s convenient to introduce another c l a s s of f i n i t e energy solutions for (1) - (4) which-iwe c a l l c l a s s s o l u t i o n s ^ tha'tiare> c h i e f l y characterized by membership i n the function space (see [4] or [ 6 ] ) . We w i l l prove that every cl a s s J- s o l u t i o n i s a c l a s s J s o l u t i o n , and that every cl a s s H s o l u t i o n 1 J o i s a c l a s s s o l u t i o n under the above-mentioned hypotheses. The l a t t e r r e s u l t i s v e r i f i e d byreomblningcouriresu'l.fe on the square-summability of a class H s o l u t i o n with a recent c h a r a c t e r i z a t i o n o of the function space J 1 obtained by Heywood [5]. -38-Let ft be any domain i n R with n _> 2. Given a £ L (ft) and f e L 2(Q^,)for a l l T > 0. We say that u i s a class s o l u t i o n of equations (1) - (4) i f and only i f for a l l T > 0, the following conditions are s a t i s f i e d : (33) u £ ^ ( Q j ) and u t e J(Q T) (34) | |u(«,t) - a| | -> 0 as t + 0 + (35) u e L 2 (Q ) and u - Au = -Vp + f holds a.e. for x.x. l o c x°° t r i 3 some scalar function p e L 2 (Q ) with Vp e L 2 (Q ). f loc 0 0 loc 0 0 Condition (35) holds i f and only i f (35)' / (u -<j> + Vu : Vtj) - f-<f>) dx dt = 0 Q T holds f o r a l l <f> e J]_(QT) and a l l T > 0. For i f (35)' holds, then u has second order • de r i v a t i v e s u E L (<Q<» (see f o r example x.x. l o c w » i J Lemma 3 of [5]) and we get from (35)', through an i n t e g r a t i o n by parts, J (u - Au - f)*<j> dx dt = 0 for a l l <)> £ D(Q ). Thus (35) i s obtained. On the other hand, i t i s an easy matter to v e r i f y the reverse implication. If a £ J ^ f t ) and f £ L 2 ( Q T ) f or a l l T > 0, one can show that equations (1) - (4) possess a class J n s o l u t i o n (and thus a cla s s J -39-s o l u t i o n by the following proposition) by the method of Galerkin's approximation. D e t a i l s are omitted. I t i s also easy to prove d i r e c t l y that the class solutions of (1) - (4) are unique. Proposition 2. Let 0. be an unbounded domain i n R n, n > 2. Suppose that a e L 2 ( f t ) , that f e L 2(Q T) for a l l T > 0, and that u i s a class s o l u t i o n of (1) - (4). Then u i s also a c l a s s J s o l u t i o n of (1) - (4). Proof. For a l l T > 0, u e J(Q T) since u e J 1 ( Q T ) a n d since J^(Q^) cz J(Q^) . Substituting ve*" f o r u i n the equation of (35) , multi p l y i n g the r e s u l t i n g equation by a function cj> E D(Q^,), T > 0, and i n t e g r a t i n g over Q^ ,, we obtain / n ( v 1-* (l ) " (A v - v)«<f>) dx dt = / fe t«tj) dx dt Integrating by parts with respect to x y i e l d s (36) fn (vt'<j) + Vv:V<J> + vi))) dx dt = fn fe-t-c{> dx dt Identity (36) holds for a l l <j> e ^ I ^ T ^ s i n c e u ( Q r p i s dense i n J, (Q T) i n the norm | | • [ j , . In p a r t i c u l a r , i t holds for a l l (j) e Applying the i d e n t i t y (Vv,V<j)) = -(v,A^ <{>) and i n t e g r a t i n g by parts with respect to t i n (36), we obtain -40-for a l l i|) E ^ with cj>(*,T) = 0. The proposition i s proved. From now on, we assume that the s p a t i a l domain ft s a t i s f i e s the following condition: ft i s an ex t e r i o r domain i n R 3 which has an ex t e r i o r subdomain.D'with a class C 2 boundary 3Dc_ ft, such that (*) the region ft - D i s covered by nonintersecting normals to 9D (we assume that the normals do not i n t e r s e c t at points of 9ft-.-) . The following c h a r a c t e r i z a t i o n of the function spaces J^(ft) and J (ft) f ° r such 52 i s due to Heywood [5]. O. oo Proposition 3. If W^(ft) denotes the completion of C (ft) i n the norms ||*||^ > and i f j|(ft) denotes the c o l l e c t i o n of a l l <|> i n Wl(ft) for which v<f> = 0, then J ^ f t ) = J*(ft). I f WQ(ft) denotes the completion of C°°(ft) i n the norm ||v*ll and i f J*(ft) denotes the c o l l e c t i o n of a l l * i n W (ft) for which V-d> = 0, then J (ft) = J* (ft) . T O T n o Now suppose that a e H Q(ft)nL 2(ft) and that f = f + f where f± e J ^ ) n L 2 ( Q T ) and f y f y e L 2(Q T) for a l l T > 0. Let u be a class H q s o l u t i o n of equations (1) - (4) with these prescribed -41-data. We have proved i n Section 3 that u, u f c e L 2 ( Q T ) for a l l T > 0, thiis-u i s continuous i n . L 2 ( f i ) as a function of t a f t e r r e d e f i n i t i o n on a set of t measure zero. Since, for every t > 0, u(',t) e J (Si) Pi L 2 ( f i ) , o ° we have u(«,t) e W^(fi) by Lemma 2 of Heywood [4] which states that O! W Q(fi) fl.L 2(fl) c W2;($) ..if'fi i s an ex t e r i o r domain. Furthermore,V'u = 0 and u i s continuous i n the norm | | • | | ^ as a function of t. Hence, i t follows from Proposition 3 that u e J i ^ x ^ a ^ T > ^* T n e r e ~ maining conditions for u to be a class s o l u t i o n are e a s i l y v e r i f i e d . We have therefore proved Theorem 3. Let fi be an ex t e r i o r domain i n R3 described above. Suppose that a e H (fi) fl L 2 (fi) and that f = f, + f„ where o 1 2 fl £ Jo (V n l 2 ( Q T } a n d f 2 ' f 2 t e L 2 (V f o r a 1 1 T > °* 111611 t h e unique class H q s o l u t i o n of equations (1) - (4) i s a class s o l u t i o n and hence also a class J sol u t i o n of (1) - (4). -42-5. The Case of Nonhomogeneous Boundary Values In t h i s section, we assume that fi i s the e x t e r i o r of a f i n i t e object i n R 3 s a t i s f y i n g condition (*) (Section 4).and that there e x i s t s an i n e r t i a l reference frame i n which the f l u i d v e l o c i t i e s tend to zero f a r from the object. I f the object moves with a v e l o c i t y -b (t) r e l a t i v e to the i n e r t i a l frame, a f i c t i t i o u s J 00 force b ( X ) t(t) w i l l appear i n the equations of motions when written i n a coordinate frame attached to the object, and a f l u i d v e l o c i t y b o o(t) w i l l be imposed at i n f i n i t y r e l a t i v e to t h i s n o n i n e r t i a l frame. Thus, the equations f o r t h i s nonstationary Stokes flow are given by (36) (37) (38) (39) u - Au u(x,0) :,u(x,t) u(x,t) -> -Vp + b 0 a(x) b Q ( x , t ) b (t) oot i n Q ^00 i n Q CO (x,t) e 3fi x [0,oo) as lxl -* oo We w i l l define a cla s s H q s o l u t i o n for ( 3 6 ) -that the uniqueness theorem holds. (39) i n such a way We assume that b Q ( x , t ) E C 2(3ft x [0,oo)),that b (t) e C2[0,oo) and that the prescribed data a, b , and b oo O oo permit the boundary values to be extended into Q as a solenoidal 00 , function b(x,t) which s a t i s f i e s : (40) b e C 2 ( Q ), Ab t e x i s t s and Ab„ e L 2 ( Q R R ) for a l l T > 0, oo t t 1 (41) a number R > 0 e x i s t s such that b(x,t) = b (t) for a l l x 00 with|x| > R and a l l t e [0,oo), and (42) a - b(.,0) e J Q(ft) . We c a l l such extension b of the boundary values admissible. Because b e C 2 ( Q m ) and because Vb, Ab, Vb vanish f o r |x| > R, one finds r e a d i l y that Vb, Ab, Vb e L 2 ( f t x [0,T)) for a l l T > 0. We c a l l u a cla s s H q s o l u t i o n of (36) - (39) i f u = v + b where b i s an admissible extension of the boundary values into QOT as described above and v i s a cla s s H Q s o l u t i o n of equations (1) - (4) with i n i t i a l data - ja.(x)-b(x,0) f o r a l l x £ ft and force f = Ab - b,_ + b ^. That i s , v s a t i s f i e s the conditions t °°t (43) v E H (Q_), and v e J (Q J for a l l 0 < e < T < » o l t o e, l (44) | |Vv(-,t) - V(a(.) - b(-,0))| | 0 as t + 0 + (45) v E L 2 (Q ) and for some sca l a r function p E L 2 (Q ) X.X. IOC ooy ^ IOC oo' i 3 with Vp £ L 2 (Q ), there holds v. - Av = -Vp + Ab - b^ . + b . a.e. r 1 nr> c o " 1- r *• ~t --44-Since b e C 2(Q o t) and b(x,t) = b^(t) for a l l |x| > R and a l l t e [0,°°), i t i s easy to see that ? = -b + b s a t i s f i e s ?, C e .L2(Q_);r'for a l l T > 0. Further, the force f s a t i s f i e s t 1 ,j f, f e L 2(Q T) for a l l T > 0 because b s a t i s f i e s Abfc z L 2(Q T> for a l l T > 0. It then follows from Theorem 3 of [2] that there e x i s t s a c l a s s H q s o l u t i o n : :of equations (36) - (39). The following lemma, needed to prove the uniqueness theorem, provides a p a r t i a l c h a r a c t e r i z a t i o n of the space H o ( f i ) . o Lemma 13. I f E e W,>(fi) and V«£ = 0 (or equivalently, K e (fi) by Proposition 3) and i f £ e L 2 ( f t ) , then• £ e H (fi). 1 1 I f , for a l l T>0, b, b f c e J ^ ) , b x x , b f c x x e L 2 ( Q T ) , and i f i i i i b(x,0) E H (fi) fl L 2 ( f i ) , then b e H Q ( Q t ) for a l l T > 0. 1 Proof. Let a(t) be a smooth function on [0,°°) such that a(0) = 0 and a(t) = 1 f o r t >^ 1. We consider the function u(x,t) = a(t)5(x). C l e a r l y , ]|u(«,t)|| + 0 as t + 0 +. I t i s also c l e a r from our hypotheses that u and u belong to J^(Q T) f ° r a l l T > 0. Thus u i s a class s o l u t i o n of equations (1) - (4) with zero i n i t i a l data and force f E a'(t)g(x) - a(t)A£(x). Since the force f sat-i s f i e s the condition f, f e L 2(Q^) f o r a l l T > 0, i t follows from Theorem 3 that u i s also a class H solution of (1) - (4). This o implies that ct(t)£(x) e H D ( Q T ) f o r a 1 1 T > 0; thus £(x) e H (-fi) . -45-The second assertion follows r e a d i l y from Theorem 3 as well i f we.observe that b i s a class s o l u t i o n of (1) - (4) i n which the i n i t i a l value b(x,0) and force b^ _ - Ab s a t i s f y the assumptions of Theorem 3. Before we proceed to prove the uniquenes's theorem, we state a lemma (see Heywood [5, Lemma 11]) which w i l l be needed i n our proof. Lemma 14. Let ft be an a r b i t r a r y open set of R n. Suppose that u e C ( f t ) , that u = 0 on 9 f t ,that u has generalized f i r s t d e r i v -a t i v e s , and that the i n t e g r a l s J^u2dx and J (Vu) 2dx are f i n i t e . °1 Q Then u e W 2(ft). Theorem 4. Suppose the prescribed data a, b , b permit O o o an admissible extension b of the boundary values into Q . Then CO equations (36) - (39) can have at most one class H q s o l u t i o n . Proof. Let b and b be any two admissible extensions of the boundary values into Q . We f i r s t note that A(b-b) e L2(Q- ) o o t T for a l l T > 0. Since b - b e C 2(Q ) and since b - b and (b - b) o o t vanish i n a neighborhood of i n f i n i t y f o r a l l t e [ 0 , o o ) , we see that, f o r every t e [ 0 , o o ) , (b - b) (• ,t) , v(b - b) (• ,t) , A(b - b) (• ,t) belong to L 2 ( f i ) , and that b - b, V(b - b), (b - b) f c and V(b - b) f c a l l belong to L 2(ft x [0,T)) f o r a l l T > 0. By Lemma 14, b - b, (b - b)^ e W2(ft x [0,T)) f o r a l l T > 0 because they are continuous on -46-ft, equal to zero on 9ft x [0,°°), and are bounded in norm l l ' H ^ uniformly i n 0 < t < T. In virtue of Proposition 3, b — b and (b - b) f c belong to J^O-j^ ^ o r a ^ T > 0. Moreover, we have, by Lemma 14 and the f i r s t part of Lemma 13, (b - b)(»,t) e H (ft) for o every t >^ 0 and in particular (b - b) (•,()) e H Q(ft). Thus the second part of Lemma 13 implies that b - b e H o(Q T) for a l l T > 0. Now let u = v + b be a class H q solution of (36) - (39) where b i s an admissible extension of the boundary values into Q CO and where v satisfies conditions (43) - (45). Let b be any admissible extension of the boundary values and set v = u - b = v + b - b. We assert that v and b satisfy conditions (43) - (45). Indeed, we have shown in the previous paragraph that b - b e H (Q,p) and (b - b) e J (Qm) for a l l T > 0, and thus v E H ( C V) and t o 1 o T v E J (Q ) for a l l 0 < e < T < » . Moreover, i t i s easy to verify that ||w('$t) - Va + Vb (• ,0) | | •> 0 as t 0 + and that v - Av = -Vp + Ab - b^ + b holds a.e. for some scalar function t t °°t p eL 2 (Q ) with Vp e L 2 (Q ). This proves our assertion. Next IOC oo' r loc » suppose that u and u are two class H q solutions of (36) - (39) and that b is any admissible extension of the boundary values. By what we have just shown, v = u - b and v = u - b both are class H o solution of equations (1) - (4) with force b - b + Ab and cot t i n i t i a l value a - b (•,()). Uniqueness of solutions of equations (1) - (4) implies that v = v and thus u E u. - 4 7 -Suppose that the prescribed i n i t i a l and boundary values a, b Q , admit an extension b into s a t i s f y i n g (40), (41) such that a - b(«,0) e L 2 ( f i ) . We c a l l u a class s o l u t i o n of (36) -(39) i f u = v + b i n which v i s a class s o l u t i o n of (1) - (4) with force . b - b^ + Ab and i n i t i a l value a - b(.,0). I t can cot t be deduced from the proof of Theorem 4 that i f u - b e J^(Q^) f o r a l l T > 0 f o r some extension b then u - b e Jj_(Q T) for a l l T > 0 for every such extension b. Again, following the same method of proof as i n Theorem 4, one can show that the clas s solutions of (36) -(39) are unique. We have Theorem 5. If the prescribed i n i t i a l and boundary values a, b Q , b^ admit an extension b into s a t i s f y i n g (40), (41) such that a - b(*,0) e L 2 ( f i ) , then equations (36) - (39) have a unique cla s s s o l u t i o n . I f , i n addition, the i n i t i a l rvalue a s a t i s f i e s a:,e'b(-,0) Ti ,H.o(fi) G L 2(fi)' , then equations (36) - (39) have a unique cla s s H q s o l u t i o n and i t i s i d e n t i c a l with the unique cl a s s s o l u t i o n . Proof. The l a s t part of t h i s theorem follows from Theorem 3 whereas the remaining assertions are obvious. In [2, Theorems 4, 5 and 6] Heywood has studied the convergence of solutions of some nonstationary flow problems to the s o l u t i o n of a steady flow problem which describes the physical - 4 8 -s i t u a t i o n of a f l u i d occupying the e x t e r i o r of an object, adhering to the object's boundary, with i t s v e l o c i t y tending to a constant prescribed vector at i n f i n i t y . Similar to h i s method, we study the a t t a i n a b i l i t y as l i m i t s of nonstationary solutions of solutions of the steady flow problem i n fi i n fi for x e 9fi as |x| -> oo ( 4 6 ) ( 4 7 ) ( 4 8 ) Aw V«w w(x) w(x) = VP 0 = w o -> w i n which w and w are constant prescribed vectors. We f i r s t define O oo the generalized solutions .-of equations ( 4 6 ) - ( 4 8 ) . Let E, be a function defined i n fi' such that ( 4 9 ) E, i s smooth, solen o i d a l , equal to Wq i n a neighborhood of 9fi, and equal to* zwro i n a neighborhood of i n f i n i t y . . We c a l l w a generalized s o l u t i o n of ( 4 6 ) - ( 4 8 ) i f and only i f (i) w = w - E, e H Q(fi) f o r some E, with the properties j u s t described, and ( i i ) Aw = Vp holds a.e. for some scalar p e L 2 (fi) with Vp e L?-' (fi). loc l o c We claim that generalized solutions are unique. C l e a r l y the difference of any two functions which s a t i s f y ( 4 9 ) belongs to D ( f i ) . Thus, i f w^ and w„ are any two generalized solutions of ( 4 6 ) - ( 4 8 ) and i f E, i s any -49-function s a t i s f y i n g (49), then w^ - £ e H Q(ft) and -w^ - £ e H Q(ft). It follows that w, - w_ e H (ft) and A(wn - w.) = VP for some p e L 2 (ft) 1 z o 1 2 loc with Vp e L 2 (ft). Hence (v(w, - w0),Vd>) =0 for a l l d> e D(ft), and loc 1 / T T there follows w^ E by taking a l i m i t f or <j> = w^ - w2- Existence of a s o l u t i o n i s also e a s i l y shown; since A£ e D(ft), one can choose w e K (ft) such that Aw = -AE . o We now show that the generalized solutions of (46) - (48) can be obtained as l i m i t s as t + » of solutions of the nonstationary flow problem (36) - (39) i n which the i n i t i a l and boundary values are prescribed to s a t i s f y : (50) a number T > 0 e x i s t s such that, for a l l t > T , b (x,t) = w-v o 0 0 0 for a l l x e 9ft and b (t) = w , and CO 00 (51) the i n i t i a l and boundary values admit an admissible extension b defined i n Q . CO We f i r s t state a r e s u l t (Theorem 4 of [2]) which concerns with the behavior of class H solutions as t •-> » . o Proposition 4. If a e J (ft) and i f f = f, + f . + f . where * o 1 2 3 (I) f± e J Q ( Q T ) for a l l T > 0, ( i i ) f y e L 2 ( Q T ) f o r a l l T > 0, and ( i i i ) f„ = Vq f o r some q e L 2 (Q ) with Vq e L 2 (Q ), then the 3 l o c V n loc 00 c l a s s H s o l u t i o n of (1) - (4) converges to zero i n L 2 (ft) as t ->• °° o loc provided / q ||vf^|| dt and J ™ | | f 2 | | 2 dt are f i n i t e . (It i s worth noting that t h i s proposition i s v a l i d for any domain ftcRn(n >_ 2).) -50-Next, suppose that £ i s a function which s a t i s f i e s (49) and that b i s admissible extension of the boundary values b , b into Q . J Q CO. \ » It i s evident that b(x,t) = (1-a(t))bCx,t) + a(t)g(x) i s also an admissible extension of b , b into Q ; here a i s a twice O oo co continuously d i f f e r e n t i a b l e real-valued function defined for a l l t >^ 0 which i s equal to 0 for t <_ -T and equal to 1 for t > T + 1 . Now suppose u i s the c l a s s H s o l u t i o n of (36) -— o o (39). Setting v = u - b, we see that v and b s a t i s f y (43) -(45). I f w = w + £ i s the general'izedls.olution of 5(46-) (-'3(48) , then v - w i s a c l a s s H q s o l u t i o n of (1) - (4) with i n i t i a l value (a(x) - b(x,0)) - w(x) f o r a l l x e ft and force f E A (b - £) -b + b . We assert that f f u l f i l l s the hypotheses of Proposition t oot 6, from which i t follows that v(',t) -> w i n Ljr Q c(ft) as t oo . F i r s t , we note that A(b - £ ) , A(b - g) f c e L 2 ( Q T ) for a l l T > 0 and that / ||A(b - £ ) | | 2 dt < oo because b(x,t) = £ (x) for a l l x e Q and a l l t > T +1. Let £ = -b^ + b . From the expression — o t °°t f o r b, one r e a d i l y finds that £(x,t) i s equal to -b t(x,t) + b (&) i f t <_ T q, equal to ct'(t)b(x,t) - i (1-a ( t ) ) b t (x,t) -a'(t)£(x) i f T < t < T + 1 , and equal to 0 i f t > T + 1 . I t i s then c l e a r o o ' H — o that £, ? t e L 2 (Q T) for a l l T > 0 and that f | | C |'|2 dt < "because b_ = b ^ = 0 for a l l t > T +1. Thus our assertion i s proved, t eo t — o r Further-r, since b(x,t) = £(x) for a l l x e ft and a l l t > T + 1, we can — o deduce that | |u(* ,t) - w| | , .-> 0 as t -»- « f o r every bounded subset ft' of ft. We have therefore proved - 51 -Theorem 6. I f a, b Q , and are prescribed to s a t i s f y (50), (51), then equations (36) - (39) have a unique cl a s s H Q s o l u t i o n u and i t converges to the generalized s o l u t i o n w of (46) - (48) as t -s- co i n the sense that ||u(»,t) - w | | ^ t -> 0 for every bounded subset fi' of Q. Corollary. I f , i n addition to the hypotheses of Theorem 6, the i n i t i a l value a s a t i s f i e s also the condition a - b(»,0) e H o ( j j ) f l L 2 ( f i ) , then the nonstationary problem (36) -(39) possesses a unique class s o l u t i o n and i t converges i n L ^ ^ ( f i ) as t to to the generalized s o l u t i o n of the stationary problem (46) - (48). Proof. This C o r o l l a r y i s a d i r e c t consequence of Theorem 5 and Theorem 6. As a p a r t i c u l a r case, Theorem 6 and i t s c o r o l l a r y model the following p h y s i c a l experiment. Suppose an object i s i n i t i a l l y at r est i n a three dimensional space f i l l e d with a Stokesian f l u i d . It i s then smoothly accelerated u n t i l a given v e l o c i t y i s attained, a f t e r which i t i s kept i n motion with the same 5.velocity. Let the object's v e l o c i t y r e l a t i v e to the i n e r t i a l frame of i n i t i a l rest be -a(t)w , where w i s a constant vector and a i s a smooth function 00 oo defined on [0,°°) such that a(0) = 0 and a(t) = 1 f o r a l l s u f f i c i e n t l y -52-large t. Then equations (36) - (39) with a(x) = 0, b Q(x,t) = 0, and b (t) = a(t)w , describe the motion of the f l u i d in a reference oo oo' frame attached to the object. This nonstationary problem has a unique class solution,which is identical with the unique class H q solution, and i t converges in L^ o c(^) as t » to the solution of the stationary problem (46) - (48) with boundary values 0 on 9fi and w^ at i n f i n i t y . Indeed, the prescribed values a, b Q and b^ satisfy conditions (50) and (51) i f we construct an admissible extension as b(x;t) = a(t)£(x), where £ is smooth, solenoidal, equal to zero in a neighborhood of o f i , and equal to w^ in a neighborhood of i n f i n i t y . Moreover, we see that a - b( i,0) = 0. Thus our.assertion follows from the corollary of Theorem 6. • -53-Refererices Finn R. -, and Gilbarg D., Three-dimensional subsonic flows, and•asymptotic estimates-for. e l l i p t i c . p a r t i a l d i f f e r e n t i a l equations. lActa Math.,-98 v(1957), 265-296. Heywood J.G., On nonstationary Stokes flow past an obstacle, Indiana Univ. Math. J . , 24 (1974), 271-284. Heywood J.G., On some paradoxes concerning two-dimensional Stokes flow past an obstacle, Indiana Univ. Math. J . , 24 (1974), 443-450. Heywood J.G., On convergence to steady state of solutions of parabolic equations i n unbounded domains, J. D i f f e r e n t i a l Equations (To appear). Heywood J.G., On uniqueness questions i n the theory of viscous flow (To appear). Ladyzhenskaya O.A., The Mathematical Theory of Viscous Incompressible Flow, second e d i t i o n , Gordon and Breach, New York, 1969. Necas J . , Les methodes d i r e c t e s en thebfie des equations i e e l l i p t i q u e s . Masson et C , Editeurs, (1967), P a r i s . -54-[8] Payne L.E. and Weinberger H.F., Note on a lemma of Finn and Gilbarg. Acta Math. 98, 297-299 (1957). [9] Du P l e s s i s N., Ari Introduction to P o t e n t i a l Theory, Oliver & Boyd, Edingburgh, 1970. [10] Poincare H., Theorie du p o t e n t i a l Newtonien, P a r i s , 1899. [11] Taylor A.E., Introduction to Functional Analysis, Wiley, New York, 1958. [12] Hochstadt H., The functions of Mathematical Physics, Wiley-Interscience, New York,"1971.
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On square summability and uniqueness questions concerning nonstationary stokes flow in an exterior domain Ma, Chun-Ming 1975
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Title | On square summability and uniqueness questions concerning nonstationary stokes flow in an exterior domain |
Creator |
Ma, Chun-Ming |
Publisher | University of British Columbia |
Date Issued | 1975 |
Description | In this thesis we investigate the square summability, uniqueness, and convergence to steady state of solutions of the nonstationary Stokes equations in an exterior domain. A class of generalized solutions (which will be called class H[sub o] solutions), whose members are required a priori to have finite Dirichlet integrals but not necessarily to have finite L² norms, has been introduced by J.G. Heywood for the purpose of studying the convergence of nonstationary solutions to stationary ones as time t → ∞. In our present work, we prove that, in the case of an exterior domain Ω, of R[sup n](n > 2) , such solutions are necessarily square-summable if both the initial data and the force are square-summable. 2 We give a partial result for Ω in R². Furthermore, we prove that if Ω = R³ the unique class H[sub o] solution is identical with the unique finite energy solution (i.e. L²(Ω)) of various classes when the data permits existence of both types of solutions. This has enabled us to show that the finite energy solutions of a particular nonstationary Stokes problem converge to solutions of steady state as t → ∞. We have also succeeded in extending the definition of class H[sub o] solutions to nonstationary Stokes problems with general nonhomogeneous boundary values in such, a way that the uniqueness theorem for such solutions is preserved. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0079511 |
URI | http://hdl.handle.net/2429/19662 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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