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The steady Navier-Stokes problem for low Reynolds' number viscous jets Chang, Huakang 1991

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T H E STEADY NAVIER-STOKES PROBLEM FOR LOW REYNOLDS' NUMBER VISCOUS JETS By Huakang Chang B. Sc. (Mathematics) Southwest Jiaotong University M . Sc. (Mathematics) Southwest Jiaotong University A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES MATHEMATICS DEPARTMENT We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA March 1991 © Huakang Chang, 1991 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 or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Mathematics Department The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Abstract The classical existence theorem for the steady Navier-Stokes equations, based on a bound for the solution's Dirichlet integral, provides little qualitative information about the so-lution. In particular, if a domain is unbounded, it is not evident that the solution will be unique even when the data are small. Inspired by the works of Odqvist for the interior problem and of Finn for the problem of flow past an obstacle, we give a potential theoretic construction of a solution of the steady Navier-Stokes equations in several domains with noncompact boundaries. We begin by studying a scalar quasilinear elliptic problem in a half space, which serves as a model problem for the development of some of the methods which are later applied to the Navier-Stokes equations. Then, we consider Navier-Stokes flow in a half space, modeling such phenomena as a jet emanating from a wall, with pre-scribed boundary values. The solution which is obtained decays like |x|~2 at infinity and has a finite Dirichlet integral. Finally, we solve the problem of flow through an aperture in a wall between two half spaces, with a prescribed net flux through the aperture, or with a prescribed pressure drop between the two half spaces. A steady solution is constructed which decays like |s| - 2 at infinity. For small data, uniqueness is proven within the class of functions which decay like |x| _ 1 at infinity and have finite Dirichlet integrals. ii Table of Contents Abstract ii List of Figures v Acknowledgement vi 1 Introduction 1 2 A Quasilinear Elliptic Model Problem In A Half Space 6 2.1 Introduction 6 2.2 Green's function for the Laplace operator 7 2.3 Existence and uniqueness 11 3 Fundamental Tensor For The Stokes Operator 21 4 The Steady Stokes And Navier-Stokes Problems In A Half Space 25 4.1 Introduction 25 4.2 Green's tensor for the Stokes operator in a half space 26 4.3 Solvability of the Navier-Stokes problem in a half space 32 ii i 4.4 Generalized solutions and uniqueness 37 5 Steady Jets Through A n Aperture In A Wall 40 5.1 Introduction 40 5.2 Stokes problem and Green's tensors 41 5.2.1 Stokes problem 41 5.2.2 Decay properties 46 5.2.3 Green's tensors . . *. 49 5.2.4 Estimates for the Green's tensors 50 5.3 Existence theorem 53 5.4 Uniqueness 59 A Inequalities 63 B Proof of Proposition 4.2 67 Bibliography 69 iv List of Figures 1.1 A P E R T U R E D O M A I N IN M3 2 5.1 D O M A I N D E C O M P O S I T I O N 47 V Acknowledgement First and foremost I am grateful to my supervisor Professor J . Heywood, not only for suggesting my dissertation topic, but also for providing invaluable advice, support, and guidance which led to the completion of this work. I would like to thank Professors A . T. Bui and C. A . Swanson for serving on my advisory committee, and for their courses in partial differential equations which I had the pleasure of taking. I am also indebted to my parents, for the tolerance and understanding they displayed when I left them to pursue my career. vi Chapter 1 Introduction Steady incompressible viscous flow is governed by the steady Navier-Stokes equations v Au- pu-Vu-Vp = f and V - u = 0, (1.1) where u(x) = u = ( u\ , U 2 , « 3 ) is a vector field representing the velocity of flow, and p(x) = p is a scalar function representing the pressure. The other quantities appearing in (1.1) are the prescribed external force / , the viscos-ity v, and the fluid density p. The variable x represents position in the fluid. The term vAu is an internal force due to deformation, and V p is an internal force due to pressure. The term pu • V u represents inertial reaction. The first equation in (1.1) expresses the equilibrium of these four forces at each point in the flow field. The second equation in (1.1) expresses the incompressiblity of the flow. In all that follows, we rewrite (1.1) by setting A = pjv and rescaling p and / . In showing that there exist unbounded domains fi C M3 such that the steady Navier-Stokes problem ( A u — V p = A u • V u and V • u = 0 for x € f l (1.2) u(x)' := 0 for x G dSl, and u(x) —> 0 as |x| —+ co t has nontrivial solutions, Heywood [14] drew attention to the problem of a steady jet through an aperture in a wall. If a fluid occupies the two half spaces fi_ and fi+ on 1 0 Chapter 1. Introduction lim v(x) = P lim p(x) = P. S k n = ( 0 , 0 , 1 J X3 f2+ Figure 1.1: APERTURE DOMAIN IN B? either side of an infinite wall, as well as a bounded region that is removed from the wall, constituting a hole that connects the two half spaces (see Figure 1.1), then there exist nontrivial solutions with either a prescribed net flux through the hole / u • n ds = Js F (1.3) or alternatively with a prescribed pressure drop from one side of the wall to the other P = P--P+= Hm p(x) - l im p(x) x 6 f i _ , |x|—»oo , x€fi+ , |X|T*OO (1.4) In the fifth chapter, fl is taken to be such a domain, with f l _ = { x : x 3 < — 1 } and fi+ = { x : X3 > 1 } . We refer to it as an aperture domain. We assume dCt € C°° . The surface S in (1.3) is S = fi D { x : £3 = 0 } . Various existence theorems for these problems based on methods of functional analysis Chapter 1. Introduction 3 and on a-priori bounds for the solutions'Dirichlet integrals ||Vu||2 = JQ |Vu|2dx were given in [14], [16], [23] and [34]. However, these existence theorems give little qualitative information about the solutions which are obtained, beyond that they belong to C°°(A), if d£l G C°° . In particular, little is known about their decay properties at infinity. Because of this, there is no information available about their uniqueness, although uniqueness is expected in the case of small data. The first existence theorem for the Navier-Stokes equations was given by Odqvist [30] in the late 1920s. He considered the first boundary value problem for the steady equations in a bounded domain. Using the Green's tensor for the Stokes equations he transformed the problem to an integral equation which, in the case of small data, he solved by means of a perturbation series. In order to prove the convergence of the series, he provided estimates for the Green's tensor which have subsequently been important to many authors. In 1933, Leray [26] studied the problem of steady flow past an obstacle. He combined new a-priori estimates for the solution's Dirichlet integral with topological methods and functional analysis to prove the existence of a solution. His existence theorem is valid even in the case of large data, when it cannot be expected to be unique or stable. However, when the data is small and the solution can be expected to be unique and stable, Leray's methods failed to show this, because they do not yield the appropriate decay properties for the solution at infinity. To obtain such decay properties, and other qualitative properties of flow past an obstacle, Finn [10] adapted the potential theoretic approach of Odqvist to the problem, using the Green's tensor for a linearization of the equations due to Oseen. The key Chapter 1. Introduction 4 estimates in Finn's analysis are estimates for this Green's tensor. Like Odqvist, Finn dealt only with the case of small data. The results on flow through an aperture of Hey wood [14], [16], Ladyzhenskaya and Solonnikov [23] and Solonnikov [34], which were mentioned above, are analogous to Leray's result for flow past an obstacle. Our objective in this work is to proceed in the manner of Finn, to use potential theoretic methods in order to obtain a solution with appropriate decay properties at infinity. For this problem, it turns out that it is appropriate to use a Green's tensor for the Stokes equations. However, as will be seen in Chapter 5, there are several such Green's tensors to be considered, depending on whether one prescribes the flux condition (1.3) or the pressure condition (1.4). We begin in Chapter 2 by studying a a scalar quasilinear elliptic equation (a variant of the steady Burgers' equation) in a half space as a model problem for the development of some of the methods which are later applied to the Navier-Stokes equations. Solutions which decays like \x\~2 at infinity are constructed in the case of small data, by potential theoretic methods. The fundamental tensor and the truncated fundamental tensor for the Stokes equations are introduced in Chapter 3. In Chapter 4, we obtain estimates for the Green's tensor for the Stokes equations in a half space, and use them to solve the boundary value problems for the steady Stokes and Navier-Stokes equations in a half space. In Chapter 5, we introduce Green's tensors for the Stokes equations in an aperture domain, corresponding to problems in which either the flux or the pressure drop is prescribed, by using methods developed by Hey wood [14] for solving the corresponding Stokes problems. We obtain estimates for these Green's tensors by comparing them with Green's tensor for a half space. Then, solutions satisfying either the prescribed flux condition or the Chapter 1. Introduction 5 prescribed pressure condition are constructed. The solutions that are obtained have finite Dirichlet integrals and decay like \x\~2 at infinity, and are unique among all such solutions, and even among all solutions which have finite Dirichlet integrals and decay like | x | - 1 at infinity. Chapter 2 A Quasilinear Elliptic Model Problem In A Half Space 2.1 Introduction In trying to develop methods which might be applied to the Navier-Stokes equations, one often studies Burgers' equation as a model problem. Burgers' equation is simpler than the Navier-Stokes equations in that there is no pressure term, and the solution is not required to be solenoidal. However, the energy estimate which is basic to much of the Navier-Stokes theory is not valid for the multidimensional Burgers' equation. In this respect, the equation ttt + 2 A u (a • Vu) = A u - f , where a = (a\, a-i, a-s) is a constant vector, and u is a scalar unknown, is more suitable than Burgers' equation as a model problem for the Navier-Stokes theory, because it does admit an energy estimate. In this chapter, we study steady solutions of this equation. Thus we consider the equation A u - 2 A u ( a - V u ) = / , which can also be written as A u - A a - V ( u 2 ) = / . (2.1) 6 Chapter 2. A Quasil'me&r Elliptic Model Problem In A Half Space 7 Our objective will be to develop an existence and uniqueness theory for the Dirichlet problem Au(x) = A [ a - V u 2 ( x ) ] in ft < u(x) = b(x) on dft (2-2) u(x) —• 0 as |x| —> oo in ft t when the domain ft is a half space in JR.3. There is an existence theory for the Navier-Stokes equations based on a bound for the solution's Dirichlet integral, going back to Leray [26], which can easily be applied to this problem. For a recent variant of the method see [16]. However, that existence theory does not give much information about the solution, beyond the boundedness of solution's Dirichlet integral. In particular, it does not give enough information to prove the solution's uniqueness, which is expected in the case of small data. Our main result here is an existence theorem by potential theoretic methods, which does provide such information. Unlike Leray's theorem, we require that the data be small. But, assuming that the data is small, our solution is shown to decay like |x |~ 2 at infinity. 2.2 Green's function for the Laplace operator Let ft = { x = (xi , x 2 , x 3 ) G JR3 : x 3 > 0 } . We will use x = ( X i , x 2 , 0) to denote a point on the boundary 5ft. The Green's function for the Laplace operator in the half space ft is G(*,y)= (r-^— - j-?-?) (2.3) 4TT v F - v l \ x - y\ J, where x' = (x t , x 2 , —X3). Chapter 2. A Quasilinear Elliptic Model Problem In A Half Space Lemma 2.1 The Green's function (2.3) satisfies 2x 3 |G (* ,y) |< *3 47r | x — y\\x' — y\ 2ir\x — y\2 2 y a < Vs 4TT \X — y\\x' — y\ ~ 2ir\x — y\2 ( 2 . 4 ) for all x, y in Q,, PROOF : Note that |x' -y\ = \x- y'\ = ^/(xi - Vi)2 + (x2 - y2)2 + (a* + Vs)2 , and 1 1 \x-y\ \x'-y\ By the triangle inequality, we have < \x-y\- \x'-y\\ \x-y\\x'-y\ \x-y\- | x ' - y\ | < | x - x ' | = 2 x 3 , \x - y\ - \x' - y\ \ = \ \x - y\ - \x - y'\ \<\y-y'\ = 2y3 \x - y\ < \x' -y\ = \x- y'\ for all x,y in f l . Together these inequalities imply ( 2 . 4 ) . Lemma 2.2 The gradient of the Green's function (2.3) satisfies | V G ( X , J / ) | < - L f L - R - + ^ J _ ] < 1 47r \ | x — j / | 2 | x ' — 3/| 2 y 2 7 r | x — y | 2 for all x,y in fi Q.E.D.. ( 2 . 5 ) P R O O F : We have Chapter 2. A Quasilinear Elliptic Model Problem In A Half Space \x'-y\ \x>-y\2 and \x-y\ < \x'-y\ = \x-y'\ , for all x, y in ft . The lemma follows. Q.E.D. Lemma 2.3 There exist positive constants C\ and C2 such that [ dy < Cl Jn \y - x0\2 \y - x\2 ~ Ix-xol, f dy < C2 Jn \y - x0\4 \y - x\2 ~ \x - x0\2 t for all x E ft and a point XQ outside of the closure of ft . (2.6) (2.7) PROOF : For convenience, we choose XQ such that dist (xo , dQ.) > 1. Using a domain decomposition, the second integral can be estimated as follows: dy where in\y-\y - x 0 | 4 |y - x | 2 = h + h + h + h , dy < f dJ. • / n i , - , i s i i - . 0 i \y~x\2\y-xo\3 8 j dy J\v-*\<\Y \x — Xo|3 167T |x — x 0 | 2 H*~*ol IV - XV Chapter 2. A Quasilinear Elliptic Model Problem In A Half Space 10 = / d y 4 j dy < < \*-*o\2 J%-xo^x-,ol |y-*o| 4 4 j dy |x-X 0| 2 . / l < | j , - x 0 | < £ | x - x o | |j/-a;o|4 16TT \x — XQ\2 h = I dv _ , ' n i » — o i . i » — i > j i « — o i . i » — o i i i » - * i ' y xo\4\y x\2 < I Jk\x-dy < ^|i-xo|<lv-x|<oo \y - x\6 16TT Ix — x 0 | 2 4 V |y-x 0 | , |y-x|>l |x-x 0 | , |y-xo|<|y-x| \y - X 0 | 4 \y - x | 2 < / ^ Jk\x-i 2 ^ - x 0 | < | y - x 0 | < o o \y — X 0 | 6 16TT  — l x — X Q | 2 Combining these, we obtain the estimate (2.7). By a similar procedure, we can obtain the estimate (2.6) with C\ < 30 7r. Q.E.D. Lemma 2.4 Let x0 = (0,0, — 1) ^ f l . For all x G f l , there holds L uTTH IV*G(X> ») I dy * TT^TT (2-8) Jn \y — x 0 | 4 | x — x0\* for some positive constant M < 32. Chapter 2. A Quasilinear Elhptic Model Problem In A Half Space 11 PROOF : By (2.5), we have Noting that \y — x0\ > 1 for all y £ ft, the lemma follows from (2.7). Q.E.D. 2.3 Existence and uniqueness The following standard notation will be used. Cfc(ft) is the set of all functions which have continuous derivatives up to the Hh order. C£(ft) is the subset of Cfc(ft) whose members have compact support in ft . C f e + Q(ft) is the subset of Cfc(ft) whose members have locally Holder continuous kih order derivatives with exponent a , 0 < a < 1. Co+ a(ft) is the intersection of C£(ft) and C f c + a (f t) . C°°(ft) is the set including all functions which have continuous derivatives of all orders, and Co°(ft) is its subset of elements which have compact support in ft. The norm in £ p(ft) is denoted by || • | | p and simply by || • || if p = 2. Wo(ft) is the completion of C£°(ft) in the Dirichlet norm | |V • | |. Under appropriate assumptions about the regularity and decay of g and b, the first boundary value problem for the Poisson equation Au(x) = g(x) in ft < u(x) = b(x) on dn (2-9) u(x) —* 0 as |x| —> oo in ft has the unique solution u(x) = / G(x,y)g(y)dy - [ Gyi(x,y) b(y) dy . (2.10) Jy3>0 Jy3=0 Chapter 2. A Quasilinear Elliptic Model Problem In A Half Space 12 Thus, the problem (2.2) is formally equivalent to the integral equation u(x) = A / G(x,y) [a-Vu2(j/)1 dy + u0(x) , (2.11) JV3>0 1 J /y3>0 where u 0(x) = - / GV3(x, y) b(y) dy . (2.12) 'V3=0 Upon integration by parts, one can formally rewrite (2.11) as u(x) = - A / [a- VyG(x,y)} u2(y) dy + u0(x) . (2.13) Lemma 2.5 Let xo = ( 0 , 0 , —1), and let b be a continuous function with compact support on the boundary dD,. Then UQ defined by (2.12) belongs to C°°(fl) D C(f i) , and satisfies 11x0(2;) 1 = 1 / Gy3(x,y)b(y)dy \Jy3=0 for some positive constant Ao. Furthermore, if the first order derivatives of 6(x) are continuous on the boundary dfl, then UQ also satisfies lx — x0|2 |x — x 0 | ; PROOF : Since b is continuous with compact support on the boundary, the properties of double layer potentials (cf . [5]) imply (2.14). The assertion that uo defined by (2.12) belongs to C°°(fl) D C(Q,) is a standard potential theoretic result (cf . [5] or [12]). The estimate (2.15) also follows from the representation of UQ as a double layer potential. Q.E.D. Chapter 2. A Quasilinear Elliptic Model Problem In A Half Space 13 Theorem 2.1 Suppose the data satisfy the assumptions of Lemma 2.5, so that (2.14) and (2.15) hold. Then, for any real number A satisfying N I A K j ^ (2-16) there exists a solution u of the integral equation (2.13) satisfying 2A0 \x — X0\ and "Ml ^  r^rh; (217> ( 2 ' 1 8 ) for all x in fl. Here B(X) will be given in Lemma 2.10, M and AQ are the constants given in Lemmas 2.4 and 2.5 respectively, |a| = y ^ L i a 2 , and x0 = (0, 0, — 1) is a point chosen from the complement of the closure of fl. Furthermore, u belongs to u e C°°(fl) n C( f 2 ) and is a solution of (2.2). We seek a solution of (2.13) in the form oo u(x) = £ un{x) A" (2.19) n=0 where UQ is given by (2.12). Setting v(x) = \x — x0\2 u(x) and vn(x) = \x — x0\2 un(x) t (2.20) we can also write oo «(*) = £ > » ( * ) A" (2.21) n=0 To obtain a recurrence relation for the coefficients u n , we can formally substitute (2.19) into (2.13), to get oo oo n . £ u„(x) A n = - A ^ A n ^ / uk(y)un.k(y) [ a • V v G(x , y) ] dy + u0(x), n=0 n=0 k-0 Chapter 2. A Quasilinear ElUptic Model Problem In A Half Space 14 or oo oo n . £ u„ + 1(x)A" + 1 = - £ £ / uk(y)un.k(y) [ a - V v G ( x , y)) dy. (2.22) n=0 n=0 Jfc=0 J V 3 > 0 By comparing the coefficients of A" on both sides, we have u n + 1(x) = - / [a• V v C7(x ,y)]J2 Mv)"*-*(y)dy, (2.23) •/V3>0 K = 0 and u n + 1(x) = - | x - x 0 | 2 / 11 ^ [a • VyG{x, y)] 1 £ vfc(y)i;„-*(y) dy. (2.24) Jv*>o {\y - x 0 | 4 J fc=0 L e m m a 2.6 The series JZnLo ^nAn, with constant coefficients, dominates the series (2.21), i f Ao is as in Lemma 2.5, and n An+1 = | a | M £ AkAn.k> n = 0, 1 , 2 , 3 ••• . (2.25) PROOF : This will be proved by induction. For n = 0, we have Ivofa)! < A 0 for all x € ft, by (2.12). Suppose that \vk(x)\ < Ak for all k < n. Then, (2.8) and (2.24) together imply K + 1 ( x ) | < | x - x 0 | 2 / |vyc7(x,y)| X: My)lk- f c(y)|rfy Jo | y - x 0 | 4 n / 1 < |a| £ AkAn.k \x-x0\2 / 1 -\VyG(x,y)\dy n < |a| M J2 Ak An-k = A n + 1 . fc=0 Q.E.D. L e m m a 2.7 For all nonnegative integers n, un and vn, defined by (2.23) and (2.24) respectively, belong to C°°(ft) n C(ft). Chapter 2. A Quasilinear Elliptic Model Problem In A Half Space 15 PROOF : Again, we proceed by induction. For n = 0, Lemma 2.5 implies that uo belongs to C°°(fl) D C(Vt). Suppose the conclusion holds for all k < n. Then, we can integrate (2.23) by parts using (2.20), Lemma 2.6 and the induction hypothesis to obtain (2.26) Q.E.D. u n + 1(x) = / G(x,y) J2 2 [a • Vufc(y)] un-k(y)dy Jyz>o k = 0 which clearly implies u n + 1 belongs to C°°(fl) nC(fV) (cf. [5], [12]). Remark : From (2.26), it is clear that un+i is the unique solution of Au = 2 ^ [ a • Vufc }un-k for x £ fl k=0 u(x) = 0 for x G <9fl, and u(x) -> 0 as |x| —•» oo . Lemma 2.8 The series 23^L0 A N A" is uniformly convergent for A satisfying (2.16), and oo £ A n | A | " < 2 A 0 . (2.27) n=0 Thus, for such A, the series (2.21) is absolutely and uniformly convergent in the variable x , and so is (2.19). The limits v , u are continuous, and satisfy \ v(x) \ < 2 AQ , and (2.17). PROOF : We consider the function ' __1 A(X) = { 2|a|MA A0 1 - - 4 | a | M A 0 A ] A ^ O , A = 0. (2.28) A(X) is analytic for A satisfying (2.16), and thus has a Taylor expansion Y^Lo A N ^ N AT A = 0 with AQ = A(0) = AQ • It is easy to see that A(A) satisfies the quadratic equation A(X) = A 0 + |a|MA[A(A)] 2 . (2.29) Chapter 2. A Quasilinear Elliptic Model Problem In A Half Space 16 Substituting J2%Lo An A n for A(X) in this equation, one obtains £ i n A n = A 0 + A | a | M n=0 T 2 n=0 = Ao + A | a | M £ A" £ AkAn-u , n=0 fc=0 from which it is evident that the coefficients An satisfy the recurrence relation (2.25), and thus An = An. For A satisfying (2.16), one can easily show that A(|A|) < 2 Ao • Lemma 2.6 implies that the series (2.21) is absolutely and uniformly convergent in the variable x , for any fixed A satisfying (2.16), and therefore so is the series (2.19). Thus, the limit v is continuous and satisfies \v(x)\ < A(\X\) < 2A0, for all x € ft. The limit u satisfies the estimate (2.17). Q.E.D. L e m m a 2.9 The series B(\) = Yin°=o Bn ^n > w ^ constant coefficients, dominates the series \x — x0\2 Y^Lo ^un(x) A n , if BQ = A0, and £ n + 1 = 2 | a | M £ BkAn.k< n = 0 , l , 2 , 3 - - - , (2.30) k=o where the { An } is as in Lemma 2.6. PROOF : The lemma will be proved by induction. For n = 0, Lemma 2.5 implies that \x — x0\2 | V u 0 ( z ) | < B0, since B0 = A0. Suppose that \x — x0\2 \Vuk(x)\ < Bk , for all k < n. Then Lemmas 2.6 and 2.7, and (2.26), imply that | V u n + 1 ( x ) | < / \VxG(x,y)\ 2\a\\Vyuk(y)\\un_k(y)\dy Jy3>o k = 0 < 2 | a | £ BkAn.k f 1 \VxG(x,y)\dy k = 0 J n \y - x°\ l |x - x 0 | 2 |x — x 0 | 2 2 | a | M £ BkAn.k k=Q Chapter 2. A Quasilinear Elliptic Model Problem In A Half Space 17 This proves the result claimed in the lemma. Q.E.D. Lemma 2.10 The series Y^'ZLo Bn Xn is convergent for X satisfying (2.16), and °° An ^ ) - S B - A - l - 2 N ^ ^ ) A , ^ where A(X) and AQ are as in Lemma 2.8. Thus, the series 23£Lo Vu„( i ) A" is absolutely and uniformly convergent in the variable x, for any fixed X satisfying (2.16). The limit Vu(x) is continuous, and satisfies (2.18). PROOF : Consider the function B(X) = A0 / [ 1 - 2 |a| M A(X) A]. Obviously, B(X) has a unique Taylor expansion £ £ L 0 BnXn for 4MA 0|a | |A | < 1, since A(|A|) < 2A0. Also, since AQ = BQ , the function B(X) satisfies B(X) = Ao + 2 |a| M A(X) B{X) X = B0 + 2 |a| M A(X) B(X) X . By a procedure similar to the proof of Lemma 2.8, we find that the series Y^=o &n ^" i s convergent to B(X) for A satisfying (2.16). It is obvious that B(X) satisfies (2.31) for A satisfying (2.16), since A(|A|) < 2 AQ . Lemma 2.9 implies that the series ££Lo Vu n ( i ) A n is absolutely and uniformly con-vergent in the variable x, for any fixed A satisfying (2.16). The limit satisfies (2.18). Since (2.19) is uniformly convergent, and every un has continuous first order derivatives, the limit u(x) has continuous derivatives Vu(r) equal to ]CnLo ^un(x) Xn . Q.E.D. PROOF OF Theorem 2.1 : Combining Lemmas 2.6 through 2.10, we obtain a solution u of the integral equation (2.13) with continuously bounded derivatives. The function Chapter 2. A Quasilinear Elliptic Model Problem In A Half Space 18 defined by the integral / n G(x,y) u2(y)dy belongs to C2+a(Q), because u 2 has bounded continuous derivatives by Lemmas 2.8 and 2.10, and therefore belongs to Ca(il). Hence, the function defined by the integral / n [a • VyG(x, y) ] u2(y) dy belongs to C 1 + Q ( f l ) (cf. [12]). Since tto belongs to C°°(fl) flC(fl), the equation (2.13) implies that u belongs to C1+a(Ct) and satisfies (2.17). Upon integration by parts, one finds that u is also a solution of the integral equation (2.11). From this point, it is readily seen that u £ C°°(f2)nC(Q), and that u is a solution of (2.2) satisfying (2.17) and (2.18). Q.E.D. The decay properties (2.17) and (2.18) of the solution u imply Corollary 2.1 The solution obtained in Theorem 2.1 belongs to Cp(0.) for all p > | , and has finite Dirichlet integral. Corollary 2.2 Assume that u , v are two solutions of (2.2) satisfying the decay property (2.18), then the difference w = u — v belongs to Wo(f2). PROOF : Since both u and v take the same boundary values, the difference w vanishes on the boundary. (2.18) implies that w has a finite Dirichlet integral, thus w 6 Wo(fi), by a theorem in [14]. Q.E.D. Theorem 2.2 For X satisfying (2.16), there exists at most one solution u(x) of (2.2) which has a finite Dirichlet integral, and satisfies \X — XQ\ where AQ is as in Lemma 2.5. Chapter 2. A Quasilinear Elliptic Model Problem In A Half Space 19 PROOF : "Since |x — x0| > 1 for all x £ ft, the solution obtained in Theorem 2.1 has a finite Dirichlet integral and satisfies the stated decay property. Suppose that there is an another solution u of the problem (2.2) having a finite Dirichlet integral and satisfying |u(x)| < 2Ao \x — x 0 | - 1 inside ft. Let w = u — u. Then w belongs to W0(ft), and |u>(x)| < AAo \x — xo|_1 for all x 6 ft. Since both u and u are solutions of the boundary value problem (2.2), we have / Vw-V(j>dx = - A / (u-u) (u + u)(a-V<£)d"x, Jn Jn for all <j> £ Wo (ft). By substituting w for <f> on the both sides of above identity, we have / |Vuf < |A||a| / fci - «| (|u| + |fi|) |Vu,| <|A||a| / - i i i - | V u , | . Jn Jn Jn \x — xo| Thus, using the Schwarz inequafity and Lemma 1 in Appendix A, we obtain | |VH| 2<4A 0 |A||a| | |Vu;| | ( / , H 2 i2dxY < 4 A0 |a| |A| ||Vu;||2, (2.32) \Jn |x — xol'' / Since M was chosen in Lemma 2.4 to be greater that 1, it follows from (2.16) that 4 A 0 |a| |A| < 1. Hence, ||Vu;|| = 0. This implies that w = 0 on ft , i.e. u = u on ft . Q.E.D. Remark : More generally, we can consider the problem Au(x) = A [a-Vu2(x)] + /(x) in ft - u(x) = b(x) on dil (2.33) u(x) —* 0 as |x| —• oo in ft ( with nonzero force / € Ca(ft). Suppose that / decays like |x | - 3 _ e at infinity, for some e > 0 , or that / = V h2 , where h is a function that decays like |x| - 1 . Suppose that the Chapter 2. A Quasilinear Elliptic Model Problem In A Half Space 20 boundary value b are continuous and decay like |x| 1 at infinity. Then, instead of (2.12), one has UQ(X) = I G{x, y) f(y) dy - I GV3 (x , y) b(y) dy , Jy3>0 «'V3=0 and by a procedure similar to that for the homogeneous problem, we obtain Theorem 2.3 If A satisfies ' a " A ' < TWAo <2-34> there exists a solution u € C2+a(Q) f) C(Q) of (2.33) satisfying W .) l < j ^ j (2.35) and i V u < * > ' 2 , < 2 - 3 6 > for some constants C(A) and e , 0 < e < | . Here |a| = ai > A) is a constant such that M * ) | < 7 - ^ - 7 |Vuo(x) |< A ° x - x 0 | lx — x 0 | 2 and N is a constant (we can take N = 15) such that Hence, u has a finite Dirichlet integral. The solution is unique in the class of all functions that satisfy (2.35) and have finite Dirichlet integrals. Chapter 3 Fundamental Tensor For The Stokes Operator For a pair { u , p} , consisting of a solenoidal vector field u(x) and a scalar function p(x), the related stress tensor Tu and its adjoint T'u are defined to be the 3 x 3 matrix functions with entries {Tu)ij{x) = -P{x)Sij + [DXjui(x) + DXiuj(x)]t (T'u)0-(x) = p(x)6ij + [D^Uito + DsiUjix)]^ for i, j = 1, 2, 3 , where £,j is the Kronecker delta notation. By means of the divergence theorem, we have / [v(Au- Vp ) - t i - (Av + Vg)] dx= I [v-Tu-u-T'v] • ds (3.2) JD JdD for any bounded domain D with smooth boundary dD, and any pairs { u, p} and { v , q } , where u , y are solenoidal vector fields, and p, q are scalar functions. Here ds is interpreted as a directed infinitesimal surface element on the boundary. In JB? , the function 0(x, y) = Q(x — y) = — \x — y |/8 ir is the fundamental solution of the biharmonic differential operator A 2 . The fundamental tensor for the Stokes operator in R3 (cf . [30], [11], [22]) is the pair { E , e } , where E = [ Ei:,} is a 3 x 3 matrix, and e = [ ti ] is a 1 x 3 vector, with components Eifay) = Eijix-y) = ^ t i A - ^ - j 0 ( x - y ) (3.3) 21 Chapter 3. Fundamental Tensor For The Stokes Operator 22 '8TT H + (xi ~ Vi)(xJ ~ Vj) \x~y\ | x - y |3 for all i , , ; = 1,2,3. Clearly, E(x,y) = E(x — y) is symmetric, E(x,y) — E(y,x), and e(x, y) = — e(y,x). By an easy calculation, one can show that the fundamental tensor { E , e } for the Stokes operator satisfies (3.5) \E(x,y)\ < | V £ ( x , y ) | < 8TT \X - y| |Ve(x , y)| < 47r |x — y\2 V6 87r |x — y\2 ^ ' v ' " n 47r | x — y | 3 for all x ^ y in JR3. The fundamental tensor is, of course, constructed to satisfy AxE(x,y) - V x e (x ,y ) = 6(x - y) I , AyE(x,y) + V y e (x ,y ) = S(x - y) I t and (3.6) V - £ ( x , y ) = 0 , x ^ y , (3.7) where £(x,y) = £(x — y) is the dirac delta function, and / = is the 3 x 3 identity matrix. Thus, it follows from (3.2) that any solution {u , p} of the Stokes equations Au - V p = / , V • u = 0 (3.8) in a bounded region D satisfies u(x) = / E(x, y) • f(y) dy + / (u(y) • ^£? (x , y) - £ ( x , y) • Tu(y)} • ds,, p(x) = I e(x,y)- f(y)dy + f {u{y) • Tye(x, y) - e(x, y) • Tu(y)} • </sy, 72? Jar; " where T'yE(x,y) is formed by interpreting the columns of E = [E\,E2,Ez\ as vector fields. Chapter 3. Fundamental Tensor For The Stokes Operator 23 In constructing Green's tensor for the flux and pressure problems in the fifth chapter, we use a truncated fundamental tensor for the Stokes operator introduced by Fujita [11]. Let ^(t) € C?(R) satisfy 1 i f \t\ < 1 , 0 i f \t\ > 2 . Let 7 be a positive parameter. A family of functions {^(t) : 7 > 0} are defined by ril(t) = (3.9) i f (0 = 7 ^ / 7 ) (3.10) Clearly, ^(t) = 1 if \t\ < 7, and ^(t) = 0 if \t\ > 27 . The truncated fundamental tensor { E~<, e 7 } is defined by ( ^ A - ^ - ) © ^ - 2 / ) , «?(*,*) = - A A 0 7 ( x - y ) , (3.11) where 0 7 ( x — y) = rj^dx — y\) 0(x — y). By the properties of T/7(|X — y\), the tensor { F/1, e 7 } vanishes for |x — y | > 2 7 and equals { E , e } for |x — y | < 7 . Also, | £ 7 | < c , | x - y t - i |e 7 | < c^x - y | " I V ^ I ^ c . l x - y r 2 , (3.12) for all x ,y in R3 , where is a positive constant depending on 7. # 7 (x) = i (3.13) A smooth 3 x 3 tensor i / 7 ( x , y ) = ii~ 7(x — y) is defined by setting 0 if x = 0 , - A 2 0 7 ( x ) 7 i f x ^ O . Since A 2 0 ( x ) = 0 for all x ^ 0 in R3 , and all the derivatives of 777(|x|) vanish outside of 7 < |x| < 27 , the tensor i7 7 (x — y) is of class C°° in all space and vanishes outside of 7 < | x — y | < 2 7 . Thus, the vector field Wv(x) = J H\x - y) • v(y) dy Chapter 3. Fundamental Tensor For The Stokes Operator 24 belongs to the class C°° , provided that v € C1. By a calculation, one can show that V-F y(i , y ) = 0 , for x / y , (3.14) and that A x £^(x ,y) - Vxe^(x,y) = 6(x - y) I - H\x, y), (3.15) A ! /^(x,y) + Vye^(x,y) = 6(x - y) I - H^(x,y). By combining (3.2), (3.15) and the properties of the delta function, one can show that any solution of (3.8) satisfies u(x) = J EP{x - y) • f(y) dy + J {x - y) • u(y) dy , (3.16) provided the distance between x and 80. exceeds 2 7 Chapter 4 The Steady Stokes And Navier-Stokes Problems In A Half Space 4.1 Introduction In a half space fl = { x 6 JR3 : X3 > 0 } , the first boundary value problem for the steady Stokes or Navier-Stokes equations is posed as follows: A u - V p = / and V • u = 0 for x G f l (4.1) u(x) = b(x) for x € dQ t and u(x) —• 0 as \x\ —• 00 t or A u — V p = A u • V u and V • u = 0 for x £ fl (4.2) u(x) = b(x) for x G dfl,, and u(x) —• 0 as |x| —» 00. Throughout this chapter, it will be assumed that b has continuous derivatives on the boundary df!, and that b(x) —• 0 as |x| —• 00 , with an appropriate rate of decay. It is well known that the existence of smooth solutions with finite Dirichlet integrals can be proven by methods of functional analysis. But, for the Navier-Stokes problem, there is not much information available about the decay of these solutions at infinity. In this chapter, we solve the problems (4.1) and (4.2) by potential theoretic methods, obtaining solutions which decay like \x\~2 at infinity. Finally, we show that the solution of the Navier-Stokes problem (4.2) which is constructed here is unique in the class of all solutions with finite Dirichlet integrals and decay like | x | _ 1 at infinity. 25 Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 26 4.2 Green's tensor for the Stokes operator in a half space To find the Green's tensor for the Stokes operator in a half space ft, we need to consider the following boundary value problem for the Stokes equations A u - V p = 0 and V • u = 0 for x G ft (4.3) u(x) = b(x) for x £ dil, and u(x) —• 0 as |x| —+ co . Let K(x, y) be the 3 x 3 matrix, and k(x, y) be the 1 x 3 vector, with components (4.4) K . x _ 3 0 * 3-y 3) (xj-yj)(xi-yi) h(x,y) = 3 VZ (4.5) 7r dxi \\x — y\3J for j , 1 = 1, 2, 3. It is obvious that the 3 x 3 matrix K(x, y) is symmetric, that is, Kij(xi y) — Kji{xi y) • By a calculation, we have Proposition 4.1 The matrix K(x, y) satisfies / K(x, y) dy-^dy2 = I 0 -I if i 3 > c , if i 3 = c, if x3 < c (4.6) where I is the 3 x 3 identity matrix. The following result can be found in [30] and [4], we include a proof in Appendix B. Proposition 4.2 If b(x) = O {\x\~e) as \x\ —* oo on the boundary of ft for some e > \ , then (4.3) has the unique solution u(x) = I b(() K(x, 0 di P(x) = / b(OHx,o4. y Ji3=o (4.7) Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 27 For fixed x , the fundamental tensor E(x, y) decays like |y| 1 at infinity. Thus the boundary value problem AyA(x, y) + V „ a ( x , y) = 0 , V„ • A(x , y) = 0 for x, y G fl A(x, y) = 2£(x, y) on df! (4.8) has the unique solution A(x,y) = I E(x,i)K(y,i)d£, J£3=0 a(x ,y ) = - / E(x,i)k(yJ)d£. (4.9) (4.10) Lemma 4.1 A(x , y) obtained by (4.9) has the symmetric properties Aij(x, y) = A t j ( y , x) = A i t ( x , y) = A i t ( y , x) , (4.11) for aii i , j ' = 1,2,3 and ali x, y G f l . PROOF : From the fact that Eij(x, y) = 2?,j(y, x) = Eji(x, y) = 2£j,(y, x ) , and Kij{xi y) = Kji{xi y) i it is not difficult to show that A(x, y) is a symmetric matrix. We need only show that A(x , y) = A(y, x ) . By a calculation, we find that K(x, y) V3=0 = 2TyE(y, x)-n(y) V3=0 = 2 r ; % ! / ) . n ( y ) (4.12) Thus, A(x, y) = / £ ( x , f) K(y, £) d( = 2 j E(x , £) T<£(£ y) • n(£) di. Let SIR be a smoothly bounded domain which is only slightly different from the semiball of radius R. Let 5Q,R — ^ 1 1 ^ with TR C d f l . We can assume that l i m ^ ^ TR — d f l , and that for any x, y G f l , can be chosen large enough so that Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 28 x, y £ HR . Using the divergence formula (3.2) on the tensors E(x, £) and E(y, £) in the domain QR , we have L R {E{x' ° ' T i E [ i ' y ) ~ m y ) • r ^ ( x ' °} • n ( ° ds< = I { E(x, 0 [ AtE(t, y) - V<e(£, y) ] - £(£, y) [ A^E(x, () + V<e(x, 0 ] } = / [£?(x ,0 /^ -y) -^ ,y) /^-* ) ]de = y) - £(x, y) = 0 . This implies that / E(x, 0 • ?>£(£, y) • n ( 0 <bc = / E((, y) • T ^ x , 0 • n(£) ds^ . JanR JdnR * Taking the limit as R —> oo , we easily see that lim / £ ( x , 0 • TtE{Z, y) • n(£) ds< = lim / y) • T^E(x, () • n(£) <fa{ = 0 . H — o o JSR fi—oo ./SR Hence, we obtain / E(x, 0 • r { £(& y) • n(0 ds< = f E(t, y) • T ^ ( x , 0 • ntf) <b( . ./sn Van This implies (4.11), since / £ ( x , 0 K(y, 0 dSi = I Eft y) K(x, () ds< . (4.13) Jan Jan Q.E.D. The tensor A(x , y) can be written as A ( x , y ) = / E(i,y)K(x,i)d£= f E(£, y) K(y , £) d£. (4.14) ^6=o ^€3=0 It is clear that the integrals in (4.9) and (4.14) are uniformly convergent for all x , y £ £2. Hence A(x, y) has the following properties: Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 29 i) A(x, y) is a symmetric tensor satisfying (4.9) and (4.14); ii) lirn r 3_ > 0+ A(x, y) = E(x, y), and limV3_*0+ A(x, y) = E(x, y); iii) A(x, y) —• 0 as \y\ —* co for any fixed x £ ft, and vice versa. Finally, we obtain the Green's tensor G(x , y) = [C7,j(x, y) ] and #(x , y) = [p,(x, y) ] , for i, j = 1,2,3, by setting g(x,y) = e (x , y) - a ( x , y ) . The Green's tensor { G , g } satisfies | A x G ( x , y) - Vxg(x, y) = S(x - y) I for x, y £ ft < A y G ( x , y) + V v ^ ( x , .y) = 6(x - y) 7 for x, y £ ft • (4.16) and has the properties i) Gu(x, y) = G,j(y, x) = Gj,(x, y) = G J t (y , x) for all x 7^  y £ ft and z, j = 1,2,3; ii) l im X 3 _ 0 + G(x ,y ) = l im w _ 0 + G(x ,y) = 0; iii) G(x , y) —• 0 as |y| —• 00 for fixed x , and vice versa. Remark : From the formula (4.7), we see that G ( x , y) = £ ( x , y ) - A ( x , y ) , (4.15) < V - G ( x , y) = 0 for x, y £ ft t tf(x,y) = 2 T ; £ ( x , y ) - n ( y ) = 7/yG( x, y) • n(y) 1/3=0 . (4.17) Lemma 4.2 There exists a positive constants C < 1 such that |G(x, y)| < C (4.18) for all x ^ y £ ft . Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 30 PROOF : From (4.15), it is clear that G ( x , y ) | < | £ ( x , y ) | + | A ( x , y ) | . (4.19) Recalling (3.5), we only need to consider the auxiliary term A(x, y). From (4.14), we have \A\2 = £ 4"= E [ £ / Kjl(xJ)Eil(i,y)di ij=i i,j=i L j = i Jt»=° -, 2 f 3 4=o £ 3 /=1 y) d£ Thus, by a calculation, |A(x, y)| < | 5 - / — di '«3=o | x - £ | 3 | y - £ | . Comparing the right side of this inequality with the auxiliary term for the Green's func-tion for the Laplace operator in the second chapter, we see that x3 2 3_ r di _ 1 > 1 7r -4,=o | x - £ | 3 | y - f | \x-y'\ ~ \x~y\ Therefore, we obtain |A(x, y)| < < 4 7r |x — y'| 4 7r |x — y| By (3.5) and (4.21), the constant C can be chosen as (36 + 2v /6)/167r < 0.82. (4.20) (4.21) Q.E.D. The following is a result of Solonnikov [33]. A n alternative proof is sketched below. Lemma 4.3 There exists a positive constant c such that the gradient of the Green's tensor G(x , y) satisfies | V G ( x , y ) | < |x - y l 2 (4.22) for all x, y € f l . Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 31 PROOF : ' B y the definition (4.15) and the estimate (3.5) for the fundamental tensor, we need to consider | V A | only. D»M*. y) = ^ t V)M . (4-23) where j? (t \ (h-yh £ & - y « c h-yi c , 3 (ft - yt) (ji - y<) (fo - yh) From (4.20), we see that for h = 1, 2, yh-xh _ d ( 1 \ _ x 3 j (6 -VK)d£ \x - y ' | 3 dy* V I* - y'\ J 2 Tr 4=o |* - £ | 3 |y - ^l3  3/3 +S3 / 1 \ _ £ l / (6 - ys) d | By comparing these with each term in (4.23), we see that there exists a constant c\ such that | V y A ( x , y ) | < C l | x - y | 2 Thus, the result claimed in the lemma has been proved. Q.E.D. Theorem 4.1 There exist constants M > 1 and N > I, such that f T^—TA | V v G ( x , y ) | d y < (4.24) Jn \y - x0\4 \x - x 0 | 2 I r ^ - H | V y G ( x , y ) | dy < (4.25) Jn \y - x 0 | 2 \x - x0\ for all x E ft and fixed x0 = (0, 0, —1) 0 . PROOF : This follows immediately from Lemmas 4.3 and 2.3. Q.E.D. Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 32 4.3 Solvability of the Navier-Stokes problem in a half space Under appropriate assumptions about the regularity and decay of / and b, the Dirichlet problem for the Stokes equations (4.1) in Q has the solution u(x) = I f(y)G(x,y)dy+ [ u(y) K(x,y) dy , (4.26) Jn Jan P(x) = I f(y)9(x,y)dy+ I u(y)k(x,y)dy . (4.27) Jn Jan Using (4.26) and (4.27), the problem (4.2) is seen to be formally equivalent to the integral equation, u(x) = X I u(y)- Vu(y)G(x,y)dy + u0(x) , (4.28) Jn with an associated pressure p{x) = \ f u(y)-Vu(y)g(x,y)dy + p0{x) , (4.29) Jn where { UQ , po } is the solution of corresponding Stokes problem (4.3) given by u0(x)= b(y)K(x,y)dy p0(x) = b(y) k(x,y) dy. (4.30) Jan Jan Furthermore, integrating by parts, we see that (4.28) is formally equivalent to u(x) = - A / u(y) • VyG(x,y)u{y)dy+ u0(x) . (4.31) Jn Lemma 4.4 Let XQ = (0, 0, — 1), and let b have compact support on the boundary dft,. Then there exists a positive constant Ao such that uo defined by (4.30) satisfies Ao Mx)\ = / Hy)K(x,y)dy \Jy3z=0 \x — X0\2 (4.32) Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 33 for all x € ft. Moreover, u 0 € C°°(ft) D C(ft), and the gradient of uo satisfies for al/ x 6 ft . PROOF : Since 6 £ C*(dft), and has compact support, the decay properties of K imply (4.32) and (4.33) (cf. [4]). By a potential theoretic argument for a double layer potential, we conclude that u0 £ C°°(ft) ("1 C(ft). Q.E.D. Lemma 4.5 Under the assumption of the Lemma 4.4, for A satisfying there exists at least one vector field u 6 C 2(ft) flC(ft) solving the integral equation (4.31) and satisfying W(x)\ < , 2 A ° „ for x £ ft . (4.35) \x — x0r PROOF : We seek a solution of (4.31) in the form u(x) = f) un(x)X\ (4.36) n=0 where UQ is given by (4.30). Setting v(x) = |x — x 0 | 2 u(x ) , and vn(x) = |x — x 0 | 2 un(x) i (4-37) one has oo » ( * ) = £ t > n ( x ) A B , n=0 (4.38) Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 34 To find v , we formally substitute (4.36) into the integral equation (4.31) obtaining oo oo - n Y un(x)Xn = - A Y A" / Y Mv) • V y G ( x , y) u n_ f c(y) dy + u0(x), n=0 n=0 J Q fc=0 or oo oo - n Yun+i(*)\n+1 = - Y A n + 1 / E «*(»)• v»G(*.»)««-fc(y)dy-n=0 n=0 J° k=0 By comparing the coefficients of A n on the both sides, we obtain un+1(x) = - / Y My) • VyG(x,y)un-k(y)dy , (4.39) and u„+i(x) = -\x - x0\2 I -Jn y - i Y My) • V y G(x,y)u n_ f c(y) Jfc=0 dy. (4.40) | y - * o | 2 By following a procedure similar to the proofs of Lemmas 2.6, 2.7 and 2.8, we find that: i) The series A(X) = Y^Lo An A" , with constant coefficients dominates the series (4.38), provided A n + i = M Y AkAn-k k=0 for n = 0, 1, 2, 3,• • • (4.41) where AQ is as in Lemma 4.2. ii) un and vn , defined by (4.39) and (4.40) respectively, belong to C°°(f2) D C(f2). iii) For A satisfying (4.34), the series A(X) converges to 1 A(X) = { and A(|A|) < 2A0. 2MX A0 1 - y7! -AMAQX if A ^ 0 i f A = 0 (4.42) Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 35 This implies that the series (4.36) and (4.38) are absolutely and uniformly convergent in the variable x. It is obvious that u is continuous and satisfies By the construction of u , it is easy to verify that u solves the integral equation (4.31). Q.E.D. Lemma 4.6 For X satisfying (4.34), the solution u obtained in Lemma 4.5 is differen-tiable and satisfies l v " W ' s r a . ( 4- 4 4 ) Hence u has a finite Dirichlet integral, and is a solution of the integral equation (4.28). PROOF : By the decay property and the regularity of un , (4.39) implies that "n+i = J2 / uk(y)-Vun-k(y)G{x,y)dy . (4.45) This shows that u n G C 2(Q) flC(fl) for all positive integers n , since u 0 £ C 2 (fl) nC(£2). Through a procedure similar to the proofs of Lemmas 2.9 and 2.10, it can be shown that i) The series B = £ £ L 0 BnXn dominates |x - x 0 | 2 £ £ L o V u „ ( i ) A n provided that Bo = Ao and Bn+1 = M J2 AkBn.k , k=0 where { An } is as in the proof of Lemma 4.5 and M is as in Theorem 4.1. ii) For A satisfying (4.34), B = £ £ L 0 £ n A n converges to B { X ) = l-MA(X)X . and by (4.42), we see that B(|A|) < 2 A0 . Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 36 iii) The series ]££Lo ^un{x) A" is absolutely and uniformly convergent in the variable x. Thus, it follows from Lemma 4.5 that u is difFerentiable. Also, V u is the limit of E~=o V u n ( x ) A" , and satisfies (4.44). From (4.45) and the properties of uniform convergence, we see that u is a solution the integral equation (4.28). Q.E.D. Theorem 4.2 For A satisfying (4.34), the solution u of the integral equation (4.28) obtained in Lemma 4.6, together with the associated scalar function p given by (4.29), solves the boundary value problem (4.2). Moreover, u satisfies for all x G f l . Here AQ is as in Lemma 4.4. Thus, u has a finite Dirichlet integral. PROOF : The decay property (4.46) is obtained from Lemmas 4.5 and 4.6. Since UQ G C 2 (fl) DC(fl) and u G C 1 (fl) DC(fi) , by a procedure similar to the proof of Theorem 2.1, the equations (4.31), (4.28), (4.29) and (4.46) imply that u G C 2(fi) n C(fi) and p G C 1 ( f l ) , and that u and p solve the problem (4.2). Obviously, the estimate (4.46) implies that u has a finite Dirichlet integral. Q.E.D. Remark : Using the same method, we can solve the nonhomogeneous problem A u - V p = A u • V u + / and V • u = 0 for x G fl (4.47) u(x) — b(x) for x G d f l , and u(x) -+ 0 as |x| —* oo . A similar result with the result in Theorem 2.3 can be obtained. Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 37 4.4 Generalized solutions and uniqueness Let X>(£2) = { <p G C ^ f i ) : V • <p = 0 } , and let Jb(ft) be the completion of V(Ct) in the Dirichlet norm ||Vc/?||. Assume that there exists a solenoidal vector field b defined in 0 and satisfying (a) b has a finite Dirichlet integral; (b) b(x) = b(x) on the boundary dil; (c) b(x) —* 0 as |x| —• oo . Definition 4.1 u is a generahzed solution of the Stokes problem (4.1) ifu — bE Jo(fl) for some b satisfying the conditions (a), (b) and (c), and / V u : Vv? dx = - / / tp dx (4.48) for all ip e V(U). Definition 4.2 u is a generalized solution of the Navier-Stokes problem (4.2) provided that u — b € Jo(£l) for some b satisfying the conditions (a), (b) and (c), and / V u : V y dx = - A / (u • Vu) • y> dx , (4.49) ./n. Jn for all ip e 2>(£2). In the Definition 4.1, the test functions can be increased from <p G T>(Cl) to all <f G Jo(£t) if the integral on the right side of (4.48) defines a continuous linear functional on if G «7o(ft). But, the test functions in Definition 4.2 can not be increased to »7o(^ ) Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 38 since the right side of (4.49) may not be defined for all if £ »7o(f2) • By a theorem in [14], we have the following Proposition 4.3 If u and v are two generalized solutions of either problem (4.1) or problem (4.2), then the difference w — u — v belongs to Jo(Q.). By a procedure similar to the proof of Lemma 4.2, one can obtain the following: Lemma 4.7 Suppose that there exists a positive constant c such that \b(x)\ < i — — r V i € dn (4.50) |x - x 0 | where xo = (0,0, — 1). Then the vector field defined by b(x)=f b(0K(x,0dhd£2 (4.51) is solenoidal, and satisfies the properties (a), (b) and (c). If {u , p) is a classical solution of either the problem (4.48) or (4.49), and u has a finite Dirichlet integral, then u is a generalized solution. Obviously, the solution u obtained in Theorem 4.2 has a finite Dirichlet integral, and is thus a generalized solution. Theorem 4.3 Let u be the solution obtained in Theorem 4.2 and let v be any generalized solution of (4.2) which decays like | x | _ 1 at infinity. Then v = u on Q. PROOF : Let w = v — u. By Proposition 4.3, w (E »7o(f!) • From the definition of generalized solution, it is easy to verify that w satisfies / V u ; : V<p dx = — I (v • V u — u • Vu) <p dx Jo. Jo — (w • Vw + w • V u + u • Viu) <f dx Jo Chapter 4. The Steady Stokes And Navier-Stokes Problems In A Half Space 39 for all (p € 2?(fi). By Lemma 1 in Appendix A , the decay properties of u and t; imply that the right side defines a continuous linear functional on LO € P ( f l ) , and hence on its completion <p € Jo(Q.) • Substituting w for tp, we have / | V u ; | 2 dx — — I (w • V u ; + w • V u + tx • Vto) • w dx . Jn Jn By integration by parts, one can verify that / u • V u ; • w — I w • V u ; • w = 0 Jn Jn and — / w • V u • w dx = I w • Vu ; • u dx Jn Jn Thus, using the Schwarz inequality and Lemma 1 in Appendix A , and (4.46) one has | |Vu ; | | 2 = A j^w-Vwu < |A| | |Vu; | | ^ jf u 2 / 2 ^ 2 f 71)2 < 2 A o | A | | | V u ; | L / i |r < 4 A 0 | A | | | V u ; | | 2 . The assumption (4.34) implies that 4 A 0 |A| < 1, and hence we conclude that | |Vu; | | = 0 . Therefore, w = 0. Q.E.D. Chapter 5 Steady Jets Through An Aperture In A Wall 5.1 Introduction Our plan in this chapter is to first obtain a solution { u 0 , pQ } of the Stokes problem A u - V p = 0 and V • u = 0 for £ € (5.1) u(x) = 0 for x £ d£l, and u(x) —• 0 as |x| —• oo , satisfying either the flux condition (1.3) or the pressure condition (1.4). Then, we seek the solution u of the corresponding nonlinear problem as a solution of the integral equation u(x) = A / u(y)-\7u(y)G{x,y)dy + uo(x), (5.2) with associated pressure p{x) - A / u(y)- S7u(y)g(x,y)dy-\-p0(x)i (5.3) where { G , g } is a Green's tensor for the Stokes equations satisfying either the flux condition / G - n d s = 0, (5.4) J s or the pressure condition lim g{x,y) = lim g(x,y) = 0. (5.5) ye«-. \y\->°° , |y|-+°° 40 Chapter 5. Steady Jets Through An Aperture In A Wall 41 The core of the existence theorem for the integral equation (5.2) lies in proving that the estimate |x - x 0 | 2 / 1 |V v G(x, ty) | dy < M < oo (5.6) Jo \y - x 0 | 4 holds uniformly for all x £ £2. The solution u is constructed by means of a perturbation series in powers of A . 5.2 Stokes p rob lem and Green's tensors 5.2.1 Stokes p rob lem We need to review some preliminary results concerning the Stokes problem A u - V p = / and V • u = 0 for x £ £2, u(x) = 0 for x £ dil, and u(x) —• 0 as |x| —• oo . Let (5.7) V(il) = {<p£C0x>(il) : V-<^ = 0 } , »7o(£2) = Completion of V(il) in the Dirichlet norm ||Vy?|| t Wo(£2) = Completion of vector function space ^(il) in norm \\V<p\\ , Jo (ty = Wo(£2) : V • ip = 0 } . Clearly, J7b(f2) is a subspace of i70*(£2). It is easy to see, using the divergence theorem, that / u-nds = 0 (5.8) Js for all u £ Z>(£2). Hence, the same follows in the sense of the traces for all u £ j7o(£2) • On the other hand, we have Chapter 5. Steady Jets Through An Aperture In A Wall 42 Proposition 5.1 There exists a vector field b(x) which belongs to »70*(fi), and which satisfies f b-nds = 1 . (5.9) Js Moreover, there exists a positive constant c such that | & ( * ) | < j ^ | V 6 ( x ) | < | ^ and | A 6 ( x ) | < ^ (5.10) A suitable function b may be constructed as follows. Let 0 be the angle between the positive X3-axis and the ray joining a point x with the origin. Let b(x) be defined by b(x) = « ( c o s 2 0 ) 2 | x | - 3 x for 0<e<\n 0 for \TT<e<lTT 4 — — 4 - ( c o s 2 0 ) 2 | x | - 3 x for f 7 r < 0 < * \ Let u> be an averaging kernel; assume that u £ Off (|x| < | ) and / | x | < i u>(x) dx = 1. We define b by setting 6(x) = & \ b(x + y) u(y) dy, J\v\<± where 8 is a normalizing constant chosen so that fs b • n ds = 1. It is easy to verify that b E «J0*(fl). Obviously, there is a positive minimum distance between the support of b and the boundary d£l. A key relation proved in [14] is the following Proposition 5.2 Suppose that u £ *70*(fi) • Then u £ Jo(ti) if and only if (5.8) holds. Thus, the quotient space j7o*(fl) / j7o(fl) has dimension dim[ J0*($l) / »7o(fi) ] = 1 • In order to arrive at a proper generalized formulation of (5.7), recall that a smooth vector function g is necessarily a gradient, i.e., g = V p for some scalar function p, if Chapter 5. Steady Jets Through An Aperture In A Wall 43 and only if /"• g • ip dx = 0 for all <p £ V(il). Thus, the equation A u — V p = / can be imposed by requiring that / V u : Vip dx = — f f • tp dx, for all tp £ 2>(Q). Under a suitable assumption on / , (5.11) below, the set of test functions can be increased from V(Q) to its completion Jo(Cl), but not to J0*(il). Testing by tp = b has an implication beyond the holding of A u — V p = / . A natural generalized formulation of the conditions V • u = 0, u = 0 , and u(x) —• 0 as \x\ —• oo is the requirement u £ J0*($l). This, of course, includes an additional condition || V u | | < co , which is accepted here. Notice, however, that the requirement u £ j7o(ft) would not be appropriate, since that would unintentionally imply the zero flux condition (5.8). The principal assumption we need concerning / , which is made in most of this section, is that I / f-<pdx <c,\\V<p\\, V ^ e J o * ( D ) . (5.11) I Jn Definition 5.1 u is agenerahzed solution of the problem (5.7) provided u £ »70*(f2) and / Vu:V(pdx = - f f-<pdx (5.12) Jn Jn for all <p £ J o ( H ) . Generalized solutions of (5.7) are not unique. If one tries to prove uniqueness, letting w — u — v be the difference of two solutions, one has w £ J^0*(fl) and / Vtu : Vy? cte = 0 , V y? £ JQ(Q,) . Jn But, this does not imply that w = 0, since w may not belong to Jo(^) • However, a well posed problem results by combining (5.7) with the auxiliary flux condition (1.3). Then, Chapter 5. Steady Jets Through An Aperture In A Wall 44 uniqueness can be proved since one will have Js w • n ds = 0, and hence w £ j7o(f!), by Proposition 5.2. One also has existence: Proposition 5.3 For any f satisfying (5.11), and any constant F, there exists a unique generahzed solution of (5.7) satisfying (1.3). The existence of a solution satisfying the flux condition (1.3) is obtained by seeking it in the form u = v + Fb, with v £ j7o(fi). Then (5.12) becomes / V u :V<pdx = - j f-ipdx-F I Vb:Vipdx (5.13) Jo Jo Jo which is to hold again for all <p £ j7o(fi) • The right side is a bounded linear functional on <p £ jTo(fl), in view of (5.11) and the properties (5.10) of b that were set out above. In place of the flux condition (1.3), solutions of (5.7) can be uniquely determined by specifying the drop in pressure, from one side of the wall to the other. To make sense of this, one must show that the pressure associated with any generahzed solution has a limit at infinity on either side of the wall. This was accomplished in [14] by proving a regularity estimate in a half space: Assuming / £ £ 2 ( f l ) , there holds / f \D2u\2 + |Vp| 2 1 dx < C j f | V u | 2 + | / | 2 l dx, (5.14) and a similar estimate for x 3 < —2. As a corollary we obtain the following proposition sharpening a corresponding result in [14]. In this proposition, it is not necessary that / satisfies (5.11), if the test function space in (5.12) is restricted to <p £ D(f l ) . Proposition 5.4 Suppose f £ £ 2 ( f i ) , and let u he any generalized solution of (5.7) with associated pressure p. Then I \D2u2 dx+ I | V p | 2 dx < oo (5.15) 1 Jo Chapter 5. Steady Jets Through An Aperture In A Wall 45 l|£2«llk , + liv«||k ) + | |u||V , '\\D*u\\l, + | | V u | | 2 + ||«||£. N(z) J V ( i ) N(x) . (5.17) Moreover, u(x) —• 0 as |x| —• oo . Suppose further that V • / € £ 2 ( f i ) . Then, there exist unique constants P- and P+ such that l im p(x) = P. , lim p(x) = P+ . (5.16) x6U_ , |x|—•(» xgU+, |r|—>oo Finally, u , p G C°°(ft), i f / G C°°(ft) and d£2 £ C°°. To prove this, one combines (5.14) with local estimates up to the boundary of Cat-tabriga [4] and Solonnikov [32] to get (5.15), and the further regularity u , p £ C°°(fj). By a Sobolev inequality (see Lemma 2 in Appendix A ) , we have u £ £ 6 (£2) , since || V u | | < oo . By another well known Sobolev inequality (cf . [12]), there holds |u(x)| < C " N 2 " 1 1 2 - J - I K 7 . . I I 2 H - I I 2 < c for all x £ ft, where A^x) = { j € (1 : |y — x| < 1 }. Because u £ £ 6 ( f t ) , and | | V u | | , | | D 2 u| | < co , one can easily show that (5.17) implies u(x) —> 0 as |x| —• oo . To show (5.16), note that V p £ £ 2 ( f t+) implies the existence of a constant P+ such that p — P+ £ £ 6 ( f t + ) . Indeed, choose p defined in all U?3 such that p = p in ft+ and | | V p | | < oo. Then there must exist u £ W 0 ( ^ 3 ) such that ( V u , Vyo) = ( V p , V<^ >), for all <p £ Wo( l ? 3 ) . Since v — p is harmonic and | | V (i; — p)|| < oo , one concludes that v — p equals a constant P+ , and that ||p — P+\\Ce(n3) < c | |Vt>||. Now (5.16), for 0+ , is obtained by using an analogue of (5.17) for the scalar function p — P+ in £2+ , and the assumption that A p = — V • / £ £ 2 (ft+). While the pressure p is determined only up to an additive constant, the pressure drop (1.4) is uniquely determined by the generalized solution u. It is natural to ask whether a generalized solution is uniquely determined by prescribing the pressure drop P . Chapter 5. Steady Jets Through An Aperture In A Wall 46 Proposition 5.5 For any f satisfying the hypotheses of Propositions 5.3 and 5.4, and for any constant P, there exists a unique generalized solution of (5.7) satisfying the prescribed pressure condition (1.4). To prove this, we use Propositions 5.2, 5.3 and 5.4. By Proposition 5.2, there exists a unique solenoidal vector field $ in Join)1 = | V G J0*(n) : J VV>: V > = 0 , Vv?G Jo(fi)} such that / s $ • n ds = 1. For any flux F, Propositions 5.3 and 5.4 imply (5.7) has a unique solution u and associated pressure p, which are smooth and satisfy the system strictly. We can write u = v + F $ , where v £ t7o(0). Then (5.7) implies •J Vv : V $ + jf V p • $ = - F jT | V $ | 2 - ^ / • $ . Noting that the first term is zero since $ £ ^ ( f l ) 1 , and using (5.16), we obtain P = P_ - P + = lim ( / pb-nds- f pb-nds) V-/N='-.^3<-l |^x|=r,x3>l / (5.18) = - / V p • 6 = - / V p $ = F / I V $ | 2 + / ./n ./n ./n. / n where 6 is as described before. Clearly, we can choose F so that P on the left side of (5.18) is the prescribed pressure drop. Thus we have shown the existence of a solution of (5.7), (1.4). Since F is uniquely determined by the prescribed pressure drop P, it is obvious that the solution is unique. 5.2.2 Decay properties We now show that the solutions uo obtained in Propositions 5.3 and 5.5 decay like \x\~2 at infinity, by considering their restrictions to the half spaces ft+ and f l _ , and identifying them with solutions of the Stokes equations constructed by double layer potentials. Chapter 5. Steady Jets Through An Aperture In A Wall 47 x0 x 3 (5.19) Figure 5.1: DOMAIN DECOMPOSITION Consider the prescribed flux problem (5.1), (1.3). Let be the unique solution of (5.1) with flux / s t>o • nds = 1 , and q0 an associated pressure. Then Vo satisfies the Stokes equations in £2+ , and its boundary values have compact support on the boundary dCl+ . From hydrodynamicai potential theory, see [30], the double layer potential wo(x) = / vo{0 K{x, 0 d£i d£2 < • ^ 3 = 1 r0(x) = / vo(OHx,OdtidZ2 also satisfies the Stokes equations in £2+ , and has boundary values Wo(x) = vo(x) on dil+ . It is not difficult to verify that w0 has a finite Dirichlet integral, using the decay properties of K. Thus both v0 and w0 are solutions of the Stokes equations in the half space £2+ , taking the same boundary values, and both have finite Dirichlet integrals. Hence vo = WQ in £2+ , by a uniqueness theorem for the Stokes equations in a half space proven in [14]. Clearly, qo = r0 + Q+ in £2+ , where Q+ is a constant. Since w0 has Chapter 5. Steady Jets Through An Aperture In A Wall 48 compact support in dfi,+ , it is easy to see that there exists a positive constant CLQ such that for all x £ fi+ , where x 0 is a point not in the closure of f l . For convenience, we choose XQ such that dist(xo, dil) = 1. Similarly, we can obtain an estimate for the restriction of { v0 , q0 } to f l _ : \v0(x)\ < , Q°~ |V«p(g)| < , a °" „ and | 9 o ( z ) - <3-| < a°~ ,x — x0\2 ^ " I x - x o l 3 , ~ ~ \x — x0\3 t for all x £ f l _ , and some positive constants and Q_ . Since VQ is smooth in all of f l , there exists a positive constant ao such that for all x £ f l . Obviously, { u 0 , Po } = { Fv0, F g 0 } is the unique solution of (5.1), (1.3). By a similar procedure, we can obtain analogous estimates for the solution {t>o, qo } of (5.1) with pressure drop P = 1. Obviously, the solution of (5.1), (1.4) can be expressed as { uo , po } = { P vo , P qo } . Combining these results we have L e m m a 5.1 The solutions { u0, po } of the Stokes problems (5.1), (1.3) and (5.1), (1.4), obtained in Propositions 5.3 and 5.5, belong to C°°(fl) and satisfy M * ) | < , a ° | F | . 2 and | V u 0 ( x ) | < Q ° | F | | 3 (5.20) \x — XoY t \x — x 0 | J t and M i)|<r^ a n d | V u o ( x ) | < T ^ i J f L (5.21) |x — Xop ( |X — Xo|J respectively, for appropriate positive constants a0 (different in (5.21) than in (5.20)). Chapter 5. Steady Jets Through An Aperture In A Wall 49 5.2.3 Green's tensors To find a Green's tensor for the system (5.1), we need to consider the Stokes problem { AxX(x, y) - V xx(x, y) = 0 and V x • X(x, y) = 0 for x, y £ il (5.22) X(x, y) — E(x, y) for x £ dil, and X(x, y) —• 0 as |x| —*• oo . Since dim [JQ(H) /<JO(H)] = 1, the Stokes problem (5.22) is not uniquely solvable in JQ(Q) . We may obtain different Green's tensors by imposing different conditions on it. Two Green's tensors for the Stokes operator in an aperture domain will be studied here. One satisfies the zero flux condition (5.4), the other satisfies the pressure condition (5.5). For any x ^ y in il, and sufficiently small 7 > 0, the truncated fundamental ten-sor { Ey, e 7 } satisfies (3.15) and equals zero on the boundary dil. Hence, the tensor { E - E~<, e - e 7 } satisfies A x [ £ ( x , y ) - £ 7 ( x , y ) ] - Vx [e(x,y) - e 7 (x,y)] = # 7 (x ,y) Ay [ E(x, y) - £ 7 ( x , y) ] + Vy [ e(x, y) - e 7(x, y) ] = # 7 (x , y) V - [ £ ( x , y ) - £ 7 ( x , y ) ] = 0, £ ( x , y ) - £ 7 ( x , y ) = £ ( x , y ) on dQ . Since i ? 7 (x ,y ) = .r77(x — y) is smooth, with compact support for fixed x, the Stokes problem AxW(x,y)- Vxw{x,y) = # 7 (x ,y ) and Vx-W{x,y) = 0, < W(x, y) = 0 for x G dil, W{x, y) -+ 0 as |x| -> 00 , (5.23) W(x, y) • n dsx = - ^ ^ ( x , y) • n dsx , has a unique smooth solution by Propositions 5.3 and 5.4. Using the representation formula (3.16) and the symmetry of H"1, one can verify that the solution also satisfies Chapter 5. Steady Jets Through An Aperture In A Wall 50 the adjoint equations AyW(x,y) + Vyw(x,y) = Hi(x,y) and Vx-W(x,y) = 0. Now, it is obvious that the pair {X , x} = {E — E~* — W, e — e 7 — w} is smooth and uniquely solves (5.22) along with the flux condition J X(x, y) • n dsx = E(x, y) -ndsx . By the uniqueness of the solution, the pair { X, x } clearly does not depend on the parameter 7. The Green's tensor satisfying the zero flux condition (5.4) is given by G(x,y) = E(x,y) -X(x,y) (5.24) g{x,y) = e(x,y)-x{x,y). Similarly, we can construct a Green's tensor satisfying the pressure condition (5.5), simply replacing the flux condition in (5.23) by the pressure condition lim w(x,y) = l im w(x,y) = 0. (5.25) i g W _ , |x|—»oo x £ W + , |x|—>oo Lemma 5.2 There exists a unique Green's tensor for the Stokes operator in the aperture domain Q, satisfying either (5.4) or (5.5), and A X G - Vxg = 6(x - y) 11 AVG + Vyg = 8{x - y) I for x, y £ f2 , (5.26) V -G (x ,3 / ) = 0, f o r x G f l , and G (x ,y) = 0 on dVl. 5.2.4 Estimates for the Green's tensors To prove the estimate (5.6), we need some preliminary estimates for the decay of the Green's tensors. We will compare the Green's tensors in the aperture domain with the Chapter 5. Steady Jets Through An Aperture In A Wall 51 Green's tensor in a half space. Let { G + , g+ } denote the Green's tensor for the Stokes operator in the half space ft+ . Lemmas 4.2 and 4.3 imply that there exist positive constants C* and Cj such that \G^x,y)\<j-^— and | V G + ( z , y ) | < (5.27) \x y\ ^ \x y\ ^ for all x, y G ft+ . We claim similar estimates for the Green's tensors in an aperture domain. Lemma 5.3 There exist two positive constants Mi and M2 , such that the Green's ten-sors obtained in Lemma 2 satisfy for all x , y G ft . PROOF : The domain ft can be written as a union ft = ft_ U ftc U ft+ , where ftc = ft \ { ft+ U ft_ } (see Figure 5.1). For x ^ y G ft, we must consider several cases: i) x, y G fi+; ii) x, y G ft- ; iii) x, y G ftc; iv) z G ft+ and y £ ft+ ; v) y G ft+ and a; 0 ft+ ; vi) x G ft- and y £ ft_ ; vii) y G ft- and x & ft_ . If z , y G ft+ , then the difference { G — G+ , g — g+} satisfies Ay[G{x,y)-G+(x,y)] + Vy[g{x,y)-g+(x,y)] = Q for y G ft+ , V „ - [ G ( z , y ) - G + ( z , y ) ] = 0 for y G ft+ , G(z ,y ) -C7+(z ,y ) = G ( z , y ) for y e dQ+ . G ( z , y) has compact support on the boundary dft+ . By an argument similar to the proof of Lemma 5.1, we conclude that the restriction of G — G+ to the half space ft+ can Chapter 5. Steady Jets Through An Aperture In A Wall 52 be expressed as a double layer potential G ( x , y ) - G + ( x , y ) = / G ( x , £ ) K(y,t) dft d(2 . The properties of the double layer potential and (5.27) imply that there exist positive constants and M 2 + such that | G ( x , y ) | < ^ and | V G ( x , y ) | < ^ - ^ (5.29) for x , y £ fi+ . Next, suppose x , y £ fic. Then, since G = .E — A \ we deduce from (3.5) and the smoothness of X that there exist positive constants M{ and M | such that i G ^ y ) ^ ^ and | V G ( x , y ) | < (5.30) for all x , y £ fic . If x £ fi+ and y ^ f l + , then the restriction of the Green's tensor to the half space fi+ is smooth in the variable x and satisfies A x G ( x , y ) - V ^ ( x , y ) = 0, and V x - G ( x , y ) = 0 for x £ Vt+ . Hence, the restriction of G to fl+ in the variable x can be expressed as a double layer potential with compactly supported boundary values. Hence there exist constants Mi and M2 such that | G ( x , y ) | < - ^ - and | V G ( x , y ) | < - ^ 2 — (5.31) | x - y | t p - y r , for all x £ fi+ and y £ fi+ . Chapter 5. Steady Jets Through An Aperture In A Wall 53 Each of the other cases can be treated similarly to one of those above. Thus, the lemma is proved. Q.E.D. Theorem 5.1 There exists a positive constant M satisfying (5.6) for all x in f l , where XQ is a point chosen from the complement of f l . For convenience, we assume that dist(x0, 80.) > 1, and that M > 1. PROOF : Using the estimate (5.28), we have / r J - 1 7 | V v G ( x , y ) | d y < / ^ -dy . Ju | y - x 0 | 4 Jn \y - x 0 | 4 |x - y\2 The theorem follows by Lemma 2.3. Q.E.D. 5.3 Existence theorem Let { u0 , po } denote the unique solution of the Stokes problem (5.1), (1.3) or (5.1),(1.4). We seek a corresponding solution of the integral equation (5.2). Let us focus atten-tion first on the flux problem (1.2), (1.3). Then we choose the Green's tensor { G , g} that satisfies the zero flux condition (5.4). Formally, upon integration by parts, (5.2) is equivalent to the following integral equation u(x) = - A / u{y)- VyG(x,y)u(y)dy + u0(x)t (5.32) •/ft since G is divergence free and equal to zero on the boundary. We seek a solution of (5.32) in the form oo u(x) = £ un(x) A" , (5.33) n=0 Chapter 5. Steady Jets Through An Aperture In A Wall 54 where «o is from Lemma 5.1. For n > 0, let un+i be defined by the recursive relation r n un+1(x) = - / V uk(v) • VvG(x,y) un_*(y) dy . (5.34) The following lemma shows that un is well defined for all nonnegative integers n. Lemma 5.4 The series \F\ Y^=o °nAn dominates the series \x — x0\2 Yin°=o un(x) An , provided n a n + 1 = \F\M Y <**Gn-fc , n = 0, 1, 2, 3 • • • , (5.35) fc=0 where do is as in (5.20), and M is as in (5.6). P R O O F : This will be proved by induction. For n = 0, Lemma 5.1 implies that \x — Xo\2 |txo(x)| < aQ. Suppose that \x — x0\2 \uk(x)\ < \F\ak for all k < n. Then (5.6) and (5.34) imply that K+i(x)| < /' J2 \uk(y)\ | V s G ( x , y ) | \un.k(y)\dy < \F\2 £ L uT—u \VyG(x,y)\dy k% J q \y-xor n M < \F\2 £ "kan-k k=0 = \F\an+1-^—-\x - x0\2 Q .E .D . \x - x0\2 Lemma 5.5 For all nonnegative integers n, un(x) defined by the recurrence relation (5.34) belongs toC°°(n). Chapter 5. Steady Jets Through An Aperture In A Wall 55 P R O O F : Again, we proceed by induction, uo belongs to C°°(£l) by Lemma 5.1. Suppose ujt € C°°(£2), for all k < n. Then, using the decay properties supplied by Lemma 5.4, we integrate (5.34) by parts to obtain (5.36) " n + l ( (x) = / ]£ uk(y) •Vun.k(y)G(x,y)dy. By the estimates of Cattabriga [4] and Solonnikov [32], it is clear that u n + i belongs to C°°(£2), because it is a solution of the Stokes problem n A u — V r = Uk • Vun_fc , and V • v = 0 , for x € ft v • n ds = 0 and |x|—»oo k=0 v — 0 for a; € dft , J v •    t  lim u(x) = 0 . (5.37) Q.E.D. Lemma 5.6 For X satisfying \F\\X\< 1 4 a 0 M tie series Yl^Lo a n ^" J S convergent and satisfies (5.38) £ an|A|n <2a 0 , (5.39) and thus the series (5.33) is absolutely and uniformly convergent in the variable x . The limit u(x) is continuous and satisfies I f vi ^  2ao\F\ \ u ( x ) \ ^ TI ZTi lx — x0|2 (5.40) P R O O F : Let us consider the function 1 A(A) = 2M\F\X 1 - yjl -Aa0M\F\X if A ^ O , if A = 0 (5.41) Chapter 5. Steady Jets Through An Aperture In A Wall 56 A(X) is analytic for A satisfying (5.38), and thus has a unique Taylor expansion at A = 0. It is easy to show that A(X) must equal Y^Lo a » "^ • By a calculation, we conclude that A(\X\) < 2a0 for all A satisfying (5.38). Thus, Lemma 5.4 implies that (5.33) is absolutely and uniformly convergent, and hence that u is continuous by Lemma 5.5. In addition, u satisfies (5.40) by (5.39) and Lemma 5.4. Q . E . D . Lemma 5.7 The series \F\ £ £ L 0 bnXn dominates the series \x - x0\2 Y^Lo Vun(x)An, provided n 6o = a 0 , bn+1 = \F\M Y akbn-k , n = 0, 1, 2, 3 ••• , (5.42) where M is as in (5.6), and { an } is as in Lemma 5.4. P R O O F : From (5.36) and hydrodynamic potential theory, we have V u n + 1 ( i ) = / £ uk(y)- Vun_fc(y) VxG(x,y)dy. (5.43) By Lemmas 5.1 and 5.3, and an induction procedure similar to the proof of Lemma 5.4, we obtain the result claimed in the lemma. Q . E . D . Lemma 5.8 For A satisfying (5.38), the series ]££Lo A " J S convergent and satisfies oo £ 6» |A | B <2a 0 . (5-44) n=0 Thus, the series ]££Lo ^ u n ( x ) ^ n i S absolutely and uniformly convergent in the variable x , and its limit is Vu(x). Moreover, Vu is continuous and satisfies Chapter 5. Steady Jets Through An Aperture In A Wall 57 P R O O F : Consider the function A(X) (5.46) where A(X) is given by (5.41). Using (5.39), it is easy to show that B(X) is analytic and satisfies £(|A|) < 2a 0 for A satisfying (5.38). By a calculation, one can see that the Taylor expansion for B(X) at A = 0 must be the series Yl^=o A n . Lemmas 5.6 and 5.7 imply that Y^=o ^un(x) A" converges to Vu( i ) uniformly and absolutely in the variable x , for A satisfying (5.38). Hence, u G C1(f2) and satisfies (5.45). Q . E . D . Theorem 5.2 For X satisfying (5.38), the prescribed flux problem (1.2), (1.3) has a solution u £ C°°(£2) satisfying 2aQ\F\ 2a0\F\ (5.47) where ao is as in (5.20). P R O O F : From (5.33), (5.34) and the properties of uniform convergence, we have oo oo U(X) - £ Un(X) ^ = «o(x) + A £ Un+l(x) A" n=0 n=0 oo . n = u 0 ( x ) - A £ A n / £ ujt(y)- VyG(x,y)un-k(y)dy n=0 J U k=Q . oo n = u 0 ( x ) - A / £ A n £ uk(y) • V y G(x ,y ) u„_ fc(y) dy n=0 fc=0 oo V y G(x ,y ) = uo(x) - A / £ u„(y) A" J A L n=0 = u 0 ( x ) - A / u(y) • V y G(x ,y ) u(y) dy . JO £ «n(y)AB n=0 dy Chapter 5. Steady Jets Through An Aperture In A Wall 58 This shows that u is a solution of the integral equation (5.32) satisfying (5.47). Using a standard argument from potential theory and (5.47), we conclude that u belongs to C 1 + °(f i ) , for some 0 < a < 1 . The decay and smoothness properties of u allow one to integrate by parts in (5.32), to show that u is a solution of the integral equation (5.2). Then, by a potential theoretic bootstrapping argument, we conclude that u belongs to C°°(fi). Finally, by the properties of the Green's tensor, it is clear that u combined with the pressure function p obtained from (5.3) is a solution of the Navier-Stokes problem (1.2). Since G satisfies (5.4), we conclude that / u • n ds = / UQ • n ds — F Js Js Thus, u satisfies (1.3). Q.E.D. The prescribed pressure drop problem (1.2), (1.4) can be treated in an analogous way by using the Green's tensor { G , g } which satisfies the pressure condition (5.5). Thus, we have Theorem 5.3 For X satisfying \P\ W < (5.48) the prescribed pressure drop problem (1.2), (1.4) has a solution u G C°°(Q) satisfying where QQ is as in (5.21), and M is as in (5.6). Chapter 5. Steady Jets Through An Aperture In A Wall 59 Remarks : 1. The estimates (5.47) and (5.49) imply that the solutions obtained in Theorems 5.2 and 5.3 belong to H J0*(Ct), for p > § . 2. Using the divergence theorem, one can see that there is no solution of (1.2) which decays faster than | x | - 2 at infinity and carries nonzero net flux through the aperture. 3. The nonhomogeneous Navier-Stokes problem with the prescribed flux condition (1.3) or the pressure condition (1.4) can be solved similarly, by considering the integral equation (5.32) with u 0 (x )+ / n f(y) G(x, y) dy substituted for u 0(x) in (5.32). 5.4 Uniqueness A generalized solution of the Navier-Stokes problem (1.2) can be defined as follows. Def ini t ion 5.2 A vector field u(x) over fi is said to be a generalized solution of the problem (1.2) provided u(x) belongs to J0'(il) and satisfies for all <p e V(il) . The test function space is T>(fl) in this definition; it can not be increased to its completion Jo(il) because the nonlinear term on the right side of (5.51) may not be well defined. A u - V p = A u • V u + / and V • u = 0 for x 6 f l (5.50) u(x) = 0 for x G dil f and u(x) —•0 as |x| —• oo (5.51) Chapter 5. Steady Jets Through An Aperture In A Wall 60 If a pair { u, p ) satisfies the Navier-Stokes problem (1.2) strictly, and u has a finite Dirichlet integral, then u is a generalized solution, by a theorem in [14]. Therefore, the solutions obtained in the last section are also generalized solutions. Since V u £ £ 2(£2) and u £ £ 6 (£2) , the estimates of Cattabriga [4] and Solonnikov [32] can be used to show that u is bounded in £2. Hence u • V u € C2 (£2), and we may conclude from Proposition 5.4 that u(x) —• 0, as |x| —• oo . By a further use of the estimates of [4], [32], one can show that A p = V u T : V u £ C2 (£2), and thus conclude from Proposition 5.4 that lim P(x) = P. , lim p(x) = P+. (5.52) xfcl2_ , |x|—>oo xGW-f. , |x|—»oo Finally, it is well known that the estimates of [4], [32] can be bootstrapped to show that u , p £ C°° (£2). T h e o r e m 5.4 Let { u , p } be the solution of (1.2), (1.3) obtained for X satisfying (5.38) in Theorem 5.2. Let { v , g } be any generalized solution of (1.2), (1.3) which decays like | x | _ 1 at infinity. Then u = v and p = q + constant. PROOF : Let u; = v — u and r = cjr — p . Since / s w • n <is = 0, it follows that w £ J7b (£2). Subtracting equations, we easily obtain J Vw.Vif = - X J ( u • Vu ; + to • V u + w • Vtu ) • (p , (5.53) for all </> £ X>(£2). Using the Hardy inequality (see Appendix A) L V^vk dx ~ 411 ' Vy G ^ ' (5"54) Chapter 5. Steady Jets Through An Aperture In A Wall 61 valid for all ift £ Jo*{to)» a n d the decay properties of w and u , the right side of (5.53) is seen to define a continuous functional on ip € X>(fi), and hence on its completion <p £ Jo (fl) . Thus, we can replace <p by w in (5.53) to obtain ||VUJ|| 2 = — A J ( u • Vw -f- to • V u -f w • Vw ) • w Integrating by parts, using the decay property of w , we see that J 0 • Vw • w = — J ifi • Vw • w = 0 for all ^ 6 Jo (to) • Therefore, | | V u ; | | 2 = - A J w-Vu-w = X j w-Vw-u < |A| | |Vu; | | ||u>u||. The decay property (5.47) and the Hardy inequality (see Appendix A) imply that | | V H | 2 < 2 a 0 | F | | A | | | V u ; | | 2 . (5.55) Since the parameter A satisfies (5.38), and since the constant M in Theorem 5.1 is greater than 1, this implies that || Vu>|| = 0. Q.E.D. Theorem 5.5 Let { u , p } be the solution of (1.2), (1.4) obtained for X satisfying (5.48) in Theorem 5.3. Let { v , q } be any generalized solution of (1.2), (1.4) which decays like | x | - 1 at infinity. Then u = v and p = q + constant. PROOF : Let {w, r } = {v — u , q — p} . Knowing that the pressure trends to a limit at infinity in each half space, (5.52) above, we can calculate the pressure drop in term of u : P = P.-P+ = - f Vpb=f Vu:Vbdx + X f u-Vu-bdx Jo Jo Jo Chapter 5. Steady Jets Through An Aperture In A Wall 62 where b is as Section 3. Subtracting this from the analogous identity for { v , q } , we get J Vw : V 6 = - A J (u-Vw + w-Vu + w -Vw)-b . (5.56) We can write w = w + F b, where w € »7o(ft) and F is a constant. Because of the decay properties of u , we can let tp tend to w in (5.51) to get / V u : V u ; dx + A / u • V u • u> dx = 0 ./n Subtracting this from the analogous identity for v , we obtain J V u ; : Vu> = — A j (u • Vw + u; • V u + u> • Vw ) • w . (5.57) Multiplying (5.56) by F and adding (5.57) we finally have || V u ; | | 2 = — A J ( u • V u ; + u; • V u + u> • Vu ; ) • w Reasoning as in the last part of the proof of Theorem 5.4, we obtain the result claimed in Theorem 5.5. Q.E.D. A relation between flux through the aperture and the pressure drop is obtained as following: Corollary 5.1 Let u be a generalized solution with an associated scalar p. If the Sow has the flux F and the pressure drop P, then we have P F = || V u | | 2 + A / (u-Vu)-udx (5.58) Jn Furthermore, if u decays like | x | _ 1 at infinity, then the last term in (5.58) vanishes. Hence, PF= | | V u | | 2 . (5.59) Appendix A Inequalities The following inequalities are well known, we sketch a short proof for them. We generally assume that fl is an open set in 1R3. Lemma 1 Let <p be a continuous vector field over JR3 . Suppose that <p has first deriva-tives almost everywhere in JR3 which are square integrable over JR.3. Then there is a constant vector <po such that for any choice of the point x in JR3 . PROOF : We will only prove the lemma for functions ip G W0(1R3). The proof for general (p can be found in Payne & Weinberger [31] and Finn [10]. First, we consider <p G Co°(5? 3). Using integration by parts, we have L j ^ ' y = JR3^y)M^-y\)dy = ~ I V ( ^ 2 ( j / ) - V ( l n | x - y | ) dy The Schwarz inequality implies \*-y\ 63 < 2 | | V d | Appendix A. Inequalities 64 which implies ( A . l ) . Since Wo(2R3) is the completion of ^ (H3) in the Dirichlet norm, (A. l ) holds for all <p in W0(M3). Q.E.D. Corollary 1 Let ft be an unbounded domain in R3 . Suppose <p is a continuous vector field over ft such that (p has first derivatives almost everywhere in ft which are square integrable over ft . Also, i) ip(x) = 0 for x & dil; ii) <p(x) — • 0 as x —• oo . Then for any choice of the point x in JR3 . PROOF : We can define the extension (p(x) of tp(x) over M3 by <p(x) = < <p(x) if x G ft 0 otherwise Lemma 1 implies since (po = 0 here. Q.E.D. Appendix A. Inequalities 65 Lemma 2 ( Sobolev inequality ) If u G W 0 (iR 3 ) , then NI*(A>) < f / f l|V«|| . (A.3) PROOF : Let tp e C 0 X > ( ^ 3 ) . From Green's identity, we get = J_ / Vy(y) • (y - *) , 47r /RS |x — y | 3 since A (—1 / [4 n |x — y|]) = 6(x — y ) . If we replace v? by (p4 in above equality and note that Vy? 4 = 4y>3 Vv? , we have = I / / M V ^ h ' - ' ) ^ 1 / k M r 2 ^ * . 7T w | x _ y | 3 7r JJRS I I |x — y | 2 and j ^ ( x ) / « 3 |x - y| | x - y | < - / *?3(y)V<^(y)dy  r ^ ^ d x Using Lemma J and Schwarz inequality, we get IMISMJH) ^ ^ l l v ^ l l 2 L V 3(y)|V<^(y)|dy < £||v>||3|M&(W ) • This implies that (A.3) holds for all functions in C^iM3). Since Wo(2R3) is the completion of C^ilR3) in the Dirichlet norm || V • | | , (A.3) holds for all those functions in W0(1R3). Q.E.D. Arguing as in Corollary 1 we obtain Appendix A. Inequalities 66 Corollary 2 ( Sobolev inequality ) Let ft be an open set in M3, and u G Wo(ft). then IHIr.(n)<c||Vu||n, (A.4) where the positive constant c can be chosen as y4/7r which is independent of ft . Appendix B Proof of Proposition 4.2 First of all, we see that the matrix K(x, y) is symmetric and Ji3=0 (B.l) where the I is the 3 x 3 identity matrix. Assume that b(x) is a continuous vector field on the boundary plane dil and bounded by that positive constant C. Then, for any 1 > rj > 0, u(x)-b(x) = / b(£)K(x,£)d£- f b(x)K(x,£)d£ Jan Jau = / [b(£)-b{x)\ K(x,£)d£ Jen 1 J = / [b(0-b(x))K(x,£)d£ •jT [bU)-b(i)} K(xJ)d£. + l±-£l>i By properties of K(x, y), we see that there exists a positive constant M such that K(x,£) \ d£<M. I*-£I<1 It is obvious that b is uniformly continuous on dil, and bounded by a constant C. For any e > 0 , there exists n > 0 such that whenever 0 < x3 = \£ — x\ < r), b(£) - b(i) 67 4 M . Appendix B. Proof of Proposition 4.2 68 So that we have that Jan Ki)-m11K(X,&\di<^- I IK{ X ,o\d(<£- (B.2) Also, < 2Cx3 ( < JCz, < 2 C x 3 K{x, 1) d( We choose S small enough such that 6 < ne/2C. Thus, 0 < x3 < 6 < ne/2C, and / | bd) - b(x) 11 K(X, i) \di<£- (B.3) \i-l\>1 Altogether, for any e > 0 , there exists a 8 > 0 such that whenever 0 < X 3 < 6, there holds \u(x) - b(x)\ < e, or lim u(x) = b(x) . (B.4) 13—>o+ It can be easily verified that K(x, y) = 2 y) • n(y) (B.5) so that the u(x) obtained by (4.7) is just a double layer hydropotential with density vector field b(x) on the boundary dft,. Of course, it satisfies the Stokes equations. Thus, (4.7) gives the solution of the boundary value problem (4.3). Q.E.D. Bibliography [1] K . I . BABENKO, On stationary solutions of the problem of flow past a body by a viscous incompressible fluid, Mat. Sb. 91 (1973), 3-26. [2] K . I . BABENKO, On properties of steady viscous incompressible fluid flows, Proc. 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