Existence, uniqueness and stability of steady flows of second and third grade fluids in an unbounded “pipe-like” domain

International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

Existence, uniqueness and stability of steady #ows of second and third grade #uids in an unbounded `pipe-likea domain Arianna Passerini*, M. Cristina Patria Dipartimento di Matematica, Universita% di Ferrara, Via Machiavelli 35, I-44100 Ferrara, Italy Received 26 January 1999; received in revised form 12 October 1999

Abstract We study steady motions of viscous incompressible third-grade #uids in unbounded channels with arbitrary shape. Such #ows exist for small #uxes, due to a pressure drop. We prove that they are asymptotically stable in time, provided the viscosity is su$ciently large, and the initial condition on the perturbation su$ciently small.  2000 Elsevier Science Ltd. All rights reserved. MSC: 35Q35; 76D99 Keywords: Non-Newtonian #uids; n-grade #uids

1. Introduction In the last years, generalizations of the Navier}Stokes model to highly non-linear constitutive laws are continuously proposed and studied (see, e.g. Refs. [2,5,21,30] and the references quoted therein), because of their interest in applications. In order to explain a lot of non-standard features such as normal stress e!ects, rod climbing, shearthinning and shear-thickening, Rivlin}Ericksen #uids of di!erential type were introduced, starting from 1955, in the following papers [4,8,32,34]. They are rather complex from the point of view of PDE theory. Nevertheless, several authors in #uid mechanics are now engaged with the equations of * Corresponding author. E-mail addresses: [email protected] [email protected] (A. Passerini)

(M.C.

Patria),

motion of non-Newtonian #uids of second and third grade. In particular, some authors are interested in studying n-grade #uids as self-consistent models and not as approximating models. Therefore, in studying dynamics they ask that all the #ows meet the Clausius}Duhem inequality and that the speci"c Helmholtz free energy of the #uid has a minimum at equilibrium: see Refs. [6,9]. On the other hand, it is just under the same hypotheses that Navier}Stokes model is studied. That is to say: we always assume that some real #uids exist such that N}S or n-grade #uids are exact models, and not `truncationsa of viscoelastic #uids. Moreover, as noted in Refs. [14,28], di!erent assumptions could heavily a!ect the stability of the rest state. Under those thermodynamical hypotheses, a lot of results concerning existence and stability have already been produced; see, for instance Refs. [12,13,15,26,27].

0020-7462/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 9 ) 0 0 0 8 1 - 5

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A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

It has to be stressed that for n-grade #uids a di$culty concerning the well-posedness of the problem can arise. Since the equations of motion are higher order than the Navier}Stokes equations, the usual no-slip boundary conditions could not be su$cient for determinacy. For a general discussion of this problem see Ref. [31]. In particular, dealing with porous boundaries, examples for non-uniqueness can be found in Refs. [16,18,29], for both bounded and unbounded domains, so that additional boundary conditions have to be suitably chosen. On the other hand, the boundary condition familiar from Navier}Stokes theory can be su$cient to avoid ill-posed problems. For instance, in the case of the cannister #ow (see Ref. [9]) and in the case of the present paper. As known, steady motions of an incompressible #uid in a channel can occur by means of a pressure drop. This topic was studied, in a very general domain, in Refs. [19,20,33] for the Navier}Stokes model. In this paper we consider, in the same kind of unbounded channel with bounded cross-section, the problem of the motion of third-grade incompressible #uids, whose Cauchy stress tensor T is related to the kinematical variables by T"!pI#kA #a A #a A #b(tr A )A ,        (1.1) where A " v#( v)2  and dA A "  #( v)2A #A v.    dt

(1.2)

scribed interval of time. If we put k a a l" , a "  , a "  ,   . . .

b p b" , n" , . . (1.5)

where . is the constant mass density, then the equations become *(v!a *v)  !l*v#v ' (v!a *v)# n  *t " ' N(v),

(1.6)

' v"0 in );(0, ¹),

(1.7)

where N(v)"a ( v)2A #(a #a )A #b(tr A )A .        (1.8) Note that in order to "nd second-grade #uids or Navier}Stokes #uids as particular cases, it is su$cient to put into Eqs. (1.6), (1.8) b"0 and a #a "0, or a "a "b"0, respectively.     Eq. (1.6) is written in such a way that on the left-hand side, besides the linear terms, the term with the third spatial derivatives (the highest ones) also appears. Our aim is to study the problem in the channel )"+(x, x )31: !R(x (#R,   x3&(x ),, (1.9)  where &(x ) is a bounded, simply connected, do main of 1 such that sup diam &(x )"M(R,  V Z1

(1.3)

Moreover, thermodynamics imposes (see Ref. [9]) the restrictions k*0, a *0, b*0, "a #a ")(24kb,    (1.4) on the (constant) material coe$cients a , a and b.   At this point, one can write the equations of motion, in the absence of body forces, and look for solutions in the prescribed domain, during a pre-

inf diam &(x )"M'0. (1.10)  V Z1 We "nd that the problem is well posed if we ask that Eqs. (1.6), (1.7) hold in );(0, ¹) with boundary and initial conditions: v" "0, /"2



V 

v d&" , 

(1.11)

v(x, 0)"v (x). (1.12)  Note that the boundary is not porous and the #ux

is constant because of Eq. (1.7). (Actually, it is

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

constant whenever the normal component of the "eld vanishes on the boundary.) Prescribing the #ux is not exactly a boundary condition, but is necessary for uniqueness. The same happens for the Navier}Stokes model and is strictly related to the shape of the domain as one can see in Ref. [17]. Some classical problems concerning Navier} Stokes theory in channels have recently been faced and solved also for second- and third-grade #uids. The di$culties are all related to the fact that a divergence-free function vanishing on the boundary, and carrying a #ux di!erent from zero, cannot have global summability properties in an in"nite channel. In particular, since of Poincare` inequality, there is no hope of getting a "nite Dirichlet integral. In fact, one has

   
  
 

2z" ""

X

\X

)c(M)

V  X

 v d& dx  

"v" d& dx 

\X V  M X " v" d& dx )c(M)  4 \X V  and the Dirichlet integral is divergent for z going to in"nity. The main example of such a feature is Poiseuille #ow in pipes. Concerning third-grade #uids, in in"nite cylinders, the existence of Poiseuille-like solutions, re#ecting the translational invariance of the domain with respect to z, can be proved in weighted Sobolev spaces following the arguments of Ref. [27]. In the same paper, the authors study for non-Newtonian #uids the problem (posed by Leray for N}S) of the asymptotic stability in space of Poiseuille #ows, with respect to localized distortions of the domain. Here, the problems can be overcome because the perturbation, carrying zero #ux, comes out summable. This helps in proving that it tends to zero far from the distorsion. A completely di!erent subject is the more general domain with no asymptotics studied in the present paper and introduced "rst by Ladyzhenskaya}Solonnikov [19,20] for N}S. They "nd steady solutions for any #ux in the form v"u#a where a is a prescribed #ux carrier vector 

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"eld whose #ux is . The #ux carrier is divergence free and vanishes on the boundary. By using their techniques, we "nd existence but with a smallness condition on the #ux, due to the highly non-linear model. Moreover, we need large viscosities even for the asymptotic stability of the rest state. In our domain, although the di!erence between v and a has zero #ux, we cannot hope to "nd global summability properties for u. The reason is that the "eld a, which is constructed independently of the equations, plays the role of a force in the problem for the di!erence. Since a is not summable, we should not expect a summable solution. However, one should note that both Poiseuille #ow and the #ux carrier a, which are not summable, are nevertheless bounded. Moreover, in the literature the boundedness of the solutions, even of weak solutions, is required in the schemes used to prove existence for n-grade #uids. Therefore, it seems rather natural to look for solutions in spaces such that boundedness is a consequence of embeddings, even if there is no (global) summability. Such spaces, which are essentially spaces of local summability endowed with a suitable norm, will be introduced in Section 3 (see also Ref. [33]). In order to deal with such spaces, some restrictions on the domain, used also in Ref. [20], are necessary. Such conditions partially compensate the lack of translational invariance with respect to cylinders. In particular, they are needed to ensure at least the translational invariance of some properties, in the sense we specify now. Such assumptions are summarized in explicitly requiring that ) enjoys the uniform CK-regularity property, as de"ned in Ref. [1]. Essentially, this means that all the local regularity properties, related to the shape of the domain, are independent on the location: no value of z is `speciala in this domain. By means of a locally xnite open cover the local properties imply global properties. Just to give an idea, let us consider a cover of ) made of sets of the kind d) "+x3): z!1(x (z,, z31. X  It is not di$cult to prove that in each d) one can X write, with constants independent of z, Poincare` inequality, Sobolev inequalities, Cattabriga estimates [3] and other classical results. Then, it is

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A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

trivial to see that Poincare` inequality actually holds in the whole of ) (with constant M/2). Further, by locally applying Sobolev embedding theorems to functions which are not globally summable, one can deduce that such functions are bounded in the whole of ), as we will see in Section 3. The paper is so organized: Section 2 is devoted to introduce the notations, the functional spaces and to describe a scheme (see Ref. [23]) very convenient to decompose systems (1.6), (1.7), (1.11), (1.12) into a system of two coupled linear problems, both for the static and the dynamical case. In Section 3, after having endowed with a suitable norm the spaces of local summability, we prove existence and uniqueness of the solution for the steady problem. This is done "rst by solving the two auxiliary problems by means of the `invading domainsa technique and then using a uniqueness argument to prove that the solution lies in `locala spaces. Finally, we succeed in "nding a unique solution for the original problem (obviously for small #uxes) by simply applying the Banach "xed point theorem. This is possible in `locala spaces, because one can use compactness results there, though the domain is unbounded. In particular, we want to stress that the spaces used here are not subcases of the weighted spaces used in Ref. [27] as one can see in Ref. [25]. In Section 4, we solve the unsteady problem. Precisely, we solve the problem for the di!erence (perturbation) between the unsteady #ow and the steady solution corresponding to the same #ux. To get a local in time existence, the scheme of the proof is essentially the same as in the steady case. Finally, under the hypothesis of large viscosities with respect to a , and with a further restriction on the size  of the initial data, we obtain simultaneously global in time existence and asymptotic stability of the steady #ow, with an exponential decay of the perturbation.

2. Notations and preliminary results Through the paper we shall use the following notations: HK()), m a non-negative integer, is the usual Sobolev space =K()) endowed with the norm

"" ' "" and the scalar product ( ' , ' ) . With these K K notations H()) denotes ¸()) with the norm "" ' "" ,"" ' "" and scalar product ( ' , ' ) ,( ' , ' ). Here   and in the sequel, for all functional spaces, the subscript zero denotes the subspaces of functions whose trace vanishes on the boundary. We also set the subdomains ) , X *) , in such a way X ) "+x3): !z(x (z,, X  *) ") !) X X X\ and for z31 we de"ne "'"

K

d) "+x3): z!1(x (z,. X  For > a Banach space with norm "" ' "" and 7 1)p(Rwe set ¸N((0, ¹); >)







2 ""v(t)""N dt(R 7  and denoted by =KN((0, ¹); >) the space of functions such that the distributional time derivatives of order up to and including m are in ¸N((0, ¹); >). For p"R, we put " v: t3(0, ¹)Pv(t)3>,

¸((0, ¹); >)





" v: t3(0, ¹)Pv(t)3>, ess sup ""v(t)"" dt(R . 7 RZ2 The norm in =I((0, ¹); HK())), is denoted by "'" for k'0 and by " ' " for k"0. The norm IK2 K2 in =I((0, ¹); CK())), is denoted by " ' " for IK2 k'0 and by " ' " for k"0. K2 The space of functions of class CI on (0, ¹) with values in > is denoted by CI((0, ¹); >).

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

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Finally, by the symbols c , A , A we denote any G   generic constant whose possible dependence on parameters will be speci"ed when necessary. If their numerical value is unnecessary to our aims, then it may have several di!erent values in a single expression. By following a scheme due to Mogilevskii} Solonnikov [23], we can write a system of two coupled linear problems whose possible solution leads to a solution of Eqs. (1.6), (1.7) in );(0, ¹) with conditions (1.11), (1.12). The systems can be obtained by coupling the original system with the auxiliary Cauchy problem

In order to show existence for the unsteady problem of third-grade #uids in );(0, ¹), we shall show existence of solutions (v, q, w) to Eqs. (2.2), (2.3). In an analogous way, but slightly changing the scheme, we shall show existence for the steady problem related to Eqs. (1.6), (1.7), (1.11):

l *q # q#v ' q " : Trv q"n, *t a  q(x, 0)"q (x), (2.1)  where the initial data are arbitrary. Of course, in the sequel, we shall suppose a '0.  Then, the original problem and the auxiliary one are equivalent to

In particular, we shall solve the two problems:

!a *v#v# q"w, 

' v"0 in );(0, ¹),

and

v" "0, /"2



V 

v d&"



' V"0 in ), V" "0, /



V  

< d&" . 

(2.2)

(2.5)

!*V# Q"W,

(2.6a)

' V"0 in ),

(2.6b)

V" "0, /



V 

< d&"



lW#a V ' W"F(V, Q) in ), 

(2.6c)

(2.7)

with F(V, Q)" ' N(V)!V ' V!a ( V)2 Q. 

and *w l # w#v ' w"F(v, q) in );(0, ¹), *t a  w(x, 0)"w (x),  where

!l*V#V ' (V!a *V)# %" ' N(V), 

(2.8)

In order to "nd from Eqs. (2.6a)}(2.6c) and (2.7) a solution of Eq. (2.5), we must obviously set (2.3)

l (2.4) F(v, q)" ' N(v)# v!( v)2 q. a  In order to "nd the equivalence, we must set w (x)"v (x)!a *v (x)# q (x),      where v (x) is given in Eq. (1.12). (Note that, be cause of this last relation, the initial data for Eq. (2.3) will be arbitrary as for Eq. (2.1).) We remark that in problem (2.2), where w, are to be considered data, the time t plays the role of parameter. Moreover, in problem (2.3) v, q are considered to be data.

%"lQ#a V ' Q. 

(2.9)

We want to underline that the change in Eq. (2.6a) induces an expression with no linear terms in Eq. (2.8), which is more convenient in order to apply a "xed point theorem, as we shall do in the next section. Finally, in order to solve systems (2.2), (2.3) we shall actually solve the problem for the di!erence between the unsteady #ow and the steady solution corresponding to the same #ux. This last will be obtained by putting V"U#a, where a3C()) is a prescribed #ux carrier vector "eld, a" b which satis"es the following

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A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

properties:

' a"0,

(2.10a)

a" "0, /

(2.10b)



(2.10c)

b d&"1,  V  supp aL),

(2.10d)

sup "DKa(x)")c ∀m*0, (2.10e) x Z ""b"" )c ∀z31, ∀m*0. (2.10f) KBX The last property is actually a consequence of the previous one. For instance, a can be taken as in Refs. [11,19]. All the equations we write have to be understood in the weak formulation, when necessary.

a "xed point, then Eqs. (3.1) and (3.2) are equivalent to the original system. Thus, we "rst need to "nd existence for both problems (3.1) and (3.2). To do this, we shall apply the technique of the `invading domainsa, that is we shall "nd the solutions as the limit of converging sequences of solutions to the problems !*UX# QX"(#*a,

' UX"0 in ) , X UX"

/X

"0

(3.5)

and lWX#a V ' WX"F in ) ,  X X

(3.6)

where z31> and, given V as in Eq. (3.3), one de"nes V "u V X X

(3.7)

with u 3C() ) such that for a "xed d3(0, 1) X X

3. Existence for the steady problem In order to prove existence for systems (2.6), (2.7), we shall use a "xed point theorem. To this end, we consider the two separated problems:

d d u (x, x )"0 if x (!z# or x 'z! , X    2 2

!*U# Q"(#*a,

u (x, x )"1 if "x "(z!d. X  

' U"0 in ),

Thus, we give two preliminary results concerning the existence of solutions for Eqs. (3.5) and (3.6).

U" "0, /



V 

; d&"0 

(3.1)

and lW#a V ' W"F in ), (3.2)  where the #ux carrier a is prescribed with properties (2.10a)}(2.10f). Let us prescribe ( in a suitable space, recall that V"U#a

(3.3)

and set F(U, Q)" ' N(U#a)!(U#a) ' (U#a) !a ( (U#a))2 Q. (3.4)  Then, we can use Eqs. (3.1) and (3.2) to de"ne a map F(()"W, in such a way that if F has

Lemma 1. For any xxed z31> and a3C()) satisfying Eqs. (2.10a)}(2.10f), let (3HK() ) be given X with m*0 and assume ) uniformly of class CK>. Then, there exists a unique solution (UX, QX)3 () );HK() ) to problem (3.5), such that  X X ""UX"" )c (""("" #""a"" ),    

(3.8a)

""UX"" #"" QX"" )c (""("" #""a"" ), K> K  K K>

(3.8b)

where c and c do not depend neither on z or on the   origin of the reference frame. If moreover, (3 C() ) and ) is of class C, then (UX, QX)3 X [C() )]. X For the proof, we can note that Eq. (3.5) is a Stokes problem in a bounded domain, so that the results concerning existence, estimate Eq. (3.8a) and

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

uniqueness are well known; for instance, see Ref. [10]. In particular the existence of QX can be proved by means of Lemma 1.1, p. 180 of Ref. [10]. The further regularity can be proved recursively starting from Eq. (3.8a) by means of Cattabriga's estimates (see Ref. [10, Theorem IV.5.1, p. 218]), so getting Eq. (3.8b). If (3C() ) and the boundary of ) is C, then X Eq. (3.8b) holds for all m and the "nal part of the thesis follows by Sobolev's embedding theorems. Moreover, it is easy to prove that the constants c and c are independent on z, and on the origin of   the x -axis.  Concerning problem (3.6), taking into account that ) automatically veri"es the strong local Lipschitz property, we give the following result. Lemma 2. Let ) be Lipschitzian and z31> xxed. Let V 3CK() ) and F3HK() ) (with m*0) be given. X  X X Then, if 2l , (3.9) "V " ( X  a (2m#1)  one has that Eq. (3.6) admits a solution WX3HK() ) X such that




""WX"" ) l!a "V " K  X 



2m#1 \ ""F"" , K 2

m"0, 1,




""WX"" ) l!a "V "  X  K

1087

which is su$cient to prove existence, because the problem is linear. For the case m"0 the proof is completely analogous. Thus, if Eq. (3.9) holds true, one can get Eq. (3.10a). Indeed, we can show Eq. (3.10b) recursively. Without loss of generality, we limit ourselves in considering the case m"2. By formally di!erentiating twice (3.6), multiplying by DWX and integrating by parts over ) , one obtains X l""DWX"" )""DF"" ""DWX""    #a sup " V (x)" ""DWX"" X    VZX #a sup "DV (x)" "" WX"" ""DWX"" . X    VZX (3.11) The last term on the right-hand side of Eq. (3.11) can be estimated by means of Eq. (3.10a). Thus, from elementary algebraic inequalities, estimate (3.10b) follows. A more detailed expression for the coe$cients of the polynomial can be found by means of the general estimate "(DK(V ' WX), DKWX)" X

(3.10a)



2m#1 \K 2

;PK\("V " )""F"" , m*2, (3.10b) X K K where PK\ indicates a monic polynomial with positive coezcients depending on m. Proof. We shall use the Galerkin method to show existence in H() ). Though the technique is stanX dard, we want to point out how condition (3.9) arises. Since V vanishes on *) , indicating with WX the X X L approximating solution, we have the estimate l""WX "" )"(F, WX ) "#a "(V ' WX , WX ) " L  L  X L L )""F"" ""WX "" #a sup " V (x)" ""WX "" , X L   L    VZX

2m#1 ) "V " ""DKWX"" X   2 #c(m)"V " ""WX"" ""DKWX"" , X K K\  where c(m)"0 for m"0 and 1, c(m)"1 for m"2, and




m c(m)" max k IK

for m'2. This can be showed by induction, so completing the proof. 䊐 Since the #ux carrier a, which is actually the only datum of the problem, has no global summability properties in ), we are lead to look for solutions lying in `locala spaces, which will be introduced now (see also Ref. [33]).

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A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

Let us set )c, HK()) " : +v3HK ()): c'0 s.t. ""v""  KBX ∀z31,, VK()) " : +v3HK()): ' v"0 weakly,. These spaces can be endowed with the norm ""v""H "min+c31>: ∀z31 ""v"" )c, K KBX and one can prove that they are Banach spaces. In particular, in order to prove that they are complete, one can see that for any v3HK()) ""v"" V )(2x #1)""v""H K   K and this is independent of the origin of the x -axis.  Therefore, for any Cauchy sequence +v , - of eleL LZ ments of HK()) one can construct the limit as a function v3HK ()) satisfying for any choice of  the origin ""v"" V )A x #A , (3.12) K     where A , A do not depend on the origin. From   the uniqueness of the limit it follows that, since ) "d) , we can change continuously the ori  gin, and each time choose x " in Eq. (3.12).   What we obtain can be written in a unique reference frame ("xed once for all), in the following way: ""v"" )(A /2#A , ∀x 31. KBV>    This would prove that v belongs to HK()). In HK()), it is possible to "nd a Poiseuille-like solution, in the sense that it will be bounded even without global summability properties. As a consequence of the assumptions on ), we "nd the embedding HK>())LCK()). In fact, since the constants of Sobolev's embedding theorems in the domains d) are bounded in z, we have X "v" "sup "v" )sup (c(m, k)""v"" ) K KBI K>BI IZIZ)c(m)sup ""v"" )c(m)""v""H . K>BI K> IZA consequence of this simple result is that, since the data V in problem (3.2) must belong to a space of continuous functions, by looking for solutions of (3.1) in HK>()) we can "nd solutions for problem

(3.2). So the application F, whose "xed point is solution of Eq. (2.5), can be well de"ned. Thus, we start by giving in local spaces, the existence result for Eq. (3.1). Theorem 1. Given (3HK()), m*0, and a3C()) satisfying Eqs. (2.10a)}(2.10f), then if ) is uniformly of class CK>, system (3.1) admits a unique solution (U, Q)3VK>());HK()) satisfying  ""U""H)c (m)(""(""H#""a""H), (3.13a)     ""U""H #"" Q""H )c (m)(""(""H #""a""H ). (3.13b) K> K  K K> Moreover, if (3C()) and ) is uniformly of class C, then (U, Q)3[C())]. Proof. Once Eq. (3.13a) is obtained, Eq. (3.13b) easily follows from Cattabriga estimates. Therefore, following the procedure of [20], one "rst proves that the solution UX of Eq. (3.5) satis"es ""UX"" V )A x #A , ∀x 3[1, z], (3.14)       where A and A do not depend on z (and, obvi  ously, on x ).  In particular, in order to show that A , A do   not depend on z we use ""("" V #""a"" V )(2x #1)(""(""H#""a""H).        (3.15) From Eq. (3.14) one can get the existence of a solution U3H ()) by means of the `invading do  mainsa technique (see Ref. [20]). The main step is then to show that this solution belongs to H()). In order to do this, we "rst remark that the hypotheses on ( and a, together with the assumptions on ), give no reason for localizing the origin of the x -axis in any special  place. We could have constructed a solution in the same way, but translating the origin along the axis, so using a di!erent family of invading domains. We could have got an estimate of the form (3.14), and we would have got exactly the same constants A and A .   Now, we note that if the solution is unique in the class of solutions lying in < ()) and satisfying   ""U"" V )A x #A , ∀x 31> (3.16)     

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

for some choice of the origin, then, by the same argument used to prove completeness of HK()) it follows that U belongs to V ()). Recalling that  A and A are proportional to ""(""H#""a""H, one     could get the estimate

Hence, it remains to prove that the solution is unique in the class of solutions lying in < ()),   constructed starting from di!erent origins and verifying Eq. (3.16). Let us call a solution U, and assume to choose a new origin in x "j'0; starting from there let  us construct a di!erent solution U of Eq. (3.1). By construction the two solutions will satisfy, respectively, ""U"" X )A z#A , ∀z31>,    ""U"" HXY )A z#A , ∀z31>,    where )H is the domain analogous to ) but `cenXY X treda in x "j, and z"z!j. Thus, for z'j#1  (because, in the proof of existence, it must be x '1) we have )H L) and  X\H X "" U"" HX\H )"" U"" X )A z#A ,     ∀z'j#1, "" U"" HX\H )A z#(A !A j),     Hence, setting z'j#1,

U"U!U,

∀z'j#1. we

infer,

for

"" U"" HX\H )A z#A .    On the other hand, if we write the equation for the di!erence, multiply it by U and integrate by parts over )H , since ' U"0, and U vanishes on *), X\H we obtain



H X\H






" U" d)#



!

H\X




X

and get



V

V \



1 *; Q; ! d&  2 *z



1 *; Q; ! d&"0,  2 *z

where Q is obviously the di!erence of the two pressures. Now, we can integrate in z3(x !1, x )  




"" U"" HX\H dz# 

H V\H








1 *; Q; ! d)  2 *z

B 1 *; Q; ! d)"0.  2 *z

#

""U""H)c (""(""H#""a""H).    

1089

BH\V>H By means of Schwarz and PoincareH inequalities, one immediately "nds

  

*; d))c"" U"" HX\H .   *z

HV\H Moreover, as in Ref. [20] one has the estimate



H V\H

"Q; " d))c"" U"" HX\H .   

 Finally, setting



s(x )" 

V

"" U"" HX\H dz  V \ and noticing that ds "" U"" HV \H " (x ),    dx   the previous estimates can be used to get the di!erential inequality ds (x ). s(x ))c  dx   This last can be integrated and furnishes s(x )*s(x )eAV \V    for j#1(x (x .   Therefore, we get, for all x 'j#1,     s(x )eAV \V )s(x ))"" U"" HV \H     )A x #A ,    which is impossible unless s(x )"0. So, the  uniqueness result follows from the arbitrariness of x and from the inequality  "" U"" HV\\H )s(x ).    The last part of the thesis again follows by Sobolev embedding theorems. )

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Before giving the existence result for Eq. (3.2), we underline that because of the construction of u , X a constant j'1 exists, not depending on z, such that for any V "V " )j"V" . X  

(3.17)

Theorem 2. Let ) be Lipschitzian. Given V3CK())  such that ' V"0, and F3HK()), with m*0, then if m"0 and 2l "V" ( ,  a j  where j is dexned in Eq. (3.17), Eq. (3.2) admits a unique solution W3H()) such that






\ l j ! "V" ""W""H)c    a 2  \ a j ; 1# "V" l!  "V" ""F""H.    2

 

Moreover, if m*1 and 2l "V" ( , (3.18)  a j(2m#1)  then Eq. (3.2) admits a unique solution W3HK()). Finally, for m"1 and m*2, respectively, we have






WX of Eq. (3.6) satis"es ""WX"" V )A x #A ,    K 

(3.19a)

l"" WX"" Q #a   !a 








; 1# "V"

 

\Q

Q

1 u < " WX" d& X 2

3 # a sup " V (x)" "" WX"" Q . (3.21) X  2  VZX We recall that, by construction, sup X " u " deVZ X pends only on a "xed d3(0, 1), and not on z. Thus, using Cauchy inequality, integrating again over (x !1, x ) and taking into account that  



BV



1 u < " WX" d) X 2



#

B\V

)"V"



1 u < " WX" d) X 2

d  dx



V

 V \

"" WX"" Q ds, 

further using Eq. (3.14) to estimate F, we arrive at V

V \

\ a j(2m#1) l!  "V"   2





1 u < " WX" d& X 2

)"" F"" Q "" WX"" Q  



\K l j(2m#1) ""W""H )c ! "V" K  a  2 

(3.20)

where A and A do not depend on x or z.    In order to do this, one can di!erentiate Eq. (3.6) and directly estimate the derivatives multiplying by DIWX, for k"1,2, m and then integrate by parts over ) , with s3(x !1, x ). If k"1, we get Q  

\ l 3j ! "V" ""W""H)c    a 2  \ 3a j ; 1# "V" l!  "V" ""F""H,    2

 

∀x 3[1, z], 

"" WX"" Q ds 

d )c dx



V

 V \ 

"" WX"" Q ds#A x   

;PK\("V" )""F""H , (3.19b) K K where c is a positive number and PK\ is dexned in  Lemma 2.

with c independent of x . As a consequence of this  inequality, the procedure used in Ref. [18] tells us how to infer Eq. (3.20) from Eq. (3.10a). From a di!erential inequality of the kind

Proof. The scheme of the proof is the same as for Theorem 1. First, we have to show that the solution

ds s(x ))c (x )#A x #A , ∀x 3[1, z],      dx  

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

we deduce the estimate s(x ))A x #A , ∀x 3[1, z],      provided the `initial conditiona s(z))A z#A   is satis"ed. The algebraic computations necessary to evaluate A and A are really boring and are omitted   here. On the other hand, to get the result for m"1 one should have added the estimate for m"0, which is completely analogous and trivial to get. Let us consider the case m*2. For m*2, inequality (3.21) has to be modi"ed, according to Eq. (3.11), by replacing the factor  with (2m#1)/2  and by adding the term c(m)"V " ""DKWX"" Q X K  ""WX"" . This term can be estimated recursively, K\Q starting from the result for m"1. Thus, also for m"2, we "nally deduce Eq. (3.20) from hypothesis (3.10b). At this stage, a weak solution W3HK ()) can be  constructed by considering the `invading domainsa technique. What is essential to "nd existence in HK()), is again a uniqueness argument. This is necessary to prove that c'0 exists such that ""W"" )c, ∀z31. KBX As in Theorem 1, the constant c can be obtained by setting x " in Eq. (3.20), which gives exactly Eqs.   (3.19a) and (3.19b). To this end, once we have found existence of solutions lying in HK ()) which satisfy an estimate  of the kind (3.20), we show uniqueness in the same class, possibly changing the origin of the x  axis. This can be done exactly as in the proof of Theorem 1. In particular, we write the equation for the di!erence and we prove that its solution, denoted again by W, is zero in H ()). In fact, if we  multiply the equation by W and integrate by parts over )H , we obtain X\H l""W"" HX\H #a   !a 



H\X



<

X

1 < "W" d& 2

1 "W" d&"0. 2

1091

Now, we can integrate in z3(x !1, x ) and get the   equation l



V

""W"" HX\H dz#a  



<

1 "W" d)"0, 2

V \ HV\H which can be used to complete the proof through the same argument used in Theorem 1. ) Finally, in order to prove the existence of steady solutions of problem (2.5), that is to say solutions of system (2.6) and (2.7), we have to show the existence of a "xed point for the map F: (3HK())P W3HK()), with m*1, de"ned as follows. Consider problem (3.1) with ( prescribed in HK()) and solve it for the pair (U, Q)3VK>());HK()).  Then, use such a pair to construct the data for Eq. (3.2), according to Eqs. (3.3) and (3.4). So, we can de"ne W"F((), with W solution of Eq. (3.2), because V3VK>())LCK()) can satisfy the hy  potheses of Theorem 2 and, still for m*1, we can show that F3HK()). To this aim, consider the estimate in Ref. [12] (Lemma 3.1, p. 78). By means of elementary calculations, starting from Eq. (1.8), it can be generalized to third grade #uids giving, for all z31 "" ' N(U#a)""

KBX )c""U#a"" (1#""U#a"" ), K>BX K>BX where c is independent of z and m*1. Moreover, recalling Eq. (3.4) and using Eq. (3.13b) we "nd ""F""H )c""U#a""H (""U#a""H K> K K> #""U#a""H #"" Q""H ) K> K )c(""(""H #""a""H )(1#""(""H #""a""H ). K K> K K> (3.22) Finally, we shall use the classical Banach "xedpoint theorem in the complete metric space G"G(D) de"ned as follows: G(D)"+(3HK>()): ""(""H )D,. K> Let us show that G(D) is complete. To our purposes, G(D) must be understood, in spite of its de"nition, as a subset of the Banach space HK()). Therefore, if G(D) is a closed subset, then it is a complete metric space, with the distance deduced by the norm of HK. In order to prove that G(D) is

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closed as subset of HK()), it is su$cient to use the compact embedding HK>(d) )6HK(d) ). In fact, X X let us assume that +( , - is a sequence in G(D), L LZ converging to (3HK()) in the norm of HK. Since the sequence is bounded in HK>()), there exists also a weak limit ', for a subsequence +( I , - , L IZ such that ∀z31 ""'"" )D. K>BX But, since d) is bounded, we have by compactness X that ∀z31 P0 for kPR. ""( I !'"" KBX L Then, since the limit is unique, ( coincides with ', and this proves that G(D) is closed in HK. In the sequel, the `radiusa D will be chosen in such a way that for all ( ,( 3G(D)   ""F(( )!F(( )""H )g""( !( ""H , g3(0, 1),   K   K as we will show in the following theorem, which is the main result of this section. Theorem 3. Let ) be uniformly of class CK>, with m*1. Then, there exists c "c (m, l, a ) such that if    " "(c , (3.23)  the system !l*V#V ' (V!a *V)# %" ' N(V), 

' V"0 in ), V" "0, /   < d&" , V   where N(V)"a ( v)2A #(a #a )A #b(tr A )A ,        admits a unique solution (V, %)3VK>());HK()),  satisfying for some c "c (m, l, a , a , b)     ""V""H #"" %""H )c " ". K> K  Proof. Let us begin by noticing that the map F is well de"ned, provided that the (unique) solution of Eq. (3.2) exists. That is to say, that the (unique) solution of Eq. (3.1) has to satisfy Eq. (3.18). To this

aim, we can "rst set D"c" " and then choose " " in a suitable way. The estimates of Theorem 1 and Eq. (2.10f) give ""U""H #"" Q""H )c (""(""H #""a""H ) K> K>  K> K> )c(D#" "))c" ". (3.24) What we need is "V" )c""V""H)c(""U""H#""a""H)     2l . (3.25) )c(" ")( a j(2m#1)  This is a "rst smallness condition on the #ux. Moreover, we have to prove that F(G(D))L G(D), for some D"c" ", such that " " veri"es Eq. (3.25). In order to do that, taking into account Eq. (3.24) we "rst rewrite Eq. (3.19b) as follows: ""W""H )c G(" "(1#c))""F""H , K>  K> where G(" "(1#c)) is a strictly positive function and is continuous when " " is bounded by Eq. (3.25). Further, G depends on the material constants too. At this point, we put together this last estimate, (3.22) and (3.24), so getting ""W""H )c G(" "(1#c))""F""H K>  K> )cG(" "(1#c))(""(""H K> #""a""H )(1#""(""H #""a""H ) K> K> K> )cG(" "(1#c))(1#c)" "(1#(1#c)" ") (c" ". This condition can be satis"ed without falling in contradiction with Eq. (3.25), because the inequality G(" "(1#c))(1#c)" "(1#(1#c)" ")!c(0 is certainly satis"ed for any su$ciently small " ", which means that we have, possibly, a further restriction on " ". Further, it remains to prove that D can be chosen in such a way that F is a contraction in G(D)LHK()). Let us take (, (3G(D), consider W"F((), W"F(() and set W!W" W, (!("(; moreover, let us set V"U#a, V"U#a the data for Eq. (3.2)

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

corresponding to (, (, respectively. The di!erence W veri"es the equation lW#V ' W"F(V)!F(V)#(U!U) ' W and therefore also the estimate ""W""H )c G(" "(1#c))(""F(V)!F(V)""H K  K #""(U!U) ' W""H ). (3.26) K Again, generalizing the estimate in Ref. [12, p. 97], one can easily "nd, for m*1 "" ' N(V)! ' N(V)""H)c""U!U""H (""V""H K K> K> #""V""H )(1#""V""H #""V""H ), K> K> K> from which ""F(V)!F(V)""H K )c""(""H (1#c)" "(1#(1#c)" "). K On the other hand, we have

study the problem for the di!erence between the unsteady #ow and the steady solution of Eq. (2.5) corresponding to the same #ux . We recall that for such a solution (V, %) we have the existence in the `locala spaces and the estimates, which are obtained through the corresponding solution (U, W, Q) of the coupled systems (2.6a)}(2.6c), (2.7). Thus, instead of solving directly Eqs. (2.2) and (2.3), we shall study the corresponding problems for the di!erence (u, q), de"ned as u(x, t)"v(x, t)!V(x), q(x, t)"q(x, t)!Q(x). Such coupled systems are !a *u#u# q"u, 

' u"0 in );(0, ¹),

(3.27)

""(U!U) ' W""H )c""U!U""H ""W""H K K> K> )cc" " ""(""H , (3.28) K which is obtained by means of the higher regularity of functions lying in G(D) with respect to HK()): W3HK>()) is necessary to obtain a contraction in HK()). In fact, using Eqs. (3.27) and (3.28), estimate (3.26) becomes ""W""H )c""(""H G(" "(1#c))(1#c) K K ;" "(1#(1#c)" ") and we note that one can obtain cG(" "(1#c))(1#c)" "(1#(1#c)" ")(1,

1093

u" "0, /"2



V  

u d&"0 

(4.1)

and *u l # u#(u#V) ' u"f(u, q,V, Q) *t a  in );(0, ¹), u(x, 0)"u (x),  where, of course,

(4.2)

u(x, t)"w(x, t)!W(x)!V(x) and f(u, q,V, Q)" ' N(u#V)! ' N(V) l # u!u ' (V!*V# Q) a  ! (u#V)2 q! u2 Q.

by imposing a (possibly) new smallness condition on " ". Thus, we have the contraction, and the theorem is completely proved by recalling the definition of the transported pressure Q in terms of the real pressure %, and directly computing the estimate for this last function. )

As for the steady case, we shall prove local in time existence for system (4.1), (4.2) by means of the Banach "xed point theorem. To this end, we consider the two separated problems

4. Existence for the unsteady problem

!a *u#u# q"t, 

' u"0 in );(0, ¹),

In this section, we shall prove existence for problems (1.6), (1.7), (1.11), (1.12). Actually, we shall

u" "0, /"2



V 

u d&"0 

(4.3)

(4.4)

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A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

and l *u # u#(u#V) ' u"f in );(0, ¹), a *t  u(x, 0)"u (x), (4.5)  where t is prescribed in a suitable space. Concerning system (4.4), we note that the time t can be considered as a parameter. Thus, we can deal with this problem exactly in the same way as we did for Eq. (3.1) in the steady case. Only, we have a resolvent-Stokes problem in place of the simple Stokes. However, as is known (see, i.e., Refs. [7,22,24]), this problem is even easier, in unbounded domains, than the Stokes one. The existence result for the `restrictiona of the problem to the subdomains ) (see Lemma 1), can easily be X obtained with the Galerkin method. Therefore, we can give immediately an existence result in the whole of ). Recalling that the norm in =I((0, ¹); HK())) is denoted by " ' " for k'0, and by " ' " for IK2 K2 k"0, we choose the `forcea t in =I((0, ¹); HK())) and denote its norm in this local space by "t"H . Then, we write IK2 Theorem 4. Given t3=I((0, ¹); HK())), k, m*0, then if ) is uniformly of class CK>, system (4.4) admits a unique solution (u, q)3=I((0, ¹); VK>()));=I((0, ¹); HK())), satisfying  "u"H )c "t"H , I2  I2 "u"H #" q"H )c "t"H . (4.6) IK>2 IK2  IK2 Moreover, if t3=I((0, ¹); C())) and ) is uniformly of class C, then (u, q)3[=I ((0, ¹); C()))]. At this stage, we can attend to the transport problem (4.5). First, we shall show a result analogous to that of Lemma 2, concerning a model problem in the bounded domains ) . X To this end, we set v "u v, z31> X X with u de"ned after Eq. (3.7), and we consider v as X X a data of the problem we are going to study. As

announced in Section 2, we denote with " ' " , IK2 where k, m*0, the norm of the spaces =I((0, ¹); CK() )). X Lemma 3. Let ) be Lipschitzian and z31> xxed. Given for m*0, v 3¸((0, ¹); CK() )), X  X f3¸((0, ¹); HK() )), and u 3HK() ), then the X  X problem *uX l # uX#v ' uX"f in ) ;(0, ¹), (4.7a) X X *t a  uX(x, 0)"u (x), (4.7b)  admits a unique solution uX3¸((0, ¹); HK() ))5=((0, ¹); HK\() )) and we have X X "uX" )c PK\(¹)(""u "" #¹"f" ), (4.8a) K2   K K2 "uX" )c (""uX"" #"f" ), m*1 K\2  K2 K\2 (4.8b) with c "c (m, l, a , ¹) and c "c (m, l, a , ¹),       while, if m'0, PK\(¹) represents a monic polynomial of degree m!1, whose (positive) coezcients depend on "v " , and we set P\(¹)"1. X K2 Moreover, if 2l , "v " ( X 2 a (2m#1)  then c does not depend on ¹. 

(4.9)

Proof. Again, we can use the Galerkin method to show existence in ¸((0, ¹); H() )). Estimates X (4.8a) and (4.8b) can be deduced in the same way as in Lemma 2. Remark 1. This result for uX3¸((0, ¹); HK() ))5 X =((0, ¹); HK\() )), can immediately be exX tended to include uX3=I((0, ¹); HK() )5 X =I>((0, ¹); HK\() ), which is actually needed X to "nd existence of classical solutions. At this point, we use the `invading domainsa technique to proof existence for problem (4.2). We give a result which is, in the unsteady case, the analogous of Theorem 2: from an estimate of linear growth of the norm in ) , and from the uniqueness X of the weak solution, we deduce the existence in local spaces.

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

Theorem 5. Let ) be Lipschitzian. Given f3¸((0, ¹); HK())), u 3HK()) and v3¸  ((0, ¹); CK())), with m*0 and ' v"0, then the  problem l *u # u#v ' u"f in );(0, ¹), *t a  u(x, 0)"u (x), (4.10)  admits a unique solution u3¸((0, ¹); HK()))5 =((0, ¹); HK\())). Further, ¹M (¹ exists such that "u"H )c (¹M )(""u ""H #¹"f"H ), ∀¹(¹M , K2Y   K K2Y where c "c (m, l, a , ¹M ) is increasing with, and    continuously depends on "v" . If moreover, K2M 2l "v" ( , 2M ja (2m#1)  then )c (¹M )(""u ""H #(¹"f"H ) "eIRu(x, t)"H K2Y K2Y   K ∀¹(¹M . (4.11) ), while where k3(0, l!ja [(2m#1)/2]"v"  2M c "c (m, l, a ,¹M ) depends continuously on    "v" , is increasing and strictly greater than 1. K2M Finally, for all ¹'0 and for any data, one has "u"H )c ("u"H #"f"H ) ∀m*1, K\2  K2 K\2 where c "c (m, l, a , ¹) depends continuously on    "v" . K2 Proof. We "rst show a linear estimate of the kind (3.20) for u in ¸((0, ¹); HK()  )), with x 3[1, z]; X V  next, we use it to proof the linear growth of the norm in ¸((0, ¹); HK()  )). V The starting point is still the same: we perform the scalar product in HK() ) (s(z), of Eq. (4.7a) with Q u . Hence, after some calculations, we can write X 1 d ""uX(t)"" Q K 2 dt






l 2m#1 ! "v " ""uX(t)"" Q X 2 K a 2  1 K # u v "D?uX(x, s, t)" d& X  2 Q ?

#



1095



K u v "D?uX(x,!s, t)" d& X  \Q ? )(""f(t)"" Q #c(m)"v " X K2 K ;""uX"" )""uX(t)"" Q , (4.12) K K\Q where the last term in the last line appears only for m*2. Clearly, once existence is proved in the spaces with m"0 and 1, the further estimates can be found by using recursively Eq. (4.12). Now, let us set 1 ! 2

l c "2 !(2m#1)j"v" , (4.13)  2 a  where, as in Eq. (3.17), j'1 is a "xed constant, depending on the construction of u but not on z, X such that for any v "v " )j"v" . X 2 2 We note that c does not depend on z.  Further, we set c ""u v " ,  X  2 which can still be bounded independently on z. Moreover, we de"ne for x 3[1, z]  V s(x , t)" ""uX(t)"" Q ds.  K V \ Then, integrating over (x !1, x ), inequality   (4.12) becomes



*s (x , t)#c s(x , t)   *t 



V

)2

V \



""f(t)"" Q ""uX(t)"" Q ds K K

K "D?uX" d)  V ? 1 V ) ""f(t)"" Q ds#es(x , t) K  e  V \ #c ""uX(t)"" V ,  K  where use has been made of the generalized Cauchy inequality, with arbitrary positive e. #c



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A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

Since by hypothesis f3¸((0, ¹); HK())) we have ""f(t)"" Q )(2s#1)"f"H )(2s#1)"f"H , K KR KR t(¹ and we can integrate using this estimate. Therefore, using Cauchy inequality and following the same line as in Theorems 1 and 2, we "nally get 2 *s *s (x , t)#c s(x , t)) x "f"H #c (x , t),    *x  e  KR *t   (4.14) where c "c !e.   Still by hypothesis, integrating the initial condition, we have s(x ,0))2x ""u ""H.    K Thus, if we multiply both sides of Eq. (4.14) by eA R and integrate over (0, t), with t(t(¹, we obtain





t s(x , t))e\A RY 2x ""u ""H# "f"H   K  e KRY 



RY *s (x , t) dt . *x    At this point, we put #c 



c "1 if c '0,   c "eA 2 if c )0   and write





t s(x , t))c 2x ""u ""H# "f"H     K e KRY





RY *s (x , t) dt . *x    Note that in this inequality we can increase the bound of integration of the last term, because (*s/*x )(x , t) is positive.   Next, we set #c 



((x ,q)" 

O

s(x , t) dt,   in order to integrate the inequality in t3(0, q), with q(¹. After this, by increasing q with ¹ on the righthand side, we infer that for all q3(0, ¹) and all

x 3[1, z] 





¹ ((x ,q))c 2x ¹""u ""H# "f"H     K 2e K2





*( #c ¹ (x ,q) . (4.15)  *x   In order to apply the already quoted result concerning di!erential inequalities, we rewrite Eq. (4.15) as follows: *( ((x , t))A (¹)x #A (¹)#C(¹) (x , t).     *x   Moreover, we need the `initial conditiona. So, recalling Eq. (4.8a), we estimate



((z, t))

R

""uX(t)"" X dt)¹ ess sup ""uX(t)"" X K K  RYZ2

)¹(c PK\(¹))(2z#1)(""u ""H#¹"f"H ).   K K2 That is to say ((z, t))A (¹)z#A (¹).   Therefore, as was done in studying the steady case, for any ¹ we can "nd A (¹) and A (¹) (which   come out continuous in ¹) such that for all x 3[1, z] and all t3[0, ¹]  ((x , t))A (¹)x #A (¹).     For the sake of brevity, we omit the computation of A (¹) and A (¹).   At this point, we take into account that ""uX(t)"" V \ )s(x , t))""uX(t)"" V  K  K  to deduce



(4.16)






2 1 ""uX(t)"" V \ dt)((x , ¹))A(¹) x # K    2  (4.17)

for all x 3[1, z], which gives the estimates for uX in  ¸((0, ¹); HK()  )). V Further, we need to go from ¸((0, ¹); HK()  )) V to ¸((0, ¹); HK()  )). V We "rst recall that the last term on the righthand side of Eq. (4.14) is ""uX(t)"" V )""uX(t)"" V . K   K 

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

Next, we recall Eq. (4.14), where in place of e we write e, because it can be chosen di!erently. Now, if we again multiply both sides of Eq. (4.14) by eA R, integrate over (0, t) (with 0(t(t(¹) and use Eq. (4.16), we obtain the estimates




t eA RYs(x , t))2x ""u ""H# "f"H  K   e KRY #c





RY 



""uX(t)"" V dt K 

(4.18)

and





t 2x ""u ""H# "f"H s(x , t))c   K   e KRY #c 

RY



""uX(t)"" V dt K 

)A (¹)x #A (¹)   



 (4.19)

for all t3[0, ¹]. Moreover, by recalling again Eq. (4.16), we see that Eq. (4.19) implies, for all x 3[0, z!1] and for  all ¹(¹M , ess sup ""uX(t)"" V K  RZ2Y )(c (¹M ))(x #)(""u ""H #¹"f"H ), (4.20)     K K2Y , is nonwhere c (¹M ), depending also on "v"  K2M decreasing, strictly greater than 1 and continuous. By means of a possibly di!erent choice of e, an analogous estimate of linear growth will lead to Eq. (4.11). It clearly should follow from Eq. (4.18) for c positive. To this end, we need e(c , and   therefore we have the condition c '0. Now we  note that the estimate easily follows from the fact that ¹((¹ for small ¹. It is of the kind ess sup ""eA R uX(t)"" V K  RZ2Y )(c (¹M ))(x #)(""u ""H #(¹"f"H ),     K K2Y (4.21) where ¹(¹M and c (¹M ) goes to in"nity for  c P0, i.e. for c P0.  

1097

Finally, we need to prove that for all x 3[0, z!1]  *uX(t)  ess sup )A (¹)x #A (¹), (4.22)    *t KV RZ2 which can be done by taking the scalar product in HK\()  ) of Eq. (4.7a) with *uX(t)/*t, and then V using Schwarz inequality. At this stage a solution u3¸((0, ¹); HK ()))5=((0, ¹); HK\())) can be constructed   by considering the `invading domainsa technique, as done in Theorems 1 and 2. Such a solution veri"es, for all x 3[0,R), estimates (4.20)}(4.22).  We remark that, though such estimates hold for small ¹ only, by construction one has that the solution actually exists also for large ¹. As in the previous cases, provided we have uniqueness, we "nd the existence of solutions for Eq. (4.10) in `locala spaces. Moreover, by recalling Eqs. (4.20) and (4.21) for ¹(¹M , and substituting x " in them, we "nd   the "rst estimate of the statement, and Eq. (4.11), respectively. The last estimate can be inferred from Eq. (4.22). Therefore, the last step of the proof consists in showing that the solution of problem (4.5) is unique in the class of solutions whose ¸-norm in ) is linearly growing with z. We shall do this X following the same procedure as for Theorems 1 and 2: let us take the equation for the di!erence d between two solutions, multiply it by d and integrate, "rst over ) , and then on s3(x !1, x ). We Q   obtain

 





1 * V l V ""d(t)"" Q ds# ""d(t)"" Q ds   2 *t  a V \  V \ 1 V # ds v "d(x, s, t)" d&(s)  2  V \ Q 1 \V # ds v "d(x, s, t)" d&(s)"0,  2 \V \ Q which implies

 










V * eJR? ""d(t)"" Q ds  *t  V \ V * )"v" eJR? ""d(t)"" Q ds . 2 *x   V \






1098

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

Since d(x, 0)"0, after having integrated twice with respect to t, the "nal result is



2 eJR? 




V

V \

)¹"v"






""d(t)"" Q ds dt 



* 2 eJR? 2 *x  

V

V \



""d(t)"" Q ds dt.  (4.23)

As in Theorems 1 and 2, the last consequence is the exponential growth of the left-hand side of Eq. (4.23) with respect to x , while by hypothesis we  "nd



2 eJR? 




V

V \



""d(t)"" Q ds dt 

)¹eJ2? ess sup ""d(t)"" V   RZ2 )ess sup A (t)x #ess sup A (t).    RZ2 RZ2 As a consequence, a growth which is at least exponential should be still less than linear. This is impossible unless d(x, t)"0. So the proof is completed. 䊐 Remark 5. In the sequel, we will see that local in time existence does not depend on the sign of c .  Nevertheless, since some of the estimates involved in the proof of local existence will appear also in the proof of global existence, in which c '0, we will  use a unique estimate for u, good for both arguments. That is to say "u"H )c (¹M )(""u ""H #(¹"f"H ), (4.24) K2Y K2Y   K which holds true for ¹(¹M (1, with c "c if   c )0, while c "max+c ; c , if c '0. Esti     mate (4.24) is used to proof Theorem 6 in place of the estimates of Theorem 5. Finally, in order to state the existence of the (unique) solution to systems (4.1), (4.2), we shall prove the existence of a "xed point for the map F : t3¸((0, ¹); HK>())) Pu3¸((0, ¹); HK>()))5=((0, ¹); HK())),

which is de"ned in a manner analogous to that of Section 3, by applying the Banach "xed point theorem. To be precise, F is a contraction when we de"ne it in the larger space ¸((0, ¹); HK())), and the "xed point will be found there. Actually, we note that though u is in ¸((0, ¹); HK>()))5=((0, ¹); HK())), we do not "nd the "xed point in the last space. Therefore, in the sequel, we understand the fact that such function belongs to =. Precisely, consider problem (4.4) (where the time t is a parameter) with t prescribed in ¸((0, ¹); HK>())), and solve it. By means of Theorem 4, we "nd the solution (u, q)3¸((0, ¹); VK>()));¸((0, ¹); HK>())).  Then, insert such a pair in the de"nition of the data of Eq. (4.5). At this point, we can de"ne F(t)"u, where u is the solution to Eq. (4.5) belonging to ¸((0, ¹); HK>())), as stated in Theorem 5. In fact, the hypothesis on v"u#V is veri"ed because on the one hand, Theorem 3 ensures us that ""V""H #"" Q""H )c " " K> K>  while from Theorem 4, estimate (4.6), we have "u"H #" q"H )c "t"H . K>2 K2  K2 Concerning the hypothesis on f in Theorem 5, we can easily verify that it is ful"lled. First, we recall de"nition (4.3), the previous estimates for V, q and u, and the estimate immediately before Eq. (3.27). Further, one can estimate the extraterms by locally applying Sobolev embedding theorems and using HoK lder inequality. For instance, one has " u2 Q"H )c"u"H "" Q""H )c" " "t"H . K2 K>2 K>2 K2 The "nal result is for m*1 "f"H )c"t"H (1#"t"H #" "). (4.25) K2 K2 K2 All the considerations before Theorem 3 hold again with the obvious changes related to the spaces ¸((0, ¹); HK()))5=((0, ¹); HK\())). So, now G"G(D) is given by G(D)"+t3¸((0, ¹); HK>())): t(x, 0)"u (x), "t"H )D,.  K>2

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

Again, to our purposes, G(D) must be understood as a subset of the Banach space ¸((0, ¹); HK())) and, as one can immediately verify, G(D) is a closed subset in ¸((0, ¹); HK())). Finally, we are able to show the existence theorem for the unsteady case. Theorem 6. Let ) be uniformly of class CK>, with m*1. Let v be given in VK>()). Then, there exists   ¹H'0 such that if " "(c ,  where c "c (m, l, a ), and if moreover,    ""v !V""H )c ,  K>  where c "c (m, l, a , " "), and V is the steady    solution corresponding to the same yux, then for all ¹3(0, ¹H), the system * (v!a *v)!l*v#v ' (v!a *v)# n   *t " ' N(v),

' v"0 in );(0, ¹),

"u"H )c (""u ""H #(¹"f"H ) K>2   K> K>2 )c ["u "H #(¹"t"H K>2   K> ;(1#"t"H #" ")] K>2 )c [""u ""H #(¹g""u ""H  K>   K> ;(1#g""u ""H #" ")]  K> (g""u ""H . (4.27)  K> Thus, from the previous inequality we see that, in order to have that the image of a ball is contained in the ball, it must be






v d&" ,   V  v(x, 0)"v (x),  where

We remark that any function in F(G(D)) has a bounded HK-norm for the time derivative. Nevertheless, it is unnecessary to prove that the map has a "xed point in G(D), thought as a closed subset of ¸((0, ¹);HK())). Hence, we start by showing that F(G(D))LG(D), at least for all ¹ less than some ¹H'0. To this end, we recall Eqs. (4.24), (4.25), together with the de"nition of G(D), we set D"g""u ""H  K> with g*1, and "nally, we require that

(4.26)

N(v)"a ( v)2A #(a #a )A #b(tr A )A        admits a unique solution v3¸((0, ¹); VK>()))5=((0, ¹); VK>())),  

n3¸((0, ¹); HK()))5=((0, ¹); HK\())). Proof. Actually, we are looking for the solution (u, q, u) of systems (4.1), (4.2). First, we assume that " " is su$ciently small to guarantee the existence of the steady solution V, found in Theorem 3, which gives the "rst smallness condition in the statement. Further, we note that the map F is well de"ned in any case.

g



1 . g(1#g""u ""H #" ")   K> Then, in order to "nd a positive number ¹H it is  clearly su$cient to "x g'c .  Now, we recall that c (¹M ) is increasing continu ously with the norm of v"u#V; therefore, by recalling Eqs. (4.6), (3.23) and (4.25), we see that one can take c (¹M ) increasing continuously with  g""u ""H #" ""y.  K> So, the condition g'c (y) is achieved if  (y!" ")/""u ""H 'c (y). From this, we deduce  K>  that ""u ""H must satisfy  K> y!" " ""u ""H ( .  K> c (y)  Since the function of y on the right-hand side has an upper bound, we "nd g'c provided that  ""u ""H is su$ciently `smalla.  K> This leads to the second smallness condition in the statement. In order to show this, it is su$cient to recall that the initial condition q (x) is the auxili ary problem (2.1) is arbitrary. As a consequence of (¹((¹H"  c

v" "0, /

1099

!1

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A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

this, in writing the coupled problems (4.1), (4.2) we can always choose

Therefore, we can apply Eq. (4.24) to this case, obtaining

q(x, 0)" q ! Q"0.  This implies

"u !u "H )c(¹"(f !f )   K2   !(u !u ) ' u "H    K2

u !*u "u    and then ""u ""H)""u ""H  I  I> for all integer k'0. Since v "u #V (and the   #ux must be small, in order to "nd the steady solution), we "nd the condition on v .  Note that ¹H has a maximum for some gH'  c '1.  Moreover, once we have "xed the initial condition, we still have freedom in the de"nition of G(D), depending on g. Hence, let us assume that we have a "xed point: one could think that it corresponds to a unique solution only if we do not change G(D). But, this is not true because whenever g (g , then   G(D )LG(D ) and the two solutions coincide in   the smallest of the two intervals of existence. Therefore, by varying g we shall always get the same solution. It remains to prove that, under suitable conditions, F is a contraction in ¸((0, ¹); HK())). To this end, let us set u "F(t ) and   u "F(t ), with t , t 3G(D). Next, let us call     (u , q ) and (u , q ) the solutions of Eq. (4.4) corre    sponding to t and t , respectively. Moreover, we   denote with f and f the forces de"ned as in Eq.   (4.3) corresponding to (u , q ) and (u , q ), respec    tively. Obviously, V and Q in Eq. (4.3) are always the same, related to the steady #ow we are perturbating. At this point, we subtract the two di!erent problems (4.5), and we see that u !u is solution of   the problem

∀¹(¹M . (4.28)

(It is understood that once D and are "xed, then c (¹M ) is a number.)  After some calculations, Eq. (4.28) gives the following estimate: "u !u "H )(¹+c[""u ""H  K>   K2 #(1#(¹M g""u ""H )(1#g""u ""H  K>  K> #" ")],"t !t "H .   K2 This means that in order to have a contraction, it is su$cient to choose ¹ less than ¹H de"ned by  (¹H+c[""u ""H #(1#(¹M g""u ""H )  K>  K>  ;(1#g""u ""H #" ")],"1.  K> Thus, the theorem is proved with ¹H"min+¹M ; ¹H; ¹H,. )   On the wake of Ref. [12], our next objective is to show that the solution identi"ed in Theorem 6 exists for t3[0,#R) (global existence), provided that the viscosity is su$ciently large with respect to a . Moreover, the perturbation decays exponenti ally with respect to t, which means that the steady solution is asymptotically stable. Theorem 7. Let v3¸((0, ¹); VK>()))5=((0, ¹); VK>())),  

n3¸((0, ¹); HK()))5=((0, ¹); HK\()))

* (u !u )#l(u !u )    *t 

with m*1, be the unique solution found in Theorem 6 for system (4.26). Let V be the solution of the steady problem corresponding to the same yux , so that v(x, t)"u(x, t)#V(x). Then, by assuming

#(u #V) ' (u !u )    "(f !f )!(u !u ) ) u ,      u(x, 0)"0 in );(0, ¹).

l c (""u ""H #"V" )#c ( , (4.29)   K>   a  where c (m)'0 and c (m)'0, it follows that   v(x, t) actually exists for all t3(0,R). Moreover, two

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

constants k, D'0 exists such that u verixes ""u(t)""H )De\IR. K>

1101

Theorem 6, then we have also (4.30)

Proof. We shall show that if l/a is su$ciently  large, the solution v can be extended to recover the whole time interval [0,R). In particular, we shall "nd that, with a given ¹ (¹H, one can rede"ne  F in ¸((n¹ ,(n#1)¹ ); HK>())), in such a way   that a "xed point can be found for any integer n. Let us recall the explicit expression of the constant l !e. c "c !e"2 !(2m#1)j"v" 2   a  Since, from Sobolev embedding theorems, from Eq. (4.6) and from the estimate of Theorem 3, we have "v" )"V" #"u" 2  2 )"V" #c"u"H )"V" #c c"u"H  2   2 and since the space of the "xed point (see the proof of Theorem 6) is such that "u"H )g""u ""H , 2  K> by taking into account that e can be arbitrarily small, we obtain 2l c *c " !(2m#1)j("V"   a   #c cg""u ""H )!e'0,   K> provided that 2l c cg""u ""H #"V" ( . (4.31)   K>  ja (2m#1)  Recalling Eq. (3.25), one can easily see that this condition does not imply the further restriction on the #ux which could arise by setting u "0 in Eq.  (4.31). On the other hand, Eq. (4.31) implies c '0.  When this condition is satis"ed we can use Eq. (4.11) for small ¹, and we can replace c with  c in it.  By comparing Eqs. (4.11) and (4.27), one can immediately infer that if u is the "xed point of

eRA ""u(t)""H )g""u ""H , (4.32) K>  K> for all t less than ¹H. Now, we want to use Eq. (4.32) to show that ¹ (¹H exists such that u(x, ¹ ) can be chosen as   a new initial condition in de"ning G(D) and imposing that F(G(D))LG(D) in the interval (¹ , ¹ #¹H). To this end, we require   ""u(t)""H )""u ""H K>  K> for some t for which the solution exists. From Eq. (4.32) we "nd the condition e\RA g)1. This condition is veri"ed for all t such that 2 t' ln(g). c  Therefore, we can choose 2 ln(g), ¹ "  c  provided that 2 ln(g)(c ,  ¹H

(4.33)

since it must be ¹ (¹H. This means that  c should be bounded from below. In fact, even for  vanishing data, the left-hand side is bounded from below. Precisely, one can see this by "xing " " and the initial data and evaluating ¹H by means of its de"nition. For instance, we can set there g" (1#p)c , where p is positive and arbitrary.  Hence, Eq. (4.33) is a condition on l/a through  c , which is certainly ful"lled if Eq. (4.29) holds  true with suitable constants. Clearly, this procedure can be repeated recursively to extend the solution to the intervals (n¹ , (n#1)¹ ), and the condition of existence is   always the same. Note that Eq. (4.29) actually `containsa also Eq. (4.31) for c "0. In order to see this, it is su$cient  to recall the argument used in Theorem 6, concerning the freedom in the choice of the initial condition q (x) of the auxiliary problem (2.1). 

1102

A. Passerini, M.C. Patria / International Journal of Non-Linear Mechanics 35 (2000) 1081}1103

Finally, from Eqs. (4.6) and (4.32) we immediately infer Eq. (4.30). )

Acknowledgements The authors thank Professor M. Padula and Professor G.P. Galdi for helpful discussions. In particular the authors express their thanks to Professor Padula for suggesting to them the proof of the uniqueness in Theorem 1. This work has been made under the auspices of GNFM of CNR and supported by the italian MURST.

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