In this chapter we present an updated account of the fundamental mathematicalresults pertaining the steady-state flow of a Navier–Stokes liquid past a rigid bodywhich is allowed to rotate. Precisely, we shall address questions of existence,uniqueness, regularity, asymptotic structure, generic properties, and (steady andunsteady) bifurcation. Moreover, we will perform a rather complete analysis ofthe longtime behavior of dynamical perturbation to the above flow, thus inferring,in particular, sufficient conditions for their stability and asymptotic stability.
1 Introduction
The motion of a rigid body in a viscous liquid represents one of the mostclassical and most studied chapters of applied and theoretical fluid mechanics.Actually, the study of this problem, at different scales, is at the foundation ofmany branches of applied sciences such as biology, medicine, and car, airplane,and ship manufacturing, to name a few. The dynamics of the liquid associated tothese problems is, of course, of the utmost relevance and, already in very elementarycases, can be quite intricate or even, at times, far from being obvious. For example,consider a rigid sphere of radius R, moving by constant translatory motion withspeed v0 and entirely immersed in a surrounding liquid, of kinematic viscosity�. Then, it is experimentally observed (see [67]) that if Re WD v0R=� . 200,the flow is steady, stable, and axisymmetric. However, if 200 . Re . 270, thisflow loses its stability, and another stable, steady, but no longer axisymmetric flowsets in, as evidenced by the loss of rotational symmetry of the wake. It is worthemphasizing the loss of symmetry of the flow, in spite of the symmetry of the data.Moreover, if 270 . Re . 300, the steady flow is unstable, and the liquid regimebecomes oscillatory, as shown by the highly organized time-periodic motion of thewake behind the sphere. The remarkable feature of this phenomenon is that theunsteadiness of the flow arises spontaneously, even though the imposed conditionsare time independent (constant speed of the body). Another significant example isfurnished if now the sphere, instead of moving by a translatory motion, rotates withconstant angular velocity, !0, along one of its diameters. Here, again in view ofthe symmetry of the data, one would guess that, at least for “small” values of j!0j(more precisely, of the dimensionless number j!0jR2=�), the flow of the liquid issteady with streamlines being circles perpendicular to and centered around the axisof rotation. Actually, this is not the case, unless the inertia of the liquid is entirelydisregarded. In fact, though the flow is steady, due to inertia, the sphere behaves likea “centrifugal fan,” receiving the liquid near the poles and throwing it away at theequator; see [12, 85].
Already from these brief considerations, one can fairly deduce that a rigorousmathematical study of the motion of a viscous liquid around an obstacle presents aplethora of intriguing problems of considerable difficulties, beginning with the veryexistence of steady-state solutions under general conditions on the data and their
Steady-State Navier–Stokes Flow Around a Moving Body 3
uniqueness going through more complicated issues such as analysis of steady andtime-periodic bifurcation and longtime behavior of time-dependent perturbations. Itis the objective of this chapter to address some of these fundamental problems, aswell as point out certain outstanding questions that still await for an answer.
In real experiments the liquid occupies, of course, a finite (though “sufficientlylarge”) spatial region. However, “wall effects” are irrelevant for the occurrence ofthe basic phenomena of the type described above. Therefore, in order not to spoiltheir underlying causes, it is customary to formulate the mathematical theory of themotion of a body in a viscous liquid as an exterior problem. This corresponds to theassumption that the liquid fills the entire three-dimensional space outside the body.It should be remarked that this assumption, though simplifying on one hand, on theother hand adds more complication to the mathematical analysis, in that classicaland powerful tools valid for bounded flow are no longer available in this case. Asit turns out, most of the questions that we shall analyze require, for their answers, asomewhat detailed analysis of the solutions at large distance from the body.
From a historical viewpoint, the mathematical analysis of the steady flow of aviscous liquid past a rigid body may be traced back to the pioneering contributionsof Stokes [105], Kirchhoff [71], and Thomson (Lord Kelvin) and Tait [106] in themid- and late 1880s. However, it was only in the 1930s that, thanks to the far-reaching and genuinely new ideas introduced by Jean Leray [81], the investigationof the problem received a substantial impulse. Leray’s results, mostly devoted tothe existence problem, were further deepened, extended, and completed over theyears by a number of fundamental researches due, mostly, to O.A. Ladyzhenskaya,H. Fujita, R. Finn, K.I. Babenko, and J.G. Heywood. It is important to observe thatthe efforts of all these authors were directed to the study of cases where the body isnot allowed to spin. The more general and more complicated situation of a rotatingbody became the object of a systematic study only at the beginning of the thirdmillennium, with the basic contributions, among others, of R. Farwig, T. Hishida,M. Hieber, Y. Shibata, the authors of the present paper, and their associates.
The main goal of this review chapter is to furnish an up-to-date state of theart of the fundamental mathematical properties of steady-state flow of a Navier–Stokes liquid past a rigid body, which is also allowed to rotate. Thus, existence,uniqueness, regularity, asymptotic structure, generic properties, and (steady andunsteady) bifurcation issues will be addressed. In addition, a rather completeanalysis of the longtime behavior of dynamical perturbation to the above solutionswill be performed to deduce, in particular, sufficient conditions for their stabilityand asymptotic stability as well.
With the exception of part of the last section, Sect. 10, this study will be focusedon the case when the translational velocity, v0, of the body is not zero and itsangular velocity is either zero or else has a nonvanishing component in the directionof v0. The reason for such a choice lies in the fact that under these assumptions,the mathematical questions listed above have a rather complete answer. On theother side, if one relaxes these assumptions, the picture becomes much less clear.The interested reader is referred to [45, §§ X.9 and XI.7] for all main propertiesknown in this case. Furthermore, for the same reason of incompleteness of results,
4 G.P. Galdi and J. Neustupa
only three-dimensional flow will be considered. An update source of informationregarding plane motions can be found, for example, in [45, Chapter XII], [36], and[56]. Finally, other significant investigations are left out of this chapter, such as themotion of the coupled system body–liquid (i.e., when the motion of the body is nolonger prescribed, but becomes part of the problem), as well as the very importantcase when the body is deformable, for which the reader is referred to [38] and [5,44],respectively.
The plan of the chapter is as follows. After collecting in Sect. 2 the main notationused throughout, in Sect. 3 it is provided the mathematical formulation of theproblem. Section 4 is dedicated to existence questions. There, one begins to recallclassical approaches and corresponding results due to Leray, Ladyzhenskaya, andFujita. Successively, improved findings obtained by the function-analytic methodintroduced by Galdi are presented, based on the degree for proper Fredholm mapsof index 0. Regularity and uniqueness questions of solutions are next addressedin Sects. 5 and 6, respectively. Section 7 is dedicated to the (spatial) asymptoticbehavior, beginning by recalling the original results of Finn and Babenko when thebody is not spinning to the more recent general contributions of Galdi and Kyed andDeuring and their associates, valid also in the case of a rotating body. Successively,in Sect. 8, one investigates the geometric structure of the solution manifold for dataof arbitrary “size.” In particular, it is shown that, generically, the number of solutionscorresponding to a given (nonzero) translational velocity and angular velocity isfinite and odd. Sect. 9 is devoted to steady and time-periodic (Hopf) bifurcationof steady-state solutions. There, it is provided necessary and sufficient conditionsfor this type of bifurcation to occur. In the final section, Sect. 10, one analyzesthe longtime behavior of time-dependent perturbations to a given steady state,providing, as a special case, sufficient conditions for attractivity and asymptoticstability. These results can be, roughly speaking, grouped in two different categories.The first one is where one assumes that the unperturbed steady state is “small insize.” In the second one, instead, one makes suitable hypothesis on the locationin the complex plane of eigenvalues of the relevant linearized operator (spectralstability).
In conclusion to this introductory section, it is worth emphasizing that throughoutthis chapter, there have been highlighted a number of intriguing unsettled questionsthat still need an answer and represent as many avenues open to the interestedmathematician.
2 Notation
The symbols N, Z and R, C stand for the sets of positive and relative integers andthe fields of real and complex numbers, respectively. We also put NC WD N\.0;1/,RC WD R \ .0;1/.
Vectors in R3 will be indicated by bold-faced letters. A base in R
3 is denoted byfe1; e2; e3g � feig and the components of a vector v in the given base by v1, v2,and v3.
Steady-State Navier–Stokes Flow Around a Moving Body 5
Unless stated otherwise, the Greek letter ˝ will denote a fixed exterior domainof R3, namely, the complement of the closure, B, of a bounded, open, and simplyconnected set of R
3. It will be assumed ˝ of class C2, and the origin O of thecoordinate system fO; eig is taken in the interior of B. Also, d is the diameter ofB, so that, settingBR WD fx 2 R
3 W .x21Cx22Cx
23/
12 < Rg,R > 0, one has B � Bd .
For R � d , the following notation will be adopted
˝R D ˝ \ BR; ˝R D ˝ �˝R;
where the bar denotes closure.One puts ut WD @u=@t , @1u WD @u=@x1, and, for ˛ a multi-index, one denotes
by D˛ the usual differential operator of order j˛j. For j˛j D 2 one shall simplywrite D2.
Given an open and connected set A � R3; Lq.A/, Lqloc.A/, 1 � q � 1;
W m;q.A/; Wm;q0 .A/ .W 0;q � W
0;q0 � Lq), W m�1=q;q.@A/, m 2 NC [ f0g,
stand for the usual Lebesgue, Sobolev, and trace space classes, respectively, of realor complex functions. (The same font style will be used to denote scalar, vector,and tensor function spaces.) Norms in Lq.A/, W m;q.A/, and W m�1=q;q.@A/ areindicated by k:kq;A, k:km;q;A, and k:km�1=q;q.@A/. The scalar product of functionsu; v 2 L2.A/ will be denoted by .u; v/A. In the above notation, the subscript A willbe omitted, unless confusion arises.
As customary, for q 2 Œ1;1� one lets q0 D q=.q � 1/ be its Hölder conjugate.By Dm;q.˝/, 1 < q < 1, m 2 NC, one denotes the space of (equivalence
classes of) functions u such that
jujm;q WDX
j˛jDm
� Z
˝
jD˛ujq� 1q<1;
and by Dm;q0 .˝/ the completion of C10 .˝/ in the norm j � jm;q . Moreover, setting
D.˝/ WD fu 2 C10 .˝/ W div u D 0g;
D1;20 .˝/ is the completion of D.˝/ in the norm j � j1;2. By D�1;20 .˝/ [D�1;20 .˝/],
one denotes the normed dual space of D1;20 .˝/ [D1;2
0 .˝/] and by h�; �i [Œ�; ��] theassociated duality pairing.
By Hq.˝/ it is indicated the completion of D.˝/ in the norm Lq.˝/, and onesimply writesH.˝/ for q D 2. Further, P is the (Helmholtz–Weyl) projection fromLq.˝/ onto Hq.˝/. Notice that, since ˝ is a sufficiently smooth exterior domain,P is independent of q.
If M is a map between two spaces, by D .M/, N .M/, and R .M/, one denotesits domain, null space, and range, respectively, and by Sp .M/ its spectrum.
In the following, B is a real Banach space with associated norm k � kB . Thecomplexification of B is denoted by BC WD BC iB . Likewise, the complexificationof a map M between two Banach spaces will be indicated by MC.
6 G.P. Galdi and J. Neustupa
For q 2 Œ1;1�, Lq.a; bIB/ is the space of functions u W .a; b/ 2 R ! B suchthat
Z b
a
ku.t/kqB dt
! 1q
<1; if q 2 Œ1;1/ I ess supt2.a;b/
ku.t/kB <1; if q D1:
Given a function u 2 L1.��; � IB/, u is its average over Œ��; ��, namely,
u WD1
2�
Z �
��
u.t/ dt:
Furthermore, one says that u is 2�-periodic, if u.t C 2�/ D u.t/, for a.a. t 2 R.Set
W22�;0.˝/ WD
nu 2 L2.��; � IW 2;2.˝/ \D1;2
0 .˝// and ut 2 L2.��; � IH.˝// Wu is 2� -periodic with u D 0
o
with associated norm
kukW22�;0WD
�Z �
��
kut .t /k22 dt�1=2C
�Z �
��
ku.t/k22;2 dt�1=2
:
One also defines
H2�;0.˝/ WDnu 2 L2.��; � IH.˝// W u is 2�-periodic with u D 0
o:
Finally, C , C0, C1, etc., denote positive constants, whose particular value isunessential to the context. When one wishes to emphasize the dependence of Con some parameter � , it will be written C.�/.
3 Formulation of the Problem
Suppose one has a rigid body, B, moving by prescribed motion in an otherwisequiescent viscous liquid, L, filling the entire space outside B. Mathematically, Bwill be taken as the closure of a simply connected bounded domain of class C2.For the sake of generality, a given velocity distribution is allowed on @B, due,for example, to a tangential motion of the boundary wall or to an outflow/inflowmechanism, as well as it is assumed that on L is acting a given body force. (Thepresence in the model of a body force other than gravity – whose contribution canbe always incorporated in the pressure term – could be questionable on physicalgrounds. However, from the mathematical point of view, it might be useful inconsideration of extending the results to more general liquid models, where nowthe “body force” would represent the contribution to the linear momentum equation
Steady-State Navier–Stokes Flow Around a Moving Body 7
of other appropriate fields.) In order to study the motion of L under these circum-stances, it is appropriate to write its governing equations in a body-fixed frame, S ,so that the region occupied by L becomes time independent. One thus gets
vt C .v � V / � rvC! v D ��v � rp C f
div v D 0
)in ˝ .0;1/: (1)
(For the derivation of these equations, we refer to [38, Section 1, eq. (1.15)].) Inthese equations, v; �p are absolute velocity and pressure fields of L, respectively,� and � its (constant) density and kinematic viscosity, and f is the body forceacting on L. Moreover,
V WD � C! x;
with � and !, in the order, velocity of the center of mass and angular velocity of Bin S . Finally, ˝ WD R
3nB is the time-independent region occupied by L that willbe assumed of class C2. (Several peripheral results continue to hold with less orno regularity at all. This will be emphasized in the assumptions occasionally.) Thesystem (1) is endowed with the following boundary condition
v D v� C V at @˝ .0;1/; (2)
with v� a prescribed field, expressing the adherence of the liquid at the boundarywalls of the body, and asymptotic conditions
limjxj!1
v.x; t/ D 0; t 2 .0;1/; (3)
representative of the property that the liquid is quiescent at large spatial distancesfrom the body.
Throughout this paper it shall be assumed that the vectors � and ! do notdepend on time. This assumption imposes certain limitations on the type of motionthat B can execute with respect to a fixed inertial frame. Precisely [45], the centerof mass of B must move with constant speed along a circular helix whose axis isparallel to !. The helix will degenerate into a circle when � � ! D 0, in which casethe motion of the body reduces to a constant rotation. Without loss of generality,we set ! D ! e1, ! � 0, and � D v0 e with e a unit vector. As indicated in theintroductory section, one is only interested in the case when the motion of the bodydoes not reduce to a uniform rotation. For this reason, unless otherwise stated, itwill be assumed
v0 ¤ 0 and e � e1 ¤ 0: (4)
By shifting the origin of the coordinate system S suitably (Mozzi–Chaslestransformation) and scaling velocity and length by v0 e � e1 and d , respectively,
8 G.P. Galdi and J. Neustupa
one can then show that (1) can be put in the following form in the shifted frame S 0(see [45, pp. 496–497]):
�vt C�vC � @1vC T .e1 x� rv � e1 v/D �v � rvCrp C f
div v D 0
9>=
>;in ˝ .0;1/
v D v� C V at @˝ .0;1/ I limjxj!1
v.x; t/ D 0; all t 2 .0;1/;
(5)
where T WD ! d2
�,
� WD
8<
:
v0 d
�e � e1; if ! ¤ 0;
v0 d
�; if ! D 0 .e � e1/;
(6)
and
V WD e1 CT�e1 x: (7)
Of course, all fields entering the equations in (5) are regarded as nondimensional.Observe also that, in the rescaled length variables, the diameter of B becomes 1.In order to simplify the presentation, the origin of the coordinate system S 0 will besupposed to lie in the interior of B. Finally, we notice that, in view of (4), it follows� ¤ 0. Since all results presented in this chapter are independent of whether � ? 0,it will be assumed throughout � > 0.
Of particular relevance to this chapter are time-independent solutions (steady-state flow) of problems (5), (6), and (7), which may occur only when f and v� arealso time independent. From (5), one thus infers that these solutions must satisfy thefollowing boundary value problem:
�vC � @1vC T .e1 x � rv � e1 v/ D �v � rvCrp C fdiv v D 0
)in ˝
v D v� C V at @˝ I limjxj!1
v.x/ D 0:
(8)
The primary objective of this chapter is to provide an updated review of somefundamental properties of solutions to (6), (7), and (8). The latter include existence,uniqueness, regularity, asymptotic structure, generic properties, and steady andunsteady bifurcation issues. Moreover, a rather complete analysis of the longtimebehavior of dynamical perturbation to these solutions will be performed that willlead, in particular, to a number of stability and asymptotic stability results, undervarious assumptions.
Steady-State Navier–Stokes Flow Around a Moving Body 9
4 Existence
The starting point is the following general definition of weak (or generalized)solution for problems (6), (7), and (8) [79].
Definition 1. Let f 2 D�1;20 .˝/ and v� 2 W 1=2;2.@˝/. A vector field v W ˝ ! R3
is a weak solution to problems (8)–(7) if the following conditions hold:
(a) v 2 D1;2.˝/ with div v D 0.(b) v satisfies the equation
�.rv;r'/C � .@1v;'/C T .e1 x � rv � e1 v;'/C � .v � r'; v/D hf ;'i; for all ' 2 D.˝/.
(9)
(c) v D v� C V at @˝ in the trace sense.
(d) limR!1
R�2Z
@BR
jvj D 0.
(Formally, (9) is obtained by taking the scalar product of both sides of (8)1 by ' andintegrating by parts over ˝. Since D1;2.˝/ � W 1;2.˝R/, R > 1, condition (c) ismeaningful.)
Remark 1. If f 2 W �1;20 .˝ 0/, for all bounded˝ 0 with˝ 0 � ˝, then to every weaksolution, one can associate a suitable corresponding pressure field. More precisely,there exists p 2 L2loc.˝/ such that
�.rv;r /C � .@1v; /C T .e1 x � rv � e1 v; /C � .v � r ; v/D �.p; div /C Œf ; �; for all 2 C10 .˝/,
where Œ�; �� stands for the duality pairing D�1;20 $ D1;20 . Notice that this equation
is formally obtained by dot-multiplying both sides of (8)1 by and integrating byparts over ˝. The proof of this property, based on the representation of elements ofD�1;20 vanishing on D1;2
0 , is given in [45, Lemma XI.1.1].
The next step is the construction of a suitable extension, U , of the boundary data.The crucial property of such extension is condition (10) given below. In fact, as willbecome clear later on, this allows one to obtain the fundamental a priori estimate forthe existence result. Now, the validity of (10) is related to the magnitude of the fluxthrough the boundary @˝, ˚ , of the field v�:
˚ WD
Z
@˝
v� � n;
10 G.P. Galdi and J. Neustupa
with n unit outer normal to @˝. For simplicity, it will be assumed ˚ D 0, eventhough all main results continue to hold also when j˚ j is sufficiently “small.” Thereader is referred to Open Problem 4.2 for further considerations about this issue.
The existence of the appropriate extension of the boundary data is provided bythe following result whose proof can be found in [45, Lemma X.4.1]
Lemma 1. Let
v� 2 W1=2;2.@˝/;
Z
@˝
v� � n D 0:
Then, for any > 0, there exists U D U .; v�;V; ˝/ W ˝ ! R3 with bounded
support such that:
(i) U 2 W 1;2.˝/.ii) U D v� C V at @˝.
(iii) divU D 0 in ˝.
Furthermore, for all u 2 D1;20 .˝/, it holds that
j.u � rU ;u/j � juj21;2: (10)
Finally, if kv�k1=2;2.@˝/ �M; for some M > 0; then
kU k1;2 � C1kv� C Vk1=2;2.@˝/ (11)
where C1 D C1.;M;˝/.
Remark 2. In view of the above result, it easily follows that the existence of a weaksolution is secured if there is u 2 D1;2
0 .˝/ satisfying
� .ru;r'/C � .@1u;'/C T .e1 x � ru � e1 u;'/C � .u � r';u/
C �Œ.U � r';u/ � .u � rU ;'/� � .rU ;r'/C � .@1U � U � rU ;'/
C T .e1 x � rU � e1 U ;'/ D hf ;'i; for all ' 2 D.˝/.(12)
In fact, setting v WD uC U ; one gets at once that conditions (a)–(c) of Definition 1are met. Moreover, since, by Sobolev theorem D1;2
0 .˝/ � L6.˝/ (e.g., [45,Theorem II.7.5]), from [45, Lemma II.6.3], it follows
limR!1
1
R32
Z
@BR
juj D 0; for all u 2 D1;20 .˝/;
Steady-State Navier–Stokes Flow Around a Moving Body 11
so that also requirement (d) is met, even with a better order of decay. In view of allthe above, we may equally refer to both v and u as “weak solution.”
4.1 Early Contributions
Classical approaches and results to the existence of weak solutions due, basically, toJean Leray [81], Olga A. Ladyzhenskaya [79], and Hiroshi Fujita [29] will be nowpresented and summarized. Besides their historical relevance and intrinsic interest,these results will also provide a further motivation for the entirely distinct approach– recently introduced in [41, 47]–that will be described in Sect. 5.
4.1.1 Leray’s ContributionIn his famous pioneering work on the steady-state Navier–Stokes equations [81,Chapitres II & III], Leray shows that for any sufficiently regular f and v�;with ˚ D 0; there is at least one corresponding solution .v; p/ to (8)1;2;3–(7),which, in addition, satisfies v 2 D1;2.˝/. (As a matter of fact, Leray requiresf � 0; [81, §3 at p. 32]. However, for his method to go through, the weakerassumption of a “smooth” f would suffice.) It is just in this weak sense that Lerayinterprets the condition at infinity (8)4. (As noticed earlier on, this condition can beexpressed in a sharper, though still weak, way; see Remark 2.) Leray’s construction,basically, consists in solving the original problem (8)1;2;3–(7) on each elements ofan increasing sequence of bounded domains f˝kgk>1 with ˝ D [1kD1˝k; underthe further condition v D 0 on the “fictitious” boundary @Bk (“invading domains”technique). In turn, on every ˝k; a sufficiently smooth solution, vk; to the system(8) is determined by combining Leray–Schauder degree theory with a uniformbound on the Dirichlet integral jvkj21;2. (It should be observed that, even though thedemonstration provided by Leray is presented in the language of Leray–Schauderfixed-point theorem, such a result was not yet available at that time; see [83, 84].)The latter is crucial, in that it allows Leray to select a subsequence that, uniformlyon compact sets, converges to a solution of the original problem, meant in a suitableintegral sense. It must be emphasized that in order to obtain the above bound, theproperty (10) of the extension is crucial. (Notice that a bound on vk can also beobtained by an alternative method, based on a contradiction argument; see [81,Chapitre II, §III]. Even though the latter is more general than the one based onthe existence of an extension satisfying (10) (see [48, Introduction]), however, itdoes not necessarily provide a uniform bound independent of k and, therefore, is ofno use in the context of the “invading domains” technique.) An important feature ofthe solution constructed by Leray is that it could be shown to satisfy the so-calledgeneralized energy inequality
formally obtained by setting ' � u in (12) and replacing “D” with “�.” A morefamiliar form of (13) can be obtained if f and v� have some more regularity. Forexample, if in addition f 2 L2.˝/ and v� 2 W 3=2;2.@˝/; then it can be shown that(13) is equivalent to the following one (see [45, Theorem XI.3.1(i)]):
�2kD.v/k22C
Z
@˝
˚.v�CV/ �T .v; p/�
�
2.v�CV/2v�
��n�hf ; vi � 0; (14)
where T .v; p/ D rv C .rv/> � p I ; I the identity matrix, is the Cauchy stresstensor. It is worth emphasizing that (14) would represent the energy balance for themotion .v; p/; provided one could replace “�” with “D.” The inequality sign in theabove formulas is, again, a consequence of the little information that this solutioncarries at large spatial distances. For the same reason, the uniqueness question isleft out.
4.1.2 Ladyzhenskaya’s ContributionLadyzhenskaya was the first to introduce the definition and the use of the term“generalized (or weak) solution” as currently used, for steady-state Navier–Stokesproblems [79, p. 78]. Her construction still employs the “invading domains”technique utilized by Leray, but the way in which she proves the existence of thesolution on each bounded domain ˝k of the sequence is somewhat simpler andmore direct. More precisely, Ladyzenskaya considers (12) with T D 0 and v� � 0and shows that it can be equivalently rewritten as a nonlinear equation in the Hilbertspace D1;2
0 .˝k/:
M.u/ WD uC �A.u/ D F (15)
where F is prescribed D1;20 .˝k/ and A is a (nonlinear) compact operator. (The
extension to the case T ¤ 0 would be straightforward.) Therefore, the operator M;
defined on the whole of D1;20 .˝k/; is a compact perturbation of a homeomorphism.
Moreover, using arguments similar to Leray’s, one can show that every solutionto (15) is uniformly bounded in D1;2
0 .˝k/; for all � 2 Œ0; �0�; arbitrary fixed�0 > 0. Then, by the Leray–Schauder degree theory, it follows that (15) hasa weak solution, uk 2 D1;2
0 .˝k/; for the given F . Since jukj1;2 is uniformlybounded in k; Ladyzhenskaya shows that a subsequence can be selected convergingto a weak solution in the sense of Definition 1; see Remark 2. It is worthemphasizing that, if ˝ is an exterior domain, the operator A is not compact(see [47, Proposition 80]) so that the “invading domains technique” is indeednecessary for the argument to work. Moreover, if u is merely in D1;2
0 .˝/; with˝ exterior domain, the very equation (22) would not be meaningful in such acase. Finally, it is important to observe that Ladyzhenskaya’s solution, as Leray’s,satisfies only the generalized energy inequality (13), and, again, the uniquenessquestion is left open because of the little asymptotic information carried by functionsfrom D
1;20 .˝/.
Steady-State Navier–Stokes Flow Around a Moving Body 13
4.1.3 Fujita’s ContributionFujita’s approach to the existence of a weak solution [29] is entirely different fromthose previously mentioned. In fact, it consists in adapting to the time-independentcase the method introduced by Eberhard Hopf for the initial value problem [64].The method referred to above is the by now classical Faedo–Galerkin method. (Also,strictly speaking, Fujita considers the case T D 0; even though the extension ofhis method to the more general case presents no conceptual difficulty.) As is wellknown, the idea is to look first for an “approximate solution” to (12), uN ; in themanifold M.N / spanned by the first N elements of a basis of D1;2
0 .˝/. This is afinite-dimensional problem whose solution, at the N -th step, is found by solvinga suitable nonlinear equation. Fujita solves the latter by means of Brouwer fixed-point theorem [29, Lemma 3.1], provided j˚ j is “small enough,” and then showsthat juN j1;2 is uniformly bounded in N . With this information in hand, one canthen select a subsequence fuN 0g that in the limit N 0 ! 1 converges (in a suitablesense) to a vector u 2 D1;2
0 .˝/ satisfying (12); see also [45, Theorem X.4.1].The advantage of Fujita’s approach, besides being more elementary, resides alsoin the fact that the solution is constructed directly in the whole domain˝. However,also in this case, solutions satisfy only the generalized energy inequality, and theiruniqueness is also left out.
4.2 A Function-Analytic Approach
The most significant aspect of solutions constructed by the above authors is thattheir existence is ensured for data of arbitrary “size,” provided only the mass fluxthrough the boundary is not too large. (Notice that, of course, kv�k1=2;2;@˝ arbitrarily“large” and j˚ j “small” are not, in general, at odds.) However, as emphasizedalready a few times, these solutions possess no further asymptotic information atlarge distances other than that deriving from the fact that v 2 L6.˝/; consequenceof the of the property v 2 D1;2.˝/ and Sobolev inequality; see Remark 2. Withsuch a little information, it is, basically, hopeless to show fundamental properties ofthe solution that are yet expected on physical grounds, such as (i) balance of energyequation, namely, (14) with the equality sign, and (ii) uniqueness for “small” data.
The main objective of this subsection is to show that, in fact, this undesiredfeature can be removed by using another and completely different approach. Theapproach, introduced in [41, 47], consists in formulating the original problem as anonlinear equation in a suitable Banach space and then using the mod 2 degree forproper Fredholm maps of index 0 to show just under the conditions on the datastated in Definition 1 the existence of a corresponding weak solution possessing“better” properties at “large” distances. As a consequence, one proves that theseweak solutions satisfy, in addition, the requirements (i) and (ii) above. Moreover,this abstract setting shows itself appropriate for the study of other importantproperties of solutions, including generic properties, and steady and time-periodicbifurcation; see Sects. 8 and 9.
14 G.P. Galdi and J. Neustupa
In order to give a precise statement of the main results, it is appropriate tointroduce the necessary functional setting. To this end, let
R.u/ WD e1 x � ru � e1 u
and set
X.˝/ D˚u 2 D1;2
0 .˝/ W @1u; R.u/ 2 D�1;20 .˝/�; (16)
where @1u 2 D�1;20 .˝/ means that there is C > 0 such that
j.@1u;'/j � C j'j1;2; for all ' 2 D.˝/;
and, therefore, by the Hahn–Banach theorem, @1u can be uniquely extended to anelement of D�1;20 .˝/ that will still be denoted by @1u. Analogous considerationshold for R.u/. It can be shown [47, Proposition 65] that when endowed with its“natural” norm
kukX WD juj1;2 C j@1uj�1;2 C jR.u/j�1;2;
X.˝/ becomes a reflexive, separable Banach space. Obviously, X.˝/ is a strictsubspace of D1;2
0 .˝/.The primary objective is to prove existence of weak solution in the space X.˝/.
In this respect, one observes that all classical approaches mentioned earlier onfurnish weak solutions in D1;2
0 .˝/ which embeds only in L6.˝/; see Remark 2.The fundamental property of X.˝/; expressed in the following lemma, is that itembeds in a much “better” space.
Lemma 2. Let ˝ � R3 be an exterior domain and assume u 2 D1;2
0 .˝/ with@1u 2 D�1;20 .˝/. Then, u 2 L4.˝/; and there is C1 D C1.˝/ > 0 such that
kuk4 � C1 j@1uj14
�1;2juj34
1;2: (17)
Thus, in particular, X.˝/ � L4.˝/.
Proof. Obviously, if u � 0; there is nothing to prove, so one shall assume u 6�0. The proof for an arbitrary exterior domain is somewhat complicated by severaltechnical issues; see [41, Proposition 1.1 with proof on pp. 8–13]. However, if ˝ �R3; it becomes simpler and will be sketched here. (The inequality proved in [41,
Proposition 1.1] is, in fact, weaker than (17). However, one can apply to Eq. (1.31)of [41] almost verbatim the argument given in the current proof after (23) and showthe stronger form (17).) For a given g 2 C10 .R
3/; consider the following problem
�' � @1' D g Crp; div' D 0; in R3; (18)
Steady-State Navier–Stokes Flow Around a Moving Body 15
where > 0. By [45, Theorem VII.4.1], problem (18) has at least one solution suchthat
' 2 Ls1.R3/ \D1;s2.R3/ \D2;s3.R3/; @1' 2 Ls3.R3/
p 2 Ls4.R3/ \D1;s3.R3/(19)
for all s1 > 2; s2 > 4=3; s3 > 1; s4 > 3=2; which satisfies the estimate
1=4j'j1;2 � Ckgk4=3; (20)
with C D C.s1; : : : ; s4/. Using (18)–(19) and recalling that by the Sobolevinequality D1;2
0 .R3/ � L6.R3/; one shows after integration by parts
.u;g/ D .u; �' � @1' � rp/ D �.ru;r'/ � .u; @1'/: (21)
The following identity is valid for all u 2 D1;20 .R
3/ with @1u 2 D�1;20 .R3/ and
2 D1;20 .R
3/ with @1 2 L65 .R3/ and can be shown by the arguments from [41,
pp. 12–13]
.u; @1 / D �h@1u; i:
In view of (19), we may use the latter in (21) to get
.u;g/ D �.ru;r'/C h@1u;'i;
which implies
j.u;g/j � .juj1;2 C j@1uj�1;2/ j'j1;2: (22)
Replacing (20) into this latter inequality, one finds
j.u;g/j � C�
34 j@1uj�1;2 C
� 14 juj1;2�kgk 4
3:
Since g is arbitrary in C10 .R3/; it follows that u 2 L4.R3/ and, furthermore,
kuk4 � C�
34 j@1uj�1;2 C
� 14 juj1;2�; for all > 0. (23)
By a simple calculation we show that the right-hand side of (23) as a function of attains its minimum at
D juj1;2=.3 j@1uj�1;2/;
16 G.P. Galdi and J. Neustupa
which once replaced in (23) proves (17), provided j@1uj�1;2 ¤ 0. To show that thisis indeed the case, suppose the contrary. Then .@1u;'/ D 0 for all ' 2 D.R3/; sothat there is p 2 D1;2.R3/ such that @1u D rp in R
3; see, e.g., [45, Lemma III.3.1].From div u D 0; we deduce �p D 0 in R
3 in the sense of distributions, which, bythe property of p; in turn furnishes @1u D rp � 0; and this contradicts the fact thatu 2 L6.R3/. The proof is thus completed. �
One is now in a position to state the following general existence result.
Theorem 1. For any � ¤ 0; T � 0; f 2 D�1;20 .˝/; and v� 2 W 1=2;2.@˝/ with˚ D 0; there exists at least one weak solution, v; to (8)–(7) that in addition satisfiesv � U 2 X.˝/; with U given in Lemma 1. Moreover, v obeys the estimate
kv � U kX � C1�jf j�1;2 C jf j
3�1;2
�C C2
�kv� C Vk1=2;2;@˝/ C kv� C Vk31=2;2.@˝/
�
(24)
whereC1 D C1.�; T ; ˝/ andC2 D C2.�; T ; ˝;M/;whenever kv�k1=2;2.@˝/ �M .
A full proof of Theorem 1 is given in [47, Theorem 86(i)]. Here it shall bereproduced the main ideas leading to the result, referring the reader to the citedreference for all missing details.
The first step is to write (12) as a nonlinear equation in the space D�1;20 .˝/. Toreach this goal, for fixed �; T ; one defines the generalized Oseen operator
O W u 2 X.˝/ 7! O.u/ 2 D�1;20 .˝/ (25)
where
hO.u/;'i WD �.ru;r'/C �h@1u;'i C T hR.u/;'i; ' 2 D1;20 .˝/: (26)
Likewise, one introduces the operators N and K from X.˝/ to D�1;20 .˝/ asfollows:
(The dependence of the relevant operators on the parameters � and T will beemphasized only when needed; see Sects. 8 and 9.) Finally, let F denote theuniquely determined element of D�1;20 .˝/ such that, for all ' 2 D1;2
0 .˝/;
hF ;'i WD .rU ;r'/ � �.@1U � U � rU ;'/ � T .R.U /;'/C hf ;'i: (28)
In view of Lemma 1 and Lemma 2 and with the help of Hölder Inequality, it iseasy to show that the operators O;N; and K and the element F are well defined.
Steady-State Navier–Stokes Flow Around a Moving Body 17
Setting
L WD OCK; (29)
the objective is to solve the following problem: For any F 2 D�1;20 .˝/; find u 2X.˝/ such that
L .u/CN.u/ D F : (30)
It is plain that, if this problem is solvable, then v D u C U is a weak solutionsatisfying the statement of Theorem 1.
The strategy to solve the above problem consists in showing that the map M WD
L CN W X.˝/ 7! D�1;20 .˝/ is surjective. To reach this goal, one may use a verygeneral result furnished in [47], based on the mod 2 degree of proper C2 Fredholmmaps of index 0 due to Smale [103]. More precisely, from [47, Theorem 59(a)], itfollows, in particular, the following.
Proposition 1. Let Z; Y be Banach spaces with Z reflexive. Let L W Z 7! Y andN W Z 7! Y and set M D LCN . Suppose:
(i) M is weakly sequentially continuous (i.e., if zn ! z weakly in Z; thenM.zn/!M.z/ weakly in Y ).
(ii) N is quadratic (i.e., there is a bilinear bounded operator B W ZZ 7! Y suchthat N.z/ D B.z; z/ for all z 2 Z).
(iii) L maps homeomorphically Z onto Y .(iv) The Fréchet derivative of N is compact at every z 2 Z.(v) There is � W RC 7! RC mapping bounded set into bounded set, with �.s/! 0
as s ! 0; such that
kzkZ � � .kM.z/kY /:
Then M is surjective.
This proposition will be applied with Z � X.˝/; Y � D�1;20 .˝/; L � L; andN � N. With this in mind, one begins to show the following.
Lemma 3. The operator N is quadratic, and M WD L CN is weakly sequentiallycontinuous.
Suppose next uk ! u weakly in X.˝/; one has to show that M.uk/ ! M.u/weakly in D�1;20 .˝/. This amounts to prove that
limk!1hM.uk/;'i D hM.u/;'i; for all ' 2 D.˝/. (33)
In fact, on one hand, being D1;20 .˝/ reflexive [45, Exercise II.6.2], the generic linear
functional acting on W 2 D�1;20 .˝/ is of the form hW ;'i; for some ' 2 D1;20 .˝/.
On the other hand, it is
jM.uk/j�1;2 � C0;
C0 > 0 independent of k; as is at once established from (26)–(27) and the uniformboundedness of kukkX . Now, to show (33), it is observed that the latter implies thatthere is M1 > 0 independent of k; such that
jukj1;2 � C:
Thus, along a subsequence fuk0g;
limk0!1
.ruk0 ;r'/ D .ru;r'/ I limk0!1
.@1uk0 ;'/ D .@1u;'/ I
limk0!1
.R.uk0/;'/ D .R.u/;'/; for all ' 2 D.˝/.(34)
Moreover, by the embedding D1;20 .˝/ � W 1;2.˝R/; R > 1; Rellich compactness
theorem, and Cantor diagonalization method, one can also show
limk0!1
kuk0 � uk4;˝R D 0 I for all R > 1; (35)
see [45, Proposition 66] for details. The desired property (33) is then a simpleconsequence of (26), (27), (34), (35), and Hölder inequality. �
The following result also holds.
Lemma 4. Let u 2 X.˝/. Then,
h@1u;ui D hR.u/;ui D 0:
Proof. If u 2 D.˝/; the proof is trivial, being a consequence of simple integrationby parts. However, if u is just in X.˝/; the claim is not obvious since it is notknown whether D.˝/ is dense in X.˝/. As a consequence, one has to arguein a different and more complicated way, especially to show the property for R.The proof becomes then lengthy, technical, and tricky. For this reason it will be
Steady-State Navier–Stokes Flow Around a Moving Body 19
omitted, and the reader is referred to [41, pp. 12–13] for the first property and to[47, Proposition 70] for the second one. �
The above lemma is crucial for the next result – a particular case of that shownin [47, Proposition 78] – ensuring the validity of condition (iii) in Proposition 1.
Lemma 5. The operator L WD OCK is a linear homeomorphism of X.˝/ ontoD�1;20 .˝/. Moreover, there is a constant C D C.�; T ; ˝/ such that
kukX � C jL .u/j�1;2: (36)
Proof. Referring to the cited reference for a full proof, here only the leading ideaswill be sketched. As shown in [76, Theorem 2.1] and [47], the generalized Oseenoperator O is a homeomorphism of X.˝/ onto D�1;20 .˝/; and, moreover,
juj1;2 C j@1uj�1;2 C jR.u/j�1;2 � C jO.u/j�1;2:
Therefore, by classical results on Fredholm operators, it is enough to show that (i)K is compact and (ii) N.L / D f0g. Let fukg � X.˝/ be a bounded sequence andlet ˝R contain the support of U . Observing that X.˝/ � W 1;2.˝R/; the Rellichcompactness theorem implies that there is a subsequence of fukg that is Cauchy inL4.˝R/. Since by (27)1 and Hölder inequality, for all ' 2 D1;2
0 .˝/
jhK .uk0/;'i � hK .uk00 /;'ij � 2� kU k4kuk0 � uk00 k4;˝R j'j1;2;
from Lemma 1(i), one infers (along a subsequence)
limk0;k00!1jK .uk0/ �K .uk00 /j�1;2 D 0;
which proves (i). To show (ii), it must be shown that
hO.u/CK .u/;'i D 0 for all ' 2 D1;20 .˝/ H) u D 0: (37)
SinceU 2 W 1;2.˝/ is of bounded support with div U D 0 and u 2 D�1;20 .˝/; by aneasily justified integration by parts, we show .U � ru;u/ D 0. So that by replacingu for ' in (37) and using this property along with (26), (27)2; and Lemma 4, onededuces
juj21;2 � �.u � rU ;u/ D 0:
As a result, (37) is a consequence of the latter and of (10) in Lemma 1. �
The following lemma guarantees condition (iv) in Proposition 1; see [47, Propo-sitions 79].
20 G.P. Galdi and J. Neustupa
Lemma 6. The Fréchet derivative, N 0.u/; of N is compact at each u 2 X.˝/.
Proof. From (27)1; it follows that
��1N 0 .u/�w D B.u;w/C B.w;u/;
with B defined in (32). Let fvkg � X.˝/ be such that
kvkkX � C;
with C independent of k 2 N; and so, by Lemma 2, one gets, in particular,
kvkk4 C jvkj1;2 � C1; (38)
with C1 D C1.˝/ > 0. Since X.˝/ is reflexive, there exist v 2 X.˝/ and asubsequence fvk0g � X.˝/ converging weakly in X.˝/ to v. As in the proof ofLemma 3, it can also be shown from (38) that (possibly, along another subsequence)
limk0kvk � vk4;˝R D 0; for all sufficiently large R ; (39)
see also [47, Proposition 66]. From (32) and Hölder inequality, one finds
jhB.u; vk0/ � B.u; v/;'ij D jhB.u; vk0 � v/;'ij��kuk4;˝Rkv � vk0k4;˝R C kuk4;˝Rkv � vk0k4;˝R
�j'j1;2;
for all sufficiently large R. Using (38) and (39) into this relation gives
limk0!1
jB.u; vk0/ � B.u; v/j�1;2 � C1kuk4;˝R ;
where C1 > 0 is independent of k0. However, R is arbitrarily large, and so, by theabsolute continuity of the Lebesgue integral, it may be concluded that
limk0!1
jB.u; vk0/ � B.u; v/j�1;2 D 0: (40)
In a completely analogous way, one shows
limk0!1
jB.vk0 ;u/ � B.v;u/j�1;2 D 0: (41)
From (40) and (41), it then follows that the operator B.u; �/; and hence N 0 .u/; iscompact at each u 2 X.˝/. �
Steady-State Navier–Stokes Flow Around a Moving Body 21
In order to apply Proposition 1 to the operator M; it remains to show condition(v), which amounts, basically, to find “good” a priori estimates for the equationM.u/ D F .
Lemma 7. There is a constant C > 0 such that all solutions u 2 X.˝/ to (30)satisfy
kukX � C .jF j�1;2 C jF j3�1;2/: (42)
Proof. Also using (26), (27)2; (10), and Lemma 4, one deduces
Using Young’s inequality in the latter allows one to deduce the validity of (42), andthe proof of the lemma is completed. �
22 G.P. Galdi and J. Neustupa
Proof of Theorem 1. The proof of the first statement follows from Proposition 1,and Lemma 3, and Lemmas 5–7. Furthermore, by property (11) of U and (28), onefinds
jF j�1;2 � jf j�1;2 C C1 kv� C Vk1=2;2.@˝/; (48)
where C1 D C1.�; �;˝;M/ whenever kv�k1=2;2.@˝/ � M . Estimate (24) is then aconsequence of Lemma 7 and (48). �
Open Problem. Property (10) of the extension U is fundamental for theestimate (46). As mentioned earlier on, (10) is only known if the flux ˚ is of“small” magnitude. While by a procedure similar to [23, 48, 61] it is probablypossible to show that such a condition on ˚ is also necessary for the existenceof an extension with the above property, one may nevertheless wonder if asmall j˚ j would indeed be necessary if the existence problem is approachedby other methods. In this respect, by combining a contradiction argument ofLeray with properties of the Bernoulli’s function in spaces of low regularity,in their deep work [72], Korobkov, Pileckas, and Russo have shown existencewithout restrictions on j˚ j; at least for flow and data that are axisymmetricalong the direction of �. Whether such a result is true in general remains open.
The following result shows an important property of weak solutions in the classX.˝/ and so, in particular, applies to those constructed in Theorem 1.
Theorem 2. Let f 2 D�1;20 .˝/ and v� 2 W 1=2;2.@˝/; and let v be a correspond-ing weak solution with v�U 2 X.˝/. Then, v satisfies the energy equality, namely,(13) with the equality sign. If, in addition, f 2 L2.˝/ and v� 2 W 3=2;2.@˝/; thenthe latter takes the form of the classical equation of energy balance:
� 2kD.v/k22 C
Z
@˝
˚.v� C V/ � T .v; p/ �
�
2.v� C V/2v�
�� n D hf ; vi (49)
Proof. From (30), with F given in (28), one deduces
L .u/;ui C hN .u/;ui D hF ;ui:
Employing in this equation (43) and (45), one obtains
juj21;2 � � .u � rU ;u/ � hF ;ui D 0;
which, recalling the definition ofF in (28), shows that u obeys (13) with the equalitysign. The second part of the theorem is shown exactly like in [45, pp. 770–771] andwill be omitted. �
Steady-State Navier–Stokes Flow Around a Moving Body 23
Open Problem. The natural question arises whether any weak solution, v;corresponding to data satisfying merely the “natural” minimal conditionsof Theorem 1, is such that v � U 2 X.˝/; and, in particular, obeys theequation of energy balance. In a remarkable work, [57] Heck, Kim, and Kozonohave shown that this is indeed the case, at least when T D 0 (the bodyis not spinning), v� � 0; and f is assumed slightly more regular, namely,f 2 D�1;20 .˝/. Whether this result continues to hold for T ¤ 0 is not known.
5 Regularity
It is expected that if the data f ; v� and the boundary @˝ are sufficiently smooth,then the corresponding weak solution is smooth as well. In this respect, one has thefollowing very general result about interior and boundary regularity.
Theorem 3. Let v be a weak solution to (8)–(7). Then, if
f 2 Wm;q
loc .˝/; m � 0;
where q 2 .1;1/ if m D 0; while q 2 Œ3=2;1/ if m > 0; it follows that
v 2 WmC2;qloc .˝/; p 2 W
mC1;qloc .˝/;
where p is the pressure associated to v in Remark 1. Thus, in particular, if
f 2 C1.˝/; (50)
then
v; p 2 C1.˝/: (51)
Assume, further, ˝ of class CmC2 and
v� 2 WmC2�1=q;q.@˝/; f 2 W m;q.˝R/;
for some R > 1 and with the values of m and q specified earlier. Then,
v 2 W mC2;q.˝R/; p 2 WmC1;q.˝R/:
Therefore, in particular, if ˝ is of class C1 and
v� 2 C1.@˝/; f 2 C1.˝R/; (52)
24 G.P. Galdi and J. Neustupa
it follows that
v; p 2 C1.˝R/: (53)
The proof of this result is rather complicated, and the interested reader is referredto [45, Theorems X.1.1 and XI.1.2]. However, if one assumes (50) [(52) and ˝ ofclass C1], then the proof of (51) [(53)] can be obtained by classical results forthe Stokes problem in conjunction with a simple bootstrap argument and will bereproduced here.
To show this, one needs the following classical regularity results for weaksolutions to the Stokes problem, a particular case of those furnished in [45,Theorems IV.4.1 and IV.5.1], to which the reader is also referred for their proofs.
Lemma 8. Let .w; �/ 2 W 1;qloc .˝/ L
qloc.˝/; 1 < q < 1; with div w D 0 in ˝;
satisfy
� .rw;r / D ŒF; � � .�; div /; for all 2 C10 .˝/. (54)
Then, if F 2 W m;qloc .˝/; m � 0; necessarily .w; �/ 2 W mC2;q
loc .˝/ WmC1;q
loc .˝/.Moreover, assume w 2 W 1;q.˝R/ for some R > 1; and w D w� at @˝. Then,if F 2 W m;q.˝R/; w� 2 W mC2�1=q;q.@˝/; necessarily .w; �/ 2 W mC2;q.˝r/
WmC1;q
loc .˝r/; for any r 2 .1; R/.
With this result in hand, it can be proved that (50) implies (51). From Remark 1,the weak solution v and the associated pressure field p satisfy (54) with
F WD �� @1v � T .e1 x � rv � e1 v/C � v � rvC f :
Then, by assumption, the embedding
D1;20 .˝/ � W
1;2loc .˝/ � L
6loc.˝/;
and the Hölder inequality one has that F 2 L3=2loc .˝/. From the first statement
in Lemma 8, it can then be deduced v 2 W 2;3=2.˝/loc; p 2 W1;3=2
loc .˝/;
and, moreover, (v; p) satisfy (8)1 a.e. in ˝. Next, because of the embeddingW
2;3=2loc .˝/ � W 1;3
loc .˝/ � Lrloc.˝/; arbitrary r 2 Œ1;1/; one obtains the improved
regularity property F 2 W 1;sloc .˝/; for all s 2 Œ1; 3=2/. Using once again Lemma 8,
one infers v 2 W3;s
loc .˝/ and p 2 W2;s
loc .˝/ which, in particular, gives furtherregularity for F. By induction, one then proves the desired property v; p 2 C1.˝/.The proof of the boundary regularity is performed by an entirely similar argumentand, therefore, will be omitted.
Steady-State Navier–Stokes Flow Around a Moving Body 25
6 Uniqueness
This section is dedicated to the investigation of the uniqueness property of weaksolutions. Basically, the main known results depend on the summability andregularity assumptions made at the outset on the data f and v�. The followingtheorem shows, in particular, that every solution in Theorem 1 is unique in its ownclass of existence, provided the size of the data is sufficiently restricted.
Theorem 4. Assume vi ; i D 1; 2; are weak solutions with vi � U 2 X.˝/;
corresponding to the same f 2 D�1;20 .˝/; v� 2 W 1=2;2.@˝/. Then, there isC D C.�; T ; ˝/ such that if
jf j�1;2 C kv� C Vk1=2;2.@˝/ < C; (55)
necessarily v1 � v2.
Proof. Setting ui D vi � U ; i D 1; 2; with U given in Lemma 1, from (25)–(28)and the assumption, one finds
L .ui /CN .ui / D F ; i D 1; 2: (56)
Therefore, from Lemma 2, Lemma 7, and (48), one infers in particular
kuik4 � C1�jf j�1;2Cjf j
3�1;2
�CC2
�kv�CVk1=2;2.@˝/Ckv�CVk31=2;2.@˝/
�; (57)
whereC1 D C1.�; T ; ˝/ andC2 D C2.�; T ; ˝;M/;whenever kv�k1=2;2.@˝/ �M .Arbitrarily fix the numberM once and for all. Setting u WD u1�u2; from (56), oneobtains
L .u/ D �B.u;u1/ � B.u2;u/; (58)
where B is defined in (32). Thus, observing that
jB.u;u1/C B.u2;u/j�1;2 � kuk4�ku1k4 C ku2k4
�
from (58), Lemma 2, and Lemma 6, one has, in particular,
kuk41 � C3
�ku1k4 C ku2k4
�� 0;
with C3 D C3.�; T ; ˝/. The result then follows from this inequality and (57). �
The natural question arises of whether the solutions constructed in Theorem 1 areunique in the class of weak solutions, that is, obeying just the requirements statedin Definition 1. The answer to this question is positive if f is assumed to possess“good” summability properties at large distances and v is sufficiently regular.
26 G.P. Galdi and J. Neustupa
Actually, this property is a particular consequence of the following result for whoseproof the reader is referred to [45, Theorem XI.5.3], once one takes into accountthat, by Sobolev inequality, L
65 .˝/ � D�1;20 .˝/ and that D�1;20 .˝/ � D�1;20 .˝/.
Theorem 5. Let
f 2 L6=5.˝/ \ L4=3.˝/; v� 2 W5=4;4=3.@˝/:
Then, there exists C D C.˝; �; T / such that, if
kf k6=5 C kv� C Vk7=6;6=5.@˝/ < C; (59)
v is the only weak solution corresponding to the above data.
Open Problem. In general, it is not known whether solutions of Theorem 1are unique in the class of weak solutions, when f and v� merely satisfy theassumptions of that theorem (and are sufficiently small).
In connection with this problem, it is worth remarking that in the special caseT D 0 and v� � 0; and with f slightly more regular (namely, f 2 D�1;20 .˝/), theresult is shown in [57, Theorem 2.3].
7 Asymptotic Behavior
As shown in previous sections, some fundamental attributes of weak solutionexpected on physical grounds, such as verifying the energy balance and beingunique for small data, can be established if one has enough information on theirsummability properties in a neighborhood of infinity, like the one provided byTheorem 1. However, there are other significant aspects that require a sharppointwise knowledge of the solution at large distances, which in principle isnot necessarily guaranteed just by the mild asymptotic information furnished inthat theorem. These aspects include, for instance, the presence of a stationary,unbounded wake region “behind” the body and a “fast” decay of the vorticity outsidethe wake region, in support of boundary layer theory. Proving (or disproving) theseproperties constituted one of the most challenging questions in mathematical fluiddynamics since the pioneering chapter of Leray.
The case T D 0 was eventually settled in the mid-1970s (about 40 years afterLeray’s work), thanks to the effort of Robert Finn, Konstantin I. Babenko, and theircollaborators. Their contributions will be briefly summarized in the following twosubsections. The case T ¤ 0 presents much more difficulties and will be treatedsuccessively in Sect. 7.3, by means of a different approach, originally introduced in
Steady-State Navier–Stokes Flow Around a Moving Body 27
[39], that allows for a rather complete description of the pointwise asymptotic flowbehavior also in that more general situation.
7.1 Finn’s Contribution
In the late 1950s/mid-1960s, in a series of remarkable papers [24–28], RobertFinn proved the following fundamental results. Let .v; p/ be any (sufficientlysmooth) solution to (8)–(7) with T D 0 and with f of bounded support, suchthat jv.x/j � C jxj�˛; some ˛ > 1=2 and all “large” jxj. Then the pointwiseasymptotic structure of .v; p/ can be sharply evaluated. In particular, combining theintegral representation of the solution, obtained via the (time-independent) Oseenfundamental tensor, E; along with a careful estimate of the latter, Finn showed thatthese solutions exhibit a paraboloidal “wake region,” R; with the property that thevelocity field, v; inside R decays pointwise slower than it does outside R. Moreprecisely, he proved that v [rv] admits an asymptotic expansion with E [rE] beingthe leading term. (Finn left open the question of the asymptotic behavior of thesecond derivatives of v [24], a problem that was finally solved another 40 yearslater by Deuring [13].) Finn called such solutions “physically reasonable” (PR) [27,Definition 5.1] and demonstrated their existence on a condition that the magnitudeof the data is sufficiently restricted [27, Theorem 4.1]. Later on, one of his students,David Clark, showed that the vorticity field of any PR solution decays exponentiallyfast outside R and far from the body [10]. Thanks to its sharp asymptotic (andlocal regularity) properties, it is easy to show that any PR solution (regardless ofthe size of the data) is also weak, namely, v 2 D1;2.˝/. However, given that thelatter is the only information that weak solutions carry in a neighborhood of infinity,the converse property is by no means obvious, to the point that some author evenquestioned its validity [59, p. 12]. All this seemed to cast profound doubts about thephysical relevance of Leray’s weak solutions.
7.2 Babenko’s Contribution
The relation between weak and PR solutions was eventually addressed by Babenko[2]. Combining Lizorkin’s multipliers theory with anisotropic Sobolev-like inequal-ities and the representation formula for the solution employed by Finn, he was ableto show that, if the body force f is of bounded support, every weak solution is,in fact, physically reasonable in the sense of Finn. Babenko’s paper can be dividedinto two main parts. In the first one, he shows, by a very elegant and straightforwardargument, that any weak solution, v; corresponding to the given data must be inL4.˝/; with corresponding pressure field in L2.˝/. The second part of the paperis aimed to show that, actually, v 2 Lq.˝/; for any q 2 .2; 4�. Once this propertyis established, then it is relatively simple to prove that the weak solution decayslike jxj�˛ for some ˛ > 1=2 and therefore is also PR. It must be noted thatBabenko’s proof of these further summability properties has aspects that are not
28 G.P. Galdi and J. Neustupa
fully transparent. Also for this reason, a distinct and more direct proof of Babenko’sresult was given later on by Galdi [35] and, successively and independently, byFarwig and Sohr [16].
7.3 A General Approach
It must be emphasized one more time that the results reported in the previous twosubsections refer to the case T D 0; that is, the body is not spinning. If oneallows T ¤ 0; then the detailed study of the asymptotic properties of a weaksolution becomes even more complicated, for several reasons. In the first place,the linear momentum equation (8)1 contains a term that grows linearly fast atlarge spatial distances. As a consequence, the fundamental tensor of the linearizedequations, T; is no longer the classical Oseen tensor E mentioned above, let alonea “perturbation” of it, but, rather, a much more complicated one; see [21, Section2]. Thus, the representation of the solution that in both contributions of Finn andBabenko plays a fundamental role in the determination of the pointwise behaviorbecomes much more involved and, actually, useless for that matter. In fact, as shownin [21, Proposition 2.1], unlike E; the tensor T does not satisfy uniform estimatesat large spatial distances.
In view of these issues, in [39] Galdi introduced a completely different approachto the study of the asymptotic structure of a weak solution, that was furthergeneralized and improved in [40, 42, 43, 46]. In this approach, the weak solutionv is viewed as limit as t !1 along sequences of the (unique) solution, w.x; t/; toa suitable initial value problem. It can be shown that, in turn, w admits a somewhatsimple space-time representation in terms of the Oseen fundamental solution to thetime-dependent Oseen equation. This fact allows one to obtain a number of sharpspatial estimates for w uniformly in time, which are thus preserved in the limitt !1; and therefore continue to hold for the weak solution v.
Referring to [45, §§X.6, X.8, XI.4, XI.6] for a full account of the (technicallycomplicated and lengthy) proofs of all the above results, here it will only be providedan outline of the main steps of the procedure used in establishing them in the caseT ¤ 0.
The first step consists in determining sharp summability properties of a weaksolution in a neighborhood of infinity, under appropriate hypothesis on the data. Tothis end, one can show the following result [45, Theorem XI.6.4].
Lemma 9. Assume, for some q0 > 3 and all q 2 .1; q0�; that
Then, every weak solution v to problems (8)–(7) corresponding to f ; v�; and theassociated pressure field p (possibly modified by the addition of a constant; see alsoRemark 1) satisfies the following summability properties:
Steady-State Navier–Stokes Flow Around a Moving Body 29
v 2 Lr.˝/ \D1;s.˝/;@v
@x12 Lt.˝/; p 2 L .˝/;
for all r 2 .2;1�; s 2 .4=3;1�; t 2 .1;1�; and 2 .3=2;1�. If, in addition,f 2 W 1;q0.˝/; v� 2 W
3�1=q0;q0 .@˝/; then we have also
v 2 D2;� .˝/; p 2 D1;� .˝/;
for all � 2 .1;1�.
The next objective is to “translate” the above global asymptotic information intoa pointwise one. For simplicity, it shall be assumed that f is of bounded support,which also implies, with the help of Theorem 3, that .v; p/ 2 C1.˝�/ forsufficiently large �.
Thus, in the second step, one uses a standard “cutoff” procedure to rewrite(suitably) (8) in the whole space R
3. More specifically, let be a smooth functionthat is 0 in the neighborhood of @˝ that contains the support of f and 1 sufficientlyfar from it. Moreover, let Z 2 C10 .˝/ such that divZ D �r � v in ˝. (Sucha field Z exists, as shown in [45, Theorem III.3.3].) From (8) one can deduce thatu WD v �Z and Qp WD p obey the following problem:
�uC � @1uC T .e1 x � ru �e1 u/D � div . v˝ v/Cr Qp C F c
div u D 0
9=
; in R3
(60)
where F c is smooth and of bounded support. At this point, the “classical” procedurewould be to write the solution u in terms of the fundamental tensor solutions, T;associated with problem (60). However, as remarked earlier on, this would not leadanywhere due to the poor properties of T. Therefore, we argue differently.
In the third step one performs a time-dependent change of coordinates whichtransforms (60) into a suitable initial value problem. To this end, for t � 0; let
Notice that equation (61)1 does not contain the linearly growing term. The solutionto the Cauchy problem (61) has the following representation:
w.y; t/ D .4�t/�3=2Z
R3
e�jy�zC� te1j2=4tu.z/ d z
�
Z t
0
Z 3
R
� .y � z; t � �/ ��Rr � ŒV ˝ V �.z; �/CH .z; �/
�d z d�;
(62)
where � .�; s/; .�; s/ 2 R3 .0;1/ is the well-known Oseen fundamental solution
to the time-dependent Stokes system [45, §VIII.3].In the final step one utilizes into (62) the summability properties for u and
V obtained from Lemma 9 along with the classical pointwise estimates of � toproduce a pointwise estimate for w.x; t/; [rw.x; t/] uniformly in t . As a result,by letting t ! 1 along sequences, the latter can be shown to provide analogousbounds for u.x/ [respectively, ru.x/], which means for the weak solution v.x/[respectively, rv.x/] for all “large” jxj.
Once the necessary asymptotic information on v is obtained, analogous estimateson the pressure field can be proved observing that from (8) it follows, for sufficientlylarge �; that
�p D r �G in ˝�;
@p
@nD g on @˝�;
where
G WD � v � rv;
g WD Œ�vC �.@1v � v � rv/C T .e1 x � rv � e1 v/� � n j@˝� :
The procedure just outlined is at the basis of the following result whose full proofis found in [45, Theorems XI.6.1–XI.6.3].
Theorem 6. Let v be a weak solution, corresponding to f of bounded support, andlet p be the corresponding pressure field associated to v by Remark 1. Then, forany ı; > 0 and all sufficiently large jxj;
Steady-State Navier–Stokes Flow Around a Moving Body 31
v.x/ D O
�jxj�1.1C � s.x//�1 C jxj�3=2Cı
�;
rv.x/ D O
�jxj�3=2.1C � s.x//�3=2 C jxj�2C
�;
p.x/ D p0 CO.jxj�2 ln jxj/; for some p0 2 R;
(63)
where s.x/ WD jxj C x1.
Remark 3. This theorem suggests, in particular, that outside any semi-infinite cone,C; whose axis coincides with the negative x1 axis, the decay is faster than insideC. This is the mathematical explanation of the existence of the wake “behind” thebody, once one takes into account that the velocity of the center of mass of the body.v0 e) is directed along the positive axis x1 (� > 0).
Remark 4. The fundamental tensor solution E.x; y/ � fEij .x; y/g of the Oseensystem (which is obtained by setting T D 0 and disregarding the nonlinear termv � rv in (8)1) is defined through the relations
Eij .x; y/ D
�ıij� �
@2
@yi@yj
�˚.x � y/; ˚.�/ WD
1
4��
Z �2 .j�jC�1/
0
1 � e��
�d�:
Now, the first terms on the right-hand side of (63)1 and (63)2 are just (sufficientlysharp) bounds for E and rE at large jxj; respectively; see [45, Section VIII.3]. Thisis suggestive of the property that a �E and a � rE; for some suitable vector a; couldbe the leading terms in corresponding asymptotic expansions. Actually, if T D 0;
this property is true, and one can show that, in such a case, the following formulaholds for all sufficiently large jxj [45, Theorem X.8.1]:
v.x/ D m � E.x/C V.x/ (64)
wherem is a constant vector coinciding with the total force, F ; exerted by the liquidon the body and
V.x/ D O�jxj�3=2Cı
�; arbitrary ı > 0:
Analogous estimate can be proven for rv.x/ [45, Theorem X.8.2].If T ¤ 0; in [78, Theorem 1.1], Kyed has shown an asymptotic formula similar
to (64) (and an analogous one for rv) where nowm D .F � e1/e1 and the quantityV is a “higher-order term” in the sense of Lebesgue integrability at large distances.A pointwise estimate (probably not optimal) for V is shown in [77, Theorem 5.3.1].(An even more detailed asymptotic structure was first shown in [21] for solutions tothe linearized (Stokes) problem and in the absence of translational motion.)
32 G.P. Galdi and J. Neustupa
This section ends with some important considerations concerning the asymptoticbehavior of the vorticity field, $ WD curl v; of a weak solution, v. In this regard,one can prove the following theorem, due to Deuring and Galdi [14], that ensuresthat $ decays exponentially fast outside the wake region and sufficiently far fromthe body.
Theorem 7. Under the same assumptions of Theorem 6, there are constantsC;R > 0 such that
j$.x/j � C jxj�3=2 e�.�=4/ .jxjCx1/=.1CR/ for all x 2 ˝R:
It is worth stressing the importance of this estimate that agrees with the necessarycondition supporting the boundary layer assumption, namely, that sufficiently farfrom the body and the wake, the flow is “basically potential.” As a matter of fact,in the case T D 0; one can prove a sharper result that provides a more accuratedescription of the asymptotic structure of the vorticity field. Precisely, in that case,one has, for all sufficiently large jxj;
$.x/ D r˚ mCO�jxj�2 e�
�2 .jxjCx1/
�(65)
where
˚.x/ D ��
4� jxje�
�2 .jxjCx1/
andm is a constant vector denoting the total force exerted by the liquid on the body[3, 10].
Open Problem. In the case T ¤ 0; it is not known whether the vorticity admitsan asymptotic expansion of the type (65), with an appropriate choice of theleading term.
8 Geometric and Functional Properties for Large Data
Theorem 1 shows that, for any set of data D WD .�; T ; v�;f /; in the specifiedspaces, there exists at least one corresponding weak solution v with the furtherproperty that u WD v�U 2 X.˝/; for a suitable extension fieldU . Also, Theorem 4shows that this is, in fact, the only weak solution in that class, provided the data aresuitably restricted, according to (55). Objective of this section is to analyze thegeometric and functional properties of the solution manifold in the space X.˝/;corresponding to data of arbitrary magnitude in the class specified in Theorem 1.In order to make the presentation simpler, throughout this section, it is set v� � 0.
Steady-State Navier–Stokes Flow Around a Moving Body 33
To reach this goal, one begins to rewrite equation (30) in an equivalent way thatemphasizes the dependence of the operator involved on the parameter p WD .�; T /.One thus writes L .p;u/ for L .u/ and N .p;u/ for N .u/ with L and N definedin (25)–(27) and (29). Moreover, let H D H.p/ denote the uniquely determinedmember of D�1;20 .˝/ such that
Thus, for a given f 2 D�1;20 .˝/; (30) can be written as
M.p;u/ D f in D�1;20 .˝/ (67)
where
M W .p;u/ 2 R2C X.˝/ 7! L .p;u/CN .p;u/CH.p/ 2 D�1;20 .˝/: (68)
The solution manifold associated to (68) is defined next:
M.f / D˚.p;u/ 2 R
2C X.˝/ satisfying (67)–(68) for a given f 2 D�1;20 .˝/
�
The main goal is then to address the following questions:
(a) Geometric structure of the manifold M DM.f /
(b) Topological properties of the associated level set
S.p0;f / WD f.p;u/ 2M.f /; p D p0g;
obtained by fixing also Reynolds and Taylor numbers.Clearly, “points” in S.p0;f / are solutions to equation (67) with a prescribed
p D p0 or, equivalently, to equation (30).
The following theorem collects the principal properties of the set S.p0;f /.
Theorem 8. The following properties hold.
(i) S.p0;f / is not empty.(ii) For any .p0;f / 2 R
2C D�1;20 .˝/; S.p0;f / is compact. Moreover, there is
N D N.p0;f / 2 N such that S.p0;f / is homeomorphic to a compact set ofRN .
(iii) For any p0 2 R2C; there is an open residual set O D O.p0/ � D�1;20 .˝/
such that, for every f 2 O; S.p0;f / is constituted by a number of points,� D �.p0f /; that is finite and odd.
(iv) The number � is constant on every connected component of O.
34 G.P. Galdi and J. Neustupa
Proof. As usual, only a sketch of some of the proofs of the above statements willbe given while referring to the appropriate reference for whatever is missing. Thestatement (i) is a consequence of Theorem 1. The proofs of the other statements arebased on two fundamental properties of the operator M WD L .p0; �/ CN .p0; �/;
namely, being (1) proper and (2) Fredholm of index 0. Now, Lemma 5 andLemma 6 guarantee that the Fréchet derivative of M at every u 2 X.˝/ is acompact perturbation of a homeomorphism, which proves the Fredholm property.Properness means that if F ranges in a compact set, K; of D�1;20 .˝/; all possiblecorresponding solutions u to M.u/ D F belong to a compact set, K�; of X.˝/.To show that this is indeed the case, one observes that in view of the continuity ofM; K� is closed, so that it is enough to show that from any sequence fung � K�;there is a subsequence (still denoted by fung) and u 2 X.˝/ such that un ! u inX.˝/. Let F n DM.un/. Since fF ng � K; one deduces (along a subsequence)
F n ! F in D�1;20 .˝/; for some F 2 K. (69)
Moreover, being fF ng bounded, by Lemma 7, it follows that fung is bounded, andso there exists u 2 X such that un ! u weakly in X.˝/. By Lemma 3 and (69),the latter implies
In view of (70), the first term on the left-hand side of (72) tends to 0 as n ! 1.Likewise, since N 0.u/ is compact, and hence completely continuous, also thesecond term on the left-hand side of (72) tends to 0 as n ! 1; so that propernessfollows from this and Lemma 7. The first property in (ii) is then a corollary ofwhat has just been proven. As for the second one, we refer to [47, Theorem 93] fora proof. We next come to show statements (iii) and (iv). In this regard, since M isproper and Fredholm of index 0, by the mod 2 degree of Smale [103], it is enough toshow that there exists F 0 2 D1;2
0 .˝/ with the following properties: (a) the equation
Steady-State Navier–Stokes Flow Around a Moving Body 35
M.u/ D F 0 has one and only one solution, u0; and (b) N.M0.u0// D f0g; see [41,Lemma 6.1]. Now, set F 0 D 0. From Lemma 7, it follows that the only solutionto M.u/ D 0 is u0 D 0. Moreover, by Lemma 3, one finds N 0.0/ � 0; so thatM0.0/ D L and condition (b) is a consequence of Lemma 5. �
Remark 5. Taking into account that the set O in Theorem 8 is dense in D�1;20 .˝/;
from Theorem 8(iii), one deduces the following interesting property of weaksolutions. Let � ¤ 0 and T � 0 be arbitrarily fixed. Given f 2 D�1;20 .˝/ and" > 0; there is g 2 D�1;20 .˝/ with jf � gj�1;2 < " such that the number of weaksolutions given in Theorem 1 corresponding to the body force g (and v� � 0) isfinite and odd.
The next result furnishes a complete generic characterization of the manifoldM.f /. Its proof, based on an infinite-dimensional version of the so-called parame-terized Sard theorem [107, Theorem 4.L], is technically involved and lengthy. Theinterested reader is referred to [47, Theorem 88].
Theorem 9. The following properties hold.
(i) There exists a dense, residual set Z � D�1;20 .˝/ such that, for any f 2 Z;the solution manifold M.f / is a two-dimensional (not necessarily connected)manifold of class C1.
(ii) For any f 2 Z; there exists an open, dense set P D P.f / � R2C such that,
for each p 2 P; equation (67) has a finite number of solutions, n D n.p;f /.(iii) The integer n D n.p;f / is independent of p on every interval contained in P.
Open Problem. It is not known whether, in the physically significant case ofvanishing body force f and boundary velocity v�; the number of correspond-ing steady-state solutions is generically finite.
9 Bifurcation
As pointed out in the introductory section, if the speed of the center of mass ofthe body, v0; reaches a critical value, it is experimentally observed, already inabsence of rotation, that the characteristic features of the original steady-state flowof the liquid may change dramatically. The outcome could be either the onset of anentirely different steady-state flow or even of a time-periodic regime. The objectiveof this section is to provide necessary conditions and sufficient conditions for theoccurrence of this phenomenon. More precisely, Sect. 9.1 will be concerned with
36 G.P. Galdi and J. Neustupa
time-independent problems, while Sect. 9.2 will deal with the time-periodic case.For the sake of simplicity, it will be assumed throughout v� � 0.
9.1 Steady Bifurcation
One is mainly interested in situations where bifurcation is generated by the“combined” action of translation and rotation of the body (provided the latter isnot zero). To this end, it is convenient to use a different nondimensionalization forthe equations (8)–(7), in order to introduce an appropriate bifurcation parameter.Precisely rescaling velocity with v0 and length with v0=!; equations (8)–(7) become
�vC ��@1vC e1 x � rv � e1 v � v � rv
�D rp C f
div v D 0
�in ˝
v D e1 C e1 x at @˝ I limjxj!1
v.x/ D 0;(73)
where now � WD v20.e � e1/=.�!/.
Remark 6. Of course, the above nondimensionalization requires ! ¤ 0. However,for future reference, it is important to emphasize that all main results presented inthis section continue to hold in exactly the same form also when ! D 0.
With the notation introduced in the previous section (see (67) and (68) with p �
�), the original equation (30) is equivalent to the following nonlinear equation
M.�;u/ WD L .�;u/CN .�;u/CH.�/ D f in D�1;20 .˝/; u 2 X.˝/ (74)
Definition 2. Let u0 2 X.˝/ be a solution to (74) with � D �0. The pair .�0;u0/is called a steady bifurcation point for (74), if there are two sequences f�k;u
.1/
k g and
f�k;u.2/
k g with the following properties:
(i) f�k;u.i/
k g; i D 1; 2 solve (74) for all k 2 N.
(ii) f�k;u.i/
k g ! .�0;u0/ in R X.˝/ as k !1; i D 1; 2.
(iii) u.1/k 6� u.2/k ; for all k 2 N.
One of the main achievements of this section is the proof that, under certainconditions that may be satisfied in problems of physical interest, bifurcation isreduced to the study of a suitable linear eigenvalue problem, formally analogousto that occurring in the study of bifurcation for flow in a bounded domain; seeTheorem 11 and Remark 8.
Steady-State Navier–Stokes Flow Around a Moving Body 37
A necessary condition in order for .�0;u0/ to be a bifurcation point is obtainedas a corollary to the following result.
Lemma 10. Let u0 2 X.˝/ be a solution to (74) with � D �0 and fixed f 2D�1;20 .˝/. If
N.M 0.�0;u0// D f0g; (75)
namely, the (linear) equation
L .�0;w/C �0B.u0;w/C B.w;u0/
D 0 in D�1;20 .˝/ (76)
has only the solution w D 0 inX.˝/; then there exists a neighborhood U.�0/; suchthat for each � 2 U.�0/ there is one and only one u.�/ solution to (74). Moreover,the map � 2 U ! u.�/ 2 X.˝/ is analytic at � D �0.
(The prime means Fréchet differentiation with respect to u.)
Proof. Consider the map
F W .�;u/ 2 U.�0/ X.˝/ 7!M.�;u/ � f :
Also using the fact that N .�; �/ is quadratic (see (31)–(32)), it easily followsthat F is analytic (polynomial, in fact) at each .�;u/. Moreover, by assumption,F .�0;u0/ D 0. Thus, the claimed property will follow from the analytic versionof the implicit function theorem provided one shows that F 0.�0;u0/ is a bijection.Now, from (31)–(32),
F 0.�0;u0/ DM 0.�0;u0/ � L .�0; �/C �0B.u0; �/C B.�;u0/
;
so that by Lemma 5 and Lemma 6, we infer that F 0.�0;u0/ is Fredholm of index0 and the bijectivity property follows from the assumption (75). �
From this result the following one follows at once.
Corollary 1. A necessary condition for .�0;u0/ to be a bifurcation point is that
dim N.M 0.�0;u0// > 0; (77)
namely, the (linear) equation
L .�0;w/C �0B.u0;w/C B.w;u0/
D 0 in D�1;20 .˝/ (78)
has a nonzero solution w 2 X.˝/.
38 G.P. Galdi and J. Neustupa
Remark 7. One can show that (77) is equivalent to the requirement that the line-arization of (73) around .�0; v0 � u0 C U / corresponding to homogeneous data,namely,
�wC �0�@1wC e1 x � rw � e1 w � v0 � rw � w � rv0
�D rp
div w D 0
)in ˝
w D 0 at @˝ I limjxj!1
w.x/ D 0;
(79)has a nontrivial solution .w; p/ 2 ŒD2;2.˝/\X.˝/�D1;2.˝/. In fact, saying that(78) has a nonzero solution w 2 X.˝/ means that there exists w 2 X.˝/ � f0gsuch that (see (25)–(30) with � D T ))
for all ' 2 D1;20 .˝/. However, by the properties of v0 and w combined with
the Hölder inequality, one shows G WD .w � rv0 C v0 � rw/ 2 L4=3.˝/;
which, in turn, by classical results on the generalized Oseen equation [45, TheoremVIII.8.1] furnishes, in particular, w 2 D2;4=3.˝/. By embedding, the latter impliesw 2 D1;12=5.˝/ \ L12.˝/; so that G 2 L12=7.˝/ which, again by [45, TheoremVIII.8.1], delivers w 2 D2;12=7.˝/ \ D1;4.˝/ \ L1.˝/. Thus, G 2 L2.˝/ andthe property follows by another application of [45, Theorem VIII.8.1]. Notice thatthe asymptotic condition in (79) is achieved uniformly pointwise.
The next objective is to provide sufficient conditions for .�0;u0/ to be abifurcation point. To this end, it will be assumed that, in the neighborhood of.�0;u0/; there exists a sufficiently smooth solution curve, that is, there is a map� 2 U.�0/ 7! u.�/ 2 X.˝/ of class C2 (say), with u.�0/ D u0 and satisfying (74)for the given f . Setting w WD u � u; one thus gets that w satisfies the equation
F.�;w/ WD L .�;w/C �ŒB.w;u.�//C B.u.�/;w/�CN .�;w/ D 0: (81)
Clearly, .�0;u0/ is a bifurcation point for (74) if and only if .�0; 0/ is a bifurcationpoint for (81). Since, as showed earlier on, F 0.�0; 0/ � L .�0; �/C�0ŒB.�;u0.�//CB.u0; �/� is Fredholm of index 0, a classical result [107, Theorem 8.A] ensures that.�0; 0/ is a bifurcation point provided the following conditions hold:
(i) dim N .F 0.�0; 0// D 1;
(ii) ŒF�w.�0; 0/�.w1/ 62 R .F 0.�0; 0//; w1 2 N .F 0.�0; 0//;
where the double subscript denotes differentiation with respect to the indicatedvariable. Condition (i) specifies in which sense the requirement of Corollary 1 mustbe met. In order to give a more explicit form to condition (ii), it is convenient tointroduce the Stokes operator:
Steady-State Navier–Stokes Flow Around a Moving Body 39
Q� W u 2 D1;20 .˝/ 7!
Q�u 2 D�1;20 .˝/; (82)
with
h Q�u;'i D �.ru;r'/; ' 2 D1;20 .˝/: (83)
As is well known, Q� is a homeomorphism [45, Theorem V.2.1]. By a straightforwardcomputation, one then shows that
ŒF�w.�0; 0/�.w1/ D �1
�0Q�w1 C �0
B.w1; Pu.�0//C B. Pu.�0/;w1/
;
(with “�” denoting differentiation with respect to �) and therefore condition (ii) isequivalent to the request that the equation
L.�0;w/C �0B.u0;w/C B.w;u0/
D �1
�0Q�w1 C �0
B.w1; Pu.�0//C B. Pu.�0/;w1/
(84)
has no solution. All the above is summarized in the following.
Theorem 10. Suppose the solution set of the equation
L .�0;w/C �0B.u0;w/C B.w;u0/
D 0 (85)
is a one-dimensional subspace of X.˝/ and let w1 be a corresponding normalizedelement. If, in addition, equation (84) has no solution w 2 X.˝/; then .�0;u0/ is abifurcation point for (74).
The assumptions of the result just proven admit a noteworthy conceptualinterpretation in the case when Pu.�0/ D 0. This happens, in particular, if u.�/is constant in a neighborhood of �0; a circumstance that may occur by a suitablenondimensionalization of the original equation [34, Section VI]. To show the above,consider the operator
L W w 2X.˝/ � D1;20 .˝/ 7! L.w/
D � Q��1Œ@1wCR.w/C B.v0;w/C B.w; v0/�2D1;20 .˝/;
where, as before, v0 WD u0 C U .The following lemma shows the fundamental properties of L. The proof is quite
involved, and, for it, the reader is referred to [47, Lemma 111].
Lemma 11. Assume u0 2 L3.˝/ \ L4loc.˝/. Then, the operator L is (graph)closed. Moreover, Sp.LC/\.0;1/ consists, at most, of a finite or countable number
40 G.P. Galdi and J. Neustupa
of eigenvalues, each of which is isolated and of finite algebraic and geometricmultiplicities, that can only accumulate at 0.
(Of course, the assumption u0 2 L4loc.˝/ is redundant if u0 2 X.˝/. Also u0 2L3.˝/ is assured by Lemma 9 if f is suitably summable at large distances.)
Combining Corollary 1, Lemma 11, and Theorem 10, one can then show thefollowing.
Theorem 11. Assume Pu.�0/ D 0; with u0 2 L3.˝/ \ L4loc.˝/. Then, a necessarycondition for .�0;u0/ to be a bifurcation point for (74) is that 0 WD 1=�0 is aneigenvalue for the operator LC. This condition is also sufficient if 0 is simple.
Proof. With the help of (25), (26), and (27) and (30), one sees that condition (77)is equivalent to assuming that the following equation has a nonzero solution w1 2X.˝/
Q�w1 C �0Œ@1w1 CR.w1/C B.v0;w1/C B.w1; v0/� D 0:
Operating with Q��1 on both sides of the latter, one concludes that 0 must bean eigenvalue of LC; which provides the first statement. Performing the sameprocedure on (85), it can be next shown that the first assumption in Theorem 10is satisfied if and only if there is a unique (normalized) w1 2 X.˝/ such that
L.w1/ D 0 w1;
that is,0 is an eigenvalue ofLC of geometric multiplicity 1. Furthermore, operatingagain with Q��1 on both sides of (84) with Pu.�0/ D 0; one gets
0 w � L.w/ D �20w1
which, by the second assumption in Theorem 10, should have no solution, whichmeans that the algebraic multiplicity of 0 must be 1 as well and the proof of theclaimed property is completed. �
Another interesting and immediate consequence of Lemma 11 and Theorem 11is the following one.
Corollary 2. Let u0 be a solution branch to (74) independent of � 2 J; whereJ is a bounded interval with J � .0;1/. Suppose, further, that u0 satisfies theassumption of the preceding theorem. Then, there is at most a finite number, m; ofbifurcation points to (74) .�k;u0/; �k 2 J; k D 1; � � � ; m.
Remark 8. It is significant to observe that the statements of Theorem 11 andCorollary 2 formally coincide with those of analogous theorems for steady
Steady-State Navier–Stokes Flow Around a Moving Body 41
bifurcation from steady solution to the Navier–Stokes equation in a boundeddomain; see, e.g., [6, Section 4.3C]. However, in the latter case, L is a compactoperator defined on the whole of D1;2
0 .˝/; whereas in the present case, L is adensely defined unbounded operator.
9.2 Time-Periodic Bifurcation
In spite of its great relevance and frequent occurrence in experimental fluidmechanics, time-periodic bifurcation in a flow past an obstacle has represented along-standing and intriguing problem from a rigorous mathematical viewpoint. Thissituation should be contrasted with flow in a bounded domain where, thanks to thepioneering and fundamental contributions of Iudovich [66], Joseph and Sattinger[68], and Iooss [65], complicated time-periodic bifurcation phenomena, like thoseoccurring in the classical Taylor–Couette experiment, could be framed in a rigorousmathematical setting.
In order to understand the reason for this uneven situation and also provide amotivation for the approach presented here, it is appropriate to briefly describe whatconstitutes a rigorous treatment of the phenomenon of time-periodic bifurcation.Suppose, as will be in fact shown later on, that the relevant time-dependent problemcan be formally written in the form
ut C L.u/ D N.u; /; (86)
where L is a linear differential operator (with appropriate homogeneous boundaryconditions) and N is a nonlinear operator depending on the parameter 2 R; suchthat N.0; / D 0 for all admissible values of . Then, roughly speaking, time-periodic bifurcation for (86) amounts to show the existence a family of nontrivialtime-periodic solutions u D u.I t / of (unknown) period T D T ./ (T - periodicsolutions) in a neighborhood of D 0 and such that u.I �/! 0 as ! 0. Setting� WD 2� t=T � ! t; (86) becomes
! u� C L.u/ D N.u; /; (87)
and the problem reduces to find a family of 2�-periodic solutions to (87) with theabove properties. If one now writes u D uC .u� u/ WD vCw; one gets that (87) isformally equivalent to the following two equations
L.v/ D N.v C w; / WD N1.v;w; /;! w� C L.w/ D N.v C w; / �N.v C w; / WD N2.v;w; /:
(88)
At this point, the crucial issue to realize is that while in the case of a bounded flow,both “steady-state” component, v; and “oscillatory” component, w; may be takenin the same (Hilbert) function space [65, 66, 68]; in the case of an exterior flow,
42 G.P. Galdi and J. Neustupa
v belongs to a space with quite less “regularity” (in the sense of behavior at largespatial distances) than w does; see also [51]. For this basic reason, as emphasizedfor the first time only recently in [49, 50], in the case of an exterior flow, it is notappropriate (or even “natural”) to investigate the bifurcation problem for (87) injust one functional setting, as done, for example, in [99]; it is instead much morespontaneous to study the two equations in (88) in two different function classes. Asa consequence, even though formally being the same as differential operators, theoperator L in (88)1 acts on and ranges into spaces different than those the operatorL in (88)2 does. With this in mind, (88) becomes
L1.v/ D N1.v;w; / I ! w� C L2.w/ D N2.v;w; /:
The above ideas will be next applied to provide sufficient conditions for time-periodic bifurcation in a viscous flow past a body. It will be assumed throughoutT D 0, leaving the case T ¤ 0 as an open question. Set
L1 W v 2 X.˝/ 7! L .�0; v/ 2 D�1;20 .˝/ (89)
with L .�0; v/ defined in (76). From Lemma 10 and Remark 6, it follows thatunder the assumption
N .L1/ D f0g; (H1)
there exists a unique weak solution analytic branch vs.�/ WD u.�/C U to (8)–(7)in a neighborhood U.�0/; with vs.�0/ D u0 C U . Thus, writing v D v.x; t I�/Cvs.xI�/; from (5), one finds that v formally satisfies the (nondimensional) problem
vtC�.v � e1/ � rvC vs.�/ � rvC v � rvs.�/
D �v � rp
div v D 0
�in ˝ R
v D 0 at @˝ R; limjxj!1
v.x; t/ D 0; t 2 R:
(90)
The bifurcation problem consists then in finding sufficient conditions for theexistence of a nontrivial family of suitably defined time-periodic weak solutionsto (90), v.t I�/; � 2 U.�0/; of period T D T .�/ (unknown as well), such thatv.t I�/! 0 as �! �0.
Following the general approach mentioned before, one thus introduces the scaledtime � WD ! t; split v and as the sum of its time average, v; over the time intervalŒ��; ��; and its “purely periodic” component w WD v � v; and set WD � � �0.In this way, problem (90) can be equivalently rewritten as the following couplednonlinear elliptic-parabolic problem
Steady-State Navier–Stokes Flow Around a Moving Body 43
�vC �0�@1v � v0 � rv � v0 � rv
�D rpCN 1.v;w; /
div v D 0
�in ˝
v D 0 at @˝; limjxj!1
v.x/ D 0(91)
and
! w� ��w � �0�@1w� v0 � rw � w � rv0
�
D r' CN 2.v;w; /div w D 0
9=
; in ˝2�
w D 0 at @˝ .��; �/; limjxj!1
w.x; t/ D 0;
(92)
where
N 1 WD � Œ@1v � vs.C �0/ � rv � v � rvs.C �0/�C�0
.vs.C �0/ � v0/ � rvC v � r.vs.C �0/ � v0/
C.C �0/hv � rvC w � rw
i (93)
and
N 2 WD Œ@1w � vs.C �0/ � rw � w � rvs.C �0/�
��0
h.vs.C �0/ � v0/ � rwC w � r.vs.C �0/ � v0/
i
C.C �0/hw � rvC v � rwC w � rw � w � rw
i;
(94)
with v0 � vs.�0/.The next step is to rewrite (91), (92), (93), and (94) in the proper functional
setting and to reformulate the bifurcation problem accordingly. To this end, onebegins to introduce the operator
L2 W w 2 D.L2/�H.˝/ 7! � P�wC �0.@1w � v0 � rw � w � rv0/
2H.˝/;
D.L2/ WD W2;2.˝/ \D1;2
0 .˝/:
(95)
The following result can be proved by the same arguments (slightly modified in thedetail) employed in [50, Proposition 4.2].
Lemma 12. Let u0 WD v0 � U 2 X.˝/. Then Sp.L2C/ \˚iR � f0g
�consists, at
most, of a finite or countable number of eigenvalues, each of which is isolated andof finite (algebraic) multiplicity, that can only accumulate at 0.
Consider, next, the time-dependent operator
44 G.P. Galdi and J. Neustupa
Q W w 2 W 22�;0.˝/ 7! !0 wt CL2.w/ 2H2�;0.˝/: (96)
Again, by a slight modification of the argument used in the proof of [50, Proposition4.3], one can show the following.
Lemma 13. Let v0 be as in Lemma 12. Then, the operator Q is Fredholm of index0, for any !0 > 0.
Finally, one needs the functional properties of the quantities N i ; i D 1; 2;
defined in (93)–(94), reported in the following lemma. The proof is, one more time,a slight modification of that given in [50, Lemma 4.5, and the paragraph after it] andwill be omitted.
Lemma 14. There is a neighborhood V.0; 0; 0/ � R X.˝/ W 22�;0.˝/ such
Also in view of Lemmas 12–14, one then deduces that (91)–(94) can be put inthe following abstract form
L1.v/ D N1.; v;w/ in D�1;20 .˝/ I ! w� CL2.w/ D N2.; v;w/ in H2�;0:
(97)
Notice that the spatial asymptotic conditions on v in (91)4 are interpreted in thesense of Remark 2, while the one in (91)4 for w holds uniformly pointwise for a.a.t 2 R; see [50, Remark 3.2].
One is now in a position to give a precise definition of a time-periodic bifurcationpoint.
Definition 3. The triple . D 0; v D 0;w D 0/ is called time-periodic bifurcationpoint for (97) if there is a sequence f.k; !k; vk;wk/g � RRCD�1;20 .˝/W 2
2�;0
with the following properties:
(i) f.k; !k; vk;wk/g solves (97) for all k 2 N.(ii) f.k; vk;wk/g ! .0; 0; 0/ as k !1.
(iii) wk 6� 0; for all k 2 N.
Moreover, the bifurcation is called supercritical [resp. subcritical] if the abovesequence of solutions exists only for k > 0 [resp. k < 0].
Steady-State Navier–Stokes Flow Around a Moving Body 45
The goal is to give sufficient conditions for the occurrence of time-periodicbifurcation in the sense specified above. This will be achieved by means of thegeneral result proved in [50, Theorem 4.1]. With this in mind, one has to show thatthe assumptions of that theorem are indeed satisfied. In this regard, supported byLemma 12, one supposes
�0 WD i!0 is an eigenvalue of multiplicity 1 of L2C;
k �0; k 2 N � f0; 1g is not an eigenvalue of L2C:(H2)
Next, consider the operator
L2./ WD L2 � S;
with
S W w2Z2;2.˝/ 7!P@1w�v0�rw�w�rv0��0
�Pvs.�0/�rwCw�r Pvs.�0/
�2H.˝/;
where, as before, “�” means differentiation with respect to �. By [108, Proposition79.15 and Corollary 79.16], one knows that for in a neighborhood of 0, there isa smooth map 7! �./; with �./ simple eigenvalue of L2C./ and such that�0 D �.0/. The following condition will be further assumed:
< Œ P�.0/� ¤ 0; (H3)
which basically means that the eigenvalue �./ must cross the imaginary axis with“nonzero speed” when �! �0.
The general result proved in [50, Theorem 3.1] can be now applied to show thefollowing time-periodic bifurcation result.
Theorem 12. Suppose (H1)–(H3) hold. Then, the following properties are valid.(a) Existence. There are analytic families
�v."/;w."/; !."/; ."/
�2 X.˝/ W 2
2�;0.˝/ RC R (98)
satisfying (97), for all " in a neighborhood I.0/ and such that
�v."/;w."/ � " v1; !."/; ."/
�! .0; 0; !0; 0/ as "! 0:
(a) Uniqueness. There is a neighborhood
U.0; 0; !0; 0/ � X.˝/ W 22�;0.˝/ RC R
such that every (nontrivial) 2�-periodic solution to (97), .z; s/; lying in U mustcoincide, up to a phase shift, with a member of the family (98).
46 G.P. Galdi and J. Neustupa
(a) Parity. The functions !."/ and ."/ are even:
!."/ D !.�"/; ."/ D .�"/; for all " 2 I.0/.
Consequently, the bifurcation due to these solutions is either subcritical orsupercritical, a two-sided bifurcation being excluded (unless � 0).
Open Problem. Sufficient conditions for the occurrence of time-periodic bifur-cation in the case when the body is also spinning (T ¤ 0) are not known.
10 Stability and Longtime Behavior of UnsteadyPerturbations
In this section, v0 will denote the velocity field of a steady-state solution to (8). Asusual, � is assumed to be positive. However, since the theories that will be describedin this section have often been equally developed for both cases � D 0 and � 6D 0;
a number of cited results also concern the case when � D 0. The function v0 issupposed to satisfy
v0 2 L3.˝/; @jv0 2 L
3.˝/ \ L3=2.˝/ (for j D 1; 2; 3): (99)
It follows from Lemma 9 that, if � ¤ 0; such a v0 exists for a large class of bodyforces f and boundary data. An associated pressure field is denoted by p0.
One is interested in the behavior of unsteady perturbations, .v0; p0/ to the solution.v0; p0/. Thus, writing v D v0 C v
0; p D p0 C p0; it follows from (5) that the
functions v0; p0 satisfy the equations
�v0t C�v0 C � @1v
0 C T .e1 x � rv0 � e1 v0/D �v0 � rv
0 C �v0 � rv0 C �v0 � rv0 Crp0
div v0 D 0
9>=
>;in ˝ .0;1/ (100)
and the conditions
v0 D 0 at @˝ .0;1/ I limjxj!1
v0.x; t / D 0; all t 2 .0;1/: (101)
For simplicity, from now on the primes are omitted in the notation above. Thus, theformal application of the Helmholtz–Weyl projection P to the first equation in (100),as formulated in Lq.˝/ (1 < q <1), yields the operator equation
Steady-State Navier–Stokes Flow Around a Moving Body 47
dv
dtD LvCNv (102)
in the space Hq.˝/. By suitably defining the domains of the operators L and N; itcan be easily seen that (102) is, in fact, equivalent to (100) and (101). To this end,let
Av WD P�v;
B1v WD P @1v
for v 2 D.A/ WD W 2;q.˝/ \D1;q0 .˝/;
B2v WD P .e1 x � rv � e1 v/;
A�;T v WD AvC �B1vC T B2v
for
v 2 D.A�;T / WD
(W 2;q.˝/ \D1;q
0 .˝/ if T D 0;˚v 2 W 2;q.˝/ \D1;q
0 .˝/I e1 x � rv 2 Lq.˝/
�if T 6D 0:
Note that A � A0;0 and A�;0 � A C �B1 are the classical Stokes and Oseenoperators, respectively. Furthermore, let
B3v WD P .v0 � rvC v � rv0/;
Lv WD A�;T vC �B3v;
Nv WD ��P .v � rv/
for v 2 D.L/ WD D.A�;T /.Obviously, the study of the stability of the solution .v0; p0/ is equivalent to
that of the zero solution of problem (100), (101) or equation (102). The propertiesof the linear operator L and, especially, those of its “leading part” A�;T play afundamental role. Thus, the next two subsections will be concerned with a detailedanalysis of these properties.
10.1 Spectrum of Operator A�,T
The following notions and definitions from the spectral theory of linear operatorswill be relevant later on.
Let X be a Banach space with norm k : k; X� be its dual, and T be a closedlinear operator in X with a domain D.T / dense in X . (This guarantees that theadjoint operator T � exists.)
48 G.P. Galdi and J. Neustupa
– The symbols nul .T / and def .T / denote the nullity and the deficiency of T;respectively. If R.T / is closed, then nul .T / D def .T �/ and def .T / D nul .T �/(see, e.g., Kato [69, p. 234]).
– The approximate nullity of T; denoted by nul 0 .T /; is the maximum integer m(m D 1 being permitted) with the property that to each � > 0; there exists anm-dimensional linear manifold M� in D.T / such that kT vk < � for all v 2 M�;
kvk D 1. The approximate deficiency of T is denoted by def 0 .T / and definedas def 0 .T / WD nul 0 .T �/. Note that nul .T / � nul0.T / and def .T / � def0.T /;the equalities holding if the range R.T / is closed. ETC. On the other hand, ifR.T / is not closed, then nul 0 .T / D def 0 .T / D1. The identity nul 0 .T / D1 isequivalent to the existence of a non-compact sequence fung on the unit sphere inX such that T un ! 0 for n!1 (see [69, p. 233]).
– T is called a Fredholm operator if both the numbers nul .T / and def .T / arefinite. This implies, in particular, that R.T / is closed in X [108, Proposition8.14(ii)]. Operator T is semi-Fredholm if the range R.T / is closed in X and atleast one of the numbers nul .T / and def .T / is finite. Consequently, T is semi-Fredholm if and only if at least one of the numbers nul 0 .T / and def 0 .T / is finite.
– The resolvent set Res.T / is the set of all � 2 C such that R.T � �I / D X andthe operator T � �I has a bounded inverse in X . Consequently, nul .T � �I / Dnul 0 .T � �I / D def .T � �I / D def 0 .T � �I / D 0 for � 2 Res.T /. Note thatRes.T / is an open subset of C.
– The point spectrum Spp.T / is the set of all � 2 C such that nul .T � �I / > 0.– The continuous spectrum Spc.T / is the set of all � 2 C such that nul .T � �I / D0; R.T � �I / is dense in X; but R.T � �I / 6D X . (In this case, R.T � �I / is notclosed in X; which implies that def .T � �I / D def 0 .T � �I / D nul 0 .T � �I / D1.)
– The residual spectrum Spr.T / is the set of all � 2 C such that nul .T � �I / D 0and the range R.T ��I / is not dense inX . The sets Spp.T /; Spc.T /; and Spr.T /
are mutually disjoint and Spp.T / [ Spc.T / [ Spr.T / D Sp.T / D C X Res.T /(the spectrum of T ).
– The essential spectrum Spess.T / is the set of all � 2 C such that T � �I is notsemi-Fredholm. Both Sp.T / and Spess.T / are closed in C and Spess.T / � Sp.T /.Obviously, Spc.T / � Spess.T /. Any point on the boundary of Sp.T / belongs toSpess.T / unless it is an isolated point of Sp.T / (see [69, p. 244]).
From [70] (if T D 0) and [102] (if T 6D 0), it follows that the operator A�;Tis closed in Hq.˝/ (1 < q < 1), and all � 2 C with a sufficiently large real partbelong to Res.A�;T /. The effective shapes and types of spectra of the operatorA�;T ;for various values of � and T ; are described in [18] (inH.˝/; the case � D 0), [19](in H.˝/; the general case � 2 R), [22] (in Hq.˝/; � D 0), and [20] (in Hq.˝/;
� 2 R). The spectrum ofA�;0; as an operator inH.˝/;was studied by K.I. Babenko[4]. Babenko’s result says that Sp.A�;0/ D Spc.A�;0/ D ��;0; where
��;0 D f� D ˛ C iˇ 2 CI ˛; ˇ 2 R; ˛ � �ˇ2=�2g (103)
Steady-State Navier–Stokes Flow Around a Moving Body 49
for � 6D 0. The set��;0 represents a parabolic region in C; symmetric about the realaxis, which shrinks to the nonnegative part of the real axis if �! 0. In fact, Sp.A/(� Sp.A0;0/) coincides with Spc.A/ and coincides with the interval .�1; 0� in R;
as mentioned, e.g., by O.A. Ladyzhenskaya in [80].The spectrum of A�;T for general T is studied in [20]. Notice that the case
T 6D 0 is qualitatively different from the case T D 0; because the magnitude ofthe coefficient of the “new” term T e1 x � rv becomes unbounded as jxj ! 1.Consequently, the operator T e1 x � r cannot be treated as a lower-orderperturbation of Stokes or Oseen operator. The main results in [20] read as follows.
Theorem 13. Let 1 < q < 1; � 6D 0 and ˝ D R3. Then the spectrum of
A�;T ; as an operator in Hq.R3/; satisfies the identities Sp.A�;T / D Spc.A�;T / D
Spess.A�;T / D ��;T ; where
��;T WD f� D ˛ C iˇ C ikT 2 CI ˛; ˇ 2 R; k 2 Z; ˛ � �ˇ2=�2g:
Note that ��;T is a union of a family of overlapping solid parabolas, whose axesform an equidistant system of half-lines f� 2 CI � D ˛C kT i; ˛ � 0; k 2 Zg. Allthe parabolas lie in the half-plane Re � � 0; and their vertices are on the imaginaryaxis.
Theorem 14. Let 1 < q < 1; � 6D 0 and ˝ � R3 be an exterior domain with
the boundary of class C1;1. Then the spectrum of A�;T lies in the left complex half-plane f� 2 CIRe � � 0g and consists of the essential spectrum Spess.A�;T / D ��;Tand possibly a set � of isolated eigenvalues � 2 C X��;T with Re � < 0 and finitealgebraic multiplicity, which can cluster only at points of Spess.A�;T /. The set � ofsuch isolated eigenvalues is independent of q 2 .1;1/.
Sketch of the proof of Theorem 13 (see [20] for the details). The proof developsalong the following steps (a)–(f).
(a) Using the definition of the adjoint operator, it can be verified that the adjointoperator A��;T to A�;T coincides with the operator A��;�T in Hq0.˝/; where1=q C 1=q0 D 1.
(b) From [17, Theorem 1.1], one can deduce that there exist constants C4 > 0
and C5 > 0 such that if u 2 D.A�;T / and f 2 Hq.˝/ satisfy the equationA�;T u D f ; then
kuk2;q C k.! x/ � rukq � C4 kf kq C C5 kukq : (104)
(c) If � 2 CX��;T ; then each solution of the resolvent equation .A�;T ��I /u D f ;for f 2 Hq.˝/; satisfies the estimate
kukq � C6 kf kq; (105)
50 G.P. Galdi and J. Neustupa
where C6 D C6.�; q/. This estimate is derived by means of the Fouriertransform, and a subtle and rather technical application of the Mikhlin–Lizorkinmultiplier theorem. Using inequalities (104) and (105), one can prove thatR.A�;T � �I / is closed and the operator A�;T � �I is injective. The samestatement also holds on the adjoint operator A��;T in Hq0.˝/. As A��;T � �Iis injective, the range R.A�;T � �I / is the whole space Hq.˝/. Consequently,� 2 Res.A�;T /. This shows that C X��;T � Res.A�;T /.
(d) We show that Spp.A�;T / D ;. Assume that � 2 ��;T and u 2 D.A�;T / satisfiesthe equation .A�;T ��I /u D 0. Applying the Fourier transform F ; this equationyields
T .e1 � � rbu/ � .� � i��1 C j�j2/bu � T e1 bv D 0; (106)
where bu D F.u/ and � D .�1; �2; �3/ denotes the Fourier variable. The case1 < q � 2 is simpler, because bu is a function from Lq
0
.˝/: if �1; r; and �denote the cylindrical coordinates in the space of Fourier variables, then onecan calculate that e1 � � rbu D @�bu. Substituting this to (106), one obtains theequation
T @�bu � .� � i��1 C j�j2/bu � T e1 bv D 0:
If O.�/ denotes the matrix of rotation about the �1-axis by angle � andbw.�1; r; �/ WD O.�/bu.�1; r; �/; then one arrives at the ordinary differentialequation
T @�bw � .� � i��1 C r2 C �21 /bw D 0:
This equation can be solved explicitly. The solution satisfies: bw.�1; r; � C2�/ D bw.�1; r; �/ e2� .��i��1Cr2C�21 /. As bw is 2�-periodic in variable � andRe � C r2C �21 D Re � C j�j2 6D 0 for a.a. � 2 R
3;bw is equal to zero a.e. in R3.
It means that u is the zero element ofHq.˝/; which implies that it cannot be aneigenfunction and � therefore cannot be an eigenvalue. The case 2 < q <1 israther more complicated becausebu is only a tempered distribution. Nevertheless,one can also arrive at the same conclusion, i.e., that any � 2 ��;T cannot be aneigenvalue of A�;T .
(e) The identity Spr.A�;T / D ; can be proven by means of the duality argument:� 2 Spr.A�;T / would imply that � 2 Spp.A
��;T /. However, the same
considerations as in step (d), applied to the adjoint operator A��;T ; show thatSpp.A
��;T / D ;.
(f) The identities Sp.A�;T / D Spc.A�;T / D Spess.A�;T / follow from the factsthat Spp.A�;T / and Spr.A�;T / are empty and Spc.A�;T / � Spess.A�;T /. Theinclusion Sp.A�;T / � ��;T follows from item (c). The inclusion ��;T �Spess.A�;T / is proven in [20] so that � is assumed to be in �ı�;T (the interiorof ��;T ), and a concrete sequence fung; such that k.A�;T � �I /unkq ! 0
Steady-State Navier–Stokes Flow Around a Moving Body 51
for n ! 1; is constructed on the unit sphere in Hq.˝/. The constructionis quite technical, so the readers are referred to [20] for the details. Thus,� 2 Spess.A�;T /; which implies that �ı�;T � Spess.A�;T /. The inclusion��;T � Spess.A�;T / now follows from the fact that Spess.A�;T / is closed. ut
Sketch of the proof of Theorem 14 (see [20] for the details.) The proof is a conse-quence of the following steps (g)–(j).
(g) One can show the same inequality as (104), applying the cutoff functiontechnique and splitting the equation A�;T u D f into an equation for theunknown u1 in a bounded domain ˝� (where � > 0 is sufficiently large) and anequation for the unknown u2 in the whole space R
3. Due to [17, Theorem 1.1],the function u2 satisfies (104), while u1 satisfies (104) because �B1u and T B2ucan be brought into the right-hand side and then one can apply the estimates ofsolutions of the Stokes problem in a bounded domain. The Lq-norms of �B1uand T B2u over˝� can be interpolated between kukq and ku1k2;q; and the normku1k2;q can be absorbed by the left-hand side. Finally, the sum of the estimatesof u1 (over ˝�) and u2 (over R3) leads to (104).
(h) The inclusion ��;T � Spess.A�;T /: assume that � 2 �ı�;T . By analogy with(f), one can construct a sequence fung on the unit sphere in Hq.˝/; such thatk.A�;T � �I /unkq ! 0 for n ! 1. Since Spp.A�;T / is not known to beempty, and it is necessary to show that � 2 Spess.A�;T /; it is important thatthe sequence fung is non-compact in Hq.˝/. The details can be found in [20],where the functions un are defined so that they have compact supports Sn in˝; and the intersection \1nD1Skn (where fSkng is any subsequence of fSng) isempty.
(i) The opposite inclusion Spess.A�;T / � ��;T : if � 2 Spess.A�;T /; then, bydefinition, nul 0 .A�;T ��I / D1 or def 0 .A�;T ��I / D1. The latter means thatnul 0 .A��;T � �I / D 1. Thus, nul 0 .A�;T � �I / may be assumed to be infinity;otherwise one can deal with the operator A��;T instead of A�;T . The identitynul 0 .A�;T � �I / D 1 enables one to construct, by mathematical induction,a sequence fung in D.A�;T / satisfying kunkq D 1; k.A�;T � �I /unkq ! 0
as n ! 1 and dist�unI Ln�1
�D 1 for all n 2 N; where Ln�1 denotes the
linear hull of the functions u1; : : : ; un�1. Using a cutoff function technique, thefunctions un can be modified so that they are all supported for jxj > � (forsufficiently large �), and the modified functions (let us denote themeun) are onthe unit sphere in Hq.˝/ and satisfy k.A�;T � �I /eunkq ! 0 for n ! 1 aswell. However, as suppeun � ˝; eun can be considered to be a function fromD..A�;T /R3 /; where .A�;T /R3 denotes the operator A�;T inHq.R
3/. This yieldsthe equality nul 0 ..A�;T /R3 � �I / D 1; which implies, due to item (c) that� 2 ��;T .
(j) The domain � 2 C X ��;T consists of points in Res.A�;T / and possibly alsoof isolated eigenvalues of A�;T with finite algebraic multiplicities, which maypossibly cluster only at points of @��;T . (See [69, pp. 243, 244].) Assume that
52 G.P. Galdi and J. Neustupa
� 2 C X ��;T is an eigenvalue of A�;T with an eigenfunction u. Applyingagain an appropriate cutoff function technique and treating the equation .A�;T ��I /u D 0 separately in a bounded domain ˝� (for a sufficiently large �) and inthe whole space R
3; one can show that u is in W 2;s.˝/ for any 1 < s < 1.(This follows from estimates valid in a bounded domain and the result fromitem (c), implying that � in the resolvent set of .A�;T /R3 .) Finally, multiplyingthe equation .A�;T � �I /u D 0 by u and integrating in ˝; one can show thatRe � < 0. ut
When q D 2; in [19] it is shown that if B is axially symmetric about the x1-axis, then Sp.A�;T / D ��;T . It means that the set of eigenvalues of A�;T ; lyingoutside��;T ; is empty. The same statement for operator A�;T inHq.˝/ for generalq 2 .1;1/ follows from Theorem 14. The proof in [19] comes from the fact that aneigenfunction u; corresponding to a hypothetic eigenvalue �; is 2�-periodic in thecylindrical variable, which is the angle ' measured about the x1-axis. Then the proofuses the Fourier expansion of u in ' and splitting of the equation .A�;T � �I /u D 0to individual Fourier modes.
Open Problem. In the general case when body B is not axially symmetric, itis not known whether the set of eigenvalues of A�;T in C X��;T is empty.
Finally, note that Sp.A0;T / can be formally obtained, by letting � ! 0 in��;T . Then ��;T shrinks to a system of infinitely many equidistant half-lines. Thespectrum of operator A0;T is studied in detail in [22].
10.2 A Semigroup, Generated by the Operator A�,T
10.2.1 The Case T D 0It is well known that the Stokes operator A generates a bounded analytic semigroup,eAt ; inHq.˝/ [55]. The fact that the Oseen operatorA�;0 � AC�B1 also generatesan analytic semigroup inHq.˝/ was proved by T. Miyakawa [89]. The main tool isthe inequality
kB1ukq � � kAukq C C.�/ kukq (107)
for all u 2 D.A/ and � > 0;which implies thatB1 is relatively bounded with respectto A with the relative bound equal to zero. Then the existence and analyticity of thesemigroup e.ACB1/t follow, e.g., from [69, Theorem IX.2.4].
The so-called Lr–Lq estimates of the semigroup eA�;0t play an important role inthe analysis of stability of steady flow. They were first derived by T. Kobayashi andY. Shibata in [70], whose main result is given next.
Steady-State Navier–Stokes Flow Around a Moving Body 53
Theorem 15. If 1 < r � q <1; then there exists C D C.�; q; r/ > 0 such that
keA�;0takq � C t� 32
�1r �
1q
�
kakr (108)
for all a 2 Hr.˝/ and t > 0. Moreover, if 1 < r � q � 3; then
jeA�;0taj1;q � C t� 32
�1r �
1q
�� 12 kakr (109)
for all a 2 Hr.˝/ and t > 0.
Sketch of the proof (see [70] for the details.) Following [70], the starting point isthe following representation formula of the semigroup
eA�;0ta D1
2� it
Z !Ci1
!�i1e�t
@
@�.A�;0 � �I /
�1a d�; ! > 0:
In order to estimate .A�;0 � �I /�1a; the Oseen resolvent problem .A�;0 � �/a D f
is split into the problem in the bounded domain ˝� (for sufficiently large �) andin the whole space R
3. The estimates in ˝� follow from the fact that the Oseenoperator in ˝� has a compact resolvent and the spectrum (which coincides withthe point spectrum) is in the left half-plane in C; with a positive distance from theimaginary axis. The estimates in R
3 are obtained by means of the Fourier transformand the Mikhlin–Lizorkin multiplier theorem. The next step is the construction ofa parametrix, which enables the authors to combine the estimates in ˝� and inR3 and obtain the estimates of j.A�;0 � �I /�1aj2;r and j�j k.A�;0 � �I /�1akr in
terms of C kakr in the exterior domain ˝. Then the limit procedure for ! ! 0C
is considered. However, due to subtle technical reasons, the limit procedure worksonly in a norm over a bounded domain and one only gets the inequality
k@mt r2eA�;0takrI˝� � C t
�3=2 kakr (110)
for t � 1n and a 2 Hr.˝/ with the support in ˝�; where C D C.m; r; �; �/. Onthe other hand, using the formula
u.x; t / D�1
4�t
�3=2 Z
R3
e�jx�t��yj2=4t a.y/ dy:
for solution of the unsteady Oseen equations
�ut C�uC � @1u D 0
div u D 0
9=
; in R3 .0;1/ (111)
54 G.P. Galdi and J. Neustupa
with the initial condition u.x; 0/ D a.x/ (for x 2 R3), one can derive the estimate
k@jt r
ku.t/kqIR3 � C t� 32
�1r �
1q
�� k2 kakr (112)
for all a 2 Hr.R3/ and t � 1. The constant C on the right-hand side depends only
on j; k; q; r; and �. Finally, combining appropriately (110) with (112), one canobtain (108) and (109). �
10.2.2 The Case T 6D 0The operator A�;T � A C �B1 C T B2 is the Oseen operator with the effect ofrotation. T. Hishida [62] considered the case � D 0 and proved thatA0;T � ACT B2generates a C0-semigroup eA0;T t in H.˝/. M. Geissert, H. Heck, and M. Hieber[53] also considered � D 0 and proved that A0;T generates a C0-semigroup eA0;T t
in Hq.˝/ for 1 < q < 1. The case � 6D 0 was studied by Y. Shibata in [102],whose main finding is given next.
Theorem 16. Let 1 < r � q < 1. The operator A�;T generates a C0-semigroupeA�;T t in Hq.˝/. Moreover, it satisfies the same inequalities (108) and (109) as thesemigroup eA�;0t .
Sketch of the proof (see [102] for the details.) One begins to study the linearCauchy problem, defined by the equations
�ut C�uC � @1uC T .e1 x � ru � e1 u/ D rp
div u D 0
9=
; in R3 .0;1/
(113)
and the initial condition u.x; 0/ D a.x/. The solution u.x; t / is expressed bymeans of the Fourier transform, and the solution of the corresponding resolventproblem with the resolvent parameter � (denoted by AR3;T ;�.�/) is expressed bythe combined Laplace–Fourier transform. Then the estimates of kAR3;T ;�.�/kqIR3and jAR3;T ;�.�/jm;qIR3 in terms of powers of j�j and kakq are derived. The solutionAR3;T ;�.�/ is then split into the part A1.�/; which “neglects” the term T .e1 x �ru � e1 u/ in (113), and A2.�/; which is a correction due to this term. Whilethe estimates of A1.�/ are shown in a similar way and for the same values of � asin the proof of Theorem 15, the estimates of A2.�/ impose sharper restrictions on �and hold only for � 2 CC WD f� 2 CI Re � > 0g. However, remarkably enough, in[102], subtle estimates of A2.�C is/ (for � > 0 and s 2 R) are derived independentof � (for 0 < � < �0), provided a has a support in BR.0/ for R > 0. The essentialrole in the expression of the solution of (113) is played by the integral of A2.�/
on the line f� D � C isI s 2 Rg; parallel to the imaginary axis. The estimatesindependent of � enable one to pass to the limit for � ! 0. Then the appropriate
Steady-State Navier–Stokes Flow Around a Moving Body 55
cutoff function procedure and the limit process for R ! 1 lead to an expressionthat confirms that u. : ; t/ depends on the initial datum a through a C0-semigroup.
One can immediately observe from the shape of the spectrum of the operatorA�;T (see Theorems 13 and 14) thatA�;T is not a sectorial operator inHq.˝/. Thus,unlike the case T D 0; the semigroup generated by A�;T is only a C0-semigroupand not an analytic semigroup.
The idea used to derive estimates analogous to (108) and (109) is similar to thatemployed in the proof of Theorem 15. However, in contrast to the case T D 0
(when, expressing the solution by the line integral on a line parallel to the imaginaryaxis, one especially needs to control the behavior of the resolvent for the valuesof the resolvent parameter � near 0), the case T 6D 0 requires the control of theresolvent “uniformly” on the whole line. This is caused by the fact that as the lineapproaches the imaginary axis in the considered limit procedure, it approaches thespectrum of operatorA�;T not only in the neighborhood of 0 but in the neighborhoodof the infinitely many points ik T ; k 2 Z. �
10.3 Existence and Uniqueness of Solutions of theInitial–Boundary Value Problem
This subsection presents a brief survey of results on the existence and uniquenessof weak and strong solutions to the initial–boundary value problem, consisting ofequation (5) and the initial condition
v.x; 0/ D a.x/ for x 2 ˝: (114)
Referring to other chapters in this handbook for a detailed analysis, here only thoseresults are recalled that are relevant to our study. The definition of the weak solution,in the unsteady case, is analogous to that provided for the steady state problems(7), (8). More precisely, v is called a weak solution to problem (5), (114) if:
(i) v 2 L1.0; T IH.˝// \ L2.0; T ID1;2.˝//; for all T > 0:(ii) lim
t!0Ckv.t/ � ak2 D 0.
(iii) v satisfies (5) in the sense of distributions.
10.3.1 The Case T D 0In the absence of rotation, existence results can be found in many works. Theymostly concern the Navier–Stokes equations, but their extension to the more generalproblem (5), (114) with T D 0 is rather straightforward.
The first results on the global in time existence of weak solutions, v; assumingthe initial velocity a 2 H.˝/; are due to J. Leray [82] (for ˝ D R
3) and E. Hopf[64] (for arbitrary open set ˝ � R
3). A more recent and detailed presentation ofthese classical results can be found, e.g., in the book [104] or in the survey paper[37]. In particular, one shows the existence of a weak solution for any a 2 H.˝/
56 G.P. Galdi and J. Neustupa
and f 2 L2.0; T I D�1;20 .˝// (and v � 0 at @˝). However, the uniqueness of suchsolutions in the same class of existence remains an open problem. The weak solutionis known to be unique if, in addition, it is in Lr.0; T I Ls.˝//; where 2 � r � 1;3 � s � 1; and 2=rC3=s � 1. More precisely, if v1 and v2 are two weak solutions,with v1 in the class Lr.0; T I Ls.˝// above and v2 satisfying the so- called energyinequality (see (119) with v0 � 0 and s D 0), then v1 D v2. As the solutions in theclass Lr.0; T I Ls.˝// satisfy the energy inequality automatically, one can speak of“uniqueness in the class Lr.0; T I Ls.˝//.”
Following [104], a weak solution v in the class Lr.0; T ; Ls.˝// with r; ands as above, is called a strong solution. In addition to be unique, strong solutionsare also known to be “smooth” (= regular), provided that the body force f iseither a potential vector field (and can be therefore absorbed by the pressure term)or “sufficiently smooth.” (See, e.g., [37] for more details.) In particular, if @˝ isof class C2 and f 2 L2.0; T I L2.˝//; then the strong solution v belongs toC..�; T /IH.˝// \ L2.�; T I W 2;2.˝// for any � 2 .0; T /. (It depends on theregularity of the initial velocity a whether � D 0 can also be considered.) Forinitial velocity a and body force f in appropriate function spaces and of “arbitrarysize,” strong solutions are known to exist in some time interval .0; T0/; but it is notknown whether one can take T0 D1; in general. If, however, the size of the data issufficiently restricted, then one can show T0 D 1. There exists a vast literatureon the subject dealing with various types of domains and different choices offunctions spaces for a and f ; starting from the pioneering and fundamental papersof A. A. Kiselev and O. A. Ladyzhenskaya, G. Prodi, and H. Fujita and T. Katoin the early 1960s and continuing with J.G. Heywood (1980), T. Miyakawa (1982),H. Amann (2000), and R. Farwig, H. Sohr, and W. Varnhorn (2009). Among thesepapers, especially [60] (by Heywood), [89] (by Miyakawa), and [1] (by Amann)deal with the Navier–Stokes problems in exterior domains. An important resultconcerning the length of the time interval .0; T0/; where a strong solution existswithout restriction on the “size” of the data, states (e.g., [1] or [59]) that if fis, e.g., in L2
�0;1I L2.˝/
�; then either T0 D 1 or else jv. : ; t/j1;2 ! 1 for
t ! T0�.
10.3.2 The Case T 6D 0The existence of a weak solution to the problem (5), (114) with T 6D 0 has beenproven by W. Borchers [7]. It also follows from a more general result proven in [94]on the existence of weak solutions in domains with moving boundaries. Recall thatthe weak solution is a function in the same class as for the case T D 0; i.e., belongsto L1.0; T I H.˝// \ L2.0; T I W 1;2
0 .˝//.Regarding the local in time existence of a strong solution to the problem
(5), (114) with T 6D 0; only a few results are available. Below the contributions ofT. Hishida [62], G. P. Galdi and A. L. Silvestre [40], and P. Cumsille and M. Tucsnak[11] are explained. They all concern the case when the motion of body B in the fluidreduces to the rotation and the translational velocity is zero. It means that the term� @1v in the momentum equation (5)1 vanishes.
Steady-State Navier–Stokes Flow Around a Moving Body 57
T. Hishida [62] assumes that the body force f is zero and the initial velocity isin D.A1=4/ and proves the existence of a solution in the class C.Œ0; T0�; D.A1=4//\C..0; T0�I D.A// for certain T0 > 0. Recall that A denotes the Stokes operator.Hishida’s proof is based on a nontrivial generalization of the semigroup method,formerly used by Fujita and Kato [30].
G. P. Galdi and A. L. Silvestre [40] also deal with the case of the zero bodyforce f . They assume that the initial velocity a in W 2;2.˝/ satisfies div a D 0
and e1 x � ra 2 L2.˝/; and they obtain a solution in C.Œ0; T0�I W 1;2.˝// \
C..0; T0/I W2;2.˝// for some T0 > 0. The proof is based on the construction of
classical Faedo–Galerkin approximations in˝R; getting a solution in˝R and lettingR ! 1. The procedure is, however, not standard because of the “troublesome”term T e1 x � rv whose influence has to be controlled.
P. Cumsille and M. Tucsnak [11] consider the equations of motion of the viscousincompressible fluid around body B in a frame in which the velocity of the fluidvanishes in infinity and the body is rotating with a constant angular velocity aboutone of the coordinate axes. Thus, the domain filled in by the fluid is time dependentand it is denoted by ˝.t/. The authors consider a body force f locally squareintegrable from .0;1/ to W 1;1.R3/ and the no-slip boundary condition for thevelocity on @˝.t/. The main theorem from [11] says that if the initial velocity ais in W 1;2
0 .˝.0// and it is divergence-free, then there exists T0 > 0 and a uniquestrong solution u 2 L2
�0; T0I W
2;2.˝.t//�\ C
�Œ0; T0�I W
1;2.˝.t//�
such thatut 2 L2
�0; T0I L
2.˝.t//�. Moreover, either T0 can be extended up to infinity or
the norm of u in W 1;2.˝.t// tends to infinity for t ! T0�. In order to obtain aproblem in a fixed exterior domain, the authors use a change of variables whichcoincides with the rotation in the neighborhood of body B; but it equals the identityfar from the body. Then they solve the problem in the fixed exterior domain ˝.Using the relations between the solutions of the equations in the frame consideredin [11] on the one hand and the body fixed frame on the other hand, the result ofCumsille and Tucsnak from [11] can be reformulated in terms of solution to theproblem (5), (114), as follows: given a 2 H.˝/ \ W 1;2
0 .˝/; there exists T0 > 0
and a unique solution v of the problem (5), (114), such that
v 2 L2�0; T0I W
2;2.˝/�\ C
�Œ0; T0�I W
1;2.˝/�;
vt � T .e1 x � rv � e1 v/ 2 L2�0; T0I L
2.˝/�:
)(115)
Cumsille and Tucsnak’s result is applied in Sect. 10.5. Since it is also used in thecase � 6D 0; we note that following the proof in [11] and using the fact that thetranslation-related term � @1v in the first equation in (5) can be considered to be asubordinate perturbation of �v; the above formulated result can be extended to thecase when it also includes the translation of B in the direction parallel to the axis ofrotation. Consequently, the result also holds for the equations in (5) with the term� @1v and the second inclusion in (115) can be modified:
vt � � @1v � T .e1 x � rv � e1 v/ 2 L2�.0; T0I L
2.˝/�: (116)
58 G.P. Galdi and J. Neustupa
10.4 Attractivity and Asymptotic Stability with SmallnessAssumptions on v0
Recall that v0 denotes (the velocity field of) a solution to problem (8) (i.e., a steady-state solution to problem (5)) and that its associated “perturbation” v satisfies (100)and (101). Since in studying longtime behavior the dependence of v on time is morerelevant than the one on spatial variables, in the following considerations, v.x; t /is often abbreviated to v.t/. Thus, for example, the initial condition (114) may bewritten in the form
v.0/ D a: (117)
10.4.1 The Case T D 0A number of results concern the longtime behavior of the unsteady perturbationsv in the class of weak solutions. The first relevant contribution in this direction isdue to K. Masuda [88], who assumes that v0 is continuously differentiable, rv0 2L3.˝/ along with the smallness condition which, according to our notation, yields
supx2˝
� jxj jv0.x/j <1
2: (118)
The perturbed unsteady solution is supposed to satisfy the momentum equationin (5) with a perturbed body force. Thus, the corresponding perturbation v satis-fies (100), (101), with an additional right-hand side f 0 in (100), representing theperturbation to the steady body force f . The Helmholtz–Weyl projected functionPf 0 is assumed to be in C1 .Œ0;1/I H.˝// \ L1 .0;1I H.˝// and such thatsupt>0
R tC1tk.d=ds/Pf 0.s/k22 ds C
R10s1=2 k.d=ds/Pf 0.s/k2 ds < 1. As for
the class of perturbations, the author assumes that v is a weak solution to theproblem (100) (with a nonzero f 0 on its right-hand side), (101), and (117), withinitial data a 2 H.˝/; and satisfies the so-called strong energy inequality, namely,
12kv.t/k22 �
12kv.s/k22 �
Z t
s
�.v.�/ � rv0; v.�//C jv.�/j
21;2 C .f
0.�/; v.�//d�;
(119)
for a.a. s > 0 (including s D 0) and all t 2 Œs; T �; arbitrary T > 0. Notice thatthe latter is formally obtained by multiplying equation (100) by v and integratingover˝ .s; t/ and relaxing the equality sign to the inequality one. Under the aboveconditions, Masuda shows that there exists T� > 0 such that v.t/ becomes regularfor t > T� and decays at the following rate
jv.t/j1;2 � Ct�1=4; kv.t/k1 � Ct
�1=8; for all t > T�: (120)
The proof uses (119) and the assumptions on the integrability of f 0 to deduce, first,that v.t/ is “small” for large t . Then, combining this with the estimates of A1=2v;
Steady-State Navier–Stokes Flow Around a Moving Body 59
one shows that v.t/ is regular and tends to zero for t ! 1 in the norm j : j1;2.The rate of decay is calculated from the energy-type inequality, satisfied by vt . Theauthor also generalizes these results to the case when the unperturbed solution v0is time dependent. It should be noted that in [88], no assumption on the size of theinitial perturbation a is used: it can be arbitrarily large. However, nothing can besaid about the behavior of v.t/ for t 2 .0; T�/.
If v0 � 0; the decay rates (120) are sharpened by J.G. Heywood in [60].The above results have been further elaborated on by P. Maremonti in [86].
Maremonti studied the attractivity of steady as well as unsteady solutions v0 toproblem (5) in the same class of weak solutions considered by Masuda with f 0 � 0.In particular, for the case v0 steady, he shows the following decay rates
kvt .t /k2 � C t�1; jv.t/j1;2 � C t
�1=2; kv.t/k1 � C t�1=2;
thus improving and extending the analogous finding of [88] and [60]. Insteadof condition (118), the author assumes that the maximum of certain variationalproblem involving v0 is not “too large.” The latter condition is certainly satisfiedif v0 is sufficiently regular and obeys (118).
The somehow more complicated question of asymptotic stability of v in the L2-norm was first addressed by P. Maremonti in [87]. In particular, he shows that allv in the class of weak solutions, with a 2 H.˝/ and satisfying the strong energyinequality (119) with f 0 � 0; must decay to 0 in the L2-norm, provided that themagnitude of v0 is restricted in the same way as specified in [86] and discussedearlier on.
An important contribution to the studies of the asymptotic stability of the steadysolution v0 was also made by T. Miyakawa and H. Sohr in [90]. The authorsshow that if the basic steady solution v0 of (5) is such that v0 2 L1.˝/;
rv0 2 L3.˝/; and the smallness condition (118) is satisfied, and if, in addition,
the perturbation f 0 to the body force f is in L2 .Œ0; T /I H.˝// for all T >
0 and in L1 .Œ0;1/I H.˝// ; then the L2-norm of each weak solution v toproblem (100), (101) satisfying the energy inequality (119) tends to 0 for t ! 1.In [90] it is also shown that the class of such weak solutions is not empty, thussolving a problem left open in [87] and partially solved in [33]. Further resultsconcerning theL2-decay of the perturbation v.t/ (as a weak solution to (100), (101))for t ! 1 are provided in the paper [8] by W. Borchers and T. Miyakawa: theauthors assume that the steady solution v0 of (5) is in L3.˝/; rv0 2 L3.˝/; thesmallness condition (118) is satisfied, and the perturbation f 0 to the body force f is
in L2loc .Œ0;1/I H.˝// \ L1 .0;1I H.˝// \ L1
�0;1I D�1;20 .˝/
�. They show
that then the L2-norm of each weak solution v to problem (100), (101) obeying(119) tends to 0 for t ! 1. Moreover, if keLtak2 D O.t�˛/ for some ˛ > 0;
then kv.t/k2 D O�.ln t /��1=2
�for any � > 0. (Here, eLt denotes the semigroup
generated by operator L; see Sect. 10.5.1.) The results of [8] are generalized by thesame authors to the case of n-space dimensions (n � 3) in [9].
60 G.P. Galdi and J. Neustupa
Even sharper rates of decay of the norms kv.t/kr (2 � r � 1) and krv.t/kr(2 � r � 3) were obtained by H. Kozono in [75], provided the perturbation f 0 tothe body force is inL1
�0;1I L2.˝/
�\C
�.0;1/I L2.˝/
�and decays like t�1 for
t !1. Kozono does not use any condition of smallness of the basic flow v0 or itsinitial perturbation a; but needs v0 in Serrin’s classLr .0;1I Ls.˝// (2=rC3=s D1; 3 < s � 1). This implies that v0 is in fact a strong Solution, and it is in asuitable sense small for large t . Obviously, the only time-independent solution inthe considered Serrin class is v0 D 0.
There exists a series of results on stability of solution v0 in the class of strongunsteady perturbations, which, unlike the cited papers [8, 9, 60, 86, 88], and [75],provide an information on the size of the perturbations at all times t > 0 and notjust for “large” t . However, on the other hand, the initial value of the perturbationis always required to be “small” as well as v0 is also supposed to be “sufficientlysmall” in appropriate norms. The first results of this kind come from the early 1970sof the twentieth century, and new results on this topic still appear.
The next paragraphs contain the sketch of the main steps to obtain a result of theabove type. Assume that v is a strong solution to problem (100), (101) in the timeinterval .0; T0/; for some T0 > 0. Multiplying the first equation in (100) (wherev0 D v) by v and integrating by parts over ˝; one obtains
1
2
d
dtkvk22 C jvj
21;2 D � .v � rv0; v/ � � jv0j1;3=2 kvk
26
� C27 � jv0j1;3=2 jvj
21;2: (121)
(The norm kvk6 has been estimated by Sobolev’s inequality: kvk6 � C7 jvj1;2;see, e.g., [45, p. 54].) Multiplying the first equation in (100) by Av � P�v andintegrating over ˝; one obtains
1
2
d
dtjvj21;2 C kAvk
22 D � ..�@1vC v0 � rvC v � rv0 C v � rv/ ; Av/ dx
�1
4kAvk22 C 4�
2�k@1vk
22 C kv0 � rvk
20 C kv � rv0k
22 C kv � rvk
22
�: (122)
The first term on the right-hand side can be absorbed by the left- hand side. Theother terms on the right-hand side can be estimated by means of the inequalities
jvj1;6 � C krvk1;2 D C�jvj21;2 C jvj
22;2
�1=2� C
�kAvk22 C jvj
21;2
�1=2;
where the first one follows from the continuous imbeddingW 1;2.˝/ ,! L6.˝/ andthe second one follows, e.g., from [45, pp. 322–323]. Thus, if YŒv� is defined by theformula
YŒv� WD kAvk22 C jvj21;2;
then
Steady-State Navier–Stokes Flow Around a Moving Body 61
k@1vk22 � 2�
2 jvj21;2;
kv0 � rvk22 � kv0k
26 jvj1;2 jvj1;6 � C jv0j
21;2 jvj1;2 YŒv�1=2
� � kAvk22 C C.�/�jv0j
21;2 C jv0j
41;2
�jvj21;2;
kv � rv0k22 � jv0j
21;3 kvk
26 � C2
7 jv0j21;3 jvj
21;2;
kv � rvk22 � kvk26 jvj
21;3 � C2
7 jvj31;2 jvj1;6 � C jvj31;2 YŒv�1=2 � C jvj21;2 YŒv�:
Employing these inequalities into (122) and choosing, e.g., � D 14; one gets
d
dtjvj21;2 C kAvk
22 � C8�
2�1C jv0j
21;2 C jv0j
41;2
�jvj21;2 C C9�
2 jvj21;2 YŒv�:(123)
Adding the inequalities (121) (multiplied by 2) and (123) (multiplied by ˛ > 0) andpassing everything to the left-hand side, one obtains
d
dt
�kvk22 C ˛ jvj
21;2
�
C jvj21;22 � 2C 2
7 � jv0j1;3=2 � C8�2˛�1C jv0j
21;2 C jv0j
41;2
�� C9�
2˛ jvj21;2
C kAvk22˛ � C9�
2˛ jvj21;2� 0:
This implies that
d
dt
�kvk22 C ˛ jvj
21;2
�C jvj21;2
2 � 2C 2
7 � jv0j1;3=2 � C8�2˛�1C jv0j
21;2 C jv0j
41;2
�
�C9�2�kvk22 C ˛ jvj
21;2
�C kAvk22
˛ � C9�
2�kvk22 C ˛ jvj
21;2
�� 0:
(124)
This inequality shows that if
2C 27 � jv0j1;3=2 C C8�
2˛�1C jv0j
21;2 C jv0j
41;2
�< 2 (125)
and kvk22 C ˛ jvj21;2 is initially so small that
C9�2�kak22 C ˛ jaj
21;2
�
< min˚2 � 2C 2
7 � jv0j1;3=2 � C8�2˛�1C jv0j
21;2 C jv0j
41;2
�I ˛�
(126)
(recall that v.0/ D a), then kv.t/k22C˛ jv.t/j21;2 is nondecreasing for t in some right
neighborhood of 0. This consideration can be simply extended, by the bootstrappingargument, to the whole interval of existence of the strong solution v (let it be .0; T0/)so that one obtains: kv.t/k22 C ˛ jv.t/j
21;2 < kak
22 C ˛ jaj
21;2 for all t 2 .0; T0/. This
62 G.P. Galdi and J. Neustupa
shows, among other things, that the norm kv.t/k1;2 cannot blow up when t ! T0�.Consequently, T0 D 1 and the inequality kv.t/k22 C ˛ jv.t/j
21;2 < kak
22 C ˛ jaj
21;2
holds for all t 2 .0;1/. Note that if
C27 � jv0j1;3=2 < 1; (127)
then one can choose ˛ > 0 so small that (125) holds. (˛ is further supposed to bechosen in this way.)
Integrating inequality (124) with respect to t; one can also derive an informationon the integrability of YŒv.t/� and on the asymptotic decay of jv.t/j1;2. Thus, alsoincluding the information on the uniqueness of strong solutions and using the resultof [87], one obtains the following Lyapunov-type asymptotic stability of v0 in theW 1;2-norm.
Theorem 17. Suppose the steady solution v0 to the problem (8) satisfies condi-tions (99) and (127) and ˛ > 0 is chosen so that (125) holds. Then, if a 2H.˝/ \ W 1;2
0 .˝/ satisfies (126), problem (100), (101) with the initial conditionv.0/ D a has a unique strong solution v on the time interval .0;1/. Furthermore,there exists C10 > 0 such that this solution satisfies
kv.t/k22 C ˛ jv.t/j21;2 C C10
Z t
0
�jv.s/j21;2 C ˛ kAv.s/k
22
�ds � kak22 C ˛ jaj
21;2
(128)
for all t > 0 and
limt!1
kv.t/k1;2 D 0: (129)
The noteworthy assumption in the above theorem is condition (127) of “sufficientsmallness” of the solution v0.
The ideas of proof described previously and similar energy-type considerationshave been applied to many other studies of stability or instability of steady-statesolutions to Navier–Stokes and related equations. Concerning flows in exteriordomains, the readers are referred, e.g., to [31, 32, 34, 58, 59].
A different approach, based on a representation of a solution by means of semi-groups generated by the operators A�;0 or L and on estimates of the semigroups,has been employed by H. Kozono and T. Ogawa [73], H. Kozono and M. Yamazaki[74], and Y. Shibata [100]. In particular, Kozono and Yamazaki [74] study the flowin an exterior “smooth” domain ˝ in R
n (n � 3), under the assumption that thetranslational velocity of the moving body is zero, which in our notation means� D 0 in the first equation in (100). The steady-state solution v0 is supposed tobelong toLn;1 .˝/\L1.˝/; and its gradient is supposed to be inLr�.˝/ for somer� 2 .n;1/. (The Lorentz-type space Lr;q .˝/ for 1 < r < 1 and 1 � q � 1
is defined by means of the real interpolation to be�Hr0.˝/;Hr1.˝/
��;q; where
Steady-State Navier–Stokes Flow Around a Moving Body 63
1 < r0 < r < r1 < 1 and 1 < � < 1 satisfy 1=r D .1 � �/=r0 C �=r1; see[74]. It is shown in [9] that Lr;q .˝/ coincides with the space of all u 2 Lr;q.˝/such that div u D 0 in ˝ in the sense of distributions and u � n D 0 on @˝in the sense of traces.) Equations (100) are treated in the equivalent form (102).The operators L and N are defined in the introductory part of this section forn D 3; but the definition in the general case n 2 N; n � 3 is analogous. In theconsidered case, L has the concrete form L D AC�B3. The operator L generatesa quasi-bounded analytic semigroup in Hq.˝/ – this is shown in [74] by means ofappropriate resolvent estimates which imply that operator L is sectorial. The strongsolution is identified with the mild solution, which satisfies the integral equation
v.t/ D eLtaC
Z 1
0
eL.t�s/Nv.s/ ds: (130)
The solution of this equation is constructed as a limit of a sequence of approxima-tions, which are defined by the equations v0 WD e�Lta and
vj .t/ WD v0.t/C
Z t
0
eL.t�s/Nvj�1.s/ ds .j D 1; 2; 3; : : : /:
The authors define Kj WD sup0<t<1 tn2
�1n�
1r�
�
kvj .t/kr� (for j D 0; 1; 2; : : : ) andshow that the sequence fKj g
1jD0 is bounded if K0 is sufficiently small. Here, the
important role is played by the estimates ofR t0
eL.t�s/Nvj .s/ ds in Lr�.˝/; whichare based on the duality argument, and Lp–Lr
0
� estimates of reL �.t�s/; whereeL �.t�s/ is the adjoint semigroup to eL.t�s/ in the dual space Hr 0
�
.˝/. Similararguments enable one to derive the inequality
where C.K0; n; r�/ is less than one for K0 “small enough.” From this, the authors
deduce that there exists v such that tn2
�1n�
1r�
�
v.t/ 2 BC..0;1/I Hr�.˝// and
sup0<t<1
tn2
�1n�
1r�
�
kvj .t/ � v.t/kr� �! 0 as j !1:
The following theorem is the main result proved in [74]:
Theorem 18. Let v0 2 Ln;1 .˝/\L1.˝/; rv0 2 Lr�.˝/ for some r� 2 .n;1/.
There exists � D �.n; r�/ > 0 such that if kv0kLn;1.˝/ � � and kakLn;1.˝/ � �;then there exists a strong solution v of the problem (102), (117) which, among otherthings, is in BC
�.0;1/I Ln;1 .˝/
�\ C ..0;1/I D.A// \ C1 ..0;1/I Hr�.˝//
and satisfies
64 G.P. Galdi and J. Neustupa
kv.t/kr � C t�n2 .
1n�
1r /; n < r � r�
for all t > 0 with a constant C depending only on n; r; and r�.
In [74] it is also shown that the better is the information on the spatial decay ofthe initial velocity a; the sharper is the asymptotic behavior of v.t/ for t !1.
The case of a nonzero translational velocity of the body (i.e., � 6D 0; in the firstequation in (100)) is dealt with by Y. Shibata [100]. Shibata’s approach strongly usesthe Lq–Lr estimates of the semigroup eA�;0t generated by the Oseen operator A�;0;provided by Theorem 15. Shibata considers the case f 2 L1.˝/; and assumingthat � and
hhf ii2ı WD supx2˝
.1C jxj/5=2 .1C jxj C x1/1=2C2ı jf .x/j
(for some 0 < ı < 14) are “small,” he at first proves the existence of a steady solution
v0 to problem (8) (with T D 0), satisfying the condition v0.x/! 0 for jxj ! 1;which is, among other things, “small” in the norm ofW 2;q.˝/ (where 3 < q <1).The author considers general Dirichlet’s boundary condition v0 D g on @˝; whereg � e1 is supposed to be “small enough” in an appropriate norm. The next theoremfollows from [100, Theorem 1.4] if g D 0.
Theorem 19. Let 3 < q < 1 and ˇ; ı be any numbers such that 0 < ı < 14
and ı < ˇ < 1 � ı. Let f 2 L1.˝/ and a 2 H3.˝/. Then there exists � D�.q; ˇ; ı/ 2 .0; 1� such that if 0 < � < �; hhf ii2ı � �ˇCı; and kak3 < �; thenthe problem (100), (101), (117) has a unique solution v 2 BC .0;1I H3.˝// suchthat
Œv�3;0;t C Œv�q;.1�3=q/=2;t C Œrv�3;1=2;t �p�;
where
Œv�q;�;t WD sup0<s<t
s� kv.s/kq: (131)
Moreover, the inequalities
Œv�r;.1�3=r/=2;t � C.r/�� C �1=2Cˇ
�;
kv.t/k1 � C.s/�� C �1=2Cˇ
� �t�1=2 C t�.1�3=2s/
�
hold for any t > 0; where 3 < r <1 and 3 < s < q.
The proof is based on solution of the integral equation
Steady-State Navier–Stokes Flow Around a Moving Body 65
v.t/ D eA�;0taCZ 1
0
eA�;0.t�s/ Œ�B3v.s/CNv.s/� ds: (132)
The author derives a series of subtle estimates of the right-hand side, which finallyenable him to solve the equation (132) by means of the contraction mappingprinciple.
10.4.2 The Case T 6D 0To the best of our knowledge, if the body is allowed to rotate, there are noresults analogous to [8, 9, 60, 86, 88], and [75] regarding the behavior of unsteadyperturbations to the steady solution v0 in the class of weak solutions.
The asymptotic stability of a steady-state solution v0 when � D 0 (body rotateswithout translating in the absence of external forces) was first proved by G. P. Galdiand A. L. Silvestre in [40]. Their approach is based on the combined use of theclassical Galerkin method (suitably adapted to the situation at hand) and the spatialasymptotic properties of v0 determined in [39]. More precisely, they show that thereexists C11 > 0 such that if kak1;2CjT j < C11 and if e1 x � ra 2 L2.˝/; then theInitial value problem (100), (101), (117) has a strong solution v on the time interval.0;1/;which is (together with its first-order and second-order spatial derivatives) inC�Œ0;1/I L2.˝/
�; and satisfies, among others, the following asymptotic property
limt!1
jv.t/j1;2 D 0: (133)
The idea of the proof is as follows: due to the local in time existential theorems,the strong solution v exists on a “short” time interval .0; T0/. Considering at firstthe equations in a bounded domain ˝R; multiplying equation (100) by v; applyingthe limit procedure for R ! 1; and assuming that T is “sufficiently small,” theauthors show that kv.t/k2 is bounded in .0; T0/ and jv.t/j21;2 is integrable over.0; T0/. Similarly, multiplying the momentum equation by Av ( differentiating withrespect to t and multiplying by vt ) and using the smallness of T ; one obtains aninequality which shows that jv.t/j1;2 is uniformly bounded in .0; T0/ and kAv.t/k22(krvt .t /k22) is integrable over .0; T0/. Since kv.t/k1;2 cannot blow up as t ! T0�;
the interval .0; T0/ can be extended to .0;1/. Since the integralsR10jvj21;2 dt andR1
0jvt .t /j
21;2 dt are finite, one can deduce that (133) holds.
In [102] and [101], Y. Shibata generalized his previous results from [100] (seeSect. 10.4.1 on the existence of a “small” steady solution and its stability) to thecase when T 6D 0. Applying the same arguments as in [100] and using the fact thatthe operator A�;T with the rotational effect generates a C0-semigroup eA�;T t whichsatisfies the same Lq–Lr estimates as the semigroup eA�;0t (see Theorem 16), heproved the following.
Theorem 20. Let 3 < q <1 and be a small positive number. Then, there exists� D �.q; / > 0 such that if a 2 H3.˝/ and
66 G.P. Galdi and J. Neustupa
kv0k3� C kv0k3C C jv0j1; 32� C jv0j1; 32C
C kak3 � �;
the problem (100), (101), (117) has a unique solution
v 2 C .Œ0;1/I H3.˝// \ C0�.0;1/I Lq.˝/ \W 1;3
0 .˝/�;
which satisfies (see (131))
Œv�3;0;t C Œv�q;.1�3=q/=2;t C Œrv�3;1=2;t �p� for any t > 0:
Note that similar results have also been obtained by T. Hishida and Y. Shibata[63] in the case when operator A�;T reduces to A0;T .
10.5 Spectral Stability and Related Results
The previous section presented, among other things, a number of results concerningthe attractivity/asymptotic stability under “smallness” assumptions on v0. Theobjective of this section is to formulate analogous result, but with more generalhypotheses that involve the spectral properties of the relevant linearization aroundv0. Recall that the perturbation v satisfies the operator equation (102), i.e.,
dv
dtD LvCNv:
Now, assume, temporarily, ˝ bounded. It is then well known that Sp.L / �
Spp.L / [96]. In a series of fundamental papers going back to the pioneering worksof G. Prodi [96] and D.H. Sattinger [97], it is shown that the zero solution of theabove equation is stable, in fact, even exponentially stable, if
9 ı > 0 W Re � � �ı; 8� 2 Sp.L /: (134)
It can be easily shown that if v0 is “small” enough (i.e., the operator B3 is a “small”perturbation of A�;T ), then (134) holds, whereas the converse is not necessarilytrue. However, if as in our case, ˝ is an exterior domain, condition (134) cannotbe satisfied. The reason is that the essential spectrum of L; located in the half-planef� 2 CI Re � � 0g; touches the imaginary axis at the point 0 if T D 0 or atinfinitely many points ikT (k 2 Z) if T 6D 0; see (103) if T D 0 and Theorem 13if T 6D 0. Now, as shown in [93], if T D 0 and in [52] if T 6D 0; the operatorB3 is relatively compact with respect to A�;T . Consequently, the operator L �
A�;T C�B3 has the same essential spectrum asA�;T ; and the spectra of L and A�;Tdiffer at most by a countable number of isolated eigenvalues which may possiblycluster only on the boundary of Spess.A�;T /; see [69, pp. 243–244]. However, asshown in [15] (by P. Deuring and J. Neustupa) and [95] (by J. Neustupa), the
Steady-State Navier–Stokes Flow Around a Moving Body 67
essential spectrum of L does not play a decisive role in the stability issue. In fact,if condition (134) is replaced by a certain assumption on the eigenvalues of L; thenone can show the stability of the zero solution of equation (102), regardless of theproperties of Spess.L/. The papers [15] and [95] both concern the case T D 0:in the whole space [15] and in exterior domain [95]. Furthermore, they use resultsfrom the previous research [93] by J. Neustupa, where the author also treats the caseT D 0 and formulates a sufficient condition for stability, under the assumption thatthe semigroup eLt ; applied to a finite family of certain functions, is L1- and L2-integrable on .0;1/ in an appropriate norm defined only over a bounded subregionof˝. Similar ideas have also been employed in papers [91] and [92] which concerna general parabolic equation in a Hilbert space or a parabolic system in an exteriordomain and in paper [52], which brings a generalization of the results from [93] tothe case T 6D 0.
The next sections present in some details the results of [15, 93, 95], and [52]. Tothis end, the cases T D 0 [15, 93, 95] and T 6D 0 [52] are considered separately.
10.5.1 The Case T D 0The paper [93] uses the following important facts and steps.
– As the operator A�;0 generates an analytic semigroup in H.˝/ and operator B3is relatively compact with respect to A�;0; the operator L � A�;0 C �B3 D
A C �B1 C �B3 generates an analytic semigroup in H.˝/ as well. (See [69,p. 498].) We denote this semigroup by eLt .
– The operator B1 is skew-symmetric in H.˝/. Set
B3sv WD P .v � .rv0/s/ ;B3av WD P .v0 � rvC v � .rv0/a/ :
The subscript s (a) denotes the symmetric (skew-symmetric = antisymmetric)part of operator B3 or of the tensor rv0.
– Let � > 0 be fixed. The operator AC .1C �/�B3s is self-adjoint in H.˝/. Thespectrum ofAC.1C�/�B3s consists of Spess .AC .1C �/�B3s/ D .�1; 0� andat most a finite set of positive eigenvalues, each of whose has a finite multiplicity.Let the positive eigenvalues be �1 � �2 � � � � � �N ; each of them beingcounted as many times as is its multiplicity. Let �1; : : : ;�N be the associatedeigenfunctions. They can be chosen in a way that they constitute an orthonormalsystem in H.˝/.
– Denote by H.˝/0 the linear hull of �1; : : : ;�N and by P 0 the orthogonalprojection ofH.˝/ ontoH.˝/0. Furthermore, denote byH.˝/00 the orthogonalcomplement to H.˝/0 in H.˝/ and by P 00 the orthogonal projection of H.˝/onto H.˝/00. Then H.˝/ admits the orthogonal decomposition H.˝/ D
H.˝/0 ˚ H.˝/00; and the operator A C .1 C �/�B3s is reduced on eachof the subspaces H.˝/0 and H.˝/00. Moreover, it is positive on H.˝/0 andnonpositive on H.˝/00.
68 G.P. Galdi and J. Neustupa
– Since�A� C .1 C �/�B3s�;�
�� 0 for all � 2 H.˝/00 \ D.A/; operator L
satisfies
.L�;�/ D�.AC �B3s/�;�
�2D
�
1C �.A�;�/
C1
1C �
�.AC �B3s C ��B3s/�;�
��
�
1C �.A�;�/ D �C12 j�j
21;2;
for all � 2 H.˝/00 \ D.A/; where C12 D �=.1C �/. This inequality expressesthe so-called essential dissipativity of L in space H.˝/00.
– All functions �1; : : : ; �N belong to D�1;20 .˝/ (the dual to D1;20 .˝/).
The main result of the paper [93] is the following.
Theorem 21. Suppose that the steady solution v0 to the problem (8) satisfiesconditions (99), and let �� > 0 be so large that jv0j1;3=2;˝�� �
18. Moreover, assume
(A) there exists a function ' 2 L1.0;1/\L2.0;1/ such that keLt�ik2I˝�� � '.t/
for all i D 1; : : : ; N and t > 0.
Then there are positive constants ı; C13; C14 such that if a 2 H.˝/ \ W 1;20 .˝/
and kak1;2 � ı; the equation (102) with the initial condition v.0/ D a has a uniquesolution v on the time interval .0;1/. The solution satisfies
kv.t/k21;2 C C13
Z t
0
�jv.s/j21;2 C kAv.s/k
22
�ds � C14 kak
21;2 (135)
(for all t > 0) and
limt!1
jv.t/j1;2 D 0: (136)
The proof is based on splitting the equation (102) into an equation in H.˝/00;where L is essentially dissipative, and a complementary equation, where one usesthe decay of the semigroup eLt following from assumption (A).
Theorem 21 tells us that the question of stability of the steady solution v0 reducesto the L1- and L2-integrability of a finite family of certain functions in the interval.0;1/; i.e., condition (A). In the paper [15], the authors consider the case ˝ D R
3
and show that condition (A) is indeed satisfied under some assumptions on thespectrum of L. The latter amounts to assume that all eigenvalues of L have negativereal parts, without any request on the essential spectrum of L. The important toolused in [15] is the fundamental solution of the Oseen equation in R
3 and theestimates for the corresponding resolvent problem.
Sufficient conditions for the stability of the null solution to (100), in terms ofeigenvalues of L and when ˝ 6� R
3; have been recently formulated in the paper
Steady-State Navier–Stokes Flow Around a Moving Body 69
[95]. Here, the author shows at first that condition (A) in Theorem 21 can be replacedby the following one
(B) Given � > 0 there exist functions ' 2 L1.0;1/\L2.0;1/ and 1; : : : ; N 2
H.˝/ \D�1;20 .˝/ such that
k�j � j k�1;2 � � for j D 1; : : : ; N; (137)ˇ�eLt�i ; j
�ˇ� '.t/ for t > 0 and i; j D 1; : : : ; N: (138)
(Condition (B) reduces to the requirement thatˇ�
eLt�i ;�j�ˇ� '.t/ if one chooses
j D �j for j D 1; : : : ; N .) In [95] it is further assumed that the steady solutionv0 is in Lr.˝/ for all r 2 .2;1� and @jv0 2 L
s.˝/; for j D 1; 2; 3 and for alls 2 . 4
3;1�. This assumption is fulfilled if the acting body force f is in Lq.˝/ for
all q 2 .1; q0�; where q0 > 3; see Lemma 9. Moreover, if f has a compact support,then v0 D E.x/ �mC v00.x/; where m is a certain constant vector, E is the Oseenfundamental tensor, and v00.x/ is a perturbation which decays faster than E.x/ forjxj ! 1; see Theorem 6. This form of v0 is used in [95], where the main resultstates the following.
Theorem 22. Let the conditions
(C1) there exists ı > 0 and a0 > 0 such that all eigenvalues � of operator L satisfyRe � < maxf�ıI �a0 .Im �/2g,
(C2) 0 is not an eigenvalue of the operator Lext
be fulfilled. Then the conclusions of Theorem 21 hold.
Here, Lext denotes the operator L with the domain extended toD2;2.˝/\D1;20 .˝/.
Condition (C1) implies that L has no eigenvalues with nonnegative real parts.
Sketch of the proof of Theorem 22 (see [95] for the details.) The proof is based onshowing that Œ(C1)^ (C2)� H) (B). The function eLt�i is expressed by the formula
eLt�i D .2� i/�1Z
� �e�t .�I �L/�1�i d�; (139)
where � � is a curve in C X Sp.L /; which depends on a parameter � > 0. Recallthat Sp.L / consists of the essential spectrum in the half-plane f� 2 CI Re � � 0g;see (103), and at most a countable number of isolated eigenvalues. Curve � � hasthree parts �1; � �
2 ; and �3; where �1 and �3 coincide with the half-lines arg.�/ D� ˙ ˛; respectively, for some fixed ˛ 2 .0; �=2/ and large j�j. Both the curves�1 and �3 lie in the half-plane f� 2 CI Re � < 0g. Since Spess.L / touches theimaginary axis at point 0; the curve � �
2 WD f� 2 CI � D �as2 C �.s20 � s2/ C
is for � s0 � s � s0g (where a > 0 and s0 > 0 are appropriate fixed positive
70 G.P. Galdi and J. Neustupa
numbers) extends into the half-plane f� 2 CI Re � > 0g. If � ! 0C; then � �2
approaches � 02 WD f� 2 CI � D �as2 C is for � s0 � s � s0g; and consequently,
� � approaches � 0 D �1[�02 [�3. (Number a > 0 is chosen so that Sp.L / lies on
the left from � 0;with the exception of point 0.) In order to verify inequality (138) incondition (B), one has to estimate
�eLt�i ; j
�. Since one can prove that the range
of L �jD.˝/ (the adjoint operator to L reduced to D.˝/) is dense in D�1;20 .˝/; inorder to satisfy (137), one can choose functions j in the form j WD L � 0j ;
where 0j 2 D.˝/ (for j D 1; : : : ; N ). Then
�eLt�i ; j
�D
1
2� i
Z
� �e�t
�.�I �L/�1�i ; j
�d�
D1
2� i
Z
� �e�t
�.�I �L/�1�i ; .L
� � �I C �I / 0j
�d�
D �1
2� i
Z
� �e�t
��i ;
0j
�d� C
1
2� i
Z
� �e�t �
�.�I �L/�1�i ;
0j
�d�: (140)
As the integrand in the first integral on the right-hand side depends on � only throughe�t ; one can consider the limit for � ! 0C and show that the integral equals theintegral on the curve � 0. A simple calculation yields that the integral on � 0; as afunction of t; is in L1.0;1/ \ L2.0;1/. (Here, it is important that � 0 � f� 2
CI Re � � 0g and � 0 touches the imaginary axis only at the point 0.) The treatmentof the second integral on the right-hand side of (140) is much more complicated. Itis necessary to derive a series of estimates of u� WD .L � �I /�1�i ; which satisfiesthe equation
.AC �B1 C �B3 � �I /u� D �i : (141)
This equation can be treated as the perturbed Oseen resolvent equation with theresolvent parameter �. Especially the estimates for � 62 Sp.L / in the neighborhoodof 0 (hence also in the neighborhood of Spess.L /) are very subtle. They finallyenable one to pass to the limit for � ! 0C and show that the integral of
e�t ��.�I �L /�1�i ;
0j
�on curve � 0 is, as a function of t; in L1.0;1/ \
L2.0;1/. The factor � plays a decisive role because it allows one to control theintegrand for � on the critical part of curve � 0; i.e., near � D 0.
Note that the assumption on the nonzero translational motion of body B in thefluid (i.e., � 6D 0) is important because it enables one to apply the theory of theOseen equation and to obtain appropriate estimates of function u� . �
Remark 9. A result similar to Theorem 22 was stated by L.I. Sazonov [98]. There,the main theorem on stability claims that the steady solution v0 is asymptoticallystable in the L3-norm if L; as an operator in H3.˝/; does not have eigenvalues inthe half-plane Re � > 0. However, the proofs of the fundamental estimates of the
Steady-State Navier–Stokes Flow Around a Moving Body 71
Oseen semigroup as well as of the main theorem do not contain all the necessarydetails, which makes it difficult to assess the validity of author’s arguments.
10.5.2 The Case T 6D 0The results of [93] are generalized to the case T 6D 0 (i.e., B is allowed to spinat constant rate) involving the rotational motion of body B; in the paper [34]by G. P. Galdi and J. Neustupa. The steady solution v0 is assumed to satisfy theproperties v0 2 L3.˝/; @jv0 2 L3.˝/ \ L3=2.˝/ (j D 1; 2; 3) and the estimatejrv0.x/j � C jxj�1 for x 2 ˝. The existence of such a solution is known fora large class of body forces f ; provided � 6D 0; see Lemma 9 and Theorem 6.The main theorem on stability of the zero solution of equation (102) is analogous toTheorem 21, that is why it is not repeated here.
The presence of the term T B2v in the operator L defined in equation (102)causes a series of new problems that one has to face and overcome. For example,unlike the case T D 0; the time derivative of v need not be an element of H.˝/.However, one can show that .dv=dt/ � �B1v � T B2v 2 H.˝/ and
Z
˝
�dvdt� �B1v � T B2v
�� v dx D
d
dt
1
2kvk22 ;
Z
˝
�dvdt� �B1v � T B2v
�� Av dx D �
d
dt
1
2jvj21;2 :
These identities play an important role in the proof of the theorem on stability.Another important step is to show that the functions r�i and ��i (i D 1; : : : ; N )are square-integrable with the weight jxj2 in ˝. This enables one to estimate thenorm kB2vk2 by C jvj1;2; which is again a crucial property in the proof of thestability result. All details can be found in [52].
Open Problem. The question whether – in analogy with the case T D 0
and paper [93] – the stability of the zero solution of equation (102) can bedetermined by the location of the eigenvalues of operator L is open.
The difficulties related to this problem are generated by the fact that now, beingT 6D 0; the operator L is no longer sectorial. Thus, even if all the eigenvalues havenegative real parts, one cannot express the eLt�i by a formula similar to (139),where the curve � � coincides with the half-lines arg.�/ D � ˙ ˛ (for some ˛ 2.0; �=2/ and large j�j) and touches or intersects the half-plane CC only in a smallneighborhood of 0. On the contrary, the curve � � must lie at the right of infinitelymany points ikT (k 2 Z) on the imaginary axis, and even if one formally passes tothe limit � ! 0 in order to obtain a curve � 0 in the half-plane f� 2 CI Re � � 0g;then � 0 must pass through the points ikT (k 2 Z). Consequently, the integral onthe right-hand side of (139) cannot be treated and estimated in the same way as inthe case T D 0.
72 G.P. Galdi and J. Neustupa
11 Conclusion
The chapter is an updated survey of important known qualitative properties ofmathematical models of viscous incompressible flows past rigid and rotating bodies.The models are based on the Navier–Stokes equations. Greatest attention is paid tosteady problems, as well as problems that are quasi-steady in the sense that thetransformed equations describing the motion of the fluid around a rotating bodyare steady in the body-fixed frame. The presented results concern the existence,regularity, and uniqueness of solutions (see Sects. 4–6). Section 7 deals with thespatial asymptotic properties of steady solutions, like the questions of the presenceof a wake behind the body and the decay of velocity and vorticity in or outside thewake in dependence on the distance from the body. Here, the case T 6D 0 (i.e., thecase when the body rotates with a nonzero constant angular velocity) is much moredifficult than the case T D 0; and the relevant results are therefore of a relativelyrecent date. The structure of the set of steady solutions for arbitrarily large given datais studied in Sect. 8 by means of tools of nonlinear analysis, like the theory of properFredholm operators, corresponding mod 2 degree, etc. One of the results assertsthat, to a given nonzero translational velocity and angular velocity, the solution setis generically finite and has an odd number of elements. Sect. 9 analyzes sufficientand necessary conditions for bifurcations of steady or time-periodic solutions fromsteady solutions. The corresponding theorems provide a theoretical explanation ofthe well-known phenomenon, i.e., that the properties and shape of a steady solutionmay considerably change if some characteristic parameters of the flow field vary.The longtime behavior of unsteady perturbations of a given steady solution v0 isfinally studied in Sect. 10. This section also brings some necessary results on theexistence and uniqueness of solutions. The core of the section are (1) the results onthe stability of v0 under the assumption that v0 is in some sense “sufficiently small”and (2) the results that do not need any condition of smallness of v0. Instead, theyuse either an assumption on a “sufficiently fast” time decay of a certain finite familyof functions related to v0 or an assumption on the position of eigenvalues of a certainassociated linear operator. (Here, one has to overcome the difficulties followingfrom the presence of the essential spectrum, having a nonempty intersection withthe imaginary axis.)
The readers find a series of references to related papers or books inside eachsection. The chapter also brings the formulation of altogether eight open problemsthat concern the discussed topics and represent a challenge for future research.
Acknowledgements The authors acknowledge the partial support of NSF grant DMS-1614011(G.P.Galdi) and the Grant Agency of the Czech Republic, grant No. 13-00522S, and Academyof Sciences of the Czech Republic, RVO 67985840 (J.Neustupa). This work was also partiallysupported by the Department of Mechanical Engineering and Materials Sciences of the Universityof Pittsburgh that hosted the visit of J. Neustupa in Spring 2015.
Steady-State Navier–Stokes Flow Around a Moving Body 73
Cross-References
�Large Time Behavior of the Navier-Stokes�Leray’s Problem on Existence of Steady State for the Navier-Stokes Flow� Stationary Navier-Stokes Flow in Exterior Domain and Landau Solutions� Stokes Semigroup, Strong, Weak and Very Weak Solutions in General Domains
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