On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid
aa r X i v : . [ m a t h . A P ] F e b On Time-Periodic Bifurcation of a Sphere Moving under Gravityin a Navier-Stokes Liquid
Giovanni P Galdi
Department of Mechanical Engineering and Materials ScienceUniversity of Pittsburgh, USA
Abstract
We provide sufficient conditions for the occurrence of time-periodic Hopf bifurcation for thecoupled system constituted by a rigid sphere, S , freely moving under gravity in a Navier-Stokesliquid. Since the region of flow is unbounded (namely, the whole space outside S ), the maindifficulty consists in finding the appropriate functional setting where general theory may apply. Inthis regard, we are able to show that the problem can be formulated as a suitable system of coupledoperator equations in Banach spaces, where the relevant operators are Fredholm of index 0. In sucha way, we can use the theory recently introduced by the author, and give sufficient conditions fortime-periodic bifurcation to take place. Keywords.
Fluid-Structure Interaction; Navier-Stokes Equations; Hopf Bifurcation; Falling Sphere.
AMS Subject Classification.
Introduction
The motion of spheres falling or rising in a viscous liquid has long been recognized as a fundamentaltopic of research, not only for its intrinsic interest but also for its role in applied sciences; see [16,18, 19, 22, 21, 25, 26] and the bibliography therein. Even though the dynamics may be differentdepending on whether the sphere is light (ascending) or heavy (falling) [17], its qualitative behavior israther similar in both cases. More specifically, let ρ S and ρ L be the density of the sphere and of theliquid, respectively, and denote by λ a suitable non-dimensional number depending on | ρ S /ρ L − | ( Galilei number ); see (1.2). Then, experimental and numerical tests [16, 18, 25, 26] show that steadyregimes occur as long as λ is not too “large.” Precisely, in a first range of Galilei numbers, the spheremerely translates (no spin) with constant translational velocity, τ , parallel to the direction of gravity, e . In this situation, the liquid flow is axisymmetric around e . For λ above a first critical value,there is a breaking of symmetry from axisymmetry to planar symmetry. However, the motion is stilltranslatory but now τ and e are no longer parallel. If λ is further increased to some higher criticalvalue, λ c (say), then a second (Hopf) bifurcation occurs: the flow ceases to be steady and becomesinstead purely time-periodic, with the constant translatory motion of the sphere giving way to anoscillating oblique movement. At even greater values of λ , a chaotic regime eventually sets in.The objective of this paper is to furnish a rigorous mathematical contribution to the interpretationof some of the above bifurcation phenomena.We recall that, from a strict mathematical viewpoint, the study of bifurcation, in fluid mechanics asin other branches of mathematical physics, presents a fundamental challenge. It consists in determining1he appropriate functional setting where the problem can be formulated in order to be addressed bygeneral abstract theory. In the situation at hand, this aspect is particularly intriguing, since the regionof flow is unbounded in all directions , which implies that 0 is a point of the essential spectrum of therelevant linearized operator [3]. This is a crucial and well–known problem [1, 2] that prevents onefrom using classical approaches that are, instead, very successful in the case of bounded flow, wherethe above spectral issue is absent [20, 14] [27, Secs. 72.7–72.9].In the past few years, we have introduced a new, general approach to bifurcation that allows us toovercome the above problem and to provide sufficient (and necessary) conditions for the occurrence ofbifurcation, in both steady-state and time-periodic cases, and for both bounded and unbounded flow[5, 8]. The basic idea of this approach is to formulate the problem not in classical Sobolev spaces but,instead, in homogeneous Sobolev spaces, characterized by the property that the various derivativesinvolved may have different summability properties in the neighborhood of spatial infinity. In sucha framework, the spectral issue mentioned earlier on is totally absent. In [7, 8, 9] we have employedthis approach to study bifurcation properties of a Navier–Stokes liquid past a fixed body, namely,when the body is kept in a given configuration by suitable forces and torques. More recently, we haveused the new method to study steady-state bifurcation of a falling (or ascending) sphere in a viscousliquid under the action of gravity [10], which is a bona fide fluid-structure interaction problem. Inparticular, we have shown that a requirement for the occurrence of the above type of bifurcation with τ parallel to e , is that the relevant linearized operator, defined in an appropriate function space, has1 as a simple eigenvalue crossing the imaginary axis at “non-zero speed” (transversality condition).Remarkably, this requirement formally coincides with the classical generic bifurcation condition for aflow in a bounded domain [20].In this article we continue and –to an extent– complete the research initiated in [10], by investigatingthe occurrence of time-periodic bifurcation of the coupled system sphere-liquid under the action ofgravity. More precisely, we show that, once the problem is formulated in the appropriate functionalsetting , we can obtain a bifurcation criterion along the lines of the classical Hopf theory. Namely,it suffices that the relevant linearized operator has a non-resonating, simple imaginary eigenvaluesatisfying the transversality condition. In order to reach this goal, we follow [8, 9] and split theunknowns into the sum of their average over a period plus an oscillatory component. In this way, theoriginal problem transforms into a coupled system of nonlinear elliptic-parabolic equations. We thenprove that such a system can be written as two coupled operator equations in suitable spaces, withthe relevant operators satisfying all the assumptions of the abstract theory introduced in [8], whichthus provides the desired bifurcation results.The plan of the paper is as follows. In Section 1 we give the precise formulation of the problem andcollect some standard notation. In Section 2, we present the abstract time-periodic bifurcation resultproved in [8]; see Theorem 2.1. Successively, in Section 3, we recollect some fundamental functionspaces introduced in [4, 23] and, for some of them, recall their relevant properties. The followingSection 4 is dedicated to the existence of a unique steady-state solution branch parametrized in theGalilei number λ . To this end, we first show, in Theorem 4.1, that for any given λ = 0 there exists acorresponding steady-state solution, s ( λ ), in a suitable homogeneous Sobolev space. Successively, weprove that if there is λ c such that the linearization, L , around s ( λ c ) is trivial, then there exists a uniqueanalytic family of steady-state solutions in a neighborhood of λ c where the translational velocity ofthe sphere is parallel to that of s ( λ c ); see Theorem 4.2. In Section 5, we investigate some importantspectral properties of L in the Lebesgue space L . Precisely, we show that the intersection of thespectrum of L with the imaginary axis is constituted, at most, by a countable number of eigenvaluesof finite multiplicity that can only cluster at 0; see Theorem 5.1. The main objective of Section 62s to establish the Fredholm property of the time-periodic linearized operator in a suitable space offunctions with zero average over a period. In particular, in Theorem 6.1 we prove that such operatoris Fredholm of index 0. With the help of the results established in the previous sections, in Section7 we then secure that the original problem is written in an abstract setting where the general theoryrecalled in Section 3 applies. Therefore, thanks to Theorem 2.1, in Theorem 7.1 we give sufficientconditions for the existence of a time-periodic branch in the neighborhood of the steady-state solution s ( λ c ). As already mentioned, these conditions amount to the request that the operator L , suitablydefined , has a non-resonating, simple imaginary eigenvalue satisfying the transversality condition. Inthe final Section 8, we consider the problem of the motion of the sphere in the time-periodic regime.In this regard, we give necessary and sufficient conditions for the occurrence of a horizontal oscillationof the center of mass, in a neigborhood of the “critical” value λ c ; see Theorem 8.1. A sphere, S , of constant density, ρ S and radius R freely moves under the action of gravity in anotherwise quiescent Navier–Stokes liquid, L , that fills the entire space outside S . We assume that S is not floating, namely, it has a non-zero buoyancy. This means that, denoting by ρ L the densityof the liquid, we take | α | := | ρ S /ρ L − | >
0. Just to fix the ideas, we shall assume α > α with − α .Let F = { O, e , e , e } be a frame with the origin at the center of S ( ≡ center of mass of S ) andthe axis e oriented along the acceleration of gravity g . The dynamics of the coupled system S ∪ L in F are then governed by the following set of non-dimensional equations [4, Section 4] ∂ t v + λ ( v − τ ) · ∇ v = ∆ v − ∇ p div v = 0 ) in Ω × (0 , ∞ ) v = τ + ̟ × x at ∂ Ω × (0 , ∞ ) , lim | x | →∞ v ( x, t ) = , t ∈ (0 , ∞ ) ,M ˙ τ + Z ∂ Ω T ( v , p ) · n = λ e , I ˙ ̟ + Z ∂ Ω x × T ( v , p ) · n = 0 in (0 , ∞ ) . (1.1)Here Ω = R \ Ω , with Ω volume occupied by S . Furthermore, v and p + e · x are (non-dimensional)velocity and pressure fields of L , while τ , ̟ stand for (non-dimensional) translational and angularvelocities of S . Moreover, λ = p α g R /ν ( >
0) (1.2)is the (dimensionless) Galilei number, with ν kinematic viscosity of L . Also, M = 4 πρ S / ρ L and I = 8 πρ S / ρ L are non-dimensional mass and central moment of inertia of S , and n is the outerunit normal to ∂ Ω. Finally, T ( v , p ) = − p + 2 D ( v ) , D ( v ) := ( ∇ v + ( ∇ v ) ⊤ ) , is the Cauchy tensor with identity tensor and ⊤ denoting transpose.Of particular significance is the subclass of solutions to (1.1) constituted by those fields ( v , p , τ , ω )3hat are time independent, namely, they solve the following boundary-value problem λ ( v − τ ) · ∇ v = ∆ v − ∇ p div v = 0 ) in Ω v = τ + ω × x at ∂ Ω , lim | x | →∞ v ( x ) = , Z ∂ Ω T ( v , p ) · n = λ e , Z ∂ Ω x × T ( v , p ) · n = . (1.3)Solutions ( v , p , τ , ω ) to (1.3) describe the so called steady free falls of the sphere in the viscousliquid, and, as explained in the introductory section, their behavior depends on the parameter λ .In mathematical terms, the time-periodic bifurcation problem can be formulated as follows. Let λ c >
0, let U = U ( λ c ) be a neighborhood of λ c , and let s ( λ ) := ( v , p , τ , ω )( λ ), λ ∈ U ( λ c ), be asufficiently smooth family of solutions to (1.3). The objective is then to prove the existence of time-periodic solutions to (1.1) “around” s ( λ c ). Since the period T := 2 π/ζ of such solutions is unknown,it is customary to scale the time by introducing the new variable s = ζ t . Therefore, writing v ( x, t ) = u ( x, t ) + v ( x ) , p ( x, t ) = p ( x, t ) + p ( x ) , τ ( t ) = γ ( t ) + τ , ̟ ( t ) = ω ( t ) + ω our bifurcation problem means that we must find a 2 π -periodic solution-branch ( u , p , γ , ω )( λ ), λ ∈ U ( λ c ), to the following set of equations ζ ∂ s u − λ τ · ∇ u + λ ( v · ∇ u + ( u − γ ) · ∇ v ) + λ ( u − γ ) · ∇ u = ∆ u − ∇ p div u = 0 ) in Ω × R , u = γ + ω × x at ∂ Ω × R , lim | x | →∞ u ( x, t ) = , t ∈ R ,M ˙ γ + Z ∂ Ω T ( u , p ) · n = , I ˙ ω + Z ∂ Ω x × T ( u , p ) · n = in R . (1.4)Our strategy to solve this problem consists in rewriting (1.4) as operator equations in suitableBanach spaces, with the involved operators satisfying a certain number of fundamental properties.Once this goal is accomplished, we will be able to employ the general theory introduced in [8] andrecalled in Section 2, and derive sufficient conditions for the existence of a time-periodic bifurcatingbranch.Before performing our study, we recall the main notation used in the paper. With the origin at thecenter of Ω , we set B R := {| x | < R } , and, for R > , Ω R := Ω ∩ B R , Ω R := Ω ∩ {| x | > R } .As customary, for a domain A ⊆ R , L q = L q ( A ) is the Lebesgue space with norm k · k q,A , and W m,q = W m,q ( A ) denotes Sobolev space, m ∈ N , q ∈ [1 , ∞ ], with norm k · k m,q,A . If u ∈ L q ( A ), v ∈ L q ′ ( A ), q ′ = q/ ( q − u, v ) A = R A u v . Furthermore, D m,q = D m,q ( A ) are homogeneousSobolev spaces with semi-norm | u | m,q,A := P | l | = m k D l u k q . In the above notation, the subscript ” A ”will be generally omitted, unless confusion arises. A function u : A × R R is 2 π -periodic , if u ( · , s + 2 π ) = u ( · s ), for a.a. s ∈ R , and we set u := π R π u ( t )d t . If B is a semi-normed real Banachspace with semi-norm k·k B , r = [1 , ∞ ], we denote by L r (0 , π ; B ) the class of functions u : (0 , π ) → B such that k u k L r ( B ) ≡ ( Z π k u ( t ) k rB d t ) r < ∞ , if r ∈ [1 , ∞ ) ;ess sup t ∈ [0 , π ] k u ( t ) k B < ∞ , if r = ∞ . W , (0 , π ; B ) = n u ∈ L (0 , π ; B ) : ∂ t u ∈ L (0 , π ; B ) , o . Unless otherwise stated, we shall write L r ( B ) for L r (0 , π ; B ), etc. By B C := B + i B we denote thecomplexification of B . If M is a map between two spaces, D [ M ], N [ M ], R [ M ] and P [ M ] will indicateits domain, null space, range, and resolvent set, respectively. Finally, by c , c , c , etc., we denotepositive constants, whose particular value is unessential to the context. When we wish to emphasizethe dependence of c on some parameter µ , we shall write c ( µ ) or c µ . Objective of this section is to recall a time-periodic bifurcation theorem for a general class of operatorequations proved in [8]. Before stating the result, however, we first would like to make some commentsthat will also provide the motivation of this approach.Many evolution problems in mathematical physics can be formally written in the form u t + L ( u ) = N ( u, µ ) , (2.1)where L is a linear differential operator (with appropriate homogeneous boundary conditions), and N is a nonlinear operator depending on the parameter µ ∈ R , such that N (0 , µ ) = 0 for all admissiblevalues of µ . Then, roughly speaking, time-periodic bifurcation for (2.1) amounts to show the existencea family of non-trivial time-periodic solutions u = u ( µ ; t ) of (unknown) period T = T ( µ ) ( T - periodic solutions) in a neighborhood of µ = 0, and such that u ( µ ; · ) → µ →
0. Setting s := 2 π t/T ≡ ζ t ,(6.16) becomes ζ u s + L ( u ) = N ( u, µ ) (2.2)and the problem reduces to find a family of 2 π -periodic solutions to (2.2) with the above properties.We now write u = u + ( u − u ) := v + w and observe that (2.2) is formally equivalent to the followingtwo equations L ( v ) = N ( v + w, µ ) := N ( v, w, µ ) ,ζ w s + L ( w ) = N ( v + w, µ ) − N ( v + w, µ ) := N ( v, w, µ ) . (2.3)At this point, the crucial issue is that in many applications –typically when the physical system evolvesin an unbounded spatial region – the “steady-state component” v lives in function spaces with quiteless “regularity” (1) than the space where the “oscillatory” component w does. For this reason, it ismuch more appropriate to study the two equations in (2.3) in two different function classes. As aconsequence, even though formally being the same as differential operators, the operator L in (2.3) acts on and ranges into spaces different than those the operator L in (2.3) does. With this in mind,(2.3) becomes L ( v ) = N ( v, w, µ ) ; ζ w s + L ( w ) = N ( v, w, µ ) . The general abstract theory that we are about to describe stems exactly from the above consider-ations.Let X , Y , be (real) Banach spaces with norms k · k X , k · k Y , respectively, and let Z be a (real)Hilbert space with norm k · k Z and corresponding scalar product h· , ·i . Moreover, denote by L : X 7→ Y , (1) Here ‘regularity’ is meant in the sense of behavior at large spatial distances. L : D [ L ] ⊂ Z 7→ Z , a densely defined, closed linear operator, with a non-empty resolvent set P ( L ). For a fixed (onceand for all) θ ∈ P ( L ) we denote by W the linear subspace of Z closed under the norm k w k W := k ( L + θ I ) w k Z , where I stands for the identity operator. We also define the following spaces Z π, := { w ∈ L (0 , π ; Z ) : 2 π -periodic with w = 0 }W π, := { w ∈ L (0 , π ; Z ) , w t ∈ L (0 , π ; Z ) : 2 π -periodic with w = 0 } . Next, let N : X × W π, × R
7→ Y ⊕ H π, be a (nonlinear) map satisfying the following properties: N : ( v, w, µ ) ∈ X × W π, × R N ( v, w, µ ) ∈ Y N := N − N : X × W π, × R
7→ Z π, . (2.4)The bifurcation problem can be then rigorously formulated as follows.Bifurcation Problem: Find a neighborhood of the origin U (0 , , ⊂ X × W π, × R such that theequations L ( v ) = N ( v, w, µ ) , in Y ; ζ w s + L ( w ) = N ( v, w, µ ) , in Z π, , (2.5) possess there a family of non-trivial π -periodic solutions ( v ( µ ) , w ( µ ; τ )) for some ω = ω ( µ ) > , suchthat ( v ( µ ) , w ( µ ; · )) → in X × W π, as µ → . Whenever the Bifurcation Problem admits a positive answer, we say that ( u = 0 , µ = 0) is a bifurcation point . Moreover, the bifurcation is called supercritical [resp. subcritical ] if the family ofsolutions ( v ( µ ) , w ( µ ; τ )) exists only for µ > µ < L is a homeomorphism ;(H2) There exists ν := i ζ , ζ > L − ν I is Fredholm of index 0, and dim N C [ L − ν I ] = 1with N C [ L − ν I ] ∩ R C [ L − ν I ] = { } . Namely, ν is a simple eigenvalue of L . Moreover, k ν ∈ P ( L ), for all k ∈ N \{ , } .(H3) The operator Q : w ∈ W π, ζ w s + L ( w ) ∈ Z π, , is Fredholm of index 0 ;(H4) The nonlinear operators N , N are analytic in the neighborhood U (0 , , ⊂ X × W π, × R ,namely, there exists δ > v, w, µ ) with k v k X + k w k W π, + | µ | < δ , the Taylorseries N ( v, w, µ ) = ∞ X k,l,m =0 R klm v k w l µ m ,N ( v, w, µ ) = ∞ X k,l,m =0 S klm v k w l µ m , Y and Z π, , respectively, for all ( v, w, µ ) ∈ U . Moreover, we assumethat the multi-linear operators R klm and S klm satisfy R klm = S klm = 0 whenever k + l + m ≤ R = R m = S m = 0, all m ≥ L ( µ ) := L + µ S , and observe that, by (H2), ν is a simple eigenvalue of L (0) ≡ L . Therefore, denoting by ν ( µ ) theeigenvalues of L ( µ ), it follows (e.g. [27, Proposition 79.15 and Corollary 79.16]) that in a neighborhoodof µ = 0 the map µ ν ( µ ) is well defined and of class C ∞ .We may then state the following bifurcation result, whose proof is given in [8, Theorem 3.1]. Theorem 2.1
Suppose (H1)–(H4) hold and, in addition, ℜ [ ν ′ (0)] = 0 , namely, the eigenvalue ν ( µ ) crosses the imaginary axis with “non-zero speed.” Moreover, let v be anormalized eigenvector of L corresponding to the eigenvalue ν , and set v := ℜ [ v e − i s ] . Then, thefollowing properties are valid. (a) Existence.
There are analytic families ( v ( ε ) , w ( ε ) , ζ ( ε ) , µ ( ε )) ∈ X × W π, × R + × R (2.6) satisfying (2.5) , for all real ε in a neighborhood I (0) of 0, and such that ( v ( ε ) , w ( ε ) − ε v , ζ ( ε ) , µ ( ε )) → (0 , , ζ , as ε → . (2.7)(a) Uniqueness. There is a neighborhood U (0 , , ζ , ⊂ X × W π, × R + × R such that every (nontrivial) π -periodic solution to (2.5) , lying in U must coincide, up to a phase shift,with a member of the family (2.6) . (a) Parity. The functions ζ ( ε ) and µ ( ε ) are even: ζ ( ε ) = ζ ( − ε ) , µ ( ε ) = µ ( − ε ) , for all ε ∈ I (0) .Consequently, the bifurcation due to these solutions is either subcritical or supercritical, a two-sidedbifurcation being excluded. (2) In this section we will introduce certain function classes along with some of their most importantproperties.Let
RRR be the class of velocity fields in a rigid motion:
RRR = { b u ∈ C ∞ ( R ) : b u = b u + b u × x , for some b u , b u ∈ R } , (2) Unless µ ≡ K = K ( R ) = { ϕ ∈ C ∞ ( R ) : in R ; ϕ ( x ) = b ϕ , some b ϕ ∈ RRR in a neighborhood of Ω }C = C ( R ) = { ϕ ∈ K ( R ) : div ϕ = 0 in R } . We shall call b ϕ , b ϕ the characteristic vectors of the function ϕ . In K we introduce the scalar product M b ϕ · b ψ + I b ϕ · b ψ + ( ϕ , ψ ) Ω , ϕ , ψ ∈ K , (3.1)and define the following spaces (3) L = L ( R ) := { completion of K ( R ) in the norm induced by (3.1) } , H = H ( R ) := { completion of C ( R ) in the norm induced by (3.1) }G = G ( R ) := { h ∈ L ( R ) : there is p ∈ D , (Ω) such that h = ∇ p in Ω , and h = − M − R ∂ Ω p n − I − (cid:0)R ∂ Ω p y × n (cid:1) × x in Ω } . (3.2)In [23, Theorem 3.1 and Lemma 3.2] the following characterization of the spaces L and H is proved. Lemma 3.1
Let Ω be Lipschitz. Then L ( R ) = { u ∈ L ( R ) : u = b u in Ω , for some b u ∈ RRR }H ( R ) = { u ∈ L ( R ) : div u = 0 } . Furthermore, by an argument entirely analogous to that employed in [23, Theorem 3.2] one shows:
Lemma 3.2
The following orthogonal decomposition holds L ( R ) = H ( R ) ⊕ G ( R ) . We next introduce the space V = V ( R ) ≡ { completion of C ( R ) in the norm k D ( · ) k } . The basic properties of the space V are collected in the next lemma, whose proof is given in [4, Lemmas9–11]. Lemma 3.3 V is a Hilbert space endowed with the scalar product [ u , w ] = Z Ω D ( u ) : D ( w ) , u , w ∈ V . (3.3) Furthermore, the following characterization holds V = { u ∈ L ( R ) ∩ D , ( R ) ; div u = 0 in R ; u ( y ) = b u , y ∈ Ω , some b u ∈ RRR } . (3.4) (3) Even though x × n = if x ∈ ∂ Ω, for notational convenience we will keep the term b h . lso, for each u ∈ V , we have k∇ u k = √ k D ( u ) k , (3.5) and k u k ≤ κ k D ( u ) k , u ∈ V , (3.6) for some numerical constant κ > . (4) Finally, there is another positive numerical constant κ suchthat | b u | + | b u | ≤ κ k D ( u ) k . (3.7)Let V − = V − ( R ) be the dual space of V , and let h· , ·i and k · k − be the corresponding dualitypair and associated norm, respectively. Denote by e ∈ R a given unit vector and consider the space X e ( R ) := { u ∈ V ( R ) : e · ∇ u ∈ V − } where e · ∇ u ∈ V − means that there is C = C ( u ) > | ( e · ∇ u , ϕ ) | ≤ C k D ( ϕ ) k , for all ϕ ∈ C ( R ) . (3.8)Actually, from (3.8), the density of C in V , and the Hahn–Banach theorem it follows that e · ∇ u canbe uniquely extended to a bounded linear functional on the whole of V , with k e · ∇ u k − := sup ϕ ∈ C ; k D ( ϕ ) k = 1 | ( e · ∇ u , ϕ ) | . Obviously, the functional k u k X e := k D ( u ) k + k ∂ u k − , defines a norm in X ( R ).In the following lemma we collect the relevant properties of the space X ( R ). Their proofs areentirely analogous to [7, Proposition 65] and [10, Lemma 2.1], and therefore will be omitted. Lemma 3.4
The space X e ( R ) endowed with the norm k · k X e is a reflexive, separable Banach space,dense in V ( R ) . Moreover, X e ( R ) is continuously embedded in L ( R ) , and there is c > such that k u k ≤ c ( k e · ∇ u k − k D ( u ) k + k D ( u ) k ) . Finally, we have h e · ∇ u , u i = 0 , for all u ∈ X e ( R ) . (3.9)We next introduce the following spaces of time-periodic functions: L ♯ = { F ∈ L (0 , π ); F is 2 π -periodic with F = 0 } W , ♯ = { χ ∈ L ♯ (0 , π ); ˙ χ ∈ L (0 , π ) }L ♯ := { f ∈ L ( L ); f is 2 π -periodic, with f = }W ♯ := { u ∈ W , ( L ) ∩ L ( W , ∩ V ); u is 2 π -periodic, with u = } (4) Recall that, in our non-dimensionalization, the sphere has radius 1. k χ k L ♯ := k χ k L ; k χ k W , ♯ := k χ k W , ≡ k ˙ χ k L + k χ k L , k u k L ♯ := k u k L ( L ) ; k u k W ♯ := k u k W , ( L ) + k u k L ( W , ) . Finally, we set P , := L ( D , ) . The goal of this section is two-fold. On the one hand, to establish existence of solutions to (1.3) insuitable function classes for all values of λ > λ c .We begin to give the definition of weak solution. Definition 4.1
A field v : R R is a weak solution to (1.3) if the following conditions hold.(i) v ∈ V ( R ), and τ ≡ b v = ;(ii) v satisfies the relation: (5) v , ϕ ] − λ b ϕ · e = λ (( τ − v ) · ∇ v , ϕ ) , for all ϕ ∈ C . (4.1) Remark 4.1
It is easy to show that, if a weak solution v is sufficiently smooth in Ω, then there existsa likewise smooth pressure field p : Ω R such that the quadruple ( v , p, τ ≡ b v , ω ≡ b v ) is a smoothsolution to (1.3). To this end, we begin to obseve that, clearly, the validity of (1.3) , comes from thefact that v ∈ V ( R ). If we integrate by parts the first term on the left-hand side of (4.1), we get2[ v , ϕ ] = − Z Ω ∆ v · ϕ + b ϕ · Z ∂ Ω D ( v ) · n + b ϕ · Z ∂ Ω x × (2 D ( v ) · n ) . (4.2)Thus, by taking, in particular, ϕ ∈ D (Ω) := { ϕ ∈ C ∞ (Ω) : div ϕ = 0 } , from (4.1) and (4.2) we get Z Ω ( − ∆ v − λ ( τ − v ) · ∇ v ) · ϕ = 0 , for all ϕ ∈ D (Ω) ,which, by well-known results, implies the existence of a (smooth) scalar field p for which (1.3) holds.Now, using (4.2) and (1.3) we deduce b ϕ · ( − λ e + Z ∂ Ω D ( v ) · n ) + b ϕ · Z ∂ Ω x × (2 D ( v ) · n ) − Z Ω ∇ p · ϕ = λ Z Ω ( τ − v ) · ∇ v · b ϕ , which, after an integration by parts leads to b ϕ · ( − λ e + Z ∂ Ω T ( v , p ) · n ) + b ϕ · Z ∂ Ω x × T ( v , p ) · n = λ Z Ω ( τ − v ) · ∇ v · b ϕ . (4.3) (5) For simplicity, we omit the subscript “0”. , v , ϕ ∈ RRR , we get ( τ ≡ b v ) Z Ω ( τ − v ) · ∇ v · b ϕ = Z Ω ( b v × x ) · ∇ ( b v × x ) · b ϕ = Z ∂ Ω b v × x · n ( b v × x · b ϕ ) − Z Ω ( b v × x ) · D ( b ϕ ) · ( b v × x ) = 0because D ( b ϕ ) = 0, and x × n = for all x ∈ ∂ Ω. From the latter, (4.3), and the arbitrariness of b ϕ wethen conclude that a smooth weak solution v and the associated pressure field p satisfy also (1.3) , .Finally, as a consequence of the next lemma, we shall see that v obeys also the asymptotic condition(1.3) .The next lemma establishes some important properties of weak solutions. Lemma 4.1
Let v be a weak solution to (1.3) . Then, the following properties hold. (a) v ∈ C ∞ (Ω R ) ∩ C ∞ (Ω) , all R > , and there is a pressure field p ∈ C ∞ (Ω R ) ∩ C ∞ (Ω) , all R > ,such that the quadruple ( v , p, τ ≡ b v , ω ≡ b v ) satisfies (1.3) ; (b) v ∈ L q ( R ) ∩ D , q ( R ) , p ∈ L q (Ω) , for all q ∈ (2 , ∞ ] ; (c) v ∈ X e ( R ) , e := τ / | τ | ; (d) v · ∇ v ∈ V − ( R ) and relation (4.1) is equivalent to the following one: v , ϕ ] − λ b ϕ · e = λ h ( τ − v ) · ∇ v , ϕ i , for all ϕ ∈ V ( R ) . (4.4)(e) v obeys the energy equality: k D ( v ) k = λ τ · e . (4.5) Proof.
Taking into account that Ω is of class C ∞ , the first statement, restricted to (1.3) , , , , , followsfrom the regularity results in [6, Theorem IX.5.1] applied to (4.1), along with Remark 4.2. Next, since τ = and v ∈ V ( R ), the results in [6, Theorem X.6.4] ensure that v , and p possess the summabilityproperties stated in (b). In particular, by [6, Theorem II.9.1], the latter imply also (1.3) . By H¨olderinequality and (3.5), (3.6) we show, for all ϕ ∈ C , | ( v · ∇ v , ϕ ) | ≤ k v k k∇ v k k ϕ k ≤ √ κ k v k k D ( v ) k k D ( ϕ ) k , (4.6)which delivers in particular, by (b), v · ∇ v ∈ V − ( R ) . (4.7)Furthermore, from (4.1), (3.7) and (4.6) we infer, for all ϕ ∈ C , | ( τ · ∇ v , ϕ ) | ≤ k D ( v ) k k D ( ϕ ) k + λ [ | b ϕ | + k v k k∇ v k k ϕ k ] ≤ [ k D ( v ) k + λ ( κ + √ κ k v k k D ( v ) k )] k D ( ϕ ) k , from which we conclude v ∈ X e ( R ). In view of the latter, (4.7), and the density of C in V ( R ), wemay thus deduce that (4.1) leads to (4.4). We now replace v for ϕ in (4.4) to obtain2 k D ( v ) k − λ τ · e = λ ( h τ · ∇ v , v i − h v · ∇ v , v i ) = − λ h v · ∇ v , v i , (4.8)11here, in the last step, we have taken into account (3.9). Now, after integrating by parts, we deduce( v · ∇ v , ϕ ) = − ( v · ∇ ϕ , v ) , for all ϕ ∈ C . (4.9)Since v ∈ L ( R ) by the H¨older inequality it is readily seen that the right-hand side of (4.9) definesa continuous functional in ϕ ∈ V (Ω). As a result, setting ϕ = v in (4.9) we conclude ( v · ∇ v , v ) = h v · ∇ v , v i = 0, and the property (e) follows from this and (4.8). The proof of the lemma is completed. (cid:3) We are now in a position to show the following existence result.
Theorem 4.1
For any given λ > , problem (1.3) has at least one weak solution v = v ( λ ) . Proof.
We shall employ the classical Galerkin method. To this end, let { ϕ k } ⊂ C be an ortho-normalbasis in V ( R ) and, for notational simplicity, set d ( ϕ k ) = ξ k , d ( ϕ k ) = σ k . Consider the linear combinations v m = m X ℓ =1 c ℓm ϕ ℓ , τ m = m X ℓ =1 c ℓm ξ ℓ , ω m = m X ℓ =1 c ℓm σ ℓ where the coefficients c ℓm are requested to be solution to the following algebraic set of equations[ v m , ϕ k ] − λ ξ k · e = λ (( τ m − v m ) · ∇ v m , ϕ k ) , k ∈ { , . . . m } . (4.10)Multiplying both sides of this relation by c km and summing over k ∈ { , . . . , m } we show2 k D ( v m ) k − λ τ m · e = λ (( τ m − v m ) · ∇ v m , v m ) . (4.11)By a simple integration by parts, we show(( τ m − v m ) · ∇ v m , v m ) = 0 , and so (4.11) becomes 2 k D ( v m ) k − λ τ m · e = 0 . (4.12)Now, let c := ( c m , . . . , c mm ) ∈ R m , and consider the map P : c ∈ R m P ( c ) ∈ R m , where [ P ( c )] k = [ v m , ϕ k ] − λ ξ k · e − λ (( τ m − v m ) · ∇ v m , ϕ k ) , k ∈ { , . . . m } . By what we have just shown, P ( c ) · c = 2 k D ( v m ) k − λ τ m · e , for all m ∈ N , and so, by (3.7) and the ortho-normal property of the base { ϕ k } , P ( c ) · c ≥ | c | ( | c | − κ )12hus, by [6, Lemma IX.3.1], the latter implies that for each m ∈ N , the algebraic system (4.10) hasat least one solution c = c ( m ). Furthermore, from (4.12) and again (3.7) we deduce the followingestimate for the sequence { v m } , uniformly in m ∈ N :2 k D ( v m ) k ≤ λ κ . (4.13)Taking into account that V ( R ) ⊂ W , ( B R ) and that W , ( B R ) is compactly embedded in L ( B R ), forall R >
0, from (4.13) we deduce the existence of v ∈ V ( R ) such that (possibly along a subsequence) v m → v , weakly in V ( R ), strongly in L ( B R ), all R > τ m → τ ≡ b v , ω m → ω ≡ b v in R . Employing these convergence properties, we may pass to the limit m → ∞ in (4.10) and (4.12) andshow, by classical arguments, that v is a solution to (4.1) that satisfies, in addition, the “energyinequality:” 2 k D ( v ) k ≤ λ τ · e . (4.14)From (4.14) we obtain, in particular that τ = because, otherwise, by Lemma 3.3, v ≡ , incontradiction with (4.1). Therefore, v is a weak solution to (1.3) in the sense of Definition 4.1, andthe proof of the theorem is completed. (cid:3) Remark 4.2
A particular subclass of weak solutions to (1.3) is the one characterized by having τ = τ e , τ > ω = , and v possessing rotational symmetry around e . The existence of suchsolutions is shown in [10, Theorem 1.1].The previous theorem proves the existence of a steady-state solution, v = v ( λ ), to (1.1) for all λ >
0. However, it is silent about the regularity of the map λ v ( λ ). In the next theorem, we shallfurnish sufficient conditions for the existence of a local, unique, analytic family of weak solutions forwhich the velocity of the center of mass is directed along the same direction. To this end, let v c be aweak solution to (1.3) corresponding to λ = λ c , denote by τ c = τ c e c the corresponding translationalvelocity, where e c := τ c / | τ c | . Moreover, define V c = V c ( R ) := { u ∈ V ( R ) : u ( x ) = b u e c + b u × x , x ∈ Ω } , and set X c ( R ) = V c ( R ) ∩ X e c ( R ) . (4.15)We can then prove that the map λ v ( λ ) is smooth in a neighborhood of λ c in the class X c , providedthe linearization of (4.1) around ( v c , λ c ) is trivial. To this end, let LLL : u ∈ X c ( R ) LLL ( u ) ∈ V − ( R ) , (4.16)where h LLL ( u ) , ϕ i = 2[ u , ϕ ] − λ c [ τ c h e c · ∇ u , ϕ i + b u h e c · ∇ v c , ϕ i +( u · ∇ v c + v c · ∇ u , ϕ )] , ϕ ∈ V ( R ) . (4.17)The following result holds. 13 heorem 4.2 Let v c be a weak solution to (1.3) corresponding to λ = λ c . Then, the operator LLL isFredholm of index 0. Furthermore, suppose that the equation LLL ( u ) = (4.18) has only the solution u = in X c ( R ) . Then, there exists a neighborhood U c of λ c , such that (1.3) has a unique family of weak solutions v ( λ ) ∈ X c ( R ) , λ ∈ U c , which is analytic at λ c and such that v ( λ c ) = v c . Proof.
Consider the map F : ( v , λ ) ∈ X c × U c
7→ F ( v , λ ) ∈ V − , where hF ( v , λ ) , ϕ i := [ v , ϕ ] − λ b ϕ · e − λ h ( b v e c − v ) · ∇ v , ϕ i , ϕ ∈ V . The map is well defined. In fact, since v ∈ X c , we have e c · ∇ v ∈ V − . Furthermore, from (4.9),H¨older inequality and Lemma 3.4 it follows that v · ∇ v ∈ V − as well. In addition, since F involvesonly cubic nonlinearities, F is analytic. We now observe that (4.4) is equivalent to F ( v , λ ) = 0. In[10, Lemmas 2.2 and 2.3] it is shown that the operator OOO : u ∈ X c OOO ( u ) ∈ V − with h OOO ( u ) , ϕ i := [ u , ϕ ] − λ c h τ c e c · ∇ u , ϕ i , ϕ ∈ V , is a homeomorphism, while the operator KKK : u ∈ X c KKK ( u ) ∈ V − with h KKK ( u ) , ϕ i := − λ c h ( b u e c − u ) · ∇ v c − v c · ∇ u , ϕ i , ϕ ∈ V , is compact. As a consequence, LLL ≡ OOO + KKK is Fredholm of index 0 and thus, under the assumptionsof the theorem, it is a homeomorphism. By Lemma 4.1(d), it is F ( v c , λ c ) = 0, while the partialFr´echet derivative of F at ( v c , λ c ), D v F ( v c , λ c ), is easily shown to satisfy D v F ( v c , λ c ) = LLL . Since LLL is a homeomorphism, the result is an immediate consequence of the analytic version of the ImplicitFunction Theorem. (cid:3) Remark 4.3
We observe that the assumption of Theorem 4.2 excludes that ( v c , λ c ) is a steady-statebifurcation point in the class X c [10]. The main objective of this section is to establish some important spectral properties of the operatorobtained by linearizing (1.3) around the solution v c . As shown later on, such properties will supportone of the basic assumptions of our bifurcation result. To this end, we begin to define the map e ∆ : w ∈ W , (Ω) ∩ L ( R ) e ∆( w ) ∈ L ( R )14here e ∆( w ) = − ∆ w in Ω , M Z ∂ Ω (2 D ( w ) · n ) + (cid:18) I Z ∂ Ω y × (2 D ( w ) · n ) (cid:19) × x in Ω , (5.1)and set AAA := P e ∆, with P the orthogonal projection of L ( R ) onto H ( R ); see Lemma 3.2. Wenext consider the operator LLL : u ∈ Z , := W . (Ω) ∩ H ( R )
7→ L ( u ) = − λ c τ c · ∇ u + AAA ( u ) ∈ H ( R ) . (5.2) L is well defined since τ c · ∇ u ∈ H ( R ) . (5.3)In fact, we observe that u ∈ Z , implies u ∈ V ( R ), so that by Lemma 3.2 and (3.1), it follows that(5.3) reduces to prove that Z Ω τ c · ∇ u · h + M \ ( τ c · ∇ u ) · b h + I \ ( τ c · ∇ u ) · b h = 0 , for all h ∈ G ( R ) . (5.4)Taking into account that in Ω it is τ c · ∇ u = τ c · ∇ ( b u × x ) = b u × τ c , while, by (3.2) , h = ∇ p inΩ with b h = − M − R ∂ Ω p n and b h = −I − R ∂ Ω p x × n , (5.4) becomes Z Ω τ c · ∇ u · ∇ p − Z ∂ Ω p b u × τ c · n = 0 , whose validity is immediately checked by integrating by parts the volume integral, and recalling thatdiv u = 0.The following preliminary result holds. Lemma 5.1
Let ζ ∈ R \{ } and ( λ c , τ c ) ∈ R × R . Then, the operator L + i ζ is a homeomorphismof Z , C onto H C ( R ) . Moreover, there is c = c (Ω , ρ S /ρ L ) , such that k D u k + | ζ | k∇ u k + | ζ | ( k u k + M | χ | + I | σ | ) ≤ c k ( LLL + i ζ )( u ) k , | ζ | ≥ max { λ c | τ c | , } , (5.5) where χ := b u , σ := b u . Proof.
The homeomorphism property can be obtained by proving that for any F ∈ H C ( R ), thereexists one and only one u ∈ Z , C such that L ( u ) + i ζ u = F . In view of Lemma 3.1, Lemma 3.2and (5.1) this is equivalent to requiring that for any ( f , F , G ) ∈ L C (Ω) × C × C , with div f = 0, theproblem ∆ u + λ c τ c · ∇ u − ∇ p = i ζ u + f div u = 0 ) in Ω , u = χ + σ × x at ∂ Ω , i ζ M χ + Z ∂ Ω T ( u , p ) · n = F , i ζ I σ + Z ∂ Ω x × T ( u , p ) · n = G , (5.6)has one and only one solution ( u , p , χ , σ ) ∈ W , C (Ω) × D , C (Ω) × C × C . Let us dot-multiply both sidesof (5.6) by u ∗ ( ∗ = complex conjugate) and integrate by parts over Ω. Taking into account (5.6) − we show2 k D ( u ) k + i ζ ( k u k + M | χ | + I| σ | ) − λ ( τ · ∇ u , u ∗ ) = ( f , u ∗ ) + F · χ ∗ + G · σ ∗ , (5.7)15here, here and in what follows, in order to simplify the notation we suppress the subscript c . Takingthe real and imaginary parts of (5.5), using (3.5) and Schwarz inequality, and observing that ℜ ( τ ·∇ u , u ∗ ) = 0, we deduce k∇ u k ≤ FFF WWW ; | ζ | k u k ≤ ( λ | τ | k∇ u k + FFF ) WWW , (5.8)where k W k := ( k u k + M | χ | + I| σ | ) WWW := k u k + M | χ | + I | σ | , FFF := k f k + M − | F | + I − | G | . Observing that
WWW ≤ k W k , from (5.8) and Cauchy-Schwarz inequality we get | ζ | k u k ≤ ( λ | τ | FFF WWW + FFF ) ≤ (cid:18) λ | τ | | ζ | + 1 (cid:19) FFF + | ζ | k W k , from which we conclude | ζ | k W k ≤ (cid:18) λ | τ | | ζ | + 2 (cid:19) FFF . (5.9)Replacing the latter into (5.8) , we infer | ζ | k∇ u k ≤ (cid:18) λ | τ | | ζ | + 2 (cid:19) FFF . (5.10)Combining the estimates (5.9), (5.10) with classical Galerkin method, we can proceed as in the proofof Theorem 4.1 and show that for any given ( f , F , G ) in the specified class and ζ = 0, there exists a(unique, weak) solution to (5.6) such that u ∈ V C (Ω) ∩ L C (Ω), satisfying (5.9), (5.10). We next write(5.6) − as the following Stokes problem∆ u = ∇ p + G div u = 0 ) in Ω , u = χ + σ × x at ∂ Ωwhere G := − λ τ · ∇ u + i ζ u + f . Since G ∈ L C (Ω) and u ∈ W , C (Ω), from classical results [6, Theorems IV.5.1 and V.5.3] it followsthat D u ∈ L (Ω), thus completing the existence (and uniqueness) proof. Furthermore, by [6, LemmaIV.1.1 and V.4.3] we get k D u k ≤ c [ k f k + ( λ | τ | + 1) k∇ u k + ( | ζ | + 1) k u k + | χ | + | σ | ] . (5.11)As a result, if | ζ | ≥ max { λ | τ | , } , the inequality in (5.5) is a consequence of (5.9)–(5.11). (cid:3) Let K : u ∈ Z , K ( u ) ∈ L ( R ) (5.12)16here ( χ := b u ), K ( u ) = ( λ c ( v c · ∇ u + ( u − χ ) · ∇ v c ) in Ω , in Ω . (5.13)From Lemma 4.1, we get, in particular, v c ∈ L ∞ (Ω) ∩ L (Ω) , ∇ v c ∈ L ∞ (Ω) ∩ D , (Ω) , (5.14)and so we easily check that the operator K is well defined. Finally, let LLL : u ∈ W , (Ω) ∩ H ( R ) LLL ( u ) = − λ c τ c · ∇ u + AAA ( u ) + P K ( u ) ∈ H ( R ) (5.15)We are now ready to show the main result of this section. Theorem 5.1
The operator
LLL + i ζ is Fredholm of index 0, for all ζ = 0 . Moreover, let σ ( LLL ) bethe spectrum of LLL . Then, σ ( LLL ) ∩ { i R \{ }} consists, at most, of a finite or countable number ofeigenvalues, each of which is isolated and of finite (algebraic) multiplicity, that can only accumulateat 0. Proof.
We show that the operator K defined in (5.12), (5.13) is compact. Let { u k } be a sequencebounded in Z , . This implies, in particular, that there is M > k such that ( χ k := d ( u k ) ) k u k k , + | χ k | ≤ M . (5.16)The latter, along with the compact embedding W , (Ω) ⊂ W , (Ω R ), for all R > u ∗ , χ ∗ ) ∈ W , (Ω) × R and subsequences, again denoted by { u k , χ k ) such that u k → u ∗ strongly in W , (Ω R ), for all R > χ k → χ ∗ in R . (5.17)Without loss, we assume u ∗ ≡ χ ∗ ≡ . By H¨older inequality and (5.12), we deduce k K ( u k ) k ≤ λ c (cid:2) ( k v c k ∞ + k∇ v c k ∞ )( k u k k , , Ω R + | χ k | ) + k v c k , , Ω R k∇ u k k , (cid:3) Therefore, passing to the limit k → ∞ in the previous inequality, and using (5.14), (5.17), (5.16) andthe embedding W , ⊂ W , , we inferlim k →∞ k K ( u k ) k ≤ M k v c k , , Ω R , which, in turn, again by (5.14) and the arbitrariness of R > c LLL ζ := LLL + i ζ is Fredholm of index 0, for all ζ = 0. The theorem is then a consequence of well-known results (e.g.[13, Theorem XVII.4.3]) provided we show that the null space of c LLL ζ is trivial, for all sufficiently large | ζ | . To this end, we observe that c LLL ζ ( u ) = 0 means LLL ( u ) + i ζ u = − K ( u ). Applying (5.5), we thusget, in particular, the following inequality valid for all sufficiently large | ζ || ζ | k∇ u k + | ζ | ( k u k + | χ | ) ≤ c k K ( u ) k , where c is independent of ζ . Also, from (5.12), (5.13), (5.14) and H¨older inequality, we have k K ( u ) k ≤ λ c k v c k , ∞ ( k u k , + | χ | ) , and so, from the last two displayed equations we deduce u ≡ , provided we choose | ζ | larger than asuitable positive constant depending only on Ω , λ c , M , and I . This completes the proof of the theorem. (cid:3) On the Linearized Time-Periodic Operator
Let L ♯ := { F ∈ L ( L ) , F is 2 π -periodic with F | Ω = b F = b F = } , H ♯ := { fff ∈ L ♯ : fff ∈ L ( H ) } W ♯ := { w ∈ W ♯ : u ∈ W , ( H ) } , and, for ζ = 0, consider the operators Q : w ∈ W ♯ ζ ∂ s w + LLL ( w ) ∈ H ♯ , and Q : w ∈ W ♯ ζ ∂ s w + LLL ( w ) ∈ H ♯ , (6.1)where L and L are given in (5.2) and (5.15), respectively.Our objective in this section is to establish some important properties for both operators. Webegin to show the following lemma. Lemma 6.1
Let τ ∈ R . Then, the boundary-value problems, i ∈ { , , } , k ∈ Z \{ } , i k h ( i ) k − τ ∂ h ( i ) k = ∆ h ( i ) k − ∇ p ( i ) k div h ( i ) k = 0 in Ω h ( i ) k = e i at ∂ Ω , h ( i )0 = (6.2) and i k H ( i ) k − τ ∂ H ( i ) k = ∆ H ( i ) k − ∇ P ( i ) k div H ( i ) k = 0 in Ω H ( i ) k = e i × x at ∂ Ω , H ( i )0 = (6.3) have unique solutions ( h ( i ) k , p ( i ) k ) , ( H ( i ) k , P ( i ) k ) ∈ W , (Ω) × D , (Ω) . These solutions satisfy the estimates k h ( i ) k k + k H ( i ) k k ≤ C k∇ h ( i ) k k + k∇ H ( i ) k k ≤ C ( | k | + 1) k D h ( i ) k k + k D H ( i ) k k ≤ C ( | k | + 1) , (6.4) where C is a constant independent of k . Moreover, for fixed k , consider the × matrices K , A , P and S defined by the components ( j, i = 1 , , ): ( K ) ji = Z Σ ( T ( h ( i ) k , p ( i ) k ) · n ) j , ( A ) ji = Z Σ ( x × T ( H ( i ) k , P ( i ) k ) · n ) j ( P ) ji = Z Σ ( x × T ( h ( i ) k , p ( i ) k ) · n ) j ( S ) ji = Z Σ ( T ( H ( i ) k , P ( i ) k ) · n ) j (6.5)18 nd define the × matrix AAA as follows
AAA := K PS A ! . Then, for any µ ∈ R , both K + i µ and A + i µ are invertible. Moreover, for every ζ ∈ C , we have i k k v k + 2 k D ( v ) k − τ ( ∂ v , v ∗ ) = ζ ∗ · AAA · ζ (6.6) where v := ζ i h ( i ) k + ζ i +3 H ( i ) k . Finally, for every ( λ, µ ) ∈ R × R , the matrix AAA + i λ I
00 i µ I ! := AAA + JJJ is invertible.
Proof.
We begin to show the estimate for h ( i ) k . Since the proof is the same for i = 1 , ,
3, we chose i = 1 and, for simplicity, omit the superscript. Let φ = φ ( | x | ) be a (smooth) cut–off function suchthat φ ( | x | ) = ( R R and set Φ ( x ) = curl ( x φ ( | x | ) e ). Clearly, div Φ = 0 and Φ ( x ) = e in a neighborhood of ∂ Ω.Moreover Φ ( x ) ≡ in Ω R . Setting v k := h k − Φ , from (6.2) we deduce that v k solves the followingboundary-value problem, for all | k | ≥ k v k − τ ∂ v k = ∆ v k − ∇ p ( i ) k + τ ∂ Φ − i k Φ + ∆ Φ div v k = 0 ) in Ω v k = at ∂ Ω . (6.7)Existence to (6.7) in the stated function class can be easily obtained by the Galerkin method combinedwith the estimate that we are about to derive. Let us dot-multiply both sides of (6.7) by v ∗ wherethe star denotes c.c. After integrating by parts as necessary, we geti k k v k k − τ ( ∂ v , v ∗ k ) + k∇ v k k = ( F k , v ∗ k ) , (6.8)where F k := τ ∂ Φ − i k Φ + ∆ Φ . We next observe that, by the properties of Φ , k F k k ≤ c ( | k | + 1) (6.9)where, here and in the rest of the proof, c denotes a generic (positive) constant independent of k .Also, by means of an integration by parts, we show ℜ ( ∂ v k , v ∗ k ) = 0 . (6.10)Thus, by taking the real part of (6.8) and using (6.9) and (6.10) we infer k∇ v k k ≤ c ( | k | + 1) k v k k . (6.11)19ikewise, taking the imaginary part of (6.8) and employing (6.9)–(6.11) along with Schwarz inequality,we obtain | k |k v k ≤ c ( k∇ v k k + | k | + 1) ≤ c ( | k | + 1) k v k k , which implies k v k ≤ c . (6.12)Taking into account that h k = Φ + v k , (6.12) proves (6.4) for h k . Similarly, replacing (6.12) into(6.11), we arrive at (6.4) . Finally, from classical estimates on the Stokes problem [15, Lemma 1] wefind k D v k k ≤ c ( k ∂ v k + k F k k + k∇ v k )and so (6.4) follows from this inequality, (6.9), (6.11) and (6.12). Concerning the fields H ( i ) k , let Ψ ( i ) = φ ( | x | ) e i × x , and set V ( i ) k := H ( i ) + Ψ ( i ) . Obviously, the support of Ψ ( i ) is contained in Ω R ,div Ψ ( i ) = 0 and Ψ ( i ) | ∂ Ω = e i × x . Thus, from (6.3) it follows that V ( i ) k is a solution to (6.7) with( V ( i ) k , Ψ ( i ) ) in place of ( v k , Φ ). Therefore, we can use exactly the same arguments used earlier in theproof to show that also H ( i ) k satisfies the stated properties. Let α ∈ C , and, for fixed k = 0, set (6) u := α i h ( i ) k , q := α i p ( i ) k . From (6.2) we then find i k u − τ ∂ u = div T ( u , q )div u = 0 ) in Ω u = α at ∂ Ω . (6.13)Dot-multiplying both sides of (6.13) by u ∗ and integrating by parts over Ω we deducei k k u k + k D ( u ) k − τ ( ∂ u , u ∗ ) = α ∗ · K · α . Now, suppose that there is b α ∈ C such that K · b α = − i µ b α , for some µ ∈ R . Then from the previousrelation we obtain i (cid:0) k k u k + µ | b α | (cid:1) − τ ( ∂ u , u ∗ ) = k D ( u ) k , which, in turn, taking into account that ℜ ( ∂ u , u ∗ ) = 0, allows us to we deduce u = in W , (Ω).The latter implies b α = and thus shows the desired property for K . In a similar manner, we provethe same property for S . Next, let ζ ∈ C and define (7) v := ζ i h ( i ) k + ζ i +3 H ( i ) k , p := ζ i p ( i ) k + ζ i +3 P ( i ) k . (6.14)Employing (6.2) and (6.3) we then deducei k v − τ ∂ v = div T ( v , p )div v = 0 ) in Ω v = ζ i e i + ( ζ i e i ) × x at ∂ Ω . (6.15) (6) Summation over repeated indices. (7)
Summation over repeated indices.
20y dot-multiplying both sides of (6.15) by v ∗ and integrating by parts over Ω, we showi k k v k + 2 k D ( v ) k − τ ( ∂ v , v ∗ ) = Z ∂ Ω v ∗ · T ( v , p ) · n . As a consequence, (6.6) follows by replacing (6.14) in this last inequality and using (6.5) and (6.15) .Finally, let ζ ∈ C such that ( AAA + JJJ ) · ζ = . From (6.6) it then follows thati[ k k v k + X i =1 ( λ | ζ i | + µ | ζ i +3 | )] + 2 k D ( v ) k − τ ( ∂ v , v ∗ ) = 0 , from which, recalling that ℜ ( ∂ v , v ∗ ) = 0, and that v is a solution to (6.15) in the space W , (Ω), weat once obtain v ≡ , implying ζ = . The proof is completed. (cid:3) Remark 6.1
Even though the results of the previous lemma are stated for Ω the exterior of a ball,the reader will check with no effort that they continue to hold –without changes in their proof– forany exterior domain of class C . Therefore, they generalize those obtained in [11, Lemma 5.1]With the help of Lemma 6.1, we are now able to show the following one. Lemma 6.2
Let τ ∈ R . Then, for any ( f , F , G ) ∈ L ♯ × L ♯ × L ♯ , the problem ∂ s w − τ · ∇ w = ∆ w − ∇ φ + f div w = 0 ) in Ω × [0 , π ] , w = χ + σ × x at ∂ Ω × [0 , π ] ,M ˙ χ + Z ∂ Ω T ( w , φ ) · n = F , I ˙ σ + Z ∂ Ω x × T ( w , φ ) · n = G in [0 , π ] , (6.16) has one and only one solution ( w , φ, χ , σ ) ∈ W ♯ × P , × W , ♯ × W , ♯ . This solution satisfies theestimate k w k W ♯ + k φ k P , + k χ k W , + k σ k W , ≤ C (cid:16) k f k L ♯ + k F k L + k G k L (cid:17) , (6.17) where C = C (Ω , τ, ρ S /ρ L ) . Proof.
Since the actual values of M and I are irrelevant to the proof, we put, for simplicity, M = I = 1.Moreover, without loss of generality, we may take τ = τ e . Let w = z + u where z and u satisfy thefollowing set of equations ∂ s z − τ ∂ z − ∆ z = −∇ r + f div z = 0 ) in Ω × [0 , π ] z | ∂ Ω = (6.18)21nd ∂ s u − τ ∂ u − ∆ u = −∇ q div u = 0 ) in Ω × [0 , π ] u | ∂ Ω = χ + σ × x ;˙ χ + Z ∂ Ω T ( u , q ) · n = F − Z ∂ Ω T ( z , r ) · n := F , in [0 , π ] , ˙ σ + Z ∂ Ω x × T ( u , q ) · n = G − Z ∂ Ω x × T ( z , r ) · n := G , in [0 , π ] . (6.19)From [12, Theorem 12], it follows that there exists a unique solution ( z , τ ) ∈ W ♯ ×P that, in addition,obeys the inequality k z k W ♯ + k r k P , ≤ c k f k L ♯ . (6.20)Furthermore, by trace theorem (8) and (6.20) we get k Z ∂ Ω T ( z , r ) · n k L + k Z ∂ Ω x × T ( z , τ ) · n k L ≤ c (cid:16) k z k W ♯ + k r k P , (cid:17) ≤ c k f k L ♯ , so that both functions F and G in (6.19) are in L ♯ (0 , π ) and satisfy k F k L + k G k L ≤ c ( k f k L ♯ + k F k L + k G k L ) . (6.21)To find solutions to (6.19), we formally expand u , q , χ and σ in Fourier series (summation overrepeated indices): u ( x, s ) = u k ( x ) e i k s , q ( x, s ) = q k ( x ) e i k s , χ ( s ) = χ k e i k s , σ ( s ) = σ k e i k s , k ∈ Z \{ } , u ≡ ∇ q ≡ χ ≡ σ ≡ , (6.22)where ( u k , q k , χ k , σ k ) solve the problem ( k = 0)i k u k − τ ∂ u k = ∆ u k − ∇ q k div u k = 0 ) in Ω u k | ∂ Ω = χ k + σ k × x , (6.23)subject to the further conditionsi k χ k + Z ∂ Ω T ( u k , q k ) · n = F k , i k σ k + Z ∂ Ω x × T ( u k , q k ) · n = G k , (6.24)with { F k } , { G k } are Fourier coefficients of F and G , respectively, and F ≡ G ≡ . For each fixed k ∈ Z \{ } , a solution to (6.23)–(6.24) is given by u k = X i =1 ( χ ki h ( i ) k + σ ki H ( i ) k ) , q k = X i =1 ( χ ki p ( i ) k + σ ki P ( i ) k ) , (6.25) (8) Possibly, by modifying r by adding to it a suitable function of time. h ( i ) k , p ( i ) k ) , ( H ( i ) k , P ( i ) k ) given in Lemma 6.1, and where χ k , σ k solve the equationsi k χ k + X i =1 Z ∂ Ω h χ ki T ( h ( i ) k , p ( i ) k ) + σ ki T ( H ( i ) k , P ( i ) k ) i · n = F k , i k σ k + X i =1 Z ∂ Ω h χ ki x × T ( h ( i ) k , p ( i ) k ) + σ ki x × T ( H ( i ) k , P ( i ) k ) i · n = G k . (6.26)Set ξ k := ( χ k , σ k ) ∈ C , FFF k := ( F k , G k ) ∈ C . Then, with the notation of Lemma 6.1, (6.26) can be equivalently rewritten as
BBB · ξ k = FFF k , (6.27)where BBB := i k III + AAA , and
III is the 6 × BBB is invertible forall k ∈ Z . Furthermore, from (6.6), for all ζ ∈ C we get ζ ∗ · BBB · ζ = i k (cid:0) | ζ | + k v k (cid:1) − τ ( ∂ v , v ∗ ) + 2 k D ( v ) k . (6.28)As a result, for any given FFF k , (6.27) has one and only one solution ξ k . We next dot-multiply bothsides of (6.27) by ξ ∗ k and use (6.28) to deducei k (cid:0) | ξ k | + k u k k (cid:1) − τ ( ∂ u k , u ∗ k ) + 2 k D ( u k ) k = ( FFF k , ξ ∗ k ) , which, by Cauchy–Schwarz inequality and (3.5) furnishes, in particular, the following estimates for all k ∈ Z \{ }k D ( u k ) k ≤ k FFF k k | ξ k || k | k u k k ≤ | τ | k∇ u k k + 2 k FFF k k | ξ k | ≤ | τ | k D ( u k ) k + 2 k FFF k k | ξ k || k | | ξ k | ≤ | τ |k∇ u k k k u k k + 2 | k | k FFF k k ≤ | τ |k D ( u k ) k k u k k + 2 | k | k FFF k k . (6.29)Replacing (6.29) into (6.29) , we obtain | k | k u k k ≤ c k FFF k k | ξ k | , (6.30)while using (6.29) and (6.30) into (6.29) along with Cauchy-Schwarz inequality implies | k | | ξ k | ≤ c k FFF k k . (6.31)Combining (6.29) , (6.30), (6.31) and (3.5), and recalling (6.4) and (6.25) we thus infer X | k |≥ (cid:2) ( | k | + 1) k u k k + k∇ u k k + k D u k k (cid:3) ≤ c X | k |≥ ( | k | + 1) | ξ k | ≤ c k FFF k L . (6.32)Therefore, we may conclude that the quadruple ( u , q , χ u ≡ χ , σ u ≡ σ ) defined in (6.22) with( u k , q k , χ k , σ k ) satisfying (6.23)–(6.24) is a solution to (6.19) in the class W ♯ × P , × W , ♯ × W , ♯ .Furthermore, (6.21) and (6.32) also entail the validity of the following inequality k u k W ♯ + k q k P , + k χ k W , + k σ k W , ≤ c (cid:16) k f k L ♯ + k F k L + k G k L (cid:17) . ∂ s w − τ ∂ w = ∆ w − ∇ p div w = 0 ) in Ω × [0 , π ] w | ∂ Ω = χ + σ ;˙ χ + Z ∂ Ω T ( w , p ) · n = , ˙ σ + Z ∂ Ω x × T ( w , p ) · n = (6.33)has only the zero solution in the specified function class. If we dot-multiply (6.33) by w , integrateby parts over Ω and use (6.33) , we get ddt ( k w ( t ) k + | χ ( t ) | + | σ ( t ) | ) + 2 k D ( w ( t )) k = 0 . Integrating both sides of this equation from 0 to 2 π and employing the 2 π -periodicity of the solutionwe easily obtain k D ( w ( t )) k ≡ V given in Lemma3.3, immediately furnishes w ≡ ∇ p ≡ . The proof of the lemma is completed. (cid:3) Remark 6.2
Concerning the generality of the domain Ω, an observation similar to that made inRemark 6.1 for Lemma 6.1, equally applies also to Lemma 6.2.Let F ∈ H ♯ where F = ( f in Ω F + G × x in Ω , and consider the operator equation Q ( w ) = F . (6.34)By Lemma 3.2, (6.34) is equivalent to the following problem (with τ := λ c τ c , χ := b w , σ := b w ) ζ ∂ s w − λ c τ c · ∇ w − ∆ w = ∇ φ + f div w = 0 ) in Ω × [0 , π ] , w = χ + σ × x at ∂ Ω × [0 , π ] ,M ˙ χ + Z ∂ Ω T ( w , φ ) · n − F + (cid:18) I ˙ σ + Z ∂ Ω y × T ( w , φ ) · n − G (cid:19) × x = in Ω × [0 , π ] . (6.35)Since x is arbitrary in Ω , we conclude that (6.35) , are equivalent to (6.16) , . Thus, in view ofLemma 6.2, we deduce the following important result Lemma 6.3
The operator Q is a homeomorphism. This lemma allows us to prove the following theorem that represents the main result of this section.
Theorem 6.1
The operator Q is Fredholm of index 0. roof. We commence to notice that Q = Q + P K . Thus, by Lemma 6.2, the stated property willfollow, provided we show that the map C : w ∈ W ♯ K ( w ) ∈ L ♯ is compact. Let { w k } be a bounded sequence in W ♯ . This implies, in particular, that there is M > k such that ( χ k := d ( w k ) ) k w k k W ♯ + k χ k k W , ≤ M . (6.36)We may then select sequences (again denoted by { w k , χ k } ) and find ( w ∗ , χ ∗ ) ∈ W ♯ × W , ♯ such that w k → w ∗ weakly in W ♯ ; χ k → χ ∗ strongly in L ∞ (0 , π ). (6.37)Without loss of generality, we may take w ∗ ≡ χ ∗ ≡ . We then have to show thatlim k →∞ Z T k K ( w k ) k , Ω = 0 . (6.38)From (6.37), the compact embeddings W , (Ω) ⊂ W , (Ω R ) ⊂ L (Ω R ) for all R >
1, and Lions-Aubinlemma we then have Z π (cid:0) k w k ( t ) k , Ω R + k∇ w k ( t ) k , Ω R (cid:1) d t → k → ∞ , for all R > Z π k v c · ∇ w k ( t ) k , Ω ≤ k v c k ∞ Z π k∇ w k ( t ) k , Ω R + k v c k , Ω R Z π − π k∇ w k ( t ) k , Ω , which, by (5.14) , (6.36), (6.39) and the arbitrariness of R furnisheslim k →∞ Z π k v c · ∇ w k ( τ ) k = 0 . (6.40)Likewise, Z π k w k ( t ) · ∇ v c k , Ω ≤ k∇ v c k ∞ Z π k w k ( t ) k , Ω R + k∇ v c k , Ω R Z π k w k ( t ) k , Ω , so that, by (5.14) , (6.37) , and (6.39) we deduce, as before,lim k →∞ Z π k w k ( t ) · ∇ v c k , Ω = 0 . (6.41)Finally, Z π k χ k · ∇ v c k , Ω ≤ π k χ k k L ∞ (0 , π ) k∇ v c k , which, by (5.14) and (6.37) furnisheslim k →∞ Z π k χ k · ∇ v c k , Ω = 0 (6.42)Combining (6.40)–(6.42) we thus arrive at (6.38), which completes the proof of the theorem. (cid:3) Sufficient Conditions for Time-Periodic Bifurcation
The first objective of this section is to rewrite (1.4) in an operator form of the type (2.5) and then,successively, employ Theorem 2.1 to provide sufficient conditions for the occurence of time-periodicbifurcation for our problem. Thus, let u ( x, s ) = u ( x ) + w ( x, s ) , p = p ( x ) + φ ( x, s ) , γ = γ + χ ( s ) , ω ( s ) = ω + σ ( s ) . Then, (1.4) can be equivalently written in terms of the two sets of unknowns ( u , p , γ , ω ) and ( w , φ, χ , σ )as follows − λ τ · ∇ u + λ ( v · ∇ u + ( u − γ ) · ∇ v ) + λ ( u − γ ) · ∇ u + λ ( w − χ ) · ∇ w = ∆ u − ∇ p div u = 0 in Ω , u = γ + ω × x at ∂ Ω , lim | x | →∞ u ( x ) = , Z ∂ Ω T ( u , p ) · n = , Z ∂ Ω x × T ( u , p ) · n = , (7.1)and ζ ∂ s w − λ τ · ∇ w + λ ( v · ∇ w + ( w − χ ) · ∇ v ) + λ [( w − χ ) · ∇ u +( u − γ ) · ∇ w + ( w − χ ) · ∇ w − ( w − χ ) · ∇ w ] = ∆ w − ∇ φ div w = 0 in Ω × [0 , π ] , w = χ + σ × x at ∂ Ω × [0 , π ] , lim | x | →∞ w ( x, s ) = , s ∈ [0 , π ] ,ζ M ˙ χ + Z ∂ Ω T ( w , φ ) · n = , ζ I ˙ σ + Z ∂ Ω x × T ( w , φ ) · n = in [0 , π ] . (7.2)where, for simplicity, we have suppressed the subscript . Let v c be the weak solution to (1.3) at λ = λ c , and let τ c be the associated translational velocity. We make the assumption that both τ and γ are directed along the direction e c := τ c / | τ c | , and write τ = τ e c , γ = γ e c . As a result, byTheorem 4.2 we know that, under the hypothesis (4.18), at λ = λ c there exists an analytic family ofweak solutions v = v ( λ ) such that v ( λ c ) = v c . Thus, setting µ := λ − λ c , (7.1) can be rewritten asfollows − ∆ u − λ c τ c e c · ∇ u + λ c ( v c · ∇ u + ( u − γ e c ) · ∇ v c ) = N ( u , w , µ ) − ∇ p div u = 0 ) in Ω , u = γ e c + ω × x at ∂ Ω , lim | x | →∞ u ( x ) = , Z ∂ Ω T ( u , p ) · n = , Z ∂ Ω x × T ( u , p ) · n = , (7.3)where N ( u , w , µ ) := − [( e µ e τ ( µ ) − λ c τ c ) e c + e µ e v ( µ ) − λ c v c ] · ∇ u − λ c ( u − γ e c ) · ∇ ( e v ( µ ) − v c ) − µ ( u − γ e c ) · ∇ e v ( µ ) − e µ [ ( u − γ e c ) · ∇ u + ( w − χ ) · ∇ w ] , (7.4)26nd e f ( µ ) := f ( µ + λ c ). Likewise, (7.2) can be rewritten as follows ζ ∂ s w − λ c τ c e c · ∇ w + λ c ( v c · ∇ w + ( w − χ ) · ∇ v c ) − ∆ w = N ( u , w , µ ) − ∇ φ div w = 0 in Ω × [0 , π ] , w = χ + σ × x at ∂ Ω × [0 , π ] , lim | x | →∞ w ( x, s ) = , s ∈ [0 , π ] ,ζ M ˙ χ + Z ∂ Ω T ( w , φ ) · n = , ζ I ˙ σ + Z ∂ Ω x × T ( w , φ ) · n = in [0 , π ] , (7.5)where N ( u , w , µ ) := − [( e µ e τ ( µ ) − λ c τ c ) e c + e µ e v ( µ ) − λ c v c ] · ∇ w − λ c ( w − χ ) · ∇ ( e v ( µ ) − v c ) − µ ( w − χ ) · ∇ e v ( µ ) − e µ [( w − χ ) · ∇ u + ( u − γ ) · ∇ w + ( w − χ ) · ∇ w − ( w − χ ) · ∇ w ] , (7.6)Let f N = ( N in Ω in Ω . The following result holds.
Lemma 7.1
The operators
NNN : ( u , w , µ ) ∈ X c ( R ) × W ♯ × R N ( u , w , µ ) ∈ V − ( R ) f NNN : ( u , w , µ ) ∈ X c ( R ) × W ♯ × R f N ( u , w , µ ) ∈ L ♯ (7.7) are well defined. Proof.
Since u , v , v c ∈ X c , it follows at once that e c ·∇ u , e c ·∇ v , e c ·∇ v c ∈ V − . Moreover, by Lemma3.4 we also have u , v , v c ∈ L ( R ), which, by using integration by parts, implies v ·∇ u , v c ·∇ u , u ·∇ v , u ·∇ v c , u · ∇ u ∈ V − ( R ) as well. Finally, observing that w ∈ L ( L ∩ L ) and χ ∈ L ∞ (0 , π ) we easilyshow that ( w − χ ) · ∇ w ∈ V − , which concludes the proof of (7.7) . By known embedding theorems[24, Theorem 2.1] it follows that w ∈ L ( D , ). Thus, the validity of (7.7) can be established alongthe same lines used to show (7.7) . We will omit the details. The proof of the lemma is completed. (cid:3) In view of Lemma 7.1, and (4.16), (4.17), and (6.1), it follows at once that, setting
NNN = P f NNN ,the coupled problem (7.3)–(7.6) can be written as operator equations: LLL ( u ) = NNN ( u , w , µ ) in V − ( R ) ,ζ ∂ s w + LLL ( w ) = NNN ( u , w , µ ) in L ♯ (7.8)which are in the form (2.5). We shall next check how the assumptions of Theorem 2.1 can be satisfiedin our case. We begin to notice that Theorem 6.1 secures (H3). In addition, both hypotheses (H1)and (H4) are verified if we assume N [ LLL ] = { } . ( H LLL is Fredholm of index 0, so that ( H
1) implies (H1). Moreover, if ( H NNN i , i = 1 ,
2, are (at most) quadratic in( u , w )– again by Theorem 4.2, we deduce the validity of (H4). Next, we assume ν := i ζ , ζ = 0, is a simple eigenvalue of LLL , and k ν ∈ P ( LLL ), for all k ∈ N \{ , } . ( H S is given by( τ c e c + v c ) · ∇ w + λ c ( τ ′ ( λ c ) e c + v ′ ( λ c )) · ∇ w + ( w − χ ) · ∇ ( v c + v ′ ( λ c )) , where the prime means differentiation with respect to µ . So, denoting by ν = ν ( µ ) the eigenvalues of LLL + µ S , by [27, Proposition 79.15 and Corollary 79.16]) we have that in a neighborhood of µ = 0the map µ ν ( µ ) is well defined and of class C ∞ . This justifies our last assumption: ℜ [ ν ′ (0)] = 0 . ( H Theorem 7.1
Suppose ( H – ( H hold. Let w be the normalized eigenvector of LLL correspondingto the eigenvalue ν , and set w := ℜ [ w e − i s ] . Then, the following properties are valid. (a) Existence.
There are analytic families ( u ( ε ) , w ( ε ) , ζ ( ε ) , µ ( ε )) ∈ X c × W ♯ × R + × R (7.9) satisfying (7.8) , for all real ε in a neighborhood I (0) of 0, and such that ( u ( ε ) , w ( ε ) − ε w , ζ ( ε ) , µ ( ε )) → (0 , , ζ , as ε → . (7.10)(a) Uniqueness. There is a neighborhood U (0 , , ζ , ⊂ X c × W ♯ × R + × R such that every (nontrivial) π -periodic solution to (7.8) , lying in U must coincide, up to a phase shift,with a member of the family (7.9) . (a) Parity. The functions ζ ( ε ) and µ ( ε ) are even: ζ ( ε ) = ζ ( − ε ) , µ ( ε ) = µ ( − ε ) , for all ε ∈ I (0) .Consequently, the bifurcation due to these solutions is either subcritical or supercritical, a two-sidedbifurcation being excluded. (9) As we mentioned in the introductory section, experimental and numerical tests show that, in thetransition from steady to time-periodic motion, the trajectory of the center of mass, G , of the spherechanges from a rectilinear to a zigzag path. Objective of this section is to study in more details the (9) Unless µ ≡ λ c , namely,for sufficiently small ε , the oscillatory part of the solution, w , behaves like the corresponding solutionto the linear problem Q ( w ) = , that is, w := ℜ [ w e − i s ], with w eigenvector of LLL correspondingto the eigenvalue − i ζ . Therefore, in such a neighborhood, the oscillatory component of the velocity of G will have the same kinematic characteristics of the translational velocity, χ , and angular velocity, σ associated to w . We now recall that the equation LLL ( w ) + i ζ w = is equivalent to thefollowing set of equations − ∆ w − λ c τ c · ∇ w + λ c L ( w ) + ∇ p + i ζ w = div w = 0 ) in Ω , w = χ + σ × x at ∂ Ω , i ζ M χ + Z ∂ Ω T ( w , p ) · n = , i ζ I σ + Z ∂ Ω x × T ( w , p ) · n = , (8.1)where L ( w ) := v c · ∇ w + ( w − χ ) · ∇ v c . (8.2)Thus, assuming the gravity directed along e , an oscillatory motion of G in the neighborhood of λ = λ c will take place if and only if ( χ ) e + ( χ ) e = . In the remaining part of this section we shallfurnish a characterization of the expression of χ and σ that, in particular, will provide the desiredproperty.Let us introduce the pairs ( h ( i ) , p ( i ) ) and ( H ( i ) , P ( i ) ) in W , (Ω) × D , (Ω), solutions to the followingproblems ( i = 1 , , − i ζ h ( i ) + λ c τ c · ∇ h ( i ) = div T ( h ( i ) , p ( i ) )div h ( i ) = 0 ) in Ω h ( i ) = e i at ∂ Ω , (8.3)and − i ζ H ( i ) + λ c τ c · ∇ H ( i ) = div T ( H ( i ) , P ( i ) )div H ( i ) = 0 ) in Ω H ( i ) = e i × x at ∂ Ω . (8.4)Moreover, consider the matrices b K , b A , b P , and b S , defined by ( i, j = 1 , , b K ) ij = Z Σ ( T ( h ( i ) ∗ , p ( i ) ∗ ) · n ) j , ( b A ) ij = Z Σ ( x × T ( H ( i ) ∗ , P ( i ) ∗ ) · n ) j ( b P ) ij = Z Σ ( x × T ( h ( i ) ∗ , p ( i ) ∗ ) · n ) j ( b S ) ij = Z Σ ( T ( H ( i ) ∗ , P ( i ) ∗ ) · n ) j , (8.5)29here, we recall, ∗ means complex conjugate. The existence of the above pairs in the specified functionclass is guaranteed by Lemma 6.1. Furthermore, again from this lemma, we know that the matrices b K + i λ and b A + i µ are invertible, for all λ, µ ∈ R , as well as the block matrix b AAA = b K + i λ b P b S b A + i µ ! . If we dot-multiply both sides of (8.1) by h ( i ) ∗ , integrate by parts over Ω, and employing (8.1) , , , wegeti ζ ( w , h ( i ) ∗ ) − λ c ( τ c · ∇ w , h ( i ) ∗ ) + 2[ D ( w ) , D ( h ( i ) ∗ )] = − i M ζ χ · e i − λ c ( L ( w ) , h ( i ) ∗ ) . (8.6)Similarly, taking first the complex conjugate of (8.3) , then dot-multiplying it by w , integrating byparts over Ω , and using (8.5), we deducei ζ ( w , h ( i ) ∗ ) − λ c ( τ c · ∇ w , h ( i ) ∗ ) + 2[ D ( w ) , D ( h ( i ) ∗ )] = [ b K · χ + b P · σ ] i . (8.7)From (8.6) and (8.7) we conclude (summation over repeated indeces) e K · χ + b P · σ = − λ c ( L ( w ) , h ( i ) ∗ ) e i := FFF , (8.8)where e K := i M ζ + b K . Likewise, we can show that e A · σ + b S · χ = − λ c ( L ( w ) , H ( i ) ∗ ) e i := GGG , (8.9)with e A := i I ζ + b A . From (8.8) and (8.9) we infer χ = H · ( FFF + e K · e A − · GGG ) , σ = M · ( GGG + e A · e K − · FFF ) , (8.10)where H := ( e K − b P · e A − · b S ) − , M := ( e A − b S · e K − · b P ) − . Notice that both H and M exist, because b AAA , e K , and e A are invertible.From (8.10) we can then derive the following result. Theorem 8.1
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