Prescribing the motion of a set of particles in a 3D perfect fluid

PRESCRIB ING THE MOTION OF A SET OF P AR TICLES IN A 3 D PER FECT FLUID O. GLASS AND T. HORSIN Abstract. W e establish a res ult concerning the s o-called Lag rangian con trollability of the Euler equation for incompressible perfect fluids in dimension 3. More pr ecisely w e consider a connected b ounded do main of R 3 and tw o smo o th co nt ractible sets of fluid particles, surro unding the same v olume. W e prov e that given any initial velocity field, one can find a boundar y control and a time interv al such that the co rresp onding solution of the Euler equation makes the first of the tw o sets approximately reach the second one. 1. Intro duction 1.1. Presen tation of the problem. In this pap er, w e are concerned with the La- grangian con t r o llabilit y o f the three dimensional Euler equation f or p erfect incompressible fluids b y means of a b oundary con tro l. The problem under view is the following. Let Ω be a smo oth b ounded domain of R 3 and let Γ be a nonempt y o p en part of it s b oundary ∂ Ω. Giv en T > 0, w e consider the classical Euler system for p erfect incom- pressible fluids in Ω: ∂ t u + ( u · ∇ ) u + ∇ p = 0 in (0 , T ) × Ω , (1) div u = 0 in (0 , T ) × Ω , (2) u | t =0 = u 0 in Ω . (3) As b oundary conditions, one usually considers the imp ermeabilit y condition on ∂ Ω: u · n = 0 on (0 , T ) × ∂ Ω , where n stands f o r the unit out w ard normal v ector field on ∂ Ω. How ev er, in the problem under view, the impermeability conditio n is merely imp osed on ∂ Ω \ Γ: (4) u · n = 0 on (0 , T ) × ( ∂ Ω \ Γ) . Consequen tly , since we do not prescribe b oundary v a lues for u on (0 , T ) × Γ , the problem stated as ab o v e is underdetermined. It is in fact an implicit control problem; in other w ords, we consider the b oundary v alues on Γ a s a con trol on this system, i.e. a w a y to influence the fluid in a prescribed w a y . If one w ants to close the system, as sho wn by Khazhik ov [14], one may consider as boundar y conditions on Γ the following ones: u · n on (0 , T ) × Γ and curl u ∧ n on { ( t, x ) ∈ (0 , T ) × Γ suc h that u · n < 0 } , that is, the normal part of the velocity field on Γ and the tangential en tering v orticity . W e refer to [14] for more details on this no t io n of b oundary v alues. Due to the complexit y of this form ulation of b oundary v alues, and as in the literature concerning the controllabilit y of the Euler equation (see e.g. [3], [4], [8], [9]), we prefer not t o refer to them explicitely and to lo ok for the solution ( u, p ) itself. Once a solution o f (1)- ( 4) is giv en, its suitable trace o n (0 , T ) × Γ can b e though t as the con tro l. Date : 17 / 08/2 0 11. 2000 Mathematics Su bje ct Classific ation. Primar y . 1 The question t hat w e raise is t he p ossibilit y of prescribing the motion of a set of part icles driv en b y a fluid go v erned b y (1) -(4). It is classical in fluid mec hanics that one can describe the motion of a fluid from t w o differen t p oin ts of view. One p ossibilit y , referred as the Eulerian description of the fluid, is to describ e the v elo cit y of fluid particles that pass “through” a give n p oin t. The other p ossibility , referred as the Lagrangian description of the fluid consists in follo wing fluid pa rticles a lo ng the flo w. T o that purp o se one has to solve the differen tial equation associated to the v elo cit y field. The notion of con trollability tha t w e consider in this pap er is differen t fro m the usual notion of controllabilit y a nd, in some sens e, closer to the Lagra ngian description of the fluid than the usual notion of con trollability . The standard sense of controllabilit y w ould refer to the p ossibilit y of driving the v elo cit y field (the state of the system), from one prescribed v alue to a fixed target. The study of the con t rollabilit y of the Euler equation has b een initiated by J.-M. Coron in [3] a nd [4], a nd then studied b y the first autho r in [8] and [9]. Also related to the b oundary con trol of the v elo cit y field, the question of asymptotic stabilization of this system around zero w a s studied b y J.- M. Coron in 2-D simply connected domains (see [5]) a nd by the first author fo r what concerns more general 2-D domains (see [10]). In this paper, we consider the problem of prescribing the displacemen t of a se t of particles, rather than the v elo city field in final time. In [11], the authors show ed that in 2-D, one can indeed prescribe approx imat ely the motion o f some sp ecific sets of fluids. The goal of the presen t pap er is to consider the case of the dimension 3. 1.2. Notations and definitions. In this section, w e fix the nota tions and giv e the basic definitions. In the sequel, smo oth curve s, surfaces or maps will mean C ∞ ones. W e will denote b y C ω the class of real analytic curve s/surfaces. The volume of a b o relian set A in R 3 will b e denoted b y | A | . Giv en a suitably regular v ector field u , w e will denote b y φ u the flow of u , defined (when p ossible) b y (5) ∂ t φ u ( t, s, x ) = u ( t, φ u ( t, s, x )) and u ( s, s, x ) = x. F or A a subs et of R 3 and η > 0, w e will denote: V η ( A ) := { x ∈ R 3 / d ( x, A ) < η } . W e will also use the notation L p (0 , T ; X (Ω( t ))) or C k ([0 , T ]; X (Ω( t ))) of time-dep enden t functions with v alues in a space X of functions on a domain dep ending on time Ω( t ) ⊂ R 3 . This r efers to the space functions that can b e extended to L p (0 , T ; X ( R 3 )) or C k ([0 , T ]; X ( R 3 )). W e no w recall some definitions from differen tial geometry (see e.g. [13]). Definition 1. A Jordan surface γ in R 3 is the image of S 2 b y some homeomorphism h . According to the Jordan-Brou we r Theorem (see [2] or [12]), R 3 \ γ has t wo connected comp onen ts, one of whic h is b ounded and will b e denoted b y in t( γ ). Moreo ver when the homeomorphism h is a diffeomorphism, w e ha ve a unit out w a rd normal vec tor field on γ , whic h w e will denote b y ν . Definition 2. Tw o Jordan surfaces γ 0 and γ 1 em b edded in R 3 are said to b e isotopic in Ω, if there exists a con tinuous map I : [0 , 1] × S 2 → Ω suc h that I (0) = γ 0 , I (1) = γ 1 and for each t ∈ [0 , 1 ] , I ( t, · ) is an homeomorphism of S 2 in to its image. When, for some k ∈ N \ { 0 } , this homeomorphism is a C k -diffeomorphism with resp ect the space v ariable, 2 I will b e said to b e a C k -isotop y , or, when k = ∞ , a smo oth isotopy . A one-pa r a meter con tinuous family of diffeomorphisms of Ω is called a diffeotop y of Ω. When their regularit y is not sp ecified, the geometrical ob jects are considered to b e smo oth in the sequel. W e are now in p o sition to define the Lag rangian con trollability (the cor r esp onding tw o-dimensional notion was introduced in [11 ]) . Definition 3. L et T > 0 b e a g iv en time. W e will say that the exact lagrang ian con- trollabilit y of (1)- (4) holds if giv en t w o Jordan surfaces γ 0 and γ 1 included in Ω suc h that γ 0 and γ 1 are isotopic in Ω , (6) | in t ( γ 0 ) | = | in t ( γ 1 ) | , (7) and giv en a regular initial dat a u 0 satisfying div u 0 = 0 in Ω , (8) u 0 · n = 0 on ∂ Ω \ Γ , (9) there exists a solution ( u, p ) of (1)-(4) suc h that one has ∀ t ∈ [0 , T ] , φ u ( t, 0 , γ 0 ) ⊂ Ω , (10) φ u ( T , 0 , γ 0 ) = γ 1 . (11) up t o reparameterization. Let us say a few w o rds concerning the regularity of u . In general, stro ng solutions of the Euler equation are considered in a H¨ older or Sob olev space with respect to x , whic h is included in the space of Lipsc hitz functions. In the sequel, the solutions will b e taken in t he H¨ older space C k ,α (Ω), k ≥ 1. If the exact Lagrangian con tro lla bilit y do es not occur, one ma y try to w eaken this definition as follow s. Definition 4. W e will sa y that the approxim ate Lagrang ia n con trollabilit y of (1 )-(4) holds in time T and in norm C k if giv en γ 0 , γ 1 and u 0 as ab ov e, a nd giv en an y ε > 0 , there exists a solution ( u, p ) of (1)-(4) suc h that (10) holds and that (12) k φ u ( T , 0 , γ 0 ) − γ 1 k C k ( S 2 ) < ε, up t o reparameterization. Remark 1. It is easy to see t ha t the conditions that γ 0 and γ 1 should b e isotopic and enclose the same volume are neces sary in order that a smo oth v olume-preserving flow driv es γ 0 to γ 1 . In particular | int( γ 0 ) | = | in t ( γ 1 ) | comes from the incompressibilit y of the fluid. Remark 2. Condition (10) allo ws to make sure that one “con trols” the fluid zone for all time by making it sta y in the domain; Conditions (11) or (12) would not ha ve a clear meaning without it. 1.3. Main result. The main result of this pap er is the follo wing, establishing an ap- pro ximate Lagra ngian con tro lla bilit y prop erty . Theorem 1. L et α ∈ (0 , 1) and k ∈ N \ { 0 } . Consider u 0 ∈ C k ,α (Ω; R 3 ) satisfying (8) - ( 9 ) and γ 0 and γ 1 two c ontr actible C ∞ emb e ddings of S 2 in Ω satisfying (6) - (7) . Then for any ε > 0 , ther e exist a time T > 0 and a solution ( u, p ) in L ∞ (0 , T ; C k ,α (Ω; R 4 )) to (1) - (4) on [0 , T ] such that (10) and ( 1 2) hold (up to r ep ar ameterization). 3 Remark 3. Let us chec k that, as in the tw o-dimensional case (see [1 1 ]), the exact la gra- gian controllabilit y do es not hold. As is classic al, in 3 -D the v orticity of the fluid (13) ω := curl u, satisfies the equation (14) ∂ t ω + ( u · ∇ ) ω = ( ω · ∇ ) u . Denoting w ( t, x ) = ω ( t, φ u ( t, 0 , x )), w e see that ∂ t w = ( w · ∇ ) u ( t, φ u ( t, 0 , x )) . Th us if initially curl u 0 = 0 in a neigh b orho o d of γ 0 then curl u = 0 in a neigh b orho o d of φ u ( t, 0 , γ 0 ) as long as (10) is true. No w, since u satisfies (2), for eac h time t , in suc h a neigh b orho o d of φ u ( t, 0 , γ 0 ) and a wa y fro m ∂ Ω, u ( t, · ) is lo cally the gradien t of a harmo nic function, and hence is real analytic . Therefore, if γ 0 is a real-ana lytic submanifold, so will be φ u ( t, 0 , γ 0 ) for eac h t ∈ [0 , T ]. It follows that t he exact Lagra ngian controllabilit y do es not hold in general, since o ne may tak e γ 1 smo oth but non analytic. Remark 4. As will b e clear fro m the pro of, the time T whose existence is gran ted b y Theorem 1 can b e made arbitrarily smal l . The result do es not require a time long enough in order for the controllabilit y prop ert y to tak e place, but a time small enough to mak e sure that no blow up phenomenon o ccurs. Let us no w briefly describ e the strategy to prov e Theorem 1. First, w e use a construc- tion due to A. B. Krygin [16] to obtain a solenoidal vec tor field X ∈ C ∞ 0 ((0 , 1) × Ω; R 3 ), suc h that ∀ t ∈ [0 , 1] , φ X ( t, 0 , γ 0 ) ⊂ Ω , φ X (1 , 0 , γ 0 ) = γ 1 . Then w e will prov e that there exist p oten tia l flo ws (that is, time-dep enden t g radien ts of harmonic functions in Ω), whic h approximate suitably the action of X on γ 0 . This g iv es us a solution ( ¯ u, ¯ p ) of (1)- ( 4) with u 0 = 0 and T = 1 suc h that k φ ¯ u (0 , t, γ 0 ) − φ X (0 , t, γ 0 ) k C k ( S 2 ) ≤ ε. Then w e will use a time-rescaling a r g umen t due to J.-M. Coron [4] (see also [8]), to get the result for non trivial u 0 and T small enough. The rest of the pap er is divided as follows . In Section 2, w e consider t he existence of the ab o v e men tionned v ector field X , and then of the p oten tial flo ws. In Section 3, w e pro ve Theorem 1 b y constructing the solution ( u, p ) taking the initial condition u 0 in to accoun t. Ac knowledgem en ts. The authors are par t ia lly support ed b y the Agence Natio nale de la Reche r che (ANR-09-BLAN-0213 -02). They thank Jean-Pierre Puel f or man y useful discussions leading to this pap er. 2. Potential flow s 2.1. A solenoidal vector field mapping γ 0 on to γ 1 . The v ector field X is obtained as a direct consequence of the following result due to A. B. Krygin. 4 Theorem 2 ([16]) . If γ 0 and γ 1 ar e as in The or em 1 , then ther e ex i s ts a volume-pr eserving diffe otopy h ∈ C ∞ ([0 , 1] × Ω; Ω) such that ∂ t h is c omp actly supp orte d in (0 , 1) × Ω , h (0 , γ 0 ) = γ 0 and h (1 , γ 0 ) = γ 1 . A direct consequence is that the smo ot h v ector field (15) X ( t, x ) := ∂ t h ( t, h − 1 ( x )) , is compactly supp orted in (0 , 1) × Ω a nd satisfies φ X (1 , 0 , γ 0 ) = γ 1 , and div X = 0 in ( 0 , 1) × Ω . 2.2. Mo ving the fluid zone b y p oten t ial flo ws. In this subsection we pro v e the follo wing. Prop osition 1. L et γ 0 b e a C ∞ c ontr actible two-spher e e m b e dde d in Ω and c onsider X ∈ C 0 ([0 , 1]; C ∞ ( Ω; R 3 )) a solenoidal ve ctor field such that (16) ∀ t ∈ [0 , 1] , φ X ( t, 0 , γ 0 ) ⊂ Ω , and let γ 1 = φ X (1 , 0 , γ 0 ) . F or any ε > 0 and k ∈ N ther e exists θ ∈ C ∞ 0 ((0 , 1) × Ω; R ) such that ∀ t ∈ [0 , 1 ] , ∆ x θ = 0 i n Ω , (17) ∂ θ ∂ n = 0 on [0 , 1 ] × ( ∂ Ω \ Γ) , (18) ∀ t ∈ [0 , 1 ] , φ ∇ θ ( t, 0 , γ 0 ) ⊂ Ω , (19) k φ ∇ θ (1 , 0 , γ 0 ) − γ 1 k C k ( S 2 ) ≤ ε, (20) up to r ep ar ameterization. As in [11] w e pro v e this prop osition by assuming first that γ 0 is an em b edded real analytic 2-sphere and that X is real analytic in x . Then w e progressiv ely reduce the assumptions t o the framew ork of Prop o sition 1. 2.2.1. Case of an an alytic 2 -spher e move d by an analytic isotopy. The goal of this para- graph is to pro v e the follo wing prop osition. Prop osition 2. The c onclusion s of Pr op osition 1 ar e satisfie d if w e assume that γ 0 is an analytic 2 -spher e (i.e. γ 0 is the image of S 2 by a r e al-analytic emb e dding f 0 ) and that X is mor e over in C 0 ([0 , 1]; C ω (Ω; R 3 )) . It is clear that in tha t case, φ X is v olume preserving real analytic isotop y b et we en γ 0 and γ 1 . The first step t o establish Prop osition 2 is the follo wing. Lemma 1. L et t 7→ γ ( t ) b e a C 0 ([0 , 1]; C ω ( S 2 )) family of c ontr actible 2 -spher es in Ω and X ∈ C ([0 , 1]; C ω ( Ω)) b e a famil y of ve ctor field such that (21) Z γ ( t ) X · ν dσ = 0 , then ther e exists η > 0 and ψ ∈ C 0 ([0 , 1]; C ∞ ( V η [ int ( γ ( t ))]; R )) s uch that ∀ t ∈ [0 , 1] , ∆ x ψ = 0 in V η [ int ( γ ( t ))] , (22) ∀ t ∈ [0 , 1] , ∂ ψ ∂ ν = X · ν o n γ ( t ) . (23) 5 In o t her w or ds, Lemma 1 expresses that t he solution of the Neumann system (24)  ∆ x ψ = 0 in in t ( γ ( t )) , ∂ ψ ∂ ν = X · ν on γ ( t ) , can b e exten ded across the b oundary γ ( t ) uniformly in t . Pr o of. An equiv a len t lemma in dimension 2 is giv en in [11]. By means of real analytical lo cal c harts (see e.g. the analytic in v erse theorem in [15]), γ ( t ) is lo cally mapp ed to the plan { x 3 = 0 } b y some φ with φ (int( γ ( t )) ) ⊂ { x 3 > 0 } . Moreov er, we can require that dφ ( ν ) is normal to x 3 = 0 on { x 3 = 0 } . T o obtain this prop erty , consider ˆ φ : ( x 1 , x 2 , x 3 ) 7→ ( x 1 , x 2 , 0) + x 3 dφ φ − 1 ( x 1 ,x 2 , 0) ( ν ) . Then ˆ φ is analytic and inv ertible (lo cally) in a neigh b orho o d of { x 3 = 0 } . Now , replace φ by ˆ φ − 1 ◦ φ to obtain the requiremen t. No w call g := ∂ x 3 ( ψ ◦ φ − 1 ); it satisfies a · ∇ 2 g + b · ∇ g + cg = 0 , with a real-analytic Diric hlet b oundary condition giv en on x 3 = 0 and analytic co efficien ts a , b and c . W e use the following result of Cauc h y-Kow alevsky t yp e (see e.g. Morrey [17, Theorem 5.7.1’]) Theorem 3. L et f , a , b and c b e r e al analytic f unc tion s o n G R := B R N (0 , R ) ∩ { ( x 1 , ..., x N ) , x N ≥ 0 } , and y ∈ H 2 ( G R ) satisfying a · ∇ 2 y + b · ∇ y + c = f in G R and y = 0 on x N = 0 . Assume that for some c onstant A and L we have (25) |∇ p a ( x ) | , |∇ p b ( x ) | , |∇ p c ( x ) | , |∇ p f ( x ) | ≤ LA | p | , for any multi-index p . Ther e exists then R ′ < R dep endin g only on N , A , L and R such that y m ay b e extende d analytic al ly on B R N (0 , r ) for any r < R ′ . Th us for eac h x ∈ γ ( t ), ∇ x ψ · ν can b e a na lytically extende d on a neigh b orho o d U x of x across γ ( t ). In tegrat ing ∇ ψ · ν a long ν w e deduce that ψ is real analytic and by unique con tin uat ion its extens io n is also harmonic on U x . Moreo v er, using the con tinuit y of γ in the v ariable t from [0 , 1] to C ω ( S 2 ) 1 (so that the co efficien t s satisfy (25) uniformly in time), w e can find U x so that ψ is analytically extended o n in U x for eac h s in some neigh b or ho o d of t . Using the compactness of ∪ t { ( t, γ ( t )) } w e see tha t w e can c ho ose η > 0 unifor m in t , suc h that for a ll t , ψ can b e analytically (and hence harmonically) extended in V η ( γ ( t )). Lemma 1 f ollo ws.  1 W e re fer to [15] for a definition of the top o lo gy on this space. 6 Back to the pr o of of Pr op osition 2. W e tak e γ ( t ) = φ X ( t, 0 , γ 0 ). Due to the regularit y of X , γ ( t ) is analytic fo r any t , and applying Lemma 1 w e deduce a function ψ . Reducing η > 0 giv en b y Lemma 1 if necessary , w e ma y assume that V η ( γ ( t )) do es not meet ∂ Ω. By compactness of [0 , 1], for a giv en ε > 0 w e can c ho ose 0 ≤ t 1 < · · · < t N ≤ 1 and δ 1 , . . . , δ N suc h that [0 , 1] ⊂ ∪ N i =1 ( t i − δ i , t i + δ i ) , ∀ t ∈ [ t i − δ i , t i + δ i ] , γ ( t ) ⊂ V η/ 2 ( γ ( t i )) , ∀ s, t ∈ [ t i − δ i , t i + δ i ] , k ψ ( s, · ) − ψ ( t, · ) k C k ( V η ( γ ( t i ))) ≤ ε. (26) F or each i = 1 , . . . , N , we consider K i := V η/ 2 [in t ( γ ( t i ))] ∪ V η/ 2 ( ∂ Ω \ Γ) , ψ i := ψ ( t i , · ) . Reducing η again if necess ary , w e may assume that the connected comp onen t of Ω \ K i in R 3 \ K i meets R 3 \ Ω. This is p ossible since the connected compo nent of Ω \ [in t( γ ( t i )) ∪ ( ∂ Ω \ Γ)] in R 3 \ K i do es meet R 3 \ Ω, since Γ 6 = ∅ . I t f o llo ws that eac h connected comp onen t of R 3 \ K i meets R 3 \ Ω. No w w e use the follow ing harmonic appro ximation theorem (see [6, Theorem 1.7]): Theorem 4. L et O b e an op en set in R N and let K b e a c omp act set i n R N such that that O ∗ \ K is c onne cte d, wher e O ∗ is the Alexandr off c omp actific ation of O . Then, for e ach function u wh ich is harm o nic on an op en set c ontaining K and e ach ε > 0 , ther e is a harmoni c function v in O such that k v − u k ∞ < ε on K . Recall that the Alexandroff compactification of O is o bt a ined b y adding a p oin t, sa y {∞} to O a nd to consider the top ology generated b y the op en sets of O , and the sets of the fo rm {∞} ∪ ( O \ K ), with K a subse t of O . Remark 5. One may state the same result with the C k norm instead of the uniform one. It suffices to consider a compact ˜ K whose in terior con t ains K , apply the ab ov e result on ˜ K and use standard prop erties of harmonic functions. W e c ho ose p o in ts Y 1 , ..., Y P in each connected comp onen t of R 3 \ K i , outside Ω. W e apply the preceding result with O = R 3 \ { Y 1 , . . . , Y P } and K = K i . In that case, O ∗ can b e though t as the quotien t of S 3 = R 3 ∪ {∞} by the iden tification of Y 1 , . . . , Y P and ∞ . Therefore, for any ν > 0, w e get a map ˆ ψ i in C ∞ ( R 3 \ { Y 1 , ..., Y P } ; R ) suc h that ˆ ψ i is harmonic on R 3 \ { Y 1 , ..., Y P } and suc h that k ˆ ψ i − ψ i k C k +2 ( V η/ 2 [ in t ( γ ( t i ))]) < ν, (27) k ˆ ψ i k C k +2 ( V η/ 2 ( ∂ Ω \ Γ)) < ν. (28) Since ˆ ψ i is harmo nic in Ω, there holds (29) Z ∂ Ω ∇ ˆ ψ i · n dσ = 0 . In o r der fo r (4) to b e satisfied w e consider d i in C ∞ ( ∂ Ω; R ) suc h that d i = ∇ ˆ ψ i · n on ∂ Ω \ Γ , (30) k d i k C k +1 ( ∂ Ω) ≤ C k∇ ˆ ψ i · n k C k +1 ( ∂ Ω) , (31) Z ∂ Ω d i dσ = 0 . (32) 7 and in tro duce the harmonic function h i in Ω b y the Neumann problem ∆ h i = 0 in Ω , (33) ∂ h i ∂ n = d i in Ω , (34) Z Ω h i = 0 . Note that in particular tha t b y standard elliptic estimates, (35) k h i k C k +1 ( ¯ Ω) ≤ C ν. W e in tr o duce (36) ˇ ψ i := ˆ ψ i − h i . T a king a partition of unit y χ i asso ciated t o the co ve ring of [0 , 1] by the interv als ( t i − δ i , t i + δ i ), w e define (37) θ ( t, x ) := N X n =1 χ i ( t ) ˇ ψ i ( x ) . Due to (30) and (34), θ satisfies (18). Moreo ver a ccording to (2 6), (2 7), (28) and (35) w e ha ve for ν small enough with resp ect to ε and fo r some C > 0 (38) sup t ∈ [0 , 1] k∇ θ − ∇ ψ k C k ( V η/ 3 [ γ ( t )]) ≤ C ε. In par ticular b y supp osing ε small enough, w e hav e a uniform estimate (39) k∇ θ k C k ( φ ∇ θ (0 ,t,γ 0 )) ≤ k∇ ψ k C k ( φ ∇ ψ (0 ,t,γ 0 )) + 1 . As long as φ ∇ θ ( t, 0 , γ 0 ) remains in V η/ 3 ( γ ( t )) one has, using Gron wall’s lemma, k φ ∇ θ ( t, 0 , γ 0 ) − φ ∇ φ ( t, 0 , γ 0 ) k ∞ ≤ k∇ θ ( t, · ) − ∇ φ ( t, · ) k C 0 ([0 , 1]; C 0 ( V η/ 3 [ γ ( t )]) exp( k∇ ψ k L 1 (0 , 1; L ip ( V η/ 3 [ γ ( t )]) ) . Then b y reducing ε if necessary , we get thanks to (38) that this is v alid for all time in [0 , 1]. Differen tia t ing φ with resp ect to x up to the order k , w e obtain in the same w ay that fo r all t ∈ [0 , 1], k φ ∇ θ ( t, 0 , γ 0 ) − φ ∇ φ ( t, 0 , γ 0 ) k C k ([0 , 1]) ≤ k∇ θ − ∇ φ k C 0 ([0 , 1]; C k ( V γ / 3 ( γ ( t ))) exp( C k∇ ψ k L 1 ((0 , 1); W k +1 , ∞ ( V η/ 3 [ γ ( t )])) ) . This ends the pro of Propo sition 2 .  2.2.2. Case of a smo oth 2 -spher e move d by a sp e c i a l analytic isotopy. In this paragraph, Prop osition 2 is extended t o the f ollo wing. Prop osition 3. The c onclusio ns of Pr op osition 1 a r e satisfie d if we ass ume that γ 0 is a s m o oth 2 -sp h er e (i.e. γ 0 is the image of S 2 by f 0 a C ∞ emb e dding) and that X is mor e over in C 0 ([0 , 1]; C ω (Ω; R 3 )) . 8 Pr o of. Due to a result of H. Whitney (see [19]), γ 0 is im b edded in a smo oth family of surfaces γ ν , ν ∈ ( − ν 0 , ν 0 ), with γ 0 = γ ν for ν = 0, γ ν ∩ γ ν ′ = ∅ fo r ν 6 = ν ′ and γ ν real analytic f or ν 6 = 0 . It follows that either for ν > 0 or for ν < 0, one has (40) γ 0 ⊂ in t[ γ ν ] . Without loss of generalit y , w e assume tha t (40) ho lds fo r ν > 0. The family γ ν b eing smo oth with resp ect its parameters, one has (41) γ ν → γ 0 in C ∞ ( S 2 ) as ν → 0 + . W e can then apply Prop osition 2 to X on γ ν for ν > 0 small. W e construct a θ ε suc h that (17), (18), (19 ) and (20) apply for γ ν instead of γ 0 . The construction also generates a family ψ satisfying (24) in in t[ φ X ( t, 0 , γ ν )]. No w we ha ve uniform b ounds on the k∇ ψ k C k ( φ ∇ ψ ( t, 0 ,γ ν )) with resp ect to ν , b ecause the constants of the elliptic estimates (see e. g. [7, Theorem 6.30 and L emma 6.5]) are uniform with respect to ν thanks to (41). W e deduce from (39) that w e ha ve uniform b ounds on k∇ θ ε k C k ( φ ∇ θ ε ( t, 0 ,γ ν )) as ν → 0 + . By Gron wall’s lemma one gets k φ ∇ θ ε ( t, 0 , ˜ γ 0 ) − φ ∇ θ ε ( t, 0 , γ 0 ) k C k ( S 2 ) ≤ C k k γ 0 − γ ν k C k ( S 2 ) exp  Z 1 0 k∇ θ ε k C k +1 ( in t ( φ ∇ θ ε ( t, 0 ,γ ν )) dt  . Hence w e deduce the claim by ta king ν small enough. Of course by reparameterizing X and th us θ in time w e can alw ays assume that θ is compactly supp orted in time.  2.2.3. Case of sm o oth emb e dde d c ontr actible two-spher e m o ve d by a smo oth is o topy. W e no w pro ve Prop osition 1. Pr o of of Pr op osition 1. In the general case, w e assume that X is merely C ∞ ( Ω; R 3 ). Let λ > 0 suc h that max t ∈ [0 , 1] dist( φ X ( t, 0 , γ 0 ) , ∂ Ω) > 2 λ. W e define U t := V λ (in t( γ ( t ))). Reducing λ if necess ary , w e can obtain that for all t , U t is diffeomorphic to a ball. No w w e can use Whitney’s approxim ation theorem (see e.g. [15, Prop osition 3.3.9]), for any µ > 0 and any k ∈ N there exists X µ ∈ C ([0 , 1]; C ω ( R 3 )) suc h that k X µ − X k C ([0 , 1]; C k +1 ( U t )) ≤ µ. Moreo ve r, w e can ask that div X µ = 0 in [0 , 1] × R 3 . T o see this, w e use the fact that U t is a top o lo gical ba ll; hence its second de Rham cohomology space is trivial, and an y div ergence-free v ector field (in particular X ) on U t is of the form curl A . Hence for eac h time t one can apply Whitney’s approxim ation theorem (at order k + 2) on ϕA where X = curl A and ϕ ( t, x ) is a smo oth cutoff function equal to 1 on U t and to 0 for d ( x, U t ) ≥ 2 λ . W e obtain an approx imating ve ctor field B µ and define X µ := curl B µ . Using a s b efore the compactness of the time interv al [0 , 1 ] and a pa r tition of unity (as for t he pro of of Prop o sition 2), w e can obtain a smo oth appro ximation uniformly in time. No w we apply Prop osition 3 with ˜ X µ , a pply Gronw all’s lemma and get the result for µ small enough.  9 3. Proof of Theorem 1 This section is dev oted t o the pro o f of Theorem 1. As in [4, 8, 11], w e start with the case where u 0 is small, and then treat the g eneral case. 3.1. Preliminaries. First we in tro duce some functions on Ω in order to t ak e its top ology in to accoun t , more precisely to describ e its first de Rham cohomology space. W e recall the fo llo wing construction. Let Σ 1 , ..., Σ g b e g smo oth manifolds with b oundaries of dimens io n n − 1 inside Ω suc h that: • for all i in { 1 , . . . , g } , ∂ Σ i ⊂ ∂ Ω and Σ i is tra nsv erse to ∂ Ω, • for all i, j in { 1 , . . . , g } 2 with i 6 = j , Σ i and Σ j in tersect transv ersally (whic h, by definition, includes the case of an empty inte rsection), • Ω \ ∪ g i =1 Σ i is simply connected. F or i = 1 , . . . , g , w e consider X i := { p ∈ H 1 (Ω \ ∪ g k =1 Σ k ) , [ p ] i = constan t , [ p ] j = 0 , j 6 = i } , where [ p ] k = p | Σ + k − p | Σ − k is the jump of p on eac h arbitrarily fixed side of Σ k in Ω. Then b y Lax-Milgram’s Theorem t here exists a unique q i ∈ X i suc h that for all p ∈ X i Z Ω ∇ q i · ∇ p dx = [ p ] i , whic h leads to the existenc e of a unique p i suc h that ∆ p i = 0 in Ω \ ∪ g k =1 Σ k , ∂ n p i = 0 on ∂ Ω , [ p i ] i = 1 and [ p i ] j = 0 for j 6 = i, [ ∂ n q i ] i = 0 , and w e ta ke Q i := ∇ p i , whic h is r egula r in Ω. Then we hav e t he follow ing result. Prop osition 4 (see e.g. [18], App endix I, Prop osition 1.1) . F o r any X ∈ L 2 (Ω; R 3 ) such that curl X = 0 in Ω , ther e exist χ in H 1 (Ω; R ) and α 1 , . . . , α g in R such that (42) X = ∇ χ + g X i =1 α i Q i . 3.2. A fixed point operator . Now w e intro duce an op erator, whose fixed p oint will giv e a solution of our problem when taking u 0 (suitably small) into accoun t. W e in tr o duce R > 0 suc h that Ω ⊂ B R := B R 3 (0 , R ) . W e in tr o duce a linear contin uous extens io n op erator π : C ( Ω; R 3 ) → C 0 ( B R ; R 3 ) suc h that ∀ k ∈ N and all β ∈ (0 , 1 ), π is contin uous from C k ,β ( Ω; R 3 ) to C k ,β 0 ( B R ; R 3 ). Let δ ∈ (0 , 1), and consider µ ∈ C ∞ 0 ([0 , + ∞ ); R ) with suppor t in [0 , δ ] and with v alue 1 on a neighborho o d of 0 . Giv en ε > 0 we denote ¯ y := ∇ θ, 10 the p ot ential flo w obtained b y Prop osition 1 with X obtained from Theorem 2. F or some ν ∈ (0 , 1) whic h will b e small in the sequ el w e define with α ∈ ( 0 , 1): X ν := n u ∈ L ∞ ((0 , 1); C k ,α (Ω; R 3 )) , div u = 0 in (0 , 1) × Ω , k u − ¯ y k L ∞ (0 , 1; C k,α (Ω)) ≤ ν o . It is straigh tforw ar d to c hec k that X ν is a closed conv ex subset o f L ∞ ((0 , 1) × Ω). W e extend ¯ y b y π and still denote ¯ y the extended f unction. No w, giv en u ∈ X ν , w e asso ciate F ( u ) as follow s. First, w e intro duce ˜ u := ¯ y + π ( u − ¯ y ) = π ( u ) . Next w e consider ω u ∈ L ∞ (0 , 1; C k − 1 ,α ( B R ; R 3 )) as the solution o f the following tr ansp ort equation: ω u ( · , 0) = curl π ( u 0 ) in B (0 , R ) , (43) ∂ t ω u + ( ˜ u · ∇ ) ω u = ( ω u · ∇ ) ˜ u − (div ˜ u ) ω u in (0 , 1) × B R . (44) Due the support of ˜ u , one sees by using ch aracteristics that the system (4 3)-(44) is we ll- p osed and that indeed ω u has the claimed regularit y (details for obtaining the regularit y can b e found in [8]). No w observing that curl ( a ∧ b ) = (div b ) a − (div a ) b + ( b · ∇ ) a − ( a · ∇ ) b, w e easily deduce that div ( ω u ) = 0 in B R , so that ω u can b e written in the form curl ˆ v in B R . Hence it is classical (since R ∂ Ω u 0 · n dσ = R ∂ Ω ¯ y · n dσ = 0 ) that there exists a unique v ∈ L ∞ (0 , 1; C k ,α ( Ω; R 3 )) suc h that curl v = ω u in [0 , 1] × Ω , (45a) div v = 0 in [0 , 1 ] × Ω , (45b) v · n = µ ( t ) u 0 · n + ¯ y · n on [0 , 1] × ∂ Ω , (45c) Z Ω v · Q i dx = 0 , i = 1 , . . . , g . (45d) According to Prop o sition 4, w e can determine g time-dep enden t functions λ 1 , . . . , λ g suc h that, if w e define (46) V := v + g X i =1 λ i ( t ) Q i , w e ha v e for a ll j = 1 , . . . , g (47) Z Ω V ( 0 ) · Q j dx = Z Ω u 0 · Q j dx = 0 , and for all t ∈ [0 , 1], (48) Z Ω V ( t, x ) · Q j ( x ) dx − Z Ω u 0 ( x ) · Q j ( x ) dx = − Z t 0 Z Ω ( u ( τ , x ) ∧ ω u ( τ , x )) · Q j ( x ) dx dτ . This is p ossible in a unique w ay since the matrix ( R Ω Q i · Q j dx ) i,j is inv ertible, a s a Gram matrix of indep enden t functions. No w w e finally define F ( u ) := V . 11 3.3. Finding a fixed p oint. Our goal is to pro ve hereafter: Prop osition 5. Given ν > 0 , if k u 0 k C j,α (Ω) is smal l en o ugh, F admits a fixe d p oint in X ν . W e will use the follo wing lemma (see e.g. [1, Theorem 3.14]): Lemma 2. L et j ∈ N , a ∈ ( 0 , 1) . L et f , v , and g b e elements of L ∞ (0 , 1; C j,α ( B R ; R 3 )) satisfying ∂ t f + ( v · ∇ ) f = g , with v a n d f (0 , · ) c om p actly supp orte d in B R . T h en for some C > 0 dep end ing on j and α on ly, ther e holds k f ( t, · ) k C j,α ( B R ) ≤ exp  C Z t 0 V ( s ) ds   k f (0 , · ) k C j,α ( B R ) + Z t 0 exp  − C Z τ 0 V ( s ) ds  k g ( τ , · ) k C j,α ( B R ) dτ  , with V ( s ) := k∇ v ( s, · ) k C j − 1 ,α ( B R ) if j ≥ 1 and V ( s ) := k∇ v ( s, · ) k L ∞ ( B R ) if j = 0 . Pr o of of Pr op osition 5. W e establish Prop osition 5 and find a fixe d po in t of F in X ν via Sc hauder’s fixed p o int theorem. Accordingly , w e prov e that, ν b eing fixed and for k u 0 k C k,α small enoug h, F sends X ν in to itself, that F is con tinuous and F ( X ν ) is relativ ely compact f o r the uniform top o logy on X ν . • Using Lemma 2, w e see that k ω u ( t, · ) k C k − 1 ,α ( B R ) ≤ exp  C Z t 0 V ( s ) ds   k ω u (0 , · ) k C k − 1 ,α ( B R ) + C Z t 0 exp  − C Z τ 0 V ( s ) ds  k ˜ u ( τ , · ) k C k,α ( B R ) k ω u ( τ , · ) k C k − 1 ,α ( B R ) dτ  , with as b efore V ( s ) := k∇ ˜ u ( s, · ) k C k − 2 ,α ( B R ) if k ≥ 2 and V ( s ) := k∇ ˜ u ( s, · ) k L ∞ ( B R ) if k = 1 . W e apply Gron w all’s lemma to t 7→ k ω u ( t, · ) k C k − 1 ,α ( B R ) exp  − C Z t 0 V ( s ) ds  , and deduce k ω u ( t ) k C k − 1 ,α ( B R ) ≤ k ω u (0) k C k − 1 ,α ( B R ) e C k ˜ u k L ∞ (0 , 1; C k,α ( B R )) . Th us with the definition of X ν and the con tinuit y of π , w e obtain tha t (49) k ω u ( t ) k C k − 1 ,α ( B R ) ≤ k ω u (0) k C k − 1 ,α ( B R ) e C ( k y k L ∞ (0 , 1; C k,α (Ω)) +1) . No w b y (4 8) w e hav e | λ i ( t ) | ≤ C ( | λ i (0) | + t k ω u k C 0 ,α ( B R ) k u k C 1 ,α ( B R ) ) , and th us, with (47), (49) and the definition of X µ , w e deduce (50) | λ i ( t ) | ≤ C k u 0 k C k,α ( B R )  1 + te C ( k y k L ∞ (0 , 1; C k,α (Ω)) +1) [ k y k L ∞ (0 , 1; C k,α (Ω)) + ν ]  . 12 Th us, b y com bining (49), (50), and the elliptic estimates given b y ( 4 5)-(46), w e infer that (51) k F ( u ) − ¯ y k L ∞ (0 , 1; C k,α ( Ω)) ≤ C ( k y k k ,α ) k u 0 k C k,α (Ω) , for some constan t C dep ending on k , α and y . It fo llo ws that for k u 0 k C k,α (Ω) small enough, F sends X ν in to itself. • That F ( X ν ) is relativ ely compact is seen easily: giv en ( u n ) ∈ X N ν , the sequence ( F ( u n )) b elongs to X ν and, fo llowing the construction, it is easy to see that ( ∂ t F ( u n )) is b ounded in L ∞ (0 , 1; C k − 1 (Ω)). The conclusion follows then fr om Ascoli’s theorem. • Finally , that F is con tin uous for the uniform top ology can b e seen as follo ws. Let us b e given ( u n ) ∈ X N ν con v erging uniformly to u ∈ X ν . The flo ws Φ n asso ciated to ˜ u n con v erge unifo r mly tow ards the flow Φ asso ciated to ˜ u . Hence one can see that ω u n con v erges uniformly to ω u . Due to the b ounds on ω u , this con v ergence also tak es place in L ∞ (0 , 1; C k − 1 ,β ( B R )) for any β < α . W e deduce in a straightforw ard manner the con v ergence of ( v n ) a nd ( λ n i ) corresp onding to u n to wards v and λ i corresp onding to u , and the conclusion follo ws. It follow s that F admits a fixed p oin t u in X ν . This concludes the pro of of Prop o sition 5.  3.4. Relev ance of the fixed p oint. Call u the fixed po in t obtained ab o ve. 1. Let us first c heck that u is a solution of the Euler equation in [0 , 1] × Ω. F rom (44), and since ˜ u = u in [0 , 1] × Ω, w e deduce tha t curl ( ∂ t u + ( u · ∇ ) u ) = 0 in [0 , 1] × Ω . F rom (4 8) and ( u · ∇ ) u = ∇ | u | 2 2 + (curl u ) ∧ u , w e see that Z Ω ( ∂ t u + ( u · ∇ ) u ) · Q i dx = 0 , whic h together with Prop osition 4 pro v es the claim. 2. Now w e pro ve that ¯ u fulfills the requiremen ts of Theorem 1 for k u 0 k C k,α small enough. Giv en x ∈ B R , w e hav e ˙ φ ¯ u ( t, 0 , x ) − ˙ φ ¯ y ( t, 0 , x ) = ¯ u ( t, φ ¯ u ( t, 0 , x )) − ¯ y ( t, φ ¯ u ( t, 0 , x )) + ¯ y ( t, φ ¯ u ( t, 0 , x )) − ¯ y ( t, φ ¯ y ( t, 0 , x )) . W e deduce easily that for a constan t C > 0 dep ending on y only , | ˙ φ ¯ u ( t, 0 , x ) − ˙ φ ¯ y ( t, 0 , x ) | ≤ ν + C | φ ¯ u ( t, 0 , x ) − φ ¯ y ( t, 0 , x ) | . Th us b y Gronw all’s lemma w e hav e | x ( t ) − y ( t ) | ≤ C ν where C dep ends only on ¯ y . Reasoning in the same w a y for the deriv at ives (up to order k ) with respect to x of the flo ws, w e obtain k φ ˜ u ( t, 0 , · ) − φ y ( t, 0 , · ) k C k ( B R ) ≤ C ν . Hence ta king ν small enough (and hence k u 0 k C k,α ev en smaller), w e can obtain (10) and (12) for T = 1. In order w ords, there exists c > 0, suc h that fo r an y u 0 in C k ,α with k u 0 k C k,α ≤ c , one can find a solution of the Euler equation for T = 1, satisfying (10) and (12). 13 3. Let us no w explain ho w w e can obtain the result without the condition of smallness of u 0 (but for T small enough). Giv en u 0 ∈ C k ,α (Ω), we rescale it by considering v 0 = ρu 0 , with ρ > 0 small enough so that v 0 satisfies k v 0 k C k,α ≤ c . Applying the abov e construc- tion to v 0 giv es us a solution ( u, p ) of (1)-(3) defined o n t ∈ [0 , 1] with u (0 , · ) = v 0 suc h that k φ u (1 , 0 , γ 0 ) − γ 1 k ∞ < ε. If w e define u ρ b y u ρ ( t, x ) = 1 ρ u  t ρ , x  , then u ρ is defined on t ∈ [0 , ρ ] and k φ u ρ ( ρ, 0 , γ 0 ) − γ 1 k < ε. This conclude s the pro of of Theorem 1. Reference s [1] H. Bahouri, J.-Y. Chemin, R. Danchin. F ourier analysis and nonline ar p artial differ ential e quations . Grundlehren der Mathematischen Wiss enschaften, 3 43. Springer, Heidelb erg, 201 1 . [2] L. E. J. B rouw er. ¨ Uber Jordansche Mannig fa ltigkeiten. Math. Ann., 71(4):5 98, 1912. [3] J.- M. Coro n. Contrˆ olabilit ´ e exacte fro n ti ` er e de l’´ equation d’Euler des fluides parfaits incompressible s bidimensionnels. C.R. Acad. Sci. Paris, 317:271 –276, 199 3 . [4] J.- M. Coron. On the controllabilit y o f 2-D incompressible p er fect fluids. J. Math. Pures Appl. (9), 75(2):155 –188 , 199 6 . [5] J.- M. Co ron. O n the n ull asymptotic stabilizatio n of the t wo-dimensional incompressible Euler eq ua- tions in a simply connected doma in. SIAM J. 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N avier-Stokes Equations and nu meric al analysis . North-Holland Publica tio ns. North- Holland, 1979. [19] H. Whitney . The imbedding o f manifolds in families o f analytic manifolds. Ann. of Math. (2), 37(4):865 –878 , 193 6 . 14 Ceremade, Universit ´ e P aris-Da uphine, Place du Mar ´ echal de La ttre de T assigny, 75775 P aris Cedex 16, France E-mail addr ess : glass@c erema de.dauphine.fr Universit ´ e de Versaill es, 45 a venue d es Et a ts-Unis, F780 30 Versailles, Fran ce E-mail addr ess : horsin@ math. uvsq.fr 15