The Two Dimensional Euler Equations on Singular Exterior Domains
aa r X i v : . [ m a t h . A P ] N ov THE TWO DIMENSIONAL EULER EQUATIONSON SINGULAR EXTERIOR DOMAINS
DAVID G´ERARD-VARET & CHRISTOPHE LACAVE
Abstract.
This paper is a follow-up of article [9], on the existence of global weak solutions to thetwo dimensional Euler equations in singular domains. In [9], we have established the existence of weaksolutions for a large class of bounded domains, with initial vorticity in L p ( p > L pc ( p >
2) and whenthe domain is the exterior of a single obstacle. The goal here is to retrieve these two restrictions: weconsider general initial vorticity in L ∩ L p ( p > Introduction
The motion of a 2D ideal incompressible flow in a domain Ω is governed by the Euler equations: ∂ t u + u · ∇ u = −∇ p ∀ ( t, x ) ∈ (0 , ∞ ) × Ω , (1.1)div u = 0 ∀ ( t, x ) ∈ [0 , ∞ ) × Ω , (1.2) u · ν = 0 ∀ ( t, x ) ∈ [0 , ∞ ) × ∂ Ω , (1.3) u (0 , x ) = u ( x ) ∀ x ∈ Ω , (1.4)where u = ( u ( t, x , x ) , u ( t, x , x )) is the velocity, p = p ( t, x , x ) is the pressure and ν the unitnormal vector to ∂ Ω pointing outside the fluid domain.Construction of solutions to these equations has been a constant concern over the last century. Letus mention the pioneering paper of Wolibner [28] on smooth solutions in smooth bounded domains(see also Kato [14] and Temam [27]). Among many others, one can also mention the work of McGrath[23], resp. Kikuchi [15] on the existence of smooth solutions in the whole plane, resp. in smooth2D exterior domains. Of course, a key feature of the 2D equations is the conservation of vorticity ω := curl u = ∂ u − ∂ u , which verifies a transport equation: ∂ t ω + u · ∇ ω = 0 ∀ ( t, x ) ∈ (0 , ∞ ) × Ω . (1.5)It propagates bounds on the vorticity, that allows to go from local to global in time solutions, and tointroduce several notions of weak solutions. The most famous result in this direction is certainly theone of Yudovich [13]: it provides existence and uniqueness of global solutions with bounded vorticity,in various domains (with smooth or without boundaries), see also Bardos [1] for another existenceproof. Weaker notions of global solutions have been developed since: solutions with L p vorticity, asintroduced by DiPerna and Majda [6], or solutions whose vorticy is a signed measure in H − , builtby Delort [4]. Note that uniqueness of these weaker solutions is unknown.We stress that in above studies, the domain Ω is always taken smooth. This is due to mathemat-ical technicalities, such as ensuring the L p continuity of Riesz transforms over Ω. This smoothnessassumption is of course natural in the context of smooth solutions, but it is a big restriction withregards to weak solutions. Indeed, in many situations, the singularities of the flow are created by theflow domain itself, for instance the irregularity of a solid obstacle. Hence, it is highly desirable toextend the theory of weak solutions to general domains.This issue has been investigated by various authors in the recent years. One can mention for instancethe work of M. Taylor [26], who established the existence of weak solutions for convex and boundedΩ. The convexity assumption is related there to regularity results for the Laplace equation ∆ ψ = ω in Date : October 15, 2018.
Ω, set with Dirichlet conditions. It is indeed well-known that for convex Ω, ω ∈ L (Ω) ⇒ ψ ∈ H (Ω).This yields the continuity of the Riesz Transform over L , and so classical methods for weak solutionscan be transfered to this case (see also [2] for further refined results in convex and bounded domains).One can also mention the work [16] by the second author, that establishes the existence of globalsolutions with L ∞ vorticity in the exterior of a smooth Jordan curve. This existence result is obtainedthrough an approximation process, considering smooth obstacles shrinking to the curve. It uses severaltools of complex analysis, notaby conformal transform, to derive uniform bounds on the sequence ofapproximate solutions. Note that another related asymptotics, namely obstacles shrinking to a pointfalls into the scope of a very active area of research, as testified for instance by the recent works[3, 11, 18, 19, 21, 22, 24].Finally, let us cite the recent papers [2, 5, 17, 20] on the Euler flow in polygons. In such polygonaldomains, one can even go further in the analysis (using elliptic theory in domains with corners), anddeal with the uniqueness of Yudovitch solutions.Still, the studies just mentioned have strong limitations on the geometry of the domain Ω. In thisrespect, a big step forward has been made in our recent paper [9]. There, we have proved the existenceof global weak solutions with L p vorticity for a large class of non-smooth open sets. Namely, we haveconsidered two kinds of sets: • Bounded domains: Ω = e Ω \ k [ i =1 C i where e Ω is any simply connected open set, and the C i ’s are disjoint obstacles. • Exterior domains Ω = R \ C for a single obstacle C .Let us point out that in [9, p. 133], obstacles are defined as compact connected subsets of positivecapacity . As will be shown below, cf Proposition 2.2, this is the same as simply saying that theconnected compact subsets are not reduced to points. Hence, the bounded domains considered in [9]are very general. The existence of solutions with L p vorticity ( p >
1) is proved using in a crucial waythe notion of γ -convergence of open sets, described extensively in [10] (see [9, App. C] for a shortsummary).However, in the case of exterior domains, the existence result of [9] is not so strong. First, weconsider only a single obstacle. Second, we must assume that the vorticity is in L pc (Ω) , p >
2. Theserestrictions are due to our method of proof, as will be explained later on.
Our goal in this paper is to get rid of these limitations. We shall show the existence of global weaksolutions for vorticities in L ∩ L p ( p > ), in the exterior of an arbitrary finite number of obstacles(not reduced to points). Again, it will improve [9] because we consider several obstacles, we do notassume that the initial vorticity is compactly supported, and we treat indices p ∈ (1 , R \ (cid:16) k [ i =1 C i (cid:17) , k ∈ N (1.6)where C , . . . , C k are disjoints connected compact subsets not reduced to points. (1.7)We consider initial data satisfying u ∈ L (Ω) , u → | x | → + ∞ , curl u ∈ L ∩ L p (Ω) (1.8) HE TWO DIMENSIONAL EULER EQUATIONS ON SINGULAR EXTERIOR DOMAINS 3 for some p ∈ (1 , ∞ ]. We want also u to be divergence-free and tangent to the boundary. Nevertheless,due to the irregularity of Ω, this tangency condition has to be expressed in a weak sense. Namely, weask Z Ω u · ∇ h = 0 for all h ∈ H (Ω) such that ∇ h ∈ L (Ω) and h ( x ) = 0 for large x. (1.9)Note that this exactly amounts to the usual divergence-free and tangency condition when u and Ωare smooth: indeed, integrating by parts, we find in such case Z Ω u · ∇ h = 0 = Z ∂ Ω u · ν h − Z Ω div u h, this identity being valid for all functions h ∈ D ( R ). This easily implies that div u = 0 and u · ν | ∂ Ω =0. We stress that the assumptions (1.8)-(1.9) is not restrictive: we will prove in Subsection 4.2 that forany function ω ∈ L ∩ L p ( R ) there exists some vector fields u verifying (1.8)-(1.9) with curl u = ω .Let us also note that such a vector field is not always square integrable at infinity (for ω compactlysupported, u has a finite energy iff the sum of the circulations around each obstacles is equal to − R ω , which is an important constraint).Similarly to (1.9), the weak form of the divergence free (1.2) and tangency conditions (1.3) on theEuler solution u reads: for almost all t , Z Ω u ( t, · ) · ∇ h = 0 for all h ∈ H (Ω) such that ∇ h ∈ L (Ω) and h ( x ) = 0 for large x. (1.10)Finally, the weak form of the momentum equation (1.1) reads:for all ϕ ∈ D ([0 , + ∞ ) × Ω) with div ϕ = 0 , Z ∞ Z Ω ( u · ∂ t ϕ + ( u ⊗ u ) : ∇ ϕ ) = − Z Ω u · ϕ (0 , · ) . (1.11)Our main theorem is Theorem 1.1.
Assume that Ω is of type (1.6) - (1.7) . Let p ∈ (1 , ∞ ] and u as in (1.8) - (1.9) . Thenthere exists u ∈ L ∞ loc ( R + ; L (Ω)) , with curl u ∈ L ∞ ( R + ; L ∩ L p (Ω)) which is a global weak solution of (1.1) - (1.4) in the sense of (1.10) and (1.11) . Moreover, the solution of the theorem has a Hodge-De Rham decomposition, the weak circulationsare conserved, and we have the two following estimates: k curl u k L ∞ ( R + ,L p (Ω)) ≤ k curl u k L p , k curl u k L ∞ ( R + ,L (Ω)) ≤ k curl u k L (1.12)More details will be given in due course. Remark 1.2. If p ≥ / , let us stress that such a solution is also a solution of the vorticity equation (1.5) in the sense of distributions, namely,for all ψ ∈ D ([0 , + ∞ ) × Ω) , Z ∞ Z Ω ( ω∂ t ψ + ωu · ∇ ψ ) = − Z Ω ω ψ (0 , · ) . (1.13) Indeed, for any ψ ∈ D ([0 , + ∞ ) × Ω) , ϕ := ∇ ⊥ ψ is a test function for which (1.11) holds true. As u is uniformly square integrable on the support of ϕ and curl u ∈ L ∞ ( R + , L / (Ω)) standard ellipticestimates imply that u belongs to W , / hence to L on the support of ϕ . Then an integration byparts implies (1.13) . Remark 1.3.
As in [9] , the main point is that we assume nothing about the regularity of the boundary(the obstacles can be as exotic as a Koch snowflake). This is made possible by our method of proof,based on the γ -convergence theory: it only requires the velocity to be in L near the boundary. Ofcourse, it is also because we deal only with existence issues. Uniqueness of weak solutions requires ingeneral more regularity (a uniform bound of k u k W ,p / ( p ln p ) for large p or a log-lipschitz estimate upto the boundary) which cannot be obtained without some stronger assumptions on the domains. Evenin bounded domains, Jerison and Kenig [12] exhibited an example of ω smooth, ∂ Ω ∈ C and where D. G´ERARD-VARET & C. LACAVE Du is not integrable (where u = ∇ ⊥ ψ with ψ solution of the Dirichlet problem ∆ ψ = ω ). Hence toprove the uniqueness, the authors in [2, 5, 17, 20] have to assume that ∂ Ω is smooth, except in a finitenumber of points. Let us eventually comment on the proof of Theorem 1.1. The basic idea is to construct a sequenceof approximate smooth domains Ω n and initial data u n , that generate smooth solutions u n = ∇ ⊥ ψ n .The point is to derive uniform bounds on ψ n , starting from the L p uniform bound on the vorticity ω n = ∆ ψ n . In [9], we used a sort of Poincar´e inequality on large balls B (0 , R ) containing the obstacle.There, the fact that the initial vorticity was supported in such a ball was crucial. Furthermore, it wasnecessary to show that the compact support of ω n was controlled uniformly in n at later times. Werelied here on an explicit representation of ψ n in terms of ω n . This representation, valid only for asingle obstacle, involved a biholomorphism T n sending the exterior of Ω n to the exterior of the unitdisk. The fact that the vorticity was zero at infinity (and in L p , p > u by some smooth Ω n and u n . Section 3 is the central one: it shows how to obtain estimates on theEuler approximations u n , uniformly in n . This allows to complete the proof of the existence of weaksolutions in Ω, through compactness arguments (Section 4).2. Geometry and initial data approximation
We begin this section by recalling that a domain of type (1.6)-(1.7) can be approximated in theHausdorff topology by smooth domains.
Proposition 2.1. [9, Prop. 1]
Let Ω be of type (1.6) - (1.7) . Then Ω is the limit of a sequence Ω n := R \ (cid:16) k [ i =1 O in (cid:17) where the O in are smooth Jordan domains, with O in converging to C i in the Hausdorff topology. A short reminder about Hausdorff topology can be found in [9, App. B]. For our analysis, we justrecall here this useful property:for any compact set K ⊂ Ω, there exists n K > K ⊂ Ω n , ∀ n ≥ n K . (2.1)Another geometrical feature which turns out to be crucial in our L - framework is the positiveSobolev capacity of the obstacles. The Sobolev H capacity of a compact set E ⊂ R is defined bycap( E ) := inf {k v k H ( R ) , v ≥ E } , with the convention that cap( E ) = + ∞ when the set at the r.h.s. is empty. We refer to [10] for anextensive study of this notion, while the basic properties are listed in [9, App. A]. In particular werecall that a point has zero capacity, whereas the capacity of a smooth Jordan arc is positive.By (1.7), our obstacles C i are compact, connected, and not reduced to a point. This is enough toensure that they have positive capacity. This is expressed by the following Proposition 2.2.
Let C be a connected compact subset of R . Then, we have the following equivalence: cap( C ) > if and only if C is not reduced to one point . Proof.
It is already well known that the capacity of a single point is zero. Conversely, assume that C is a connected compact set with at least two different points x = y . Let Π [ x,y ] ( C ) the projection of C over the segment [ x, y ]. It is connected, being the continuous image of C , and contains x and y . Thus,Π [ x,y ] ( C ) = [ x, y ], and it has in particular Hausdorff dimension 1. As the projection Π [ x,y ] is Lipschitz, HE TWO DIMENSIONAL EULER EQUATIONS ON SINGULAR EXTERIOR DOMAINS 5 and as Lipschitz functions decrease the Hausdorff dimension, we deduce that the Hausdorff dimensionof C is greater than 1. This implies that the capacity of C is positive, see [7]. (cid:3) The positive capacity of our obstacles is important notably because it provides a uniform Poincar´einequality for functions that vanish at the (regularized) obstacles. More precisely:
Lemma 2.3. [9, Lem. 1]
Let C be a connected compact subset of R not reduced to a point, and let O n the closures of smooth Jordan domains, converging to C in the Hausdorff sense. For any ρ > such that C ⊂ B (0 , ρ ) , there exists C ρ and N ρ such that k ϕ k L ( B (0 ,ρ ) \ O n ) ≤ C ρ k∇ ϕ k L ( B (0 ,ρ ) \ O n ) , ∀ ϕ ∈ C ∞ c ( R \ O n ) , ∀ n ≥ N ρ . Now, we focus on the initial data approximation. Let u verifying (1.8)-(1.9). We write ω := curl u and consider after truncation and convolution a sequence ω n such that ω n ∈ C ∞ c (Ω n ) , k ω n k L p (Ω n ) ≤ k ω k L p (Ω) , ω n → ω strongly in L ( R ) ∩ L p ( R ) for finite p, in L q ( R ) for all finite q if p = ∞ . (2.2)Here, the functions defined on Ω n are implicitly extended by zero, so that convergence results arestated in R . Let us note that the assumption ω n ∈ C ∞ c (Ω n ) is easy to achieve thanks to (2.1): anyfunction compactly supported in Ω is compactly supported in Ω n for n large enough.When the domain is not simply connected, the vorticity is not sufficient to determine uniquely thevelocity: we also need to specify the circulations around the obstacles. Due to our irregular domainΩ, we need to define these circulations in a weak sense. Therefore, we introduce some smooth cutofffunctions χ i,ε such that χ i,ε ≡ C i,ε , χ i,ε ≡ R \ C i, ε , where C i,ε := { x, d ( x, C i ) ≤ ε } . Again by (2.1), we can fix ε and n ( ε ) such that for all n ≥ n ( ε ): χ i,ε ≡ O in , χ i,ε ≡ O jn ∀ j = i. For brevity, we drop the subscript ε . Then, for any v ∈ L (Ω) with curl v ∈ L , we define the weakcirculation of v around C i by: γ i ( v ) := − Z Ω χ i curl v − Z Ω v · ∇ ⊥ χ i . (2.3)Let us note that this definition is independent of the choice of χ i . Also, when Ω and v are smoothenough, this weak definition coincides with the standard definition of the circulation around C i : γ i ( v ) = I ∂ C i v · τ. The above integral is considered in the counter clockwise sense, hence τ = − ν ⊥ for ν the normalvector pointing inside the obstacles.Finally, we remark that by our regularity assumption on u , γ i ( u ) is well-defined. It is nowa classical result that, given any vorticity ω n ∈ C ∞ c (Ω n ) and real numbers γ i , i = 1 . . . k , thereexists a unique divergence free and tangent vector field u n over Ω n such that curl u n = ω n and H ∂O in u n · τ ds = γ i ( u ) for all i .Actually, we will prove in Subsection 4.2 that this kind of characterization also holds for moreirregular domains and data: for a given ω ∈ L ∩ L p (Ω) and γ ∈ R k , there exists a unique u verifying(1.8)-(1.9) such that curl u = ω in D ′ (Ω) , γ i ( u ) = γ i ∀ i = 1 . . . k. After these preliminary considerations, the main point will be to show that the Euler flow u n generated by u n in Ω n converges to an Euler flow u generated by u in Ω. More precisely, our maintheorem concerning existence of global weak solution for initial velocity u (resp. ω , γ ) will be adirect consequence of the following stability result: Theorem 2.4.
Assume that Ω is of type (1.6) - (1.7) . Let p ∈ (1 , ∞ ] and u as in (1.8) - (1.9) (resp.let ω ∈ L ∩ L p (Ω) and γ ∈ R k ). For any sequences: D. G´ERARD-VARET & C. LACAVE a) Ω n of smooth domains (as in Proposition 2.1) converging to Ω ; b) ω n ∈ C ∞ c (Ω n ) , uniformly bounded in L p ( R ) such that ω n → ω = curl u strongly in L ( R ) ; c) γ n ∈ R k such that γ n → γ ( u ) ;we consider the unique strong solution ( u n , ω n = curl u n ) of the Euler equations on Ω n with initialvorticity ω n and initial circulations γ n (Kikuchi [15] ). Then we can extract a subsequence such that ω n ⇀ ω weak- ∗ in L ∞ ((0 , ∞ ); L q ( R )) for any q ∈ [1 , p ] ; u n ⇀ u weak- ∗ in L ∞ ((0 , ∞ ); L (Ω)) ; ω = curl u in D ′ ( R + × Ω) and u is a global weak solution of (1.1) - (1.4) in the sense of (1.10) and (1.11) with initial data u (resp. ω , γ ). The convergences are considered by extending u n and ω n by zero on ∪ O in . Of course, if Ω is smoothand p = ∞ , the uniqueness result of Yudovich [13] allows us to state that any subsequence convergesto the unique weak solution, hence the previous theorem holds without extraction of a subsequence.The two following sections are dedicated to the proof of Theorem 2.4.3. Uniform estimates
Let Ω n , ω n and γ n verifying the assumptions of Theorem 2.4. By results of Kikuchi [15], there existsa unique global strong solution ( u n , ω n ) of the Euler equations (1.1)-(1.4). In this smooth setting, thetransport of vorticity through equation (1.5) implies that • the L q norm of the vorticity is conserved: k ω n ( t, · ) k L q (Ω n ) = k ω n k L q (Ω n ) , ∀ q ∈ [1 , ∞ ] , ∀ t ∈ R + ; (3.1) • the mass of the vorticity is conserved: Z Ω n ω n ( t, · ) = Z Ω n ω n , ∀ t ∈ R + . (3.2)Moreover, the Kelvin’s theorem gives the conservation of the circulation: I ∂O in u n ( t, · ) · τ ds = γ in , ∀ t ∈ R + , ∀ i = 1 , . . . , k. (3.3)By assumption b) in Theorem 2.4 and by (3.1), we have easily that ω n is uniformly bounded in L ∞ ( R + ; L ∩ L p ( R )) . (3.4)3.1. Biot-Savart decomposition.
We reconstruct here the velocity in terms of the vorticity andthe circulations. By results related to the Hodge-De Rham theorem, there exists a unique vector field u n ( t, · ) which is tangent to the boundary, divergence free, tending to zero at infinity and whose curlis ω n ( t, · ) and circulations γ n : u n ( t, x ) = ∇ ⊥ ψ n ( t, x ) + k X i =1 α in ( t ) ∇ ⊥ ψ in ( x ) . (3.5)In this decomposition, ψ n satisfies the Dirichlet problem:∆ ψ n = ω n in Ω n , ψ n | ∂ Ω n = 0 , ψ n ( x ) = O (1) as x → ∞ whereas ψ in are harmonic functions, that satisfy:∆ ψ in = 0 in Ω n , ∂ τ ψ in | ∂ Ω n = 0 , I ∂ O jn ∇ ⊥ ψ in · τ = δ i,j for j = 1 , . . . , k,ψ in ( x ) = 12 π ln | x | + O (1) as x → ∞ . As this vector field can be chosen up to a constant, we can further assume that ψ in = 0 on ∂O in . HE TWO DIMENSIONAL EULER EQUATIONS ON SINGULAR EXTERIOR DOMAINS 7
In particular, the circulation conditions on the ψ in ( i = 1 , . . . , k ) together with (3.3) lead to: α in ( t ) = γ in − Z ∂O in ∇ ⊥ ψ n · τ, (3.6)where the circulation at the r.h.s. can be expressed in a weak form similar to (2.3): Z ∂O in ∇ ⊥ ψ n · τ = − Z Ω n χ i ω n − Z Ω n ∇ ψ n · ∇ χ i = − Z R χ i ω n − Z R ∇ ψ n · ∇ χ i . (3.7) Remark 3.1.
The behavior at infinity of the stream functions is somehow classical. It can be deducedfrom the link between 2D harmonic vector fields and holomorphic functions. Namely, if v = ∇ ⊥ ψ fora harmonic stream function ψ in an open set U , then the mapping f : z = x + iy v ( x, y ) − iv ( x, y ) (= − ∂ y ψ ( x, y ) − i∂ x ψ ( x, y )) is holomorphic in U . The behaviour at infinity of ψ follows from the Laurent expansion of f at z = ∞ .We refer to [15] for more details on ψ in , i ≥ . As regards ψ n , the fact that ω n is compactly supportedimplies that ψ n is also harmonic outside a disk. Moreover, the associated holomorphic function admitsa Laurent expansion at infinity whose first term is O (1 /z ) , because ∇ ψ n ∈ L (Ω n ) (Lax-Milgram).We note here that if the vorticity is no longer compactly supported, the behavior at infinity is far lessclear. It explains the difficulties to be met in Subsection 4.2. In the case of only one obstacle ( k = 1) treated in [9], some explicit formula could be used. Itinvolved the unique Riemann mapping T n which sends the exterior of O n to the exterior of the unitdisk and satisfies T n ( ∞ ) = ∞ , T ′ n ( ∞ ) > T n ( z ) = βz + O (1) with β ∈ R + ) . In this case, we could write ψ n ( t, x ) = 12 π Z ( O n ) c ln |T n ( x ) − T n ( y ) ||T n ( x ) − T n ( y ) ∗ ||T n ( y ) | ω n ( t, y ) dy, ψ n ( t, x ) = 12 π ln |T n ( x ) | , (3.8)with notation z ∗ = z | z | . Moreover, we could use in [9] a result related to the Caratheodory theoremon the convergence of T n : Proposition 3.2. [9, Prop. 15]
Let Π be the unbounded connected component of R \ C . There is aunique biholomorphism T from Π to ( B (0 , c , satisfying T ( ∞ ) = ∞ , T ′ ( ∞ ) > . Moreover, as O n converges to C (in the Hausdorff sense), one has the following convergence properties: i) T − n converges uniformly locally to T − in ( B (0 , c . ii) T n (resp. T ′ n ) converges uniformly locally to T (resp. to T ′ ) in Π . iii) |T n | converges uniformly locally to in Ω \ Π . This proposition together with the explicit formula (3.8) was the key in [9] to get uniform estimateson the velocity. A problem that we solve below is to extend such estimates to the case of severalobstacles: k >
Harmonic part.
Let us fix i in { , . . . , k } , and let us look for local uniform estimates on ψ in .Let K ⋐ Ω. Property (2.1) states that there exists n K such that K ⋐ Ω n for any n ≥ n K .Denoting by T in the unique biholomorphism from ( O in ) c to the exterior of the unit disk satisfying T in ( ∞ ) = ∞ , ( T in ) ′ ( ∞ ) >
0, we infer from (3.8) that˜ ψ in ( x ) := 12 π ln |T in ( x ) | verifies∆ ˜ ψ in = 0 in ( O in ) c , ˜ ψ in | ∂O in = 0 , I ∂O in ∇ ⊥ ˜ ψ in · τ = 1 , ˜ ψ in ( x ) = 12 π ln | x | + O (1) as x → ∞ . By this explicit formula and thanks to Proposition 3.2, it is obvious that we have for any compactsubset K ′ of Ω: ˜ ψ in and ∇ ˜ ψ in are bounded uniformly in x ∈ K ′ , n ≥ n K ′ . (3.9) D. G´ERARD-VARET & C. LACAVE
Next, we introduce ˜ χ i := 1 − X j = i χ j (with χ j defined in (2.3)), which vanishes in a small neighborhoodof O jn for j = i . Then, we define ˆ ψ in := ψ in − ˜ ψ in ˜ χ i . As ∂ τ ψ in = 0 on ∂ Ω n , it follows that ˆ ψ in is constant on ∂O jn for any j = 1 , . . . , k (in particular ˆ ψ in = 0on ∂O in ), ∆ ˆ ψ in = − ∇ ˜ ψ in · ∇ ˜ χ i − ˜ ψ in ∆ ˜ χ i , I ∂O jn ∇ ⊥ ˜ ψ in · τ ds = 0 for all j = 1 , . . . , k , ˆ ψ in = O (1) and ∇ ˆ ψ in = O ( | x | ) at infinity. Thanks to these properties, we can perform an energy estimate: k∇ ˆ ψ in k L (Ω n ) = − Z Ω n ˆ ψ in ∆ ˆ ψ in − k X j =1 ˆ ψ in I ∂O jn ∇ ⊥ ˆ ψ in · τ ds = − Z Ω n ˆ ψ in ∆ ˆ ψ in ≤ C k ˆ ψ in k L (supp ∇ ˜ χ i ) where we have used (3.9) with K ′ = supp ∇ ˜ χ i . Thanks to the Dirichlet condition on ˆ ψ in at ∂O in , wecan use Lemma 2.3 with ρ such that supp ∇ ˜ χ i ⊂ B (0 , ρ ), hence k∇ ˆ ψ in k L (Ω n ) is bounded uniformly in n. (3.10)Applying again Lemma 2.3 with ρ such that K ⊂ B (0 , ρ ), we deduce that k ˆ ψ in k H ( B (0 ,ρ )) is bounded uniformly in n, which implies with (3.9) that: k ψ in k H ( K ) is bounded uniformly in n. This ends the proof of the local estimate of ψ in away from the boundary: ψ in belongs to H (Ω) uniformly in n and i = 1 , . . . , k . (3.11)Eventually, let us show a uniform H bound on ψ in near the boundary. To this end, we introducea smooth function χ with compact support, such that χ = 1 in a neighborhood of all obstacles C i .Then, we consider the function χ ψ in , n large enough. It satisfies∆( χ ψ in ) = 2 ∇ χ · ∇ ψ in + (∆ χ ) ψ in , from which we deduce Z Ω n |∇ ( χψ in ) | ≤ k ∇ χ · ∇ ψ in + ∆ χ ψ in k L k χ ψ in k L . Note that there is no boundary term at ∂ Ω n : indeed, we have χ = 1 in a neighborhood of ∂ Ω n , and Z ∂ Ω n ( χψ in ) τ · ∇ ⊥ ( χψ in ) = X j ψ in | ∂O jn Z ∂O jn τ · ∇ ⊥ ψ in = 0 , by the zero circulation around O jn j = i and by the Dirichlet condition on ∂O in . Moreover, in theinequality above, the first factor at the r.h.s. is supported away from the boundary of Ω n . It is thereforebounded uniformly in n , by (3.11). Extending ψ in inside all obstacles O jn by their (constant) valuesat ∂O jn we can apply Lemma 2.3 to the second factor to state that k χ ψ in k L (Ω n ) ≤ k χ ψ in k L (( O n ) c ) ≤ C k∇ ( χ ψ in ) k L (Ω n ) and we end up with k∇ ( χ ψ in ) k L (Ω n ) ≤ C This yields a uniform control of ∇ ψ in in L in a vicinity of the obstacles, and still by Lemma 2.3, auniform control of ψ in in H in a vicinity of the obstacles. Combining with (3.11), we get that ψ in is bounded uniformly in n in H (Ω) , i = 1 , . . . , k. Actually, if we consider the extension of ψ in inside all obstacles O jn by their (constant) values at ∂O jn , Lemma 2.3 gives that ψ in is bounded uniformly in n in H ( R ) , i = 1 , . . . , k. HE TWO DIMENSIONAL EULER EQUATIONS ON SINGULAR EXTERIOR DOMAINS 9
Kernel part.
As before, to get estimates on ψ n in the formula (3.5), we introduce the similarproblem in the case of only one obstacle (see (3.8)): let ˜ ψ n defined by˜ ψ n ( t, x ) := 12 π Z ( O n ) c ln |T n ( x ) − T n ( y ) ||T n ( x ) − T n ( y ) ∗ ||T n ( y ) | ω n ( t, y ) dy (3.12)which verifies ∆ ˜ ψ n = ω n in ( O n ) c , ˜ ψ n | ∂O n = 0 , ˜ ψ n ( x ) = O (1) as x → ∞ . Uniform estimates.
We do not assume that ω is compactly supported and that p > ψ n far away the boundary, thanks to theexplicit formula (3.12) and Proposition 3.2. Lemma 3.3.
For any compact subset K ′ of R \ C , we have ˜ ψ n is bounded uniformly in n , t and x ∈ K ′ . Proof.
For K ′ fixed, there exists n K ′ such that K ′ ⊂ R \ O n (see (2.1)). We will prove the uniformestimate thanks to the formula (3.12).First, we recall that |T n ( y ) ∗ | ≤ T n converges uniformly to T on K ′ (see Proposition 3.2).Moreover, there exists C = C ( T , K ′ ) such that 1 + 1 /C ≤ |T ( x ) | ≤ C on K ′ . Therefore, thereexists N such that we have for all x ∈ K ′ , n ≥ N : |T n ( x ) − T n ( y ) ∗ | ≤ |T n ( x ) | + |T n ( y ) ∗ | ≤ C + 1and |T n ( x ) − T n ( y ) ∗ | ≥ |T n ( x ) | − |T n ( y ) ∗ | ≥ C − C ) (cid:16) − |T n ( x ) ||T n ( y ) | (cid:17) ≤ |T n ( x ) − T n ( y ) ||T n ( x ) − T n ( y ) ∗ ||T n ( y ) | ≤ C (cid:16) |T n ( x ) ||T n ( y ) | (cid:17) C ) (cid:16) − C |T n ( y ) | (cid:17) ≤ |T n ( x ) − T n ( y ) ||T n ( x ) − T n ( y ) ∗ ||T n ( y ) | ≤ C (cid:16) C (cid:17) for all x ∈ K ′ , n ≥ N . We split ( O n ) c in three parts: A := T − n ( B (0 , / (4 C )) \ B (0 , , A := T − n ( B (0 , C )) \ B (0 , / (4 C ))) ,A := T − n ( B (0 , C )) c ) . In the third subdomain, it is obvious that |T n ( y ) | ≥ C ) and then (cid:12)(cid:12)(cid:12) Z A ln |T n ( x ) − T n ( y ) ||T n ( x ) − T n ( y ) ∗ ||T n ( y ) | ω n ( t, y ) dy (cid:12)(cid:12)(cid:12) ≤ C k ω n ( t, · ) k L for any x ∈ K ′ and n ≥ N . In the first subdomain, it is clear that |T n ( x ) − T n ( y ) | ≥ |T n ( x ) | − |T n ( y ) | ≥ / (4 C ) for all n ≥ N , hence12(1 + C ) 1 / (4 C )1 + 1 / (4 C ) ≤ |T n ( x ) − T n ( y ) ||T n ( x ) − T n ( y ) ∗ ||T n ( y ) | ≤ C (1 + C )which implies that (cid:12)(cid:12)(cid:12) Z A ln |T n ( x ) − T n ( y ) ||T n ( x ) − T n ( y ) ∗ ||T n ( y ) | ω n ( t, y ) dy (cid:12)(cid:12)(cid:12) ≤ C k ω n ( t, · ) k L . In the second subdomain, we note that (cid:12)(cid:12)(cid:12) ln |T n ( x ) − T n ( y ) ||T n ( x ) − T n ( y ) ∗ ||T n ( y ) | (cid:12)(cid:12)(cid:12) ≤ C + (cid:12)(cid:12)(cid:12) ln |T n ( x ) − T n ( y ) | (cid:12)(cid:12)(cid:12) so (cid:12)(cid:12)(cid:12) Z A ln |T n ( x ) − T n ( y ) ||T n ( x ) − T n ( y ) ∗ ||T n ( y ) | ω n ( t, y ) dy (cid:12)(cid:12)(cid:12) ≤ C k ω n ( t, · ) k L + k ω n ( t, · ) k L p k ln |T n ( x ) − T n ( y ) |k L p ′ ( A )0 D. G´ERARD-VARET & C. LACAVE with p ′ the conjugate of p : 1 = p + p ′ . The last norm is easy to estimate by changing variable ξ = T n ( y ): k ln |T n ( x ) − T n ( y ) |k p ′ L p ′ ( A ) = Z B (0 , C )) \ B (0 , / (4 C )) | ln |T n ( x ) − ξ || p ′ | det D T − n ( ξ ) | dξ ≤ C Z B (0 , C )) | ln | η || p ′ dη ≤ C where we have used that ( T − n ) ′ converges uniformly to ( T − ) ′ on B (0 , C )) \ B (0 , / (4 C ))(see Proposition 3.2 and the Cauchy formula) and that ( T − ) ′ is smooth on this annulus.Putting together all the estimates gives that for all n ≥ N we have k ˜ ψ n ( t, · ) k L ∞ ( K ′ ) ≤ C ( k ω n ( t, · ) k L + k ω n ( t, · ) k L p )and (3.4) allows us to end the proof. (cid:3) If there is only one obstacle, ˜ ψ n = ψ n and the previous lemma gives a uniform estimate of ψ n . Letus now extend this estimate to the case of several obstacles ( k ≥ ψ n := ψ n − ˜ ψ n , which verifies: ∆ ψ n = 0 in Ω n , ψ n | ∂ Ω n = − ˜ ψ n | ∂ Ω n , ψ n ( x ) = O (1) as x → ∞ . (3.13)To derive a uniform control on ψ n , we rely on the following maximum principle in unbounded domains: Lemma 3.4.
Let ψ n the harmonic function verifying (3.13) . Then, | ψ n ( t, x ) | ≤ sup ∂ Ω n | ˜ ψ n ( t, · ) | , ∀ t, ∀ x ∈ Ω n . Proof of Lemma 3.4.
The idea is to use an inversion mapping in order to send the unbounded domainto a bounded one. Without loss of generality, let us assume that B (0 , ρ n ) ⊂ O n with some ρ n > i ( x ) := x/ | x | maps Ω n to a bounded domain e Ω n included in B (0 , /ρ n ). Sucha function has some interesting properties which can be found e.g. in [19, Lem. 3.7]. In particular,from the properties of ψ n ( t, · ) (namely, the limit at infinity and the harmonicity), one can check that(for any fixed t ) the function g n := ψ n ( t, · ) ◦ i − verifies:∆ g n = 0 in e Ω n , g n | ∂ e Ω n = ψ n ( t, · ) ◦ i − | ∂ Ω n . Of course, a key point is that ψ n ( t, · ) has a limit at infinity, so that after inversion, g n is continuousat zero. It is then harmonic in a vicinity of zero, for instance because it satisfies there the mean-valueformula. Finally, the lemma follows from the standard maximum principle:sup Ω n | ψ n ( t, · ) | = sup e Ω n | g n | = sup ∂ e Ω n | g n | = sup ∂ Ω n | ˜ ψ n ( t, · ) | . (cid:3) By the Hausdorff convergence and the disjointness of the obstacles, there exists a compact set K of R \ C and N such that ∂O in ⊂ K for all i = 2 , . . . , k and n ≥ N . Hence, by Lemma 3.3 and˜ ψ n | ∂O n = 0, we infer that sup ∂ Ω n | ˜ ψ n | is uniformly bounded. Therefore, for any compact subset K ′ ofΩ n , Lemmas 3.3 and 3.4 imply that ψ n is bounded uniformly in n , t and x ∈ K ′ . (3.14) HE TWO DIMENSIONAL EULER EQUATIONS ON SINGULAR EXTERIOR DOMAINS 11 H estimates. From (3.14) we deduce a local H estimate away from the boundary. Indeed, forany compact subset K ′ of Ω n , we can combine (3.4), (3.14) with elliptic regularity for the equation∆ ψ n = ω n in K ′ . We get that ψ n ( t, · ) belongs to W ,p ( K ′ ) uniformly in n , t. which obviously gives the local H estimate away from the boundary: ψ n ( t, · ) belongs to H (Ω) uniformly in n and t . (3.15)Then, let us show a uniform H bound near the boundary. As for the harmonic function, we use asmooth function χ with compact support, such that χ = 1 in a neighborhood of all obstacles C i . Thefunctions χ ψ n is compactly supported and satisfies∆( χ ψ n ) = χω n + 2 ∇ χ · ∇ ψ n + (∆ χ ) ψ n , χ ψ n | ∂ Ω n = 0 , from which we infer Z Ω n |∇ ( χψ n ) | ≤ k ω n k H − (supp χ ) k χ ψ n k H + k ∇ χ · ∇ ψ n + ∆ χ ψ n k L k χ ψ n k L . For n large enough, ∇ χ is supported away from the boundary of Ω n . Therefore, we can use (3.4),(3.15) and Lemma 2.3 to conclude that k∇ ( χψ n )( t, · ) k L (Ω n ) ≤ C. Still by Lemma 2.3, we deduce a uniform control of ψ n in H in a vicinity of the obstacles. Combiningwith (3.15), we get that ψ n ( t, · ) is bounded uniformly in n and t in H (Ω) , hence extending by zero ψ n ( t, · ) is bounded uniformly in n and t in H ( R ) . Compactness
By the uniform estimates of the previous section, we can assume up to a subsequence that ψ in ⇀ ψ i weakly in H ( R ) , i = 0 , . . . , k and ψ n ⇀ ψ weakly-* in L ∞ ( R + , H ( R )) . Here, we have implicitly extended the streamfunctions ψ in inside all obstacles O jn by their (constant)values at ∂O jn . Also, extending ω n by zero, we can assume that ω n ⇀ ω weakly-* in L ∞ ( R + , L ∩ L p ( R )) . As a by-product, we can obtain the convergence of the α in weakly in L ∞ ( R + ), see (3.6)-(3.7): α in ⇀ γ i ( u ) + Z R χ i ω + Z R ∇ ψ · ∇ χ i =: α i . (4.1)where we have used assumption c) in Theorem 2.4. Finally, back to (3.5), we obtain the weak-*convergence of u n to a limit field u in L ∞ ( R + , L ( R )). This vector field has a decomposition of thetype: u = ∇ ⊥ ψ + k X i =1 α i ∇ ⊥ ψ i . It is clear that div u = 0 and curl u = ω in D ′ ( R + × Ω) . To conclude the proof of Theorem 2.4, we still need: • to prove that u satisfies (1.10). • to prove convergence of the approximate initial data u n to u . • to prove that u satisfies the momentum equation (1.11). Remark 4.1.
If we consider initial vorticities such that k ω n k L p (Ω n ) ≤ k ω k L p (Ω) (see (2.2) ), thentaking the liminf of the relation k ω n k L ∞ ( R + ,L q ( R )) = k ω n k L q ( R ) we get (1.12) . Tangency condition.
Let h ∈ H (Ω), such that ∇ h ∈ L (Ω) and h ( x ) = 0 for large x . Wemust prove that R Ω u · ∇ h = 0. Let B be a ball containing the obstacles and the support of h . Weshall prove that(1) For any χ ∈ D ( R ) with χ = 1 near all obstacles, for almost all tχ ψ ( t, · ) ∈ H (Ω) . (2) There exist constants C i,j , i, j = 1 , . . . , k such that for any χ j ∈ D ( R ) with χ j = 1 near C j and χ j = 0 near the other obstacles, χ j ( ψ i − C i,j ) ∈ H (Ω) . These statements easily imply the result. Indeed, take χ = 1 near B , χ j such that P χ j = 1 near B .Then, we can write u = ∇ ⊥ χψ + X i,j α i χ j ( ψ i − C i,j ) + X i,j α i C i,j ∇ ⊥ χ j over B. Note that the vector field in the second term of the r.h.s. vanishes identically near the obstacles.Moreover, the first term of the r.h.s. belongs to H (Ω) for a.e. t . For these times t , we introduce ψ n ∈ D (Ω) such that ψ n → χψ + X i,j α i χ j ( ψ i − C i,j ) in H (Ω)and defining v n := ∇ ⊥ ψ n + X i,j α i C i,j ∇ ⊥ χ j we get u = lim v n in L ( B ), where v n is divergence-free and zero near all obstacles. Finally, Z Ω u · ∇ h = lim n → + ∞ Z Ω v n · ∇ h = 0by a standard integration by parts. Let us now indicate how to prove properties (1) and (2).(1) The proof relies on the notion of γ -convergence of open sets. It was already the key ingredientin our former paper [9], and we refer to Appendix C in this paper for a reminder. Let Ω ′ a big opendisk such that supp χ and all obstacles are included in Ω ′ . As the number of connected componentsof R \ (Ω ′ ∩ Ω n ) remains constant,Ω ′ ∩ Ω n γ -converges to Ω ′ ∩ Ω , as n → + ∞ . Let now ϕ = ϕ ( t ) ∈ L ( R + ). The function f ϕn := Z R + ϕ ( s ) χψ n ( s, · ) ds belongs to H (Ω ′ ∩ Ω n ) , and its extension by zero converges weakly in H ( R ) to f ϕ := Z R + ϕ ( s ) χψ ( s, · ) ds. By the previous γ -convergence property, it follows that f ϕ ∈ H (Ω ′ ∩ Ω) and so f ϕ ∈ H (Ω) (rememberthat χ is supported in Ω ′ ). Taking ϕ = r [ t − r,t + r ] for any 0 < r < t , we find that F ( t ) := 12 r Z t + rt − r χψ ( s, · ) ds ∈ H (Ω) HE TWO DIMENSIONAL EULER EQUATIONS ON SINGULAR EXTERIOR DOMAINS 13 for any 0 < r < t . For all Lebesgue points t of s ψ ( s, · ) (in particular for a.e. t ) we get: χψ ( t, · ) = lim r → r Z t + rt − r χψ ( s, · ) ds ∈ H (Ω) . (2) We know that ψ in is constant at ∂O jn , and denote this constant by C i,jn .We claim that C i,jn is bounded in n . Indeed, let Ω ′ a bounded open set such that supp χ j ⋐ Ω ′ ,disjoint from the obstacles O kn , k = j . The stream function ψ in satisfies the Dirichlet problem∆ ψ in = 0 in Ω ′ \ O jn , ψ in | ∂O jn = C i,jn , ψ in | ∂ Ω ′ = ϕ in where ϕ in ( := ψ in | ∂ Ω ′ ) can be seen as a given data, uniformly bounded in H / ( ∂ Ω ′ ) (because ψ in isbounded in W − /p,p ( ∂ Ω ′ )). Then, we can write ψ in = C i,jn φ n + ψ n where φ n and ψ n are defined as the solutions of the following systems:∆ φ n = 0 in Ω ′ \ O jn , φ n | ∂O jn = 1 , φ n | ∂ Ω ′ = 0and ∆ ψ n = 0 in Ω ′ \ O jn , ψ n | ∂O jn = 0 , ψ n | ∂ Ω ′ = ϕ in . A standard energy estimate yields that ψ n is bounded in H (Ω ′ ) (after extension by zero in O jn ).Hence, C i,jn φ n = ψ in − ψ n is also bounded in H (Ω ′ ). In particular, the sequence of real numbers C i,jn k∇ φ n k L (Ω ′ ) is bounded uniformly in n . Finally, arguing along the lines of [9, pages 139-141], onecan show that lim inf n → + ∞ k∇ φ n k L (Ω ′ ) > C j ). Thus, C i,jn isbounded in n .Then, clearly, χ j ( ψ in − C i,jn ) is bounded in H (Ω ′ ∩ Ω n ) and up to a subsequence, its extension byzero weakly converges to χ j ( ψ i − C i,j ) in H ( R ). We conclude as in (1) (it is even simpler here, asthere is no time dependence).4.2. Convergence of the initial data.
The bounds and weak convergence results that we havedescribed for u n ( t, · ) are in particular true at t = 0. Thus, up to a subsequence, u n ⇀ ˜ u weakly in L ( R ), where ˜ u has the form:˜ u ( x ) := ∇ ⊥ ψ , ( x ) + k X i =1 α i, ∇ ⊥ ψ i ( x ) . (4.2)The goal of this subsection is to prove that ˜ u = u . We remind that, up to subsequences: ψ , = lim n → + ∞ ψ n | t =0 , ψ i = lim n → + ∞ ψ in , i ≥ , where convergence holds weakly in H . Moreover, as in (4.1) we have α i, = γ i ( u ) + Z R χ i ω + Z R ∇ ψ , · ∇ χ i . We claim that γ i (˜ u ) = γ i ( u ). First, we recall that ψ , | ∂ Ω = 0, in the sense of Subsection 4.1: χψ , ∈ H (Ω) . In particular, α i, = γ i ( u ) + Z Ω χ i ω + Z Ω ∇ ψ , · ∇ χ i = γ i ( u ) − γ i ( ∇ ⊥ ψ , ) , where the weak circulation is defined in (2.3). From this identity and formula (4.2), it is enough toshow that γ i ( ∇ ⊥ ψ i ) = δ ij , that is Z Ω ∇ ψ i · ∇ χ j = − δ ij . (4.3)With the same notations as in Subsection 4.1, we compute Z Ω ∇ ψ i · ∇ χ j = Z Ω ∇ (cid:0) ψ i − C i,j (cid:1) · ∇ χ j = Z R ∇ (cid:0) ψ i − C i,j (cid:1) · ∇ χ j = Z R ∇ ψ i · ∇ χ j . Meanwhile, with the same kind of computations δ ij = I ∂O jn ∇ ⊥ ψ in · τ = − Z Ω n ∇ ψ in · ∇ χ j = − Z R ∇ (cid:0) ψ in − C i,jn (cid:1) · ∇ χ j = − Z R ∇ ψ in · ∇ χ j → − Z R ∇ ψ i · ∇ χ j as n → + ∞ . Combining the last equalities, we find (4.3).We conclude that γ (˜ u ) = γ ( u ). Moreover, we have clearlydiv ˜ u = 0 and curl ˜ u = ω in D ′ (Ω) . Also, the reasoning of the previous subsection can be applied to show that ˜ u verifies the tangencycondition (1.9).To be able to conclude that ˜ u = u , we still need to deal with the behavior of these vector fieldsat infinity. By assumption (1.8) we know that u → | x | → + ∞ , but we do not have yet anyinformation about ˜ u . Therefore, we establish the following lemma: Lemma 4.2.
There exists an harmonic function h on Ω such that: ˜ u = K R [ ω ] + h and h ( x ) = O (1 / | x | ) when x → ∞ , where K R [ ω ] is the usual Biot-Savart formula in the full plane: K R [ ω ] = x ⊥ π | x | ∗ ω .Proof. As ω n is compactly supported, it is obvious that K R [ ω n ] = O (1 / | x | ), depending on the sizeof the support of ω n . Moreover, still using the compact support of ω n , we have noticed in Remark3.1 that u n = O (1 / | x | ). Therefore, h n := u n − K R [ ω n ] behaves as O (1 / | x | ) at infinity. Moreover, itis a harmonic vector field on Ω n . This means that it is divergence-free and curl-free, or equivalentlythat x + iy h n, ( x, y ) − ih n, ( x, y ) is holomorphic over Ω n . Now, by assumption b) in Theorem 2.4,we know that there is q ∈ (1 ,
2) such that k ω n − ω k L q ( R ) →
0. So the Hardy-Littlewood-Sobolevtheorem (see e.g. [25, Theo. V.1] with α = 1) implies that: k K R [ ω n − ω ] k L q ∗ ( R ) ≤ C q k ω n − ω k L q ( R ) → , where q = q ∗ + . As q ∗ belongs to (2 , ∞ ), we get that K R [ ω n ] → K R [ ω ] strongly in L ( R ) . Therefore, h n converges weakly in L ( R ) to h := ˜ u − K R [ ω ]. From the harmonicity of h n and h (identifying these harmonic fields with their holomorphic counterparts), standard application ofthe mean-value theorem yields that h n → h strongly in C (Ω). We can write a Laurent expansionfor h , and as we can compute the coefficients of this expansion from the value of h on any circle ∂B (0 , R ) (with R large enough), the strong limit implies that h = O (1 / | x | ). This ends the proof ofthe lemma. (cid:3) Notice from the previous proof that K R [ ω ] − K R [ ω n ] belongs to L q ∗ ( R ) for some q ∗ >
2. For n fixed, it is also clear that K R [ ω n ] ∈ L q ∗ ( R ), hence K R [ ω ] belongs to L q ∗ ( R ). Eventually, ˜ u is thesum of a function in L q ∗ ( R ) and a function h which goes to zero at infinity.We can now show that ˜ u = u As the difference v := ˜ u − u is curl free and has zero circulation,we infer from the same arguments as in [9, page 145] that v = ∇ p for some smooth p inside Ω.Moreover, v is the sum of a function which tends to zero at infinity plus a function in L q ∗ ( R ). HE TWO DIMENSIONAL EULER EQUATIONS ON SINGULAR EXTERIOR DOMAINS 15
From the harmonicity of v and the mean value theorem, it is easy to prove that v goes to zeroat infinity. Again, ¯ v := v − iv is holomorphic and admits a Laurent expansion P n ∈ Z c k z − k where c k = iπ R ∂B (0 ,R ) ¯ vz k − dz . First, taking the limit R → ∞ in | c k | ≤ k ¯ v k L ∞ ( ∂B (0 ,R )) R k , we infer that c k = 0 for any k ≤
0. Second, we use the standard formula2 iπc = Z ∂B (0 ,R ) ( v − iv ) dz = Z ∂B (0 ,R ) v · τ ds − i Z ∂B (0 ,R ) v · n ds. The first right hand side integral vanishes because v is a gradient. The second is also zero becauseof the tangency condition (1.9). Indeed let χ = P χ i a smooth cutoff function which is equal to 1 ina neighborhood of ∂ Ω and zero on ∂B (0 , R ) (for R large enough) then Z ∂B (0 ,R ) v · n ds = Z B (0 ,R ) ∩ Ω v · ∇ (1 − χ ) = − Z Ω v · ∇ χ = 0 . Therefore, we have c = 0 and v = O (1 / | x | ). In particular, it implies that v belongs to G (Ω) := { w ∈ L (Ω) , w = ∇ h for some h ∈ H (Ω) } . Eventually, we claim that v ∈ G (Ω) ⊥ . Indeed, let h ∈ G (Ω), and χ ∈ C ∞ c ( R ) such that χ = 1 on abig open ball B containing all obstacles. By condition (1.9), Z Ω v · ∇ h = Z Ω v · ∇ ( χh ) + Z Ω v · ∇ ((1 − χ ) h ) = Z B c v · ∇ g where g := (1 − χ ) h satisfies: g ∈ H ( B c ), ∇ g ∈ L ( B c ), g = 0 over ∂B in the trace sense. Wededuce from [8, Theorem II.7.3, page 104] that there exists g n ∈ C ∞ c ( B c ) such that ∇ g n → ∇ g in L ( B c ). From there: Z B c v · ∇ g = lim n → + ∞ Z Ω v · ∇ g n = 0still by (1.9).Hence, v = 0, ˜ u = u , so that u n ⇀ u weakly in L ( R ).4.3. Convergence in the momentum equation.
Again, the arguments of [9, page 158] can beapplied stricto sensu . In short, we take some smooth domain Ω ′ ⋐ Ω, and decompose u n = P Ω ′ u n + ∇ q n in Ω ′ where P ′ Ω is the Leray projector over divergence-free fields, tangent to ∂ Ω ′ . As u n is already divergencefree, the remaining term is the gradient of a harmonic function q n .Using the momentum equation for u n , we then get easily a uniform bound on ∂ t P Ω ′ u n in somenegative Sobolev space. The strong compactness of P Ω ′ u n follows by Aubin-Lions lemma. Eventually,to pass to the limit in the convective term div ( u n ⊗ u n ), we must show that the annoying weak productdiv ( ∇ q n ⊗ ∇ q n ) converges to 0 when integrated against smooth divergence-free fields ϕ with compactsupport in Ω ′ . This convergence follows from the algebraic identity Z Ω ′ ∇ q ⊗ ∇ q : ∇ ϕ = − Z Ω ′ ∇|∇ q | · ϕ + ∆ q ∇ q · ϕ = 0valid for all harmonic functions q . For all details, see [9, page 158]. Acknowledgements.
The authors are partially supported by the Project “Instabilities in Hydrody-namics” financed by Paris city hall (program “Emergences”) and the Fondation Sciences Math´ematiquesde Paris. The authors are also grateful to Thierry de Pauw for the short proof of Proposition 2.2.
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Universit´e Paris-Diderot (Paris 7), Institut de Math´ematiques de Jussieu - Paris RiveGauche, UMR 7586 - CNRS, Bˆatiment Sophie Germain, Case 7012, 75205 PARIS Cedex 13, France.
E-mail address : [email protected] (C. Lacave) Universit´e Paris-Diderot (Paris 7), Institut de Math´ematiques de Jussieu - Paris RiveGauche, UMR 7586 - CNRS, Bˆatiment Sophie Germain, Case 7012, 75205 PARIS Cedex 13, France.
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