On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid
Sur le mouvement d’un corps rigide immergé dans un fluide parfait incompressible bidimensionnel
Jaime Ortega
a,b, Lionel Rosier
c,∗, Takéo Takahashi
caDepartamento de Ciencias Básicas, Universidad del Bío-Bío, Avda. Andres Bello S/N, Campus Fernando May, Chillan, Chile bCentro de Modelamiento Matemático (CMM), Universidad de Chile (UMI CNRS 2807), Avenida Blanco Encalada 2120, Casilla 170-3,
Correo 3, Santiago, Chile
cInstitut Élie Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, 54506 Vandœuvre-lès-Nancy cedex, France
Received 26 September 2005; received in revised form 16 December 2005; accepted 19 December 2005 Available online 11 July 2006
Abstract
We consider the motion of a rigid body immersed in a bidimensional incompressible perfect fluid. The motion of the fluid is governed by the Euler equations and the conservation laws of linear and angular momentum rule the dynamics of the rigid body.
We prove the existence and uniqueness of a global classical solution for this fluid–structure interaction problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid–structure interaction problem obtained by incorporating some viscosity.
©2006 Elsevier Masson SAS. All rights reserved.
Résumé
Nous étudions le mouvement d’un corps rigide immergé dans un fluide parfait incompressible bidimensionnel. Le mouvement du fluide est modélisé par les équations d’Euler, et la dynamique du corps rigide est régie par les lois de conservation des moments linéaires et angulaires. Nous prouvons l’existence et l’unicité d’une solution globale classique pour ce problème d’interaction fluide–structure. La preuve repose essentiellement sur des estimées à poids pour la vorticité associée à la solution forte d’un problème d’interaction fluide–structure obtenu en incorporant de la viscosité.
©2006 Elsevier Masson SAS. All rights reserved.
Keywords:Euler equations; Fluid–rigid body interaction; Exterior domain; Classical solutions Mots-clés :Equations d’Euler ; Interaction fluide–corps rigide ; Domaine extérieur ; Solutions classiques
* Corresponding author.
E-mail addresses:jortega@dim.uchile.cl (J. Ortega), rosier@iecn.u-nancy.fr (L. Rosier), takahash@iecn.u-nancy.fr (T. Takahashi).
0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2005.12.004
1. Introduction
In this paper we continue our investigation of the Cauchy problem for the system describing the motion of a rigid body immersed in an incompressible perfect fluid. In [27], the global existence and uniqueness of aclassical solution were established when the rigid body was aball. Here, the rigid body may take anarbitraryform. To be more precise, we assume that the rigid body fills a bounded, simply connected domainS(t )⊂R2of class C1 and piecewiseC2and which is different from a ball, and that it is surrounded by a perfect incompressible fluid. For the sake of simplicity, both the fluid and the solid are assumed to be homogeneous. The domain occupied by the fluid is denoted byΩ(t )=R2\S(t ). The dynamics of the fluid is described by the Euler equations, whereas the motion of the rigid body is governed by the balance equations for linear and angular momentum (Newton’s laws). The equations modelling the dynamics of the system read then
∂u
∂t +(u· ∇)u+ ∇p=0 inΩ(t )× [0, T], (1.1)
divu=0 inΩ(t )× [0, T], (1.2)
u·n=
h+r(x−h)⊥
·n on∂S(t )× [0, T], (1.3)
mh=
∂S(t )
pndΓ in[0, T], (1.4)
J r=
∂S(t )
(x−h(t ))⊥·pndΓ in[0, T], (1.5)
u(x,0)=a(x) ∀x∈Ω, (1.6)
h(0)=0∈R2, h(0)=b∈R2, r(0)=c∈R. (1.7)
In the above equationsu(resp.,p) is the velocity field (resp., the pressure) of the fluid, and h(resp.,r) denotes the position of the center of mass (resp., the angular velocity) of the rigid body,y⊥=(−y2, y1)ify=(y1, y2), and
∂S(t )=∂Ω(t ). Note that we have assumed the center of mass of the solid to be located at the origin at timet=0.
We have denoted bynthe unit outward normal to∂Ω(t ). The continuity equation for the velocity (1.3) means that the normal component of the velocity is the same for the fluid and the rigid body on∂S(t ). In other words, the fluid does not enter into the rigid body. The (positive) constantsmandJ are respectively themassand themoment of inertiaof the rigid body. They are defined by
m=
S
γdx, J=
S
γ|x|2dx,
whereγ denotes the (uniform) density of the rigid body. In Newton’s law (1.4) (resp., (1.5)), we notice that the only exterior force (resp., torque) applied to the rigid body is the one resulting from the fluid pressure integrated along the boundary∂S(t ). For a derivation of (1.1)–(1.5), we refer e.g. to [13].
As for many fluid–structure interaction problems, the main difficulties come from the fact that the system (1.1)–
(1.7) is nonlinear, strongly coupled and that the domain of the fluid is an unknown function of time. Several papers devoted to the study of this kind of systems have been published in the last decade. More precisely, when the dynamics of the fluid is modelled by the Navier–Stokes equations, the existence of solutions has been studied in [5,6,2,16,19, 20,15,28,7,8,33] when the fluid fills a bounded domain, and in [29,21,30,34,12] when the fluid fills the whole space.
The stationary problem was studied in [29] and in [9]. The asymptotic behavior of the solutions has been investigated (with simplified models) in [37] and in [26].
When the fluid is perfect, theonlyavailable result is the one by the authors [27] when the solid is a ball and the fluid fillsR2. Notice, however, that a theory providingclassicalsolutions to this kind of problems seems desirable for control purposes, as most of the control results for the Euler flows involve classical solutions. (See e.g. Coron [3,4], and Glass [14].)
Fig. 1. Configuration att=0.
In order to write the equations of the fluid in afixeddomain, we perform a change of variables. Denoting bySthe set occupied by the solid att=0 (see Fig. 1) and byΩ=R2\ Sthe initial domain occupied by the fluid, we set
θ (t )= t 0
r(s)ds, Q(t )=
cosθ (t ) −sinθ (t ) sinθ (t ) cosθ (t )
, (1.8)
and ⎧
⎪⎨
⎪⎩
v(y, t )=Q(t )∗u
Q(t )y+h(t ), t , q(y, t )=p
Q(t )y+h(t ), t , l(t )=Q(t )∗h(t ).
(1.9) Then, the functions(v, q, l, r)satisfy the following system
∂v
∂t +
v−l−ry⊥
· ∇
v+rv⊥+ ∇q=0 inΩ× [0, T], (1.10)
divv=0 inΩ× [0, T], (1.11)
v·n=
l+ry⊥
·n on∂S× [0, T], (1.12)
ml=
∂S
qndΓ −mrl⊥ in[0, T], (1.13)
J r=
∂S
qn·y⊥dΓ in[0, T], (1.14)
v(y,0)=a(y) ∀y∈Ω, (1.15)
l(0)=b, r(0)=c. (1.16)
The study of the Cauchy problem for the system (1.10)–(1.16) is more tricky than for the system considered in [27]
(rigid ball). When comparing both systems, we first notice the presence of the additional terms−r(y⊥· ∇)vandrv⊥ in (1.10),ry⊥·nin (1.12) and−mrl⊥in (1.13). Moreover, the angular velocityr fails to be constant here, and its dynamics, which is governed by (1.14), has to be taken into account. Besides some modifications in the computations and in the analysis (see below Lemmas 2.1, 6.1 and the section devoted to the uniqueness of the solution), the main difficulty comes from the presence of the term−r(y⊥· ∇)vin (1.10), which looks difficult to control as|y| → ∞.
The idea is to first replacey⊥by a truncated vectoryR⊥in (1.10), and next to derive appropriate estimates to pass to the limit in the modified equation. As a matter of fact, the theory of weighted estimates for singular integrals (see e.g.
[32]) does not provide any estimate of the form |y|∇v
L2(R2)Cf
|y| curlv
L2(R2),
for any choice of the weight functionf. The key observation thanks to which we shall be able to control the term
−r(y⊥· ∇)v is that|y|∇v∈L2(Ω)whenever curlv∈L∞(Ω)∩L1θ(Ω)(withθ >2)and v∈L2(Ω)(see below Proposition 2.2).
A Navier–Stokes based system similar to (1.10)–(1.16) has been recently studied in [18,11,10], but it should be noticed that the global existence of strong solutions in the 2D case has not been proved because of the termr(y⊥·∇)v.
Before stating the main result of the paper, we introduce some notations borrowed from Kikuchi [23]. IfV denotes any scalar-valued function space andu=(u1, u2)is any vector-valued function, we shall say thatu∈V ifui ∈V for alli, for the sake of simplicity. LetT be any positive number, and letQT =Ω×(0, T ).B(Ω)(resp.,B(QT)) is the Banach space of all real-valued, continuous and bounded functions defined onΩ (resp.QT), endowed with theL∞ norm. For anyθ >0,L1θ(Ω)denotes the space of (class of) measurable functionsωonΩsuch that
ωL1
θ(Ω):=
Ω
ω(y)|y|θdy <∞.
Finally, for anyλ∈(0,1),Cλ(Ω)(resp.,Cλ,0(QT)) is the space of all the functionsω∈B(Ω)(resp.,ω∈B(QT)) which are uniformly Hölder continuous in y with exponent λ on Ω (resp., on QT). Br(y)will denote the open ball inR2with centery and radiusr. At any pointy∈∂Ω (=∂S),n=(n1, n2)will denote the unit outer normal vector to∂Ωandτ =(τ1, τ2)will denote the unit tangent vectorτ= −n⊥. For any scalar-valued functionω, we set curlω=(∂ω/∂y2,−∂ω/∂y1)and∇ω=(∂ω/∂y1, ∂ω/∂y2), while for any vector-valued functionv=(v1, v2), we set curlv=∂v2/∂y1−∂v1/∂y2, divv=∂v1/∂y1+∂v2/∂y2and∇v=(∂vi/∂yj)1i,j2. The main result in this paper is the following one.
Theorem 1.1.Letθ >2,0< λ <1,a∈B(Ω)∩H1(Ω),b∈R2, andc∈R. Assume thatdiva=0,(a−b−cy⊥)· n|∂S=0,lim|y|→+∞a(y)=0, andcurla∈L1θ(Ω)∩Cλ(Ω). Then there exists a solution(v, q, l, r)of(1.10)–(1.16) such that
v,∂v
∂t,∇v∈B QT
, ∇q∈C QT
, v∈C1
[0, T], L2(Ω)
∩C
[0, T], H1(Ω) , y⊥· ∇v∈C
[0, T], L2(Ω)
, q∈C
[0, T],H1(Ω)
, l∈C1 [0, T]
and r∈C1 [0, T]
. Such a solution is unique up to an arbitrary function oftwhich may be added toq.
In the above theorem, we have denoted byH1(Ω)the homogeneous Sobolev space H1(Ω)=
q∈L2loc
Ω ∇q∈L2(Ω) ,
whereq∈L2loc(Ω)means thatq∈L2(Ω∩B0)for any open ballB0⊂R2withB0∩Ω= ∅. Notice that, with the above regularity, the solutionvsatisfies the following property
|ylim|→∞v(y, t )=0 (1.17)
uniformlywith respect tot∈ [0, T]. Indeed,v∈W1,∞(QT)∩C([0, T];L2(Ω)), which implies (1.17) thanks to a simple modification of Barbalat’s lemma.
Thekinetic energyof the system is given by E(t )=1
2ml(t )2+1
2Jr(t )2+1 2
Ω
v(y, t )2dy.
A great role will be played in the sequel by the scalar vorticityω:=curlv, which will be proved to be bounded in L1θ(Ω)∩L∞(Ω). (The initial vorticityω0:=curla∈L1θ(Ω)∩L∞(Ω)by assumption.) An integral term of the form
Ωf (ω(y, t ))dy, wheref:R→Ris any continuous function such thatf (ω)is integrable, is termed ageneralized enstrophy.
Using the regularity of the solution provided by Theorem 1.1 and the incompressibility of the flow associated with v−l−ry⊥(see below), we readily obtain the following result.
Corollary 1.2.Let(a, b, c)be as in Theorem1.1. Then the kinetic energy and all the generalized enstrophies of the solution given in Theorem1.1remain constant.
In particular, anyLp-norm of the vorticity is conserved along the flow.
A large part of the proof of Theorem 1.1 rests on the machinery developed in [23] to prove the existence of classical solutions to the Euler system in an exterior domain. However, unlike [23], a fixed-point argument cannot be applied directly to the Euler system, due to a lack of pressure estimate. On the other hand, when we compare the assumptions of our main result to those required in [23], we note that
(1) no additional assumption has to be made here in order to insure the uniqueness of the solution;
(2) the initial velocityahas to belong toH1(Ω).
The intrusion of anL2-estimate in a classical theory, which may look awkward at first sight, is nevertheless neces- sary. Indeed, the boundedness of the speed of the rigid body cannot be proved without the aid of the conservation of the kinetic energy of the system solid+fluid. Thus, a feature of the problem investigated here is that we need estimates both inL∞(Ω)and inL2(Ω).
To prove Theorem 1.1 we proceed in three steps. In the first step, we construct a strong solution of an approximated system in which the Euler equations have been replaced by the Navier–Stokes equations (with suitable boundary conditions and with y⊥ replaced by a truncated vector yR⊥ depending on some parameter R). In the second step, we demonstrate that the vorticity associated with the strong solution of the Navier–Stokes system is bounded in L∞(Ω)∩L1θ(Ω), uniformly with respect to the viscosity coefficientν and to the parameterR. These estimates, combined with a standard energy estimate, provide the velocity estimates needed to pass to the limit asR ∞and ν→0. In the final step, we prove that the solution to (1.10)–(1.16) has the regularity depicted in Theorem 1.1.
The paper is outlined as follows. Section 2 contains some preliminary results. Section 3 is devoted to the existence of strong solutions to the approximated Navier–Stokes system. In Sections 4 and 5, we prove some energy and vorticity estimates needed to pass to the limit asR→ ∞andν→0. Finally, the proof of Theorem 1.1 is given in Section 6.
2. Preliminaries
2.1. Extension of the velocity field to the plane
In the system (1.10)–(1.16), we can extendvtoR2by settingv(y, t )=l(t )+r(t )y⊥for ally∈Sand allt0.
Then divv=0 inR2× [0, T]andD(v)=0 inS× [0, T], where D(v)k,l=1
2 ∂vk
∂yl +∂vl
∂yk
.
We are led to introduce the following spaces H=
φ∈L2
R2 div(φ)=0 inR2, D(φ)=0 inS
(2.1) and
V=
φ∈H|φ|Ω∈H1(Ω)
. (2.2)
We define a scalar product inL2(R2)which is equivalent to the usual one (u, v)γ :=
Ω
u·vdx+γ
S
u·vdx.
The spacesL2(R2)andHare clearly Hilbert spaces for the scalar product(·,·)γ. Notice that for everyu∈Hthere exists a unique(lu, ru)∈R2×Rsuch thatu=lu+ruy⊥inS(see e.g. [35, Lemma 1.1, pp. 18]). It follows that for allu, v∈H
(u, v)γ =
Ω
u·vdx+mlu·lv+J rurv.
The spaceVis also a Hilbert space for the scalar product (u, v)V:=(u, v)γ+
Ω
∇u: ∇vdx.
A first technical result is the following
Lemma 2.1.Let u, v∈V and suppose thatu|Ω ∈H2(Ω)and thatcurlu=0 on∂S. Then we have the following identity
∂S
v·∂u
∂ndΓ =
∂S
κ
v−lv−rvy⊥
·
u−lu−ruy⊥ dΓ +
∂S
(rvu·τ+ruv·τ+rurvy·n)dΓ, (2.3) whereκdenotes the curvature of∂S.
Proof. For the sake of simplicity we assume that the domainSis of classC2, the extension to the general framework being straightforward. We may extendnas a vector field of classC1on a neighborhood of∂S. Since divu=0 and divv=0 inD(R2), we have that
u−lu−ruy⊥
·n=
v−lv−rvy⊥
·n=0 on∂S.
By using the above equations, we deduce that v−lv−rvy⊥
· ∇
u−lu−ruy⊥
·n
=0 on∂S. (2.4)
Since curlu=0 on∂S, we infer that v−lv−rvy⊥
·(∇n)∗
u−lu−ruy⊥ +
v−lv−rvy⊥
· ∂u
∂n+run⊥
=0 on∂S hence
v·∂u
∂n=κ
v−lv−rvy⊥
·
u−lu−ruy⊥ +
lv+rvy⊥
·∂u
∂n−ru
v−lv−rvy⊥
·n⊥. (2.5)
On the other hand, since divu=0 and curlu=0 on∂S, we have that
∂u
∂n = − ∂u
∂τ ⊥
whereτ = −n⊥. The above equation implies that
∂S
lv+rvy⊥
·∂u
∂ndΓ =
∂S
rvu·τdΓ. (2.6)
Gathering (2.5) and (2.6), we obtain the result. 2 2.2. Velocity versus vorticity
The following result, which relates the velocity of the fluid to the vorticity, the velocity of the rigid body and the circulation of the flow along∂S, will play a great role later.
Proposition 2.2.Letl∈R2,r∈R,C∈Randω∈L1(Ω)∩L∞(Ω). Then there exists a unique vector fieldv∈B(Ω) fulfilling
curlv=ω inΩ, (2.7)
divv=0 inΩ, (2.8)
v·n=
l+ry⊥
·n on∂S, (2.9)
∂S
v·τdΓ =C and (2.10)
|y|→+∞lim v(y)=0. (2.11)
Furthermore,v∈Lp(Ω)∀p∈(2,+∞],∇v∈Lp(Ω)∀p∈(1,+∞)and there exist some positive constantsKp, Kp such that
vLp(Ω)Kp
|l| + |r| + |C| + ωL1(Ω)+ ωL∞(Ω)
∀p∈(2,+∞], (2.12)
∇vLp(Ω)Kp
|l| + |r| + |C| + ωLp(Ω)
∀p∈(1,+∞). (2.13)
If in addition v∈L2(Ω)and ω∈L1θ(Ω)withθ >2, then
Ωωdy = −C,|y|∇v∈L2(Ω)and there exists some positive constantKsuch that
|y|∇v
L2(Ω)K
|l| + |r| + ωL∞(Ω)+ ωL1
θ(Ω)
. (2.14)
Proof. As the proof is very similar to the one of [27, Proposition 2.3], we limit ourselves to pointing out the main differences.
First Step: Reduction to the casel=0,r=0, andC=0.
Let us introduce R0:= sup
y∈∂S
|y|.
(i)Reduction to the casel=0andr=0.
We need the following lemma.
Lemma 2.3.Letl∈R2andr∈R. Then there exists a vector fieldd1∈C∞(R2,R2)such thatdivd1=0onR2and d1(y)=
l+ry⊥ if|y|R0, 0 if|y|R0+1.
Proof of Lemma 2.3. It is sufficient to pick any functionθ∈C∞(R+,R+)such that θ (s)=
1 ifsR0, 0 ifsR0+1,
and to setd1(y):=curl(θ (|y|) y·l⊥)+rθ (|y|)y⊥. 2
Settingv1:=v−d1, we see that (2.7)–(2.11) is changed into
curlv1=ω1:=ω−curld1 inΩ, (2.15)
divv1=0 inΩ, (2.16)
v1·n=0 on∂S, (2.17)
∂S
v1·τdΓ =C1:=C−r
∂S
y⊥·τdΓ and (2.18)
|y|→+∞lim v1(y)=0. (2.19)
Notice thatω1∈L1(Ω)∩L∞(Ω), as curld1∈C0∞(R2).
(ii)Reduction to the caseC=0.
We need the following lemma.
Lemma 2.4.There exists a vector fieldd2∈C1(R2,R2)such thatdivd2=0,d2(y)=0for|y|R0+1,d2·n=0 on∂Sand
∂Sd2·τdΓ =1.
Proof of Lemma 2.4. We pick a functionψ∈C02(R2,R+)such thatS= {y∈R2|ψ (y)1},ψ (y)=1 and∇ψ≡0 on∂S, andψ (y)=0 for|y|R0+1. Thend2(y)=curlψfulfills all the requirements of the lemma, except possibly
∂Sd2·τdΓ =1. As
∂Sd2·τdΓ =0, the last condition may be satisfied thanks to a normalization. 2 The change of unknown function
v2:=v1−C1d2 transforms (2.15)–(2.19) into
curlv2=ω2:=ω1−C1curld2 inΩ, (2.20)
divv2=0 inΩ, (2.21)
v2·n=0 on∂S, (2.22)
∂S
v2·τdΓ =0 and (2.23)
|y|→+∞lim v2(y)=0. (2.24)
Notice thatω2∈L1(Ω)∩L∞(Ω), asd2(y)=0 for|y|R0+1.
Second Step: Construction of a solution to (2.20)–(2.24).
Proceeding as in [27, Proposition 2.3], one obtains the existence and the uniqueness of the solutionv=d1+C1d2+ v2∈B(Ω)of (2.7)–(2.11).
Third Step:Lp-estimates.
The estimates (2.12) and (2.13) may be proved as in [27, Proposition 2.3]. Assume now thatv∈L2(Ω)and that ω∈L1θ(Ω), withθ >2. For eachR > R0letΩR:=Ω∩BR(0). The following result is needed.
Lemma 2.5.Letv:Ω→R2be a function such thatv∈H1(ΩR)for anyR > R0andcurlv∈L1(Ω). Assume further thatv∈L2(Ω). Then the following Stokes’ formula holds true
Ω
curlvdy= −
∂S
v·τdΓ. (2.25)
Proof of Lemma 2.5. An application of the usual Stokes’ formula inΩRyields
ΩR
curlvdy= −
|y|=R
v·τdΓ −
∂S
v·τdΓ, (2.26)
whereτ := −n⊥ andn denotes the unit outward normal to∂ΩR. Asv∈L2(Ω), there exists a sequenceRn ∞ such thatεn:=Rn
|y|=Rn|v|2dΓ →0 asn→ ∞. Hence, by Cauchy–Schwarz inequality,
|y|=Rn
v·τdΓ 2
|y|=Rn
|v|2dΓ ·
|y|=Rn
|τ|2dΓ =2π εn→0.
LettingRn→ ∞in (2.26) yields (2.25), since curlv∈L1(Ω). 2 It follows from Lemma 2.5 and (2.7), (2.10) that
Ω
ωdy= −
∂S
v·τdΓ = −C, hence
Ωω2(y)dy=0 and|C|Const(ωL∞(Ω)+ ωL1
θ(Ω)).
We now turn to the estimate (2.14) forv=d1+C1d2+v2. Asd1(y)=d2(y)=0 for|y|R0+1, we only have to prove the following result.
Lemma 2.6.Letω2∈L∞(Ω)∩L1θ(Ω)with
Ωω2(y)dy=0, and letv2denote the solution of (2.20)–(2.24). Then there exists a constantK2>0 (independent ofω2)such that
|y|∇v2
L2(Ω)K2
ω2L∞(Ω)+ ω2L1
θ(Ω)
. (2.27)
Proof of Lemma 2.6. Let us introduce the following weights onR2 ρ(y)=
1+ |y|21/2
and lg(y)=ln
2+ |y|2
. (2.28)
Sinceω2∈L1θ(Ω)∩L∞(Ω)(withθ >2), we obtain (with the notations of [1]) that ω2∈W10,p(Ω):=
w∈D(Ω)|ρ(y)w(y)∈Lp(Ω)
∀p∈ [2, θ] and
ω2∈W0−1,2(Ω)=
W01,2(Ω) where
W01,2(Ω):=
w∈D(Ω)|
ρ(y)lg(y)−1
w∈L2(Ω)and∇w∈L2(Ω) .
It follows then from [1, Remark 2.11] that there exists a functionψ2∈W12,2(Ω), with ψ2∈W12,p(Ω)for allp∈ (2, θ], such that−ψ2=ω2inΩ andψ2=0 on∂S. Recall that, with the notations of [1], a functionwbelongs to W12,p(Ω)withp >2 (resp.p=2) ifρ(y)−1w(y)∈Lp(Ω)(resp.,(ρ(y)lg(y))−1w∈L2(Ω)),∂w/∂yi∈Lp(Ω)and ρ(y)∂2w/∂yi∂yj∈Lp(Ω)for alli, j. It follows thatv¯2:=curlψ2belongs toW1,p(Ω) (⊂B(Ω))for allp∈(2, θ] and it fulfills (2.20)–(2.22) and (2.24). Asv¯2∈L2(Ω)and
Ωω2(y)dy=0, we infer from Lemma 2.5 that (2.23) holds as well forv¯2, hencev2= ¯v2by [23, Lemma 2.14]. We conclude that|y|∇v2∈L2(Ω), and that (2.27) holds true. This completes the proof of Lemma 2.6 and of Proposition 2.2. 2
Remark 2.7.It may occur that|y|∇v /∈L2(Ω)whenv /∈L2(Ω). Indeed, letψ (y)= −2π1 ln|y|denote the classical fundamental solution of Laplace’s equation inR2, and letv(y):=curlψ (y)=2π1 |yy⊥|2 fory∈Ω:=R2\B1(0). Then (2.7)–(2.11) are fulfilled withω=0,l=(0,0) r=0 andC=1. It is easy to see thatv∈Lp(Ω)if and only ifp >2, and that|y|∇v∈Lp(Ω)if and only ifp >2. Note that (2.25) also fails to be true forv.
3. Navier–Stokes approximation for the fluid
To solve (1.10)–(1.16), we follow an idea of P.-L. Lions ([25]). Namely, we replace the Euler equations by the Navier–Stokes equations and we supplement the system with the boundary condition rotv=0 on∂S. As the term ry⊥· ∇vmay be unbounded withy, we first study an approximated system in which the (unbounded) vectory⊥is replaced by the (bounded) vectoryR⊥, which is defined for each numberR > R0by
yR⊥=
y⊥ if|y|R,
|Ry|y⊥ if|y|R.
We then consider the following system
∂v
∂t +
v−l−ryR⊥
· ∇
v+rv⊥−νv+ ∇q=0 inΩ× [0, T], (3.1)
divv=0 inΩ× [0, T], (3.2)
v·n=(l+ry⊥)·n on∂S× [0, T], (3.3)
curlv=0 on∂S× [0, T], (3.4)
ml=
∂S
qndΓ −mrl⊥ in[0, T], (3.5)
J r=
∂S
qn·y⊥dΓ in[0, T], (3.6)
v(y,0)=a(y) y∈Ω, (3.7)
l(0)=b, r(0)=c. (3.8)
Proceeding as in [27], we may prove the following result.
Proposition 3.1.Leta∈H1(Ω),b∈R2andc∈Rbe such that diva=0 inΩ,
a·n=(b+cy⊥)·n on∂S.
Then for anyT >0the system(3.1)–(3.8)admits a unique solution(vνR, qνR, lνR, rνR)with vνR∈L2
0, T;H2(Ω)
∩C
[0, T];H1(Ω)
∩H1
0, T;L2(Ω) , qνR∈L2
0, T;H1(Ω)
, lνR∈H1
0, T;R2
, rνR∈H1(0, T;R).
4. First passage to the limit
In this section, we pass to the limit asR→ ∞.
4.1. Some estimates
We first prove an energy estimate for the system (3.1)–(3.8).
Proposition 4.1.Leta∈H1(Ω)be a function satisfying
diva=0 inΩ and a·n=(b+cy⊥)·n on∂S. (4.1)
Then there exists a positive constantC=C(S, m, J,κL∞(∂S))independent ofRandνsuch that the unique strong solution(vRν, qνR, lνR, rνR)of(3.1)–(3.8)satisfies
Ω
vRν(y, t )2dy+mlRν(t )2+JrνR(t )2+ν t 0
Ω
∇vνR(y, s)2dyds
eCνt
Ω
a(y)2dy+m|b|2+J|c|2
∀t∈ [0, T]. (4.2)
Proof. In this proof, we drop the sub and superscripts(v=vRν)for the sake of readability.
Multiplying (3.1) byvand integrating overΩ×(0, t )for anyt < T we get t
0
Ω
∂v
∂t ·vdyds+ t 0
Ω
v−l−ryR⊥
· ∇ v
·vdyds− t 0
ν
Ω
v·vdyds+ t 0
Ω
∇q·vdyds
=I1+I2−I3+I4=0.
After some integrations by parts we obtainI2=0 and I4=
m 2|l|2+J
2r2 t
0
, hence
Ω
v(y, t )2dy+ml(t )2+Jr(t )2+2ν t 0
Ω
|∇v|2dyds
Ω
a(y)2dy+m|b|2+J|c|2+2ν t 0
∂S
∂v
∂n·vdΓ. (4.3)
According to Lemma 2.1 we have that
∂S
v·∂v
∂ndΓ =
∂S
κ|v−l−ry⊥|2dΓ +2
∂S
rv·τdΓ +
∂S
r2y·ndΓ, hence there exists a positive constantC1=C1(S,κL∞(∂S))such that
∂S
v·∂v
∂ndΓ C1
∂S
|v|2dΓ + |l|2+r2
.
Using a trace inequality, we see that there exists a positive constantC2=C2(S)such that
∂S
|v|2dΓ C2
Ω
|v|2dy+ 1 2C1
Ω
|∇v|2dy.
It follows that there exists a positive constantC=C(S, m, J,κL∞(∂S))such that t
0
∂S
v·∂v
∂ndΓ dsC
2 t
0
Ω
|v|2dyds+m t 0
|l|2ds+J t 0
r2ds
+1 2
t 0
Ω
|∇v|2dyds which, combined to (4.3), yields
Ω
v(y, t )2dy+ml(t )2+Jr(t )2+ν t 0
Ω
|∇v|2dyds
Ω
a(y)2dy+m|b|2+J|c|2+νC t
0
Ω
|v|2dyds+m t 0
|l|2ds+J t
0
r2ds
. (4.4)
An application of Gronwall’s lemma gives then
Ω
v(t )2dy+ml(t )2+Jr(t )2+ν t 0
Ω
|∇v|2dydseCνt
Ω
a(y)2dy+m|b|2+J|c|2
.
The proof is completed. 2
Let us now introduce the vorticityωνR:=curlvνR. Then ωRν ∈L2
0, T;H01(Ω)
∩C
[0, T];L2(Ω)
∩H1
0, T;H−1(Ω) . Taking the “curl” in (3.1) results in
∂ωRν
∂t −νωνR+
vRν −lνR−rνRyR⊥
· ∇ωRν −rνRDR(y): ∇vRν =0 inΩ× [0, T] (4.5) where
DR(y)
i,j:=R1|y|>R
yiyj
|y|3. (4.6)
Eq. (4.5) has to be supplemented with the boundary conditionωRν =0 on∂S×[0, T]and the initial conditionωRν(0)= curlainΩ.
The next result asserts thatωRν remains bounded inL2(0, T;H01(Ω))uniformly with respect toR.
Proposition 4.2.Leta∈H1(Ω)be a function fulfilling(4.1). Then there exists a positive constantCindependent of Rsuch that
Ω
ωRν(y, t )2dy+2ν t 0
Ω
∇ωRν(y, s)2dydsC ∀t∈ [0, T]. (4.7)
Proof. Scaling in (4.5) byω, we obtain after some integrations by parts t
0
∂ωνR
∂t , ωνR
H−1×H01
ds+ν t 0
Ω
∇ωνR2dyds− t 0
Ω
rνR
DR(y): ∇vνR
ωRν(y, s)dyds=0.
Then (4.7) follows from (4.2) and (4.6). 2 4.2. Passage to the limitR→ ∞
In what follows, we fixν >0 and we letR→ +∞. According to Propositions 4.1 and 4.2, the functionsvνR and ωRν are bounded inL2(0, T;H1(Ω))(asν >0 is kept constant) and the functionslνRandrνRare bounded inL∞(0, T ).
Therefore, there exist a sequenceRk ∞and some functionsvν∈L2(0, T;H1(Ω)),ων ∈L2(0, T;H01(Ω)),lν ∈ L∞(0, T;R2)andrν∈L∞(0, T )such that
vνRk vν inL2
0, T;H1(Ω) , ωνRk ων inL2
0, T;H01(Ω) , lνRk lν inL∞
0, T;R2 -weak∗, rνRk rν inL∞(0, T )-weak∗ ask→ +∞. Clearly
divvν=0 inΩ× [0, T] (4.8)
and
vν·n=
lν+rνy⊥
·n on∂S× [0, T]. (4.9)
We now aim to take the limit in (4.5). For anyf ∈L2(0, T;L2(Ω)), we have thatf DRk →0 inL2(0, T;L2(Ω)), hence
rνRkDRk(y): ∇vνRk0 inL2
0, T;L2(Ω) . Since div(vνRk−lRνk−rνRkyR⊥
k)=0, we obtain that vνRk−lνRk−rνRkyR⊥
k
· ∇ωRνk=div ωRνk
vνRk−lRνk−rνRkyR⊥
k
. (4.10)
Pick anyR >0. It follows from (4.5) that the sequence(∂ωRνk/∂t )is bounded inL2(0, T;H−2(ΩR)). An application of Aubin’s lemma gives that (for a subsequence)
ωνRk→ων inL2
0, T;L2(ΩR) , hence
ωνRk
vνRk−lνRk−rνRkyR⊥
k
→ων
vν−lν−rνy⊥
inD
ΩR×(0, T ) and therefore, using (4.10),
vνRk−lνRk−rνRkyR⊥
k
· ∇ωRνk→
vν−lν−rνy⊥
· ∇ων inD
ΩR×(0, T ) ask→ +∞. It follows thatων fulfills the equation
∂ων
∂t −νων+
vν−lν−rνy⊥
· ∇
ων=0 inD
Ω×(0, T ) .
Clearly, the equationsων=curlvνandων|∂S=0 are satisfied. We now turn to the initial condition. Let us introduce the Hilbert space
H:=
ϕ∈H01(Ω)2|divϕ=0 and
Ω
ϕ(y)2ρ(y)2dy <∞
