On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data
aa r X i v : . [ m a t h . A P ] D ec On the motion of a rigid body in a two-dimensional ideal flow withvortex sheet initial data
Franck Sueur ∗† November 10, 2018
Abstract
A famous result by Delort about the two-dimensional incompressible Euler equations is the existence of weaksolutions when the initial vorticity is a bounded Radon measure with distinguished sign and lies in the Sobolevspace H − . In this paper we are interested in the case where there is a rigid body immersed in the fluid movingunder the action of the fluid pressure. We succeed to prove the existence of solutions `a la Delort in a particularcase with a mirror-symmetry assumption already considered by [10], where it was assumed in addition that therigid body is a fixed obstacle. The solutions built here satisfy the energy inequality and the body accelerationis bounded. We consider the motion of a body S ( t ) in a planar ideal fluid which therefore occupies at time t the set F ( t ) := R \ S ( t ). We assume that the body is a closed disk of radius one and has a uniform density ρ >
0. The equationsmodelling the dynamics of the system read ∂u∂t + ( u · ∇ ) u + ∇ p = 0 for x ∈ F ( t ) , (1.1)div u = 0 for x ∈ F ( t ) , (1.2) u · n = h ′ ( t ) · n for x ∈ ∂ S ( t ) , (1.3) mh ′′ ( t ) = Z ∂ S ( t ) p n ds, (1.4) u | t =0 = u , (1.5)( h (0) , h ′ (0)) = (0 , ℓ ) . (1.6)Here u = ( u , u ) and p denote the velocity and pressure fields, m = ρπ denotes the mass of the body while thefluid is supposed to be homogeneous of density 1, to simplified the equations, n denotes the unit outward normalon F ( t ), ds denotes the integration element on the boundary ∂ S ( t ) of the body. In the equation (1.4), h ( t ) is theposition of the center of mass of the body.The equations (1.1) and (1.2) are the incompressible Euler equations, the condition (1.3) means that theboundary is impermeable, the equation (1.4) is Newton’s balance law for linear momentum: the fluid acts on thebody through pressure force.In the system above we omit the equation for the rotation of the rigid ball, which yields that the angular velocityof the rigid body remains constant when time proceeds, since the angular velocity is not involved in the equations(1.1)-(1.6). ∗ CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France † UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France .2 Equations in the body frame We start by transferring the previous equations in the body frame. We define: v ( t, x ) = u ( t, x + h ( t )) ,q ( t, x ) = p ( t, x + h ( t )) ,ℓ ( t ) = h ′ ( t ) . so that the equations (1.1)-(1.6) become ∂v∂t + [( v − ℓ ) · ∇ ] v + ∇ q = 0 x ∈ F , (1.7)div v = 0 x ∈ F , (1.8) v · n = ℓ · n x ∈ ∂ S , (1.9) mℓ ′ ( t ) = Z ∂ S q n ds (1.10) v (0 , x ) = v ( x ) x ∈ F , (1.11) ℓ (0) = ℓ . (1.12)where S denotes the closed unit disk, which is the set initially occupied by the solid and F := R \ S is the oneoccupied by the fluid. Such a problem has been tackled by [12] in the case of a smooth initial data with finite kinetic energy, by [5] inthe case of Yudovich-like solutions (with bounded vorticities) and by [4] in the case where the initial vorticity ofthe fluid has a L pc vorticity with p >
2. The index c is used here and in the sequel for “compactly supported”.These works provided the global existence of solutions. Actually the result of [12] was extended in [11] to the caseof a solid of arbitrary form, for which rotation has to be taken into account, and the works [5] and [4] deal with anarbitrary form as well. Furthermore we will address in a separate paper the case of an initial vorticity in L pc with p >
1, in order to achieve the investigation of solutions “`a la DiPerna-Majda”, referring here to the seminal work[2] in the case of a fluid alone.It is therefore natural to try to extend these existence results to the case, more singular, of vortex sheet initialdata. In the case of a fluid alone, without any moving body, vortex sheet motion is a classical topic in fluiddynamics. Several approaches have been tried. Here we will follow the approach initiated by J.-M. Delort whoproved global-in-time existence of weak solutions for the incompressible Euler equations when the initial vorticity isa compactly supported, bounded Radon measure with distinguished sign in the Sobolev space H − . The pressuresmoothness in Delort’s result is very bad so that it could be a-priori argued that the extension to the case of animmersed body should be challenging since the motion of the solid is determined by the pressure forces exertedby the fluid on the solid boundary. However the problem (1.7)–(1.12) admits a global weak formulation where thepressure disappears. The drawback is that test functions involved in this weak formulation do not vanish on theinterface between the solid and the fluid, an unusual fact in Delort’s approach, where the solution rather satisfiesa weak formulation of the equations which involves some test functions compactly supported in the fluid domain(which is open) and the boundary condition is prescribed in a trace sense.Yet in the paper [9], the authors deal with the case of an initial vorticity compactly supported, bounded Radonmeasure with distinguished sign in H − in the upper half-plane, superimposed on its odd reflection in the lowerhalf-plane. The corresponding initial velocity is then mirror symmetric with respect to the horizontal axis. In thecourse of proving the existence of solutions to this problem, they are led to introduce another notion of weak solutionthat they called boundary-coupled weak solution, which relies on a weak vorticity formulation which involves sometest functions that vanish on the boundary, but not their derivatives. They have extended their analysis to thecase of a fluid occupying the exterior of a symmetric fixed body in [10].2 .4 Mirror symmetry In this paper, we assume that the initial velocities ℓ and v are mirror symmetric with respect to the horizontalaxis given by the equation x = 0. Our setting here can therefore be seen as an extension of the one in [10] fromthe case of a fixed obstacle to the case of a moving body.For the body velocity ℓ ∈ R the mirror symmetry entails that ℓ is of the form ℓ = ( ℓ , , F , ± := { x ∈ F / ± x > } and Γ ± := ∂ F , ± . If x = ( x , x ) ∈ F , ± then we denote ˜ x := ( x , − x ) ∈ F , ∓ . To avoid any confusion let us say here that fora smooth vector field u = ( u , u ), the mirror symmetry assumption means that for any x ∈ F , ± , ( u , u )(˜ x ) =( u , − u )( x ).This assumption has two important consequences. First the vorticity ω := curl v := ∂ v − ∂ v is odd withrespect to the variable x and therefore its integral over the fluid domain F vanishes. The other one is that thecirculation of the initial velocity around the body vanishes. In such a case, it is very natural to consider finiteenergy velocity.We will explain why in the next section by considering the link between velocity and vorticity.Let us also mention that the analysis performed here can be adapted to the case of a body occupying a smooth,bounded, simply connected closed set, which is symmetric with respect to the horizontal coordinate axis, andwhich is not allowed to rotate (for instance because of the action of an exterior torque on the body preventing anyrotation, or because the angular mass is infinite). We denote by G ( x, y ) := 12 π ln | x − y || x − y ∗ | | y | , where y ∗ := y | y ∗ | , the Green’s function of F with Dirichlet boundary condition. We also introduce the functions H ( x ) := ( x − y ) ⊥ π | x − y | and K ( x, y ) := H ( x − y ) − H ( x − y ∗ ) , (1.13)with the notation x ⊥ := ( − x , x ) when x = ( x , x ), which are the kernels of the Biot-Savart operators respectivelyin the full plane and in F . More precisely we define the operator K [ ω ] as acting on ω ∈ C ∞ c ( F ) through theformula K [ ω ]( x ) = Z F K ( x, y ) ω ( y ) dy. We will extend this definition to bounded Radon measures in the sequel but let us consider here the smooth casefirst to clarify the presentation. We also define the hydrodynamic Biot-Savart operator K H [ ω ] by K H [ ω ]( x ) = Z F K H ( x, y ) ω ( y ) dy with K H ( x, y ) := K ( x, y ) + H ( x ) . One easily verifies that lim | x | + | y |→ + ∞ K H ( x, y ) = 0 (1.14)and that H and K H [ ω ] satisfydiv H = 0 , curl H = 0 in F , H · n = 0 in ∂ S , Z ∂ S H · n ⊥ ds = − , lim | x |→ + ∞ H = 0 , (1.15)div K H [ ω ] = 0 , curl K H [ ω ] = ω in F , K H [ ω ] · n = 0 in ∂ S , Z ∂ S K H [ ω ] · n ⊥ ds = 0 , lim | x |→ + ∞ K H [ ω ] = 0 . (1.16)3et us also define the Kirchhoff potentials Φ i ( x ) := − x i | x | , which satisfies − ∆Φ i = 0 for x ∈ F , Φ i → x → ∞ , ∂ Φ i ∂ n = n i for x ∈ ∂ S , (1.17)for i = 1 ,
2, where n and n are the components of the normal vector n . Let us also observe ∇ Φ i is in C ∞ ( F ) ∩ L ( F ), and that the derivatives of higher orders of ∇ Φ i are also in L ( F ).Then we have the following decomposition result : Lemma 1.
Let ω ∈ C ∞ c ( F ) , ℓ := ( ℓ , ℓ ) ∈ R and γ ∈ R . Then there exists one only smooth divergence freevector field u such that u · n = ℓ · n on ∂ S , R ∂ S u · n ⊥ ds = γ , curl u = ω in F and such that u vanishes at infinity.Moreover u = K [ ω ] + ℓ ∇ Φ + ℓ ∇ Φ + ( α − γ ) H , where α := R F ωdx .Proof. Combining (1.15), (1.16) and (1.17) we get the existence part. Regarding the uniqueness, it is sufficient toapply [7, Lemma 2.14].Now our point is that considering some mirror symmetric velocities u and ℓ , assuming again that u is smoothwith ω := curl u in C ∞ c ( F ), one has R ∂ S u · n ⊥ ds = 0 and R F ωdx = 0, so that, according to the previouslemma, u = K [ ω ] + ℓ ∇ Φ . One then easily infers from the definitions above that u ∈ L ( F ). The kinetic energy mℓ + R F u dx of the system “fluid+body” is therefore finite. Let us also stress that u can also be written as u = K H [ ω ] + ℓ ∇ Φ . Here the advantage of using K H [ ω ] rather than K [ ω ] is that we will make use of (1.14), whichis not satisfied by K ( x, y ). Let us now define properly the Cauchy data we are going to consider in this paper. For a subset X of R wewill use the notation BM ( X ) for the set of the bounded measures over X , BM + ( X ) for the set of the positivemeasures over X , BM c ( X ) the subspace of the measures of BM ( X ) which are compactly supported in X and,following the terminology of [10], we will say that a ω ∈ BM ( F ) is nonnegative mirror symmetric (NMS) if it isodd with respect to the horizontal axis and if it is nonnegative in the upper half-plane. This means that for any φ ∈ C c ( F ; R ), Z F φ ( x ) dω ( x ) = − Z F φ (˜ x ) dω ( x ) , (1.18)with the notation of Section 1.4.We now extend the operator K [ · ] to any ω ∈ BM ( F ) by defining K [ ω ] ∈ D ′ ( F ) through the formula ∀ f ∈ C ∞ c ( F ) , < K [ ω ] , f > = Z F G ∗ curl f dω. Let ℓ , ∈ R and ℓ = ( ℓ , , ω , + ∈ BM c, + ( F , + ) and ω , − the corresponding measure in F , − obtainedby odd reflection. We then denote ω := ω , + + ω , − which is in BM ( F ) and is NMS. We define accordingly theinitial fluid velocity by v := K [ ω ] + ℓ , ∇ Φ . Let us now give a global weak formulation of the problem by considering -for solution and for test functions- avelocity field on the whole plane, with the constraint to be constant on S . We introduce the following space H = (cid:8) Ψ ∈ L ( R ); div Ψ = 0 in R , ∇ Ψ = 0 in S (cid:9) , u, v ) ρ := Z R ( ρχ S + χ F ) u · v = m ℓ u · ℓ v + Z F u · v, (1.19)where the notation χ A stands for the characteristic funtion of the set A , ℓ u ∈ R and u ∈ L ( F ) denote respectivelythe restrictions of u to S and F . Let us stress here that because, by definition of H , u is assumed to satisfythe divergence free condition in the whole plane, the normal component of these restrictions have to match on theboundary ∂ S . We will denote k · k ρ the norm associated to ( · , · ) ρ . Let us also introduce H T the set of the testfunctions Ψ in C ([0 , T ]; H ) with its restriction Ψ | [0 ,T ] ×F to the closure of the fluid domain in C c ([0 , T ] × F ). Definition 2 (Weak Solution) . Let be given v ∈ H and T > . We say that v ∈ C ([0 , T ]; H − w ) is a weaksolution of (1.7) – (1.12) in [0 , T ] if for any test function Ψ ∈ H T , (Ψ( T, · ) , v ( T, · )) ρ − (Ψ(0 , · ) , v ) ρ = Z T ( ∂ Ψ ∂t , v ) ρ dt + Z T Z F v · (( v − ℓ v ) · ∇ ) Ψ dx dt (1.20)Definition 2 is legitimate since a classical solution of (1.7)–(1.12) in [0 , T ] is also a weak solution. This followseasily from an integration by parts in space which provides( ∂ t v, Ψ) ρ = Z F v · (( v − ℓ v ) · ∇ ) Ψ dx, and then from an integration by parts in time. Our main result is the following.
Theorem 3.
Let be given a Cauchy data v ∈ H as described in Section 1.6. Let T > . Then there exists a weaksolution of (1.7) – (1.12) in [0 , T ] . In addition this solution preserves the mirror-symmetry and satisfies the energyinequality: for any t ∈ [0 , T ] , k v ( t, · ) k ρ k v k ρ . Moreover the acceleration ℓ ′ of the body is bounded in [0 , T ] . Let us slightly precise the last assertion. Actually the proof will provide a bound of k ℓ ′ k L ∞ (0 ,T ) which onlydepends on the body mass m and on the initial energy k v k ρ , but not on T .Let us also stress that it is straightforward, by an energy estimate, to prove that the weak solution above enjoysa weak-strong uniqueness property. Then, applying Th. 1 of [13], it follows that uniqueness holds for a G δ densesubset of H endowed with its weak topology.The rest of the paper is devoted to the proof of Theorem 3. A general strategy for obtaining a weak solution is to smooth out the initial data so that one gets a sequence ofinitial data which launch some classical solutions, and then to pass to the limit with respect to the regularizationparameter in the weak formulation of the equations.
Let ( η n ) n be a sequence of even mollifiers. We therefore consider the sequence of regularized initial vorticities( ω n ) n given by ω n := ( ω ) ∗ η n and some corresponding initial velocities ( v n ) n in H with v n := ℓ in S and v n := v n := K [ ω n ] + ℓ , ∇ Φ in F . Then the ( ω n ) n are smooth, compactly supported in F (at least for n large enough), NMS and bounded in L ( F ),and ( v n ) n converges weakly in H to v . 5et ( v n ) n in C ([0 , T ]; H ) be the classical solutions of (1.7)–(1.12) in [0 , T ] respectively associated to the sequence( v n ) n of initial data (cf. [12]). According to Lemma 1 the restriction v n of v n to F splits into v n = u n + ∇ Φ n where u n := K [ ω n ] and Φ n := ℓ n Φ . (2.1)Observe in particular that from now on we denote ℓ n for ℓ v n .Moreover these solutions preserve, for any t in [0 , T ], the mirror symmetry (this follows from the uniqueness ofthe Cauchy problem for classical solutions), the kinetic energy: k v n ( t, · ) k ρ = k v n k ρ , (2.2)and the L norm of the vorticity on the upper and lower half-planes: k ω n ( t, · ) k L ( F , ± ) = k ω n k L ( F , ± ) . (2.3)This last property can be obtained from the vorticity equation: ∂ t ω n + ( v n − ℓ n ) · ∇ ω n = 0 . (2.4)As already said before a classical solution is a fortiori a weak solution, thus for any test function Ψ in H T ,(Ψ( T, · ) , v n ( T, · )) ρ − (Ψ(0 , · ) , v n ) ρ = Z T ( ∂ Ψ ∂t , v n ) ρ dt + Z T Z F v n · (( v n − ℓ n ) · ∇ ) Ψ dx dt. (2.5)Using the bounds (2.2) and (2.3), we obtain that there exists a subsequence ( v n k ) k of ( v n ) which converges to v in L ∞ ((0 , T ); H ) weak* and such that ( ω n k ) k converges to ω weak* in L ∞ ((0 , T ); BM ( F )). In particular wehave that ( ℓ n k ) k converges to ℓ in L ∞ (0 , T ) weak* and that ( v n k ) k converges to v weak* in L ∞ ((0 , T ); L ( F ))weak*, where ℓ and v denote respectively the restrictions to S and F of v , and are mirror symmetric, so thatthe vector ℓ is of the form ( ℓ , ω is NMS. In particular ω has a vanishing total mass, thatis ω ( t, · )( F ) = 0 for almost every t ∈ (0 , T ). One should wonder whether or not the oddness holds in F as well,that is if (1.18) also holds true for φ ∈ C c ( F ). Actually we will see later that, for almost every time, the measure ω of the boundary vanishes, what implies a positive answer.Our goal now is to prove that the limit obtained satisfies the weak formulation (1.20). Unfortunately, the weakconvergences above are far from being sufficient to pass to the limit. We will first improve these convergences withrespect to the time variable. More precisely in the next section we will give an estimate of the body accelerationwhich will allow to obtain strong convergence in C ([0 , T ]) of a subsequence of the solid velocities. Then wewill give an estimate of the time derivative of the vorticity which will allow to obtain strong convergence in C ([0 , T ]; BM ( F ) − w ∗ ) of a subsequence of the vorticities. Finally we will pass to the limit thanks to an argumentof no-concentration of the vorticity, up to the boundary. The goal of this section is to prove the following.
Lemma 4.
The sequence (( ℓ n ) ′ ) n is bounded in L ∞ (0 , T ) .Proof. Let ℓ be in R . Then we define Ψ in H by setting Ψ = ℓ in S and Ψ = ∇ ( ℓ Φ + ℓ Φ ) in F . Therefore, v n being a classical solution of the system (1.7)–(1.12), one has( ∂ t v n , Ψ) ρ = Z F v n · (( v n − ℓ n ) · ∇ ) Ψ dx. (2.6)By using the definition of the scalar product in (1.19), (1.17) and the boundary condition (1.9) we obtain( ∂ t v n , Ψ) ρ = ℓ T M ( ℓ n ) ′ , with M := mId + ( Z F ∇ Φ i · ∇ Φ j dx ) i,j , × ∇ Ψ is in L ( F ) ∩ L ∞ ( F ) and (2.2) to get that the right hand side of (2.6) is boundeduniformly in n . Therefore ( ℓ n ) ′ is bounded in L ∞ (0 , T ).In particular we deduce from this, (2.2) and Ascoli’s theorem that there exists a subsequence, that we stilldenote ( ℓ n k ) k , which converges strongly to ℓ in C ([0 , T ]). Moreover by weak compactness, we also have that(( ℓ n k ) ′ ) k converges to ℓ ′ in L ∞ (0 , T ) weak*. The main difficult term to pass to the limit into (2.5) is the third one because of its nonlinear feature. We first use(2.1) to obtain for any test function Ψ in H T , Z T Z F v n · (( v n − ℓ n ) · ∇ ) Ψ dx dt = T n + T n + T n where T n := Z T Z F u n · (( u n ) · ∇ ) Ψ dx dt,T n := Z T Z F u n · (( ∇ Φ n − ℓ n ) · ∇ ) Ψ dx dt,T n := Z T Z F ∇ Φ n · (( u n + ∇ Φ n − ℓ n ) · ∇ ) Ψ dx dt. From what precedes we infer that ( T n k ) k and ( T n k ) k converge respectively to T and T , where T := Z T Z F u · (( ∇ Φ − ℓ ) · ∇ ) Ψ dx dt, T := Z T Z F ∇ Φ · (( u + ∇ Φ − ℓ ) · ∇ ) Ψ dx dt, where Φ := ℓ Φ .The term T n is more complicated. We would like to use vorticity to deal with this term, as in Delort’s methodwhere ruling out vorticity concentrations (formation of Dirac masses) allows to deal with the nonlinearity. Howeverthere is a difference here: the test function Ψ involved in the term T n is not vanishing in general in the neighborhoodof the boundary ∂ F . We will use several arguments to fill this gap. In the next section we point out the roleplayed by the normal trace of test functions. Let us start with the following lemma.
Lemma 5.
Let ω be smooth compactly supported in F such that u := K [ ω ] is in L ( F ) . Then, for Ψ ∈ C c ( F ) divergence free, Z F u · ( u · ∇ Ψ) dx = − Z ∂ S | u · n ⊥ | Ψ · n + Z F ωu · Ψ ⊥ . (2.7) Assume in addition that ω is NMS, then, also for Ψ ∈ C c ( F ) divergence free, Z F ± u · ( u · ∇ Ψ) dx = − Z Γ ± | u · n ⊥ | Ψ · n + Z F ± ωu · Ψ ⊥ . (2.8) Proof.
Let us focus on the proof of (2.8); the proof of (2.7) being similar. First we observe that u is smooth,divergence free, in L ( F ) and is tangent to Γ ± (since ω is NMS). Now, using that u and Ψ are divergence free, weobtain u · ( u · ∇ Ψ) = u ⊥ · ∇ (Ψ ⊥ · u ) + Ψ · ∇ ( 12 | u | ) . (2.9)7herefore integrating by parts, using that u is tangent to Γ ± , that div u ⊥ = − ω and that Ψ is divergence free, weget the desired result.Let us first recall what happens when Ψ is in C c ( F ). This will already provide some useful informations inthe next section. Lemma 6.
Let ω in BM ( F ) , diffuse (that is ω ( { x } ) = 0 for any x ∈ F ), with vanishing total mass, such that u := K [ ω ] ∈ L ( F ) . Let Ψ ∈ C c ( F ) divergence free. Then Z F u · ( u · ∇ Ψ) dx = − Z Z F ×F H Ψ ⊥ ( x, y ) dω ( x ) dω ( y ) , (2.10) where H f ( x, y ) := f ( x ) · K H ( x, y ) + f ( y ) · K H ( y, x ) . When ω is smooth, the previous lemma follows from Lemma 5: it suffices to plug the definition of the Biot-Savartoperator in the second term of the right hand side of (2.7) and to symmetrize. The gain of this symmetrizationis that the auxiliary function H f ( x, y ) is bounded, whereas the Biot-Savart kernels K ( x, y ) and K H ( x, y ) are not.More precisely it also follows from the analysis in [14] that: Proposition 7.
There exists a constant M depending only on F such that | H f ( x, y ) | M k f k W , ∞ ( F ) ∀ x, y ∈ F , x = y. (2.11) for any f ∈ C c ( F ; R ) . Proposition 7 is also true if one substitutes K ( x, y ) to K H ( x, y ) in the definition of H f above. However thechoice of K H ( x, y ) seems better since it implies the extra property that for any f ∈ C c ( F ; R ), H f is tending to0 at infinity, thanks to (1.14).Using this, one infers that Lemma 5 also holds true for any diffuse measure by a regularization process. Let usrefer again here to [14] for more details, or to the sequel of this paper where we will slightly extend this. We have the following.
Lemma 8.
There exists a subsequence ( v n k ) k of ( v n ) n which converges to v in C ([0 , T ]; H − w ) , and such that ( ω n k ) k of ( ω n ) n converges to ω := curl v in C ([0 , T ]; BM ( F ) − w ∗ ) .Proof. Let us consider a divergence free vector field Ψ in C ∞ c ( F ), so that Z F Ψ · ∂ t v n dx = (Ψ , ∂ t v n ) ρ = T n + T n + T n , where, thanks to Lemma 6, T n := − Z Z F ×F H Ψ ⊥ ( x, y ) ω n ( x ) ω n ( y ) dxdy. We can infer from Proposition 7 and (2.2) that | Z F Ψ · ∂ t v n dx | C k Ψ k H ∩ W , ∞ ( F ) . Moreover using that for any φ ∈ C ∞ c ( F ) then Ψ = ∇ ⊥ φ is a divergence free vector field in C ∞ c ( F ), we get | Z F φ · ∂ t ω n dx | = | Z F Ψ · ∂ t v n dx | C k Ψ k H ∩ W , ∞ ( F ) C k φ k H ∩ W , ∞ ( F ) . It is therefore sufficient to use the Sobolev embedding theorem and the following version of the Aubin-Lions lemmawith
M > X = L ( F ), Y = H M ( F ), the completion of C ∞ c ( F ) in the Sobolev space H M ( F ), and f n = v n ; and with2. X = C ( F ) and Y = H M +10 ( F ) and f n = ω n . Lemma 9.
Let X and Y be two separable Banach spaces such that Y is dense in X . Assume that ( f n ) n isa bounded sequence in L ∞ ((0 , T ); X ′ ) such that ( ∂ t f n ) n is bounded in L ∞ ((0 , T ); Y ′ ) . Then ( f n ) n is relativelycompact in C ([0 , T ]; X ′ − w ∗ ) . The proof of Lemma 9 is given in appendix for sake of completeness.A first consequence of the previous result is that we can pass to the limit the left hand side of (2.5): for anytest function Ψ in H T , as k → + ∞ ,(Ψ( T, · ) , v n k ( T, · )) ρ − (Ψ(0 , · ) , v n k ) ρ → (Ψ( T, · ) , v ( T, · )) ρ − (Ψ(0 , · ) , v ) ρ . Let us go back to the issue of passing to the limit the equation (2.5) for a general test function Ψ in H T . The onlyremaining issue is to pass to the limit into the term involving T n . We are going to use the following generalizationsof Lemma 6 and Proposition 7. We will denote C c,σ ( F ) the subspace of the functions in C c ( F , R ) which aredivergence free and tangent to the boundary ∂ F . Proposition 10.
There exists a constant M depending only on F such that (2.11) holds true for any f ∈ C c ( F ; R ) normal to the boundary. Proposition 7 can be proved thanks to the formula (1.13). Actually it can also be seen as a particular case of[10], Theorem 1. An extension to the case of several obstacles is given in [6].Using Proposition 10, we can obtain the following.
Lemma 11.
Let ω in BM ( F ) , diffuse (that is ω ( { x } ) = 0 for any x ∈ F ), with vanishing total mass, and suchthat u := K [ ω ] ∈ L ( F ) . Then (2.10) holds true for any Ψ ∈ C c,σ ( F ) .Proof. Let Ψ ∈ C c,σ ( F ). By mollification there exists a sequence of smooth functions ω ε , with vanishing totalmass, converging to ω weakly-* in BM ( F ) and such that u ε := K [ ω ε ] converges strongly to u in L ( F ). Moreover,for any ε , it follows from Lemma 5 that Z F u ε · ( u ε · ∇ Ψ) dx = − Z Z F ×F H Ψ ⊥ ( x, y ) dω ε ( x ) dω ε ( y ) . (2.12)As ε →
0, the left-hand side (2.12) converges to the one of (2.10). On the other hand, we use the following lemma,borrowed from [3], with X = F × F , µ ε = ω ε ⊗ ω ε , f = H Ψ ⊥ , F = { ( x, x ) / x ∈ F } , to pass to the limit the righthand side. Lemma 12.
Let X be a locally compact metric space. Let ( µ ε ) ε be a sequence in BM ( X ) converging to µ weakly-*in BM ( X ) and ( ν ε ) ε be a sequence in BM + ( X ) converging to ν weakly-* in BM ( X ) , with, for any ε , | µ ε | ν ε .Let F be a closed subset of X with ν ( F ) = 0 . Let f be a Borel bounded function in X tending to at infinity,continuous on X \ F . Then R X f dµ ε → R X f dµ . A proof of Lemma 12 is provided as an appendix for sake of completeness.Yet the test function Ψ in T n is not normal to the boundary so that we still cannot apply Lemma 11. Thefollowing Lemma, which is somehow reminiscent of the fake layer constructed in [15], allows to correct this withan arbitrarily small collateral damage. Lemma 13.
Let Ψ ∈ H . Then there exists ( ˜Ψ ε ) <ε some smooth compactly supported divergence free vectorfields on F such that ˜Ψ ε = ℓ Ψ on ∂ S and such that k∇ ˜Ψ ε k L ∞ ( F ) → when ε → + . roof. Let ξ be a smooth cut-off function from [0 , + ∞ ) to [0 ,
1] with ξ (0) = 1, ξ ′ (0) = 0 and ξ ( r ) = 0 for r > x ∈ F and 0 < ε − ˜Ψ ε ( t, x ) := ∇ ⊥ (cid:16) ξ ( ε ( | x | − ℓ ⊥ Ψ · x ) (cid:17) = ξ ( ε ( | x | − ℓ Ψ + Σ ε ( εx ) . (2.13)where we have denoted, for X ∈ R with | X | > ε ,Σ ε ( X ) := ℓ ⊥ Ψ · X ξ ′ ( | X | − ε ) 1 | X | X ⊥ . It is not difficult to see that Σ ε and ˜Ψ ε are smooth and compactly supported, and that ( k Σ ε ( · ) k Lip ( F ) ) <ε isbounded. Now that ˜Ψ ε is divergence free follows from the first identity in (2.13). Let us now use the second one.First it shows that for x in ∂ F , that is for | x | = 1, ˜Ψ ε ( x ) = ℓ Ψ ( x ). Finally, we infer from the chain rule that k∇ ˜Ψ ε k L ∞ ( F ) → ε → + . Let Ψ be in H T . Lemma 6 provides a family ( ˜Ψ ε ) <ε , the time t being here a harmless parameter. Let us alsointroduce ˇΨ ε := Ψ − ˜Ψ ε , which is in C ([0 , T ]; H ) and satisfies ˇΨ ε · n = 0 on ∂ S . We split T n into T n = ˇ T n,ε + ˜ T n,ε withˇ T n,ε := Z T Z F u n · (( u n ) · ∇ ) ˇΨ ε dx dt and ˜ T n,ε := Z T Z F u n · (( u n ) · ∇ ) ˜Ψ ε dx dt. We are going to prove that T n converges to T = ˇ T ε + ˜ T ε withˇ T ε := Z T Z F u · ( u · ∇ ) ˇΨ ε dx dt, ˜ T ε := Z T Z F u · ( u · ∇ ) ˜Ψ ε dx dt. Thanks to (2.2) and Lemma 13, lim sup n | ˜ T n,ε | + | ˜ T ε | → ε → + , so that in order to achieve the proofof Theorem 3 it is sufficient to prove that for ε >
0, ˇ T n,ε → ˇ T ε when n → ∞ .Actually we are going to first prove that for ε >
0, when n → ∞ , Z T Z Z F ×F H ( ˇΨ ε ) ⊥ ( x, y ) ω n ( x ) ω n ( y ) dxdy dt → Z T Z Z F ×F H ( ˇΨ ε ) ⊥ ( x, y ) ω ( x ) ω ( y ) dxdy dt, (2.14)and that ω ( t, · ) is diffuse for almost every time t ∈ (0 , T ). Then we will apply Lemma 11 to f = ( ˇΨ ε ) ⊥ to getˇ T n,ε = − Z T Z Z F ×F H ( ˇΨ ε ) ⊥ ( x, y ) ω n ( x ) ω n ( y ) dxdy dt. ˇ T ε = − Z T Z Z F ×F H ( ˇΨ ε ) ⊥ ( x, y ) ω ( x ) ω ( y ) dxdy dt, Therefore in order to achieve the proof of Theorem 3 it is sufficient to prove the following result, which is inspiredby the analysis in [10].
Lemma 14.
If for any compact K ⊂ F there exists C > such that for any < δ < , for any n , Z T sup x ∈ K Z B ( x,δ ) ∩F | ω n ( t, y ) | dydt C | log δ | − / , (2.15) then (2.14) holds true. roof. Let β be a smooth cut-off function in C ∞ c ( R ) such that β ( x ) = 1 for x β ( x ) = 0 for x >
2. Thendefine for δ > β δ ( x ) := β ( x/δ ). We split ˇ T n,ε intoˇ T n,ε = I n,εδ + J n,εδ when I n,εδ := − Z T Z Z F ×F (1 − β δ ( | x − y | )) H ( ˇΨ ε ) ⊥ ( x, y ) ω n ( x ) ω n ( y ) dxdy dt,J n,εδ := − Z T Z Z F ×F β δ ( | x − y | ) H ( ˇΨ ε ) ⊥ ( x, y ) ω n ( x ) ω n ( y ) dxdy dt. Let us start with I n,εδ . We are going to prove that it converges, as n → ∞ , to I εδ := − Z T Z Z F ×F (1 − β δ ( | x − y | )) H ( ˇΨ ε ) ⊥ ( x, y ) ω ( x ) ω ( y ) dxdy dt. For k >
1, let Σ k := { x ∈ F / dist( x, S ) < /k } and χ k ∈ C ∞ c ( F ; [0 , χ k ( x ) = 1 in F \ Σ k and χ k ( x ) = 0 in Σ k . We decompose, for k > I n,εδ − I εδ = D n,ε,k + D n,ε,k + D ε,k where D n,ε,k := − Z T Z Z F ×F f k,ε,δ ( x, y ) (cid:16) ω n ( x ) ω n ( y ) − ω ( x ) ω ( y ) (cid:17) dxdy dtD n,ε,k := − Z T Z Z F ×F g k,ε,δ ( x, y ) ω n ( x ) ω n ( y ) dxdy dt,D ε,k := 12 Z T Z Z F ×F g k,ε,δ ( x, y ) ω ( x ) ω ( y ) dxdy dt, with f k,ε,δ ( x, y ) := χ k ( x ) χ k ( y )(1 − β δ ( | x − y | )) H ( ˇΨ ε ) ⊥ ( x, y ) ,g k,ε,δ ( x, y ) := (1 − χ k ( x ) χ k ( y ))(1 − β δ ( | x − y | )) H ( ˇΨ ε ) ⊥ ( x, y ) . Thanks to Lemma 8 we get that ( ω n k ⊗ ω n k ) k converges to ω ⊗ ω in C ([0 , T ]; BM ( F × F ) − w ∗ ). Since f k,ε,δ ∈ C ( F × F ) we obtain that D n,ε,k converges, as n → ∞ , to 0.Now, using Proposition 10 and that supp(1 − χ k ( x ) χ k ( y )) ⊂ (cid:16) F × Σ k (cid:17) ∪ (cid:16) Σ k × F (cid:17) we obtain | D ε,k | C | ω | (Σ k ) | ω | ( F ) and | D n,ε,k | C | ω | ( F ) sup n Z T sup x ∈ K Z Σ k | ω n ( t, y ) | dydt, which both converge to 0 when k → ∞ , thanks to (2.15).Using again Proposition 10, we obtain that, for ε >
0, sup n J n,εδ and J εδ := − Z T Z Z F ×F β δ ( | x − y | ) H ( ˇΨ ε ) ⊥ ( x, y ) ω ( x ) ω ( y ) dxdy dt converges to 0 when δ →
0. This entails (2.14).
Remark 15.
We did not succeed to prove that ( ω n k ) k of ( ω n ) n converges to ω := curl v in C ([0 , T ]; BM ( F ) − w ∗ ) ,so that above we have adapted Lemma 12 rather than applied it. Let us now explain how to obtain (2.15). We will here also follow closely [10].
Lemma 16.
Let φ be a smooth function on F , + with bounded derivatives up to second order. Then there exists C > which depends only on φ , on k v n k ρ and on k ω n k L ( F ) such that Z T Z Γ + | u n · n | ∇ φ · n ⊥ ds C. roof. Using Eq. (2.4) and an integration by parts, we have ∂ t Z F , + φω n = Z F , + φ∂ t ω n = − Z F , + φ ( v n − ℓ n ) · ∇ ω n , = Z F , + ∇ φ · ( v n − ℓ n ) ω n . Using now the decomposition (2.1) we get ∂ t Z F , + φω n = I n + I n , (2.16)where I n := Z F , + ∇ φ · u n ω n I n := Z F , + ∇ φ · ( ℓ n ∇ Φ − ℓ n ) ω n . Using Lemma 5 with Ψ := ∇ ⊥ φ , we get I n = 12 Z Γ + | u · n ⊥ | ∇ φ · n ⊥ ds − Z F , + u n · ( u n · ∇ Ψ) . We now integrate in time (2.16) to get12 Z T Z Γ + | u n · n ⊥ | ∇ φ · n ⊥ ds − Z T I n + Z F , + φω n ( T, · ) − Z F , + φω n + Z T Z F , + u n · ( u n · ∇ Ψ) . It remains to use trivial bounds and (2.2)-(2.3) to conclude.Using a smooth perturbation of arctan( x ) instead of φ yields that for any compact K ⊂ Γ + , there exists C > n , Z T Z K | u n · n | ds C. (2.17)Then one infers (2.15) from (2.2) and (2.17) following exactly the proof of Lemma 2. in [10]. This achieves theproof of Theorem 3. Appendix
Proof of (2.9)
Let us denote by L := − Ψ · ∇ ( | u | ) + u · ( u · ∇ Ψ) and by R := u ⊥ · ∇ (Ψ ⊥ · u ). We extend L and R into L = X i =1 L i = − Ψ u ∂ u − Ψ u ∂ u − Ψ ( ∂ u ) u − Ψ ( ∂ u ) u + u ∂ Ψ + u u ∂ Ψ + u u ∂ Ψ + u ∂ Ψ ,R = X i =1 R i = u ( ∂ Ψ ) u + u Ψ ∂ u − u ∂ Ψ − u Ψ ∂ u − u ∂ Ψ − u Ψ ∂ u + Ψ u ∂ u + u u ∂ Ψ , and observe that L = R , L = R , L = R , L = R , L = R , L = R , L = R , L = R , where we use that u is divergence free for the first and third equalities, and that Ψ is divergence free for the fifth and last equalities.12 roof of Lemma 9 We follow the strategy of Appendix C. of [8]. Let B (0 , R ) be a ball of X ′ containing all the values f n ( t ) for all t ∈ [0 , T ], for all n ∈ N . Since X is separable, this ball is a compact metric space for the w ∗ topology. Moreover,since Y is dense in X , one distance on this metric space is given as follows: let ( φ k ) k > be a sequence of Y densein X , and define, for f, g in B (0 , R ), d ( f, g ) := X k > k | < f − g, φ k > X ′ ,X | | < f − g, φ k > X ′ ,X | . Let ε > k such that k < ε . For any t, s ∈ [0 , T ], for all n ∈ N , d ( f n ( t ) , f n ( s )) sup j k | < f n ( t ) − f n ( s ) , φ j > X ′ ,X | + ε. But using now that ( ∂ t f n ) n is bounded in L ∞ ((0 , T ); Y ′ ) we get thatsup j k | < f n ( t ) − f n ( s ) , φ j > X ′ ,X | → t − s → . Therefore the sequence ( f n ) n is equicontinuous in C ([0 , T ]; B (0 , R ) − w ∗ ). Thanks to the Arzela-Ascoli theoremwe deduce the desired result. Proof of Lemma 12
Let us denote by I ε := R X f dµ ε − R X f dµ . Let η >
0. We are going to prove that for ε small enough, | I ε | η . Let M := ν ( X ) + sup ε ν ε ( X ) which is finite by the Banach-Steinhaus theorem. Since f is assumed to be decreasing atinfinity, there exists a compact subset K of X such that | f | η/M on X \ K . Let us decompose I ε into I ε = I ε + I ε with I ε := Z X \ K f dµ ε − Z X \ K f dµ and I ε := Z K f dµ ε − Z K f dµ. First we have | I ε | η thanks to the previous choice of K . It therefore remains to prove that for ε small enough, | I ε | η .Now, let us introduce a smooth cut-off function ξ on R such that ξ ( x ) = 1 for | x | ξ ( x ) = 0 for | x | > δ > x ∈ X , β δ ( x ) := ξ ( dist( x,F ) δ ). We decompose I ε into I ε = I ε,δ + I ε,δ , where I ε,δ := Z K β δ f dµ ε − Z K β δ f dµ and I ε,δ := Z K (1 − β δ ) f dµ ε − Z K (1 − β δ ) f dµ. We have | I ε,δ | k f k ∞ ( Z K β δ dν ε + Z K β δ dν ) k f k ∞ Z K β δ dν + η for ε small enough, by weak-* convergence. Since ν ( F ) = 0 there exists δ > k f k ∞ R K β δ dν η .Now using for this δ that (1 − β δ ) f is continuous on X and that ( µ ε ) ε is converging to µ weakly-* in BM ( X ),we get | I ε,δ | η for ε small enough.Gathering all the estimates yields the result. Acknowledgements.
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