1

Introduction

We consider in this paper the incompressible α–Euler equations: X ∂t (u − α∆u) + u · ∇(u − α∆u) + (u − α∆u)j ∇uj = −∇π, div u = 0, (1.1) j

on a 2D smooth domain Ω and assuming Dirichlet boundary conditions: u ∂Ω = 0.

(1.2)

The material constant α is assumed to be positive: α > 0. We will not discuss in detail the significance of the α–Euler equations as this was addressed in many other papers. We will simply mention three important facts: • The α–Euler equations are the vanishing viscosity case of the second grade fluids found in [11]; • Like the incompressible Euler equations, the α–Euler equations describe geodesic motion on the group of volume preserving diffeomorphisms for a metric containing the H 1 norm of the velocity, see [18]. • The α–Euler equations can also be obtained via an averaging procedure in the Euler equations, see [18]. When setting α = 0 in (1.1) we formally obtain the incompressible Euler equations ∂t u + u · ∇u = −∇π,

1

div u = 0.

(1.3)

However the Dirichlet boundary conditions (1.2) are not compatible with the Euler equations. Instead we must use the no-penetration boundary conditions u · n ∂Ω = 0 (1.4) where n is the unit exterior normal to ∂Ω. A natural question is whether we have convergence of the solutions of (1.1)– (1.2) towards solutions of (1.3)–(1.4) when α → 0. The main problem in showing this convergence is that the boundary condition (1.2) is more restrictive than (1.4). Therefore boundary layers are expected to appear when passing to the limit α → 0 from (1.1)–(1.2) to (1.3)–(1.4). In addition, we cannot have strong estimates uniform in α for the solutions of (1.1). More precisely, the solutions of (1.1)–(1.2) cannot be bounded in any space where the trace to the boundary ∂Ω is well-defined. Indeed, if the solutions are bounded in such a space, then the Dirichlet condition (1.2) passes to the limit so any limit of these solutions must belong to this space and vanish on the boundary. But the solutions of (1.3)–(1.4) do not verify (1.2) even if it is imposed at the initial time, unless some very special situation occurs. Let us mention at this point that in the case of Navier boundary conditions the boundary layers are weaker and we were able to show in [4] the expected convergence, see also [5] for the case of the dimension three. The case of the Dirichlet boundary conditions was only recently dealt with, and only in dimension two and for a simplyconnected domain. More precisely, the authors of [23] were able to adapt the Kato criterion for the vanishing viscosity limit, see [20], to the case of the α → 0 limit obtaining the following result. Theorem 1.1 ([23]). Assume that Ω ⊂ R2 is simply-connected. Let u0 ∈ H 3 (Ω) be divergence free and tangent to the boundary. Assume that uα0 verifies • uα ∈ H 3 (Ω), div uα = 0 and uα = 0; 0

0

0 ∂Ω

1 2

3 • α k∇uα0 kL2 (Ω) → 0 and α 2 kuα0 kH 3 (Ω) is bounded as α → 0; • uα0 → u0 in L2 (Ω) as α → 0. Then the unique global H 3 solution of (1.1)–(1.2) with initial data

uα0 converges in 2 3 L∞ loc ([0, ∞); L (Ω)) as α → 0 towards the unique global H solution of (1.3)–(1.4) with initial data u0 . Due to the method of proof, the Kato criterion, it seems that the approach of [23] can only prove convergence towards a H 3 solution of the incompressible Euler equation. But other solutions of the Euler equations exist: the Yudovich solutions with bounded vorticity, the weak solutions with Lp vorticity and the vortex sheet solutions where the vorticity is a measure. Our first aim in this paper is to prove that convergence still holds when the limit solution is a weak solution of the Euler equation with Lp vorticity. A secondary aim is to be able to consider multiplyconnected domains and also to construct weak solutions of (1.1)–(1.2). Our main result reads as follows. Theorem 1.2. Let Ω be a smooth bounded domain of R2 and 1 < p < ∞. Let u0 ∈ W 1,p (Ω) be divergence free and tangent to the boundary. Let uα0 be such that

2

• uα0 ∈ W 3,p (Ω), div uα0 = 0 and uα0 ∂Ω = 0; 1

• α 2 k∇uα0 kL2 (Ω) and k curl(uα0 − α∆uα0 )kLp (Ω) are bounded independently of α; • uα0 → u0 in L2 (Ω) as α → 0. Then there exists a global solution uα ∈ L∞ (R+ ; W 3,p (Ω)) of (1.1)–(1.2). Moreover, there exists a subsequence of solutions uαk and a global solution u of the Euler equations (1.3)–(1.4) with initial data u0 and vorticity bounded in Lp (Ω), s,p (Ω)) for all s < 1 as curl u ∈ L∞ (R+ ; Lp (Ω)), such that uαk → u in L∞ loc ([0, ∞); W p αk → 0. A few remarks are needed to understand this result. Let us observe first that the conclusion of this theorem cannot be true for s > p1 . More precisely, the solutions uα 0

are in general unbounded in Laloc ([0, ∞); W s ,p (Ω)) for any s0 > s0 ,p

1 p

and a > 1. Indeed,

assume by absurd that is bounded in (Ω)) for some s0 > p1 and a > 1. Then the subsequence uαk from Theorem 1.2 will also converge weakly in 0 Laloc ([0, ∞); W s ,p (Ω)). It is well-known that the trace operator is well defined on 0 W s ,p (Ω) if s0 > p1 , see for example [26]. Since the trace of uαk on ∂Ω vanishes thanks uα

Laloc ([0, ∞); W

0

to (1.2), the weak convergence of uαk to u in Laloc ([0, ∞); W s ,p (Ω)) implies that the limit solution u must vanish on the boundary too. This is not true for solutions of the Euler equations unless some very special situation arises. We conclude that, in 0 general, uα is unbounded in Laloc ([0, ∞); W s ,p (Ω)) for any s0 > p1 and a > 1. A second remark is that, by Sobolev embeddings, we have that uαk → u in the 2 space L∞ loc ([0, ∞); L (Ω)) as in the result of [23]. More generally, we obtain that 1 1 s uαk → u in L∞ loc ([0, ∞); H (Ω)) for all s < min(1 − p , 2 ). A third remark is that even though we assume p < ∞ in Theorem 1.2, it is quite easy to obtain a similar result for the case p = ∞. Modifying slightly the conclusion, we can prove in this case convergence towards the Yudovich solution of the Euler equation. Moreover, the Yudovich solutions are unique so we get convergence of the whole sequence uα and not only for a subsequence uαk . More details can be found in Remark 5.1. A last remark is that for any divergence free and tangent to the boundary vector field u0 ∈ W 1,p (Ω), one can construct a family of initial data uα0 verifying the hypothesis of Theorem 1.2, see Proposition 3.5 below. Therefore, a consequence of Theorem 1.2 is that any W 1,p solution of the incompressible Euler equation is the limit when α → 0 of a sequence of W 3,p solutions of the α–Euler equations with Dirichlet boundary conditions. Let us comment now on the existence part of Theorem 1.2. The main improvement about existence of solutions is that we allow the domain Ω to be multiplyconnected and moreover we construct weak solutions. As far as we know, all previous global existence results for α–Euler or second grade fluids with Dirichlet boundary conditions are given for simply-connected domains or with some conditions on the coefficient α and the initial data, see [9, 7, 14, 15]. Here we deal with multiplyconnected domains by keeping track of the circulations of u − α∆u on the connected components of the boundary and by exploiting the transport equation that the vorticity q = curl(u − α∆u) verifies. Even for simply-connected domains the existence

3

part of Theorem 1.2 is new, although in the absence of boundaries [24] shows global existence of solutions in the full plane if the initial vorticity curl(u0 − α∆u0 ) is a bounded measure. Our initial goal was to improve the result of [23]. We achieved that in several aspects. The most important one is that we allow for much weaker solutions of the Euler equations, i.e. we prove convergence towards weak solutions with Lp vorticity instead of H 3 solutions. The second improvement is that we prove stronger convergence, i.e. we prove strong convergence in W s,p (Ω), s < p1 , uniformly in time. The third improvement is that we allow for multiply-connected domains. Our approach is quite different from that of [23]. In [23], the authors make a direct estimate of the L2 norm of uα − u via energy estimates. Here, we use compactness methods and we obtain the required estimates uniform in α by using the analyticity of the Stokes semi-group. The plan of the paper is the following. In the next section we introduce some notation, we recall some basic facts about the Stokes operator and prove some preliminary results. In Section 3 we prove our main estimates with constants independent of α. These estimates rely on the analyticity of the Stokes semi-group. In Section 4 we prove the existence part of Theorem 1.2. In Section 5 we pass to the limit α → 0 and complete the proof of Theorem 1.2. We end this paper with Section 6 where we extend Theorem 1.2 to the case of second grade fluids.

2

Notation and preliminary results

Throughout this paper C denotes a generic constant independent of α (except in Section 4 where it is allowed to depend on α) whose value may change from one relation to another. The constant 1 < p < ∞ is fixed once and for all. All function spaces are defined on Ω unless otherwise specified. We denote by Lp = Lp (Ω), H 1 = H 1 (Ω), W s,p = W s,p (Ω) the usual Sobolev spaces with the usual norms where for s non-integer W s,p is defined by interpolation. We shall also use the Hα1 norm defined by 1 kukHα1 = kuk2L2 + αk∇uk2L2 2 . The space Lpσ = Lpσ (Ω) is the subspace of Lp formed by all divergence free and tangent to the boundary vector fields. We endow Lpσ with the Lp norm. Recall that the divergence free condition allows to define the normal trace of an Lp vector field on the boundary, see [13]. We denote by P the standard Leray projector, that is the L2 orthogonal projection from L2 to L2σ . It is well-known that P extends by density to a bounded operator from Lp to Lpσ . We denote by A = −P∆ the classical Stokes operator that we view as an unbounded operator on Lpσ . It is well-known that the domain of A is D(A) = {u ∈ W 2,p ; div u = 0 and u ∂Ω = 0}. We know that for any λ ∈ C \ (−∞, 0) the operator λ + A is invertible and for any f ∈ Lpσ we have that (λ + A)−1 f ∈ D(A), see for example [16, Proposition 2.1].

4

A property that will be crucial in what follows is the analyticity of the Stokes semi-group which can be expressed in terms of the following inequality proved in [16, Theorem 1]: Theorem 2.1 ([16]). For any ε > 0 there exists a constant Cε such that for all λ ∈ C \ {0}, | arg λ| 6 π − ε, and for all f ∈ Lpσ the following inequality holds true: k(λ + A)−1 f kLp 6

Cε kf kLp . |λ|

We will also need to characterize the domains of the powers of A. The following proposition is a consequence of [17, Theorem 3] and of the results of [12]. s

s

Proposition 2.2. Let 0 6 s 6 2. We have that D(A 2 ) ,→ W s,p where D(A 2 ) is s endowed with the norm kuk = kA 2 ukLp . Moreover, s

D(A 2 ) = {f ∈ W s,p ; div f = 0 and f is tangent to the boundary } and s D(A 2 ) = {f ∈ W s,p ; div f = 0 and f ∂Ω = 0}

if

if

s<

1 p

1 s> . p

We assume that Ω is a smooth domain with holes. The boundary ∂Ω has a finite number of connected components which are closed curves. We denote by Γ the outer connected component and by Γ1 , . . . , ΓN the inner connected components. In other words, we have that ∂Ω = Γ ∪ Γ1 ∪ · · · ∪ ΓN where Γ, Γ1 , . . . , ΓN are smooth closed curves and Γ1 , . . . , ΓN are located inside Γ. We denote by n the unit outer normal to ∂Ω. We continue with a remark on the circulations of v = u − α∆u on Reach connected component of the boundary. The circulation of v on Γi is defined by Γi v · n⊥ where n⊥ = (−n2 , n1 ). Lemma 2.3. Let u ∈ L∞ ([0, T ]; W 3,p (Ω)) be a solution of (1.1)–(1.2). Then for every i ∈ {1, . . . , n} the circulation of v = u − α∆u on Γi is conserved in time. Proof. The vector field v verifies the following PDE: X ∂t v + u · ∇v + vj ∇uj = −∇π. j

We multiply by n⊥ and integrate on Γi . Recalling that u vanishes on the boundary we get Z Z Z d ⊥ ⊥ v·n + n · ∇u · v = − n⊥ · ∇π dt Γi Γi Γi P ⊥ ⊥ ⊥ where n · ∇u · v = i,j (n )i ∂i uj vj . Since n · ∇ is a tangential derivative and u vanishes on the boundary we have that n⊥ · ∇u = 0 at the boundary so the second term above vanishes. Finally, using that n⊥ is the unit tangent vector field and recalling that Γi is a closed curve we infer that the term on the right-hand side vanishes too. This completes the proof.

5

We recall now some well-known facts about the harmonic vector fields associated to the domain Ω introduced above, we refer to [21, 2, 1] and the references therein for details. We call harmonic vector field an Lp vector field defined on Ω which is divergence free, curl free and tangent to the boundary of Ω. A harmonic vector field is smooth and the space of all harmonic vector fields is finite dimensional of dimension N . A harmonic vector field is uniquely determined by its circulations on Γ1 , . . . , ΓN . A basis of the space of harmonic vector fields is given by {Y1 , . . . , YN } where Yi is the unique harmonic vector field with vanishing circulation on all Γ1 , . . . , ΓN except on Γi where the circulation must be 1. If f is a divergence free vector field tangent to the boundary we define fe to be the unique vector field of the form fe = f −

N X

ai Yi

(2.1)

i=1

where the ai are constants and fe has vanishing circulation on all Γ1 , . . . , ΓN . Equivalently, the constant ai is the circulation of f on Γi . We conclude this preliminary section with a Poincar´e-like inequality. Lemma 2.4. Suppose that f ∈ W 1,p is a divergence free vector field tangent to the boundary such that its circulation on each Γ1 , . . . , ΓN vanishes. Then there exists some constant C that depends only on Ω and p such that kf kW 1,p 6 Ck curl f kLp .

(2.2)

Proof. Since f is divergence free and tangent to the boundary, we know from classical elliptic estimates that the following inequality holds true: kf kW 1,p 6 C(kf kLp + k curl f kLp )

(2.3)

so in order to prove (2.2) it suffices to show that kf kLp 6 Ck curl f kLp .

(2.4)

Assume by absurd that (2.4) fails to be true. Then (2.4) fails for C = n so there exists a sequence fn of divergence free vector fields tangent to the boundary with vanishing circulation on each Γ1 , . . . , ΓN and such that kfn kLp = 1

and k curl fn kLp <

1 . n

Using the estimate (2.3) for fn we see that fn is bounded in W 1,p . Using the compact embedding of W 1,p into Lp we deduce that there exists a subsequence fnk and some f ∈ W 1,p such that fnk → f weakly in W 1,p and strongly in Lp . In particular kf kLp = 1. Moreover, fnk being divergence free, tangent to the boundary with vanishing circulation on each Γi and the weak convergence in W 1,p imply that so is f . Moreover, since curl fnk → 0 we have that f is also curl free. So f is a harmonic vector field with vanishing circulation on each Γi . This implies that f = 0 which is a contradiction because kf kLp = 1. This completes the proof.

6

3

Main estimate

In this section we consider some vector field u ∈ W 3,p which is divergence free and vanishing on the boundary ∂Ω. We define v = u − α∆u. The aim of this section is to estimate u as best as possible in terms of kukHα1 and of k curl vkLp with constants independent of α. To do that we will distinguish two parts in u: one part that comes from curl v and another part which comes from the circulations of v on Γ1 , . . . , ΓN . We observe first that v and Pv have the same circulation on each Γi . Indeed, the Leray decomposition says that v and Pv differ by a gradient v = Pv + ∇π 0 so

Z

⊥

Z

n ·v− Γi

Z

⊥

n · Pv = Γi

(3.1) n⊥ · ∇π 0 = 0

Γi

n⊥

where we used that is the unit tangent vector field and Γi is a closed curve. f as in relation (2.1), that is Let us define Pv f = Pv − Pv

N X

γi Yi

(3.2)

i=1

where γi is the circulation of v on Γi : Z

n⊥ · v.

γi = Γi

f Recall that the operator 1 + αA is invertible. Let us introduce u e = (1 + αA)−1 Pv so that f u e + αAe u = Pv. (3.3) Now, let us consider some scalar function q ∈ Lp . By the Biot-Savart law, there exists some divergence-free vector field f ∈ W 1,p tangent to the boundary such that curl f = q. Adding a suitable linear combination of Yi , we can further assume that f has vanishing circulation on each Γ1 , . . . , ΓN . Moreover, f is the unique vector field verifying all these properties. We denote f = S(q). This allows us to define the vector field T(q) = (1 + αA)−1 f = (1 + αA)−1 S(q). With the notation introduced above, we remark that if q = curl(u − α∆u) then T(q) = u e. Indeed, we have that f q = curl(u − α∆u) = curl v = curl Pv = curl Pv where we used (3.1), (3.2) and the fact that the Yi are curl free. The vector field f is divergence free, tangent to the boundary, has vanishing circulation on each Pv f = S(q) and the definition of T(q) allows to Γ1 , . . . , ΓN and its curl is q. So Pv −1 f=u conclude that T(q) = (1 + αA) Pv e.

7

We assume in the sequel that q = curl(u − α∆u). Let us apply the Leray projector P to the relation v = u − α∆u. We get Pv = u + αAu that is u = (1 + αA)−1 Pv. Let us apply (1 + αA)−1 to (3.2). We obtain f+ u = (1 + αA)−1 Pv = (1 + αA)−1 Pv

N N X X (1 + αA)−1 γi Yi = T(q) + γi Yiα (3.4) i=1

i=1

where we defined Yiα = (1 + αA)−1 Yi . We will estimate separately the part due q, i.e. T(q), and the Pto the vorticity α. γ Y part due to the circulations γ1 , . . . , γN , i.e. N i=1 i i We start by estimating T(q). Proposition 3.1. Let q ∈ Lp . Then T(q) ∈ W 3,p with kT(q)kW 3,p 6 C(α)kqkLp . Moreover, for any ε > 0 there exists a constant C that depends only on Ω, p and ε but not on α such that kT(q)k p1 −ε,p 6 CkqkLp , (3.5) W

kT(q)kW 1,p 6 Cα kT(q)kH 1 6 Cα

1 − 12 + 2p −ε

kqkLp ,

1 min(− 2p ,− 14 )−ε

(3.6)

kqkLp

(3.7)

and kT(q)k

H

1 , 1 )−ε min(1− p 2

6 CkqkLp .

(3.8)

Proof. Let f = S(q), i.e. f is the unique vector field which is divergence-free, tangent to the boundary, with vanishing circulation on each Γ1 , . . . , ΓN and such that curl f = q. Lemma 2.4 implies that (3.9) kf kW 1,p 6 CkqkLp . Recall that T(q) = (1 + αA)−1 f . Since f ∈ W 1,p we deduce from the classical regularity results for the (elliptic) Stokes operator (see [6]) that T(q) ∈ W 3,p . In addition, we have the bound kT(q)kW 3,p 6 C(α)kf kW 1,p 6 C(α)kqkLp . Let s ∈ (0, p1 ) (the value of s will be chosen later). We deduce from Proposition s

2.2 that f ∈ D(A 2 ) and moreover s

kA 2 f kLp 6 Ckf kW s,p 6 Ckf kW 1,p 6 CkqkLp . Recall that T(q) = (1 + αA)−1 f so T(q) + αAT(q) = f.

(3.10)

s

s

Clearly T(q) ∈ D(A 2 ) and since f also belongs to D(A 2 ) we infer that AT(q) ∈ s s D(A 2 ). We can therefore apply the operator A 2 to relation (3.10) to obtain that s

s

s

A 2 T(q) + αAA 2 T(q) = A 2 f.

8

(3.11)

s

We used above that the operators A and A 2 commute. This relation can also be written under the form s 1 1 s ( + A)A 2 T(q) = A 2 f α α or, equivalently s s 1 1 A 2 T(q) = ( + A)−1 A 2 f α α We use now the analyticity of the Stokes semi-group stated in Theorem 2.1 with λ = 1/α to deduce that s s kA 2 T(q)kLp 6 CkA 2 f kLp Going back to (3.11), we can also bound s

s

s

s

kαAA 2 T(q)kLp 6 kA 2 f kLp + kA 2 T(q)kLp 6 CkA 2 f kLp . Putting together the above estimates we get that s

s

s

kA 2 T(q)kLp + αkA1+ 2 T(q)kLp 6 CkA 2 f kLp 6 CkqkLp . s 2

According to Proposition 2.2 we have the embedding D(A ) ,→ further deduce that

W s,p

(3.12) so we can

s

kT(q)kW s,p 6 CkA 2 T(q)kLp 6 CkqkLp . This proves relation (3.5). Next, let us observe that we have the following estimate: s

kT(q)kW 2+s,p 6 CkA1+ 2 T(q)kLp . s

s

Indeed, if T(q) ∈ D(A1+ 2 ) then AT(q) ∈ D(A 2 ) so Proposition 2.2 implies again that AT(q) ∈ W s,p . The classical regularity results for the Stokes operator (see [6]) imply then that T(q) ∈ W 2+s,p with the required inequalities. Therefore relation (3.12) also yields C kT(q)kW 2+s,p 6 kqkLp . α Next, we infer by interpolation that 0 −s

1− s

s0 −s

kT(q)kW s0 ,p 6 CkT(q)kW s,p2 kT(q)kW22+s,p 6 Cα

s−s0 2

kqkLp

(3.13)

for all s0 ∈ [s, 2 + s]. We first choose s0 = 1 and s = p1 − 2ε in (3.13) and we get (3.6). We prove now the bound (3.8). If p 6 2 this relation follows from (3.5) and from 1 −ε,p 1− 1 −ε ,→ H p (for ε sufficiently small). If p > 2 we the Sobolev embedding W p p 2 have that q ∈ L ,→ L and (3.8) follows from the relation (3.5) written for p = 2. Finally, let us prove (3.7). Assume first p 6 2. Choose s0 = 2/p in (3.13) (which is 2 ,p possible because s < 1/p). Recalling the Sobolev embedding W p ,→ H 1 we deduce that s −1 kT(q)kH 1 6 CkT(q)k p2 ,p 6 Cα 2 p kqkLp . W

Choosing s = p1 − 2ε implies (3.7). The case p > 2 follows from the case p = 2 since if q belongs to Lp then it also belongs to L2 so one can use (3.6) for p = 2. This completes the proof.

9

We continue with the estimate of the part of u due to the circulations γ1 , . . . , γN . Proposition 3.2. Let u ∈ W 3,p be divergence free and tangent to the boundary. There exists a constant C that depends only on Ω and p such that N X

|γi | 6 C(kukHα1 + kqkLp ).

i=1

Proof. Let us denote g =u−u e=

N X

γi Yiα

i=1

and X=

N X

γi Yi

i=1

so that g + αAg = X.

(3.14)

e. Now we estimate the Hα1 norm of g. This requires to estimate the Hα1 norm of u To do that, let us multiply (3.3) by u e and integrate. We obtain Z Z f·u Pv e. (3.15) u·u e= ke uk2L2 + α Ae Ω

Ω

We have that

Z Ω

Z Ae u·u e=

Ω

1

1

1

A2 u e · A2 u e = kA 2 u ek2L2 = k∇e uk2L2

by the usual properties of the Stokes operator. We infer from (3.15) that Z 2 f L2 ke f·u f L2 ke ukHα1 ukL2 6 kPvk Pv e 6 kPvk ke ukHα1 = Ω

so that f L2 . ke ukHα1 6 kPvk The Sobolev embedding W 1,p (Ω) ,→ L2 (Ω) together with the bound given in (3.9) f imply (recall that f = Pv) f L2 6 CkPvk f W 1,p 6 CkqkLp . ke ukHα1 6 kPvk In the end we get kgkHα1 = ku − u ekHα1 6 kukHα1 + ke ukHα1 6 kukHα1 + CkqkLp . This implies kgkL2 6 kukHα1 + CkqkLp and

1

1

1

kA 2 gkL2 = k∇gkL2 6 α− 2 kgkHα1 6 α− 2 (kukHα1 + CkqkLp ).

10

(3.16) (3.17)

1

We apply now A− 2 to (3.14) and we take the L2 norm to obtain 1

1

1

1

1

kA− 2 XkL2 = kA− 2 g + αA 2 gkL2 6 kA− 2 gkL2 + αkA 2 gkL2 .

(3.18) 1

1

Since g = (1 + αA)−1 X and X ∈ Lpσ we have that g ∈ D(A). So A− 2 g ∈ D(A 2 ) 1 and from Proposition 2.2 we deduce that A− 2 g vanishes on the boundary. Therefore we can apply the Poincar´e inequality to deduce that 1

1

1

1

kA− 2 gkL2 6 Ck∇A− 2 gkL2 = CkA 2 A− 2 gkL2 = CkgkL2 .

(3.19)

Using relations (3.16), (3.17) and (3.19) in (3.18) yields 1

1

kA− 2 XkL2 6 C(kukHα1 +kqkLp )+α 2 (kukHα1 +CkqkLp ) 6 C(kukHα1 +kqkLp ). (3.20) Because the vector fields Y1 , . . . , YN are linearly independent, one can easily check that the application R

N

3 (a1 , . . . , aN ) 7→ kA

− 12

N X ( ai Yi )kL2 i=1

is a norm on RN . Because all norms on RN are equivalent, there exists some constant C such that N X

1

|γi | 6 CkA− 2 (

i=1

N X

1

γi Yi )kL2 = CkA− 2 XkL2 6 C(kukHα1 + kqkLp )

i=1

where we used (3.20). This completes the proof. Putting together Propositions 3.1 and 3.2 allows to estimate the full velocity. Proposition 3.3. Let u ∈ W 3,p be divergence free and vanishing on the boundary and let ε > 0. There exists a constant C that depends only on Ω, p and ε but not on α such that kuk p1 −ε,p 6 C(kqkLp + kukHα1 ), (3.21) W

kukW 1,p 6 Cα kukH 1 6 Cα

1 − 12 + 2p −ε

(kqkLp + kukHα1 ),

1 ,− 14 )−ε min(− 2p

(kqkLp + kukHα1 )

(3.22) (3.23)

and kuk

H

1 , 1 )−ε min(1− p 2

6 C(kqkLp + kukHα1 ).

(3.24)

Remark 3.4. The important thing to observe is that the power of α in (3.23) can be made strictly larger than − 21 (even if it is only slightly larger than − 12 when p is close to 1). The significance of this will be obvious later in Section 5. Indeed, we will need to show that the terms of the form α∇u∇u converge to 0 as α → 0 when kukHα1 and 1

kqkLp are bounded. The trivial bound kukH 1 6 α− 2 kukHα1 only shows that α∇u∇u is bounded while (3.23) with a power of α strictly larger than − 12 implies that α∇u∇u goes to 0.

11

Proof. We already know from Proposition 3.1 that relations (3.21)–(3.24) hold true with u replaced e on the left-hand side. It remains to show these relations with Pby u α u replaced by N i=1 γi Yi on the left-hand side. Thanks to Proposition 3.2 it suffices to show that kYiα k

W

1 p −ε,p

6 C,

kYiα kW 1,p 6 Cα

1 − 12 + 2p −ε

,

kYiα kH 1 6 Cα

1 min(− 2p ,− 14 )−ε

and kYiα k

H

1 , 1 )−ε min(1− p 2

6 C.

Since the vector fields Yi are smooth, tangent to the boundary and independent of α, these bounds can be proved exactly as in the proof of Proposition 3.1 by reasoning on Yi instead of f . The only issue is that, unlike the f of that proof, the vector fields Yi do not have vanishing circulation on each Γ1 , . . . , ΓN . Nevertheless, the only place where the vanishing of the circulations of f is used is to deduce relation (3.9). But the vector fields Yi are smooth and do not depend on α, so we can replace relation (3.9) by kYi kW 1,p 6 C. From this point the proof of Proposition 3.1 goes through by replacing everywhere kqkLp by 1, T(q) by Yiα = (1 + αA)−1 Yi and f by Yi . We end this section with a result showing that for any divergence free and tangent to the boundary vector field u0 ∈ W 1,p (Ω), one can construct a family of initial data uα0 verifying the hypothesis of Theorem 1.2. Let us denote by A2 the Stokes operator defined as an unbounded operator on L2σ . Proposition 3.5. Let u0 ∈ W 1,p (Ω) be divergence free and tangent to the boundary. Let us define uα0 = (1 + αA2 )−1 u0 . Then uα0 verifies the hypothesis of Theorem 1.2: • uα ∈ W 3,p (Ω), div uα = 0 and uα = 0; 0

• •

0

1 2

0 ∂Ω

α k∇uα0 kL2 (Ω) and k curl(uα0 − uα0 → u0 in L2 (Ω) as α → 0.

α∆uα0 )kLp (Ω) are bounded independently of α;

Proof. The classical regularity results for the (elliptic) Stokes operator (see [6]) imply that uα0 ∈ W 3,p . Moreover, uα0 is divergence free and vanishes on the boundary because it belongs to the domain of A2 . Next, we write uα0 = (1 + αA2 )−1 u0 under the form uα0 + αA2 uα0 = u0

(3.25)

which in turn implies that uα0 − α∆uα0 = u0 + ∇π0 for some π0 . Taking the curl above implies that curl(uα0 − α∆uα0 ) = curl u0 , so k curl(uα0 − α∆uα0 )kLp (Ω) is bounded independently of α.

12

Multiplying (3.25) by uα0 , integrating on Ω and using the self-adjointness of A2 yields Z 1 kuα0 k2L2 (Ω) +αkA22 uα0 k2L2 (Ω) = u0 ·uα0 6 ku0 kL2 (Ω) kuα0 kL2 (Ω) 6 ku0 k2L2 (Ω) +kuα0 k2L2 (Ω) Ω

so

1

αkA22 uα0 k2L2 (Ω) 6 ku0 k2L2 (Ω) . 1

1

Recalling that kA22 uα0 kL2 (Ω) = k∇uα0 kL2 (Ω) we infer that α 2 k∇uα0 kL2 (Ω) is bounded independently of α. Finally, let us prove that uα0 → u0 in L2 . It is well-known that there exists an orthonormal basis {Xk }k∈N of L2σ such that each Xk is an eigenvector of A2 . Let λk be the corresponding eigenvalue. We have that X ak Xk u0 = k∈N

for some sequence (ak )k∈N ∈ `2 (N). Then uα0 = (1 + αA2 )−1 u0 =

X

ak (1 + αA2 )−1 Xk =

k∈N

X k∈N

ak Xk . 1 + αλk

We conclude that kuα0 − u0 k2L2 (Ω) =

X ak − k∈N

2 ak 2 X 2 αλk = |a | →0 k 1 + αλk 1 + αλk

as α → 0

k∈N

by the dominated convergence theorem. This completes the proof.

4

Construction of the solution for fixed α

In this section the parameter α is fixed and the constants are allowed to depend on α. Our aim in this part is to prove the existence part of Theorem 1.2. More precisely, we will show the following result. Theorem 4.1. Suppose that u0 ∈ W 3,p is divergence free and vanishing on the boundary. There exists a unique global W 3,p solution u ∈ Cb0,w (R+ ; W 3,p ) of (1.1)– (1.2) with initial data u0 . Above Cb0,w stands for weakly continuous bounded functions. We now proceed with the proof of Theorem 4.1. The uniqueness part of this theorem is quite easy once we observe that, by the Sobolev embedding W 3,p ,→ W 1,∞ , the solution is Lipschitz. One can subtract the PDEs for two solutions and multiply by the difference of the solutions to observe that one can estimate the Hα1 norm of the difference and conclude by the Gronwall

13

inequality that the two solutions are equal. The argument is quite standard so we leave the details to the reader. To prove the existence of the solution, we will first find an equivalent formulation of the equations. Let γi be the circulation of v0 = u0 − α∆u0 on Γi . We know from Lemma 2.3 that γi is conserved in time. Taking the curl of (1.1) implies that the vorticity q = curl(u − α∆u) verifies the following transport equation: ∂t q + u · ∇q = 0. (4.1) Conversely, if (4.1) holds true and the circulations γi are conserved then (1.1) holds true. Indeed, let F denote the left-hand side of (1.1). The fact that (4.1) holds true means that curl F = 0. Going back to the proof of Lemma 2.3 we observe that the circulations γi being conserved means that the circulations of F on each Γ1 , . . . , ΓN vanish. From the properties of the Leray projector we know that PF and F differ by a gradient: PF − F = ∇π. Taking the curl implies that PF is curl free. But PF is also divergence free and tangent to the boundary so it must be a harmonic vector field. Since the circulations of F on Γi vanish and recalling that a gradient has vanishing circulation on Γi we deduce that PF has vanishing circulation on each Γ1 , . . . , ΓN . Since it is a harmonic vector field it must therefore vanish. We conclude that F = −∇π and (1.1) holds true. Recalling (3.4) we infer that (1.1)–(1.2) is equivalent to the following PDE in the unknown q: N X ∂t q + u · ∇q = 0 with u = T(q) + γi Yiα . (4.2) i=1

Lp ,

Above, the quantity q will belong to some 1 < p < ∞. From the construction of T(q) and of Yiα given in Section 3, we observe that u is (1 + αA)−1 applied to some Lpσ vector field. So u must vanish at the boundary, hence (1.2). To complete the proof of Theorem 4.1 it suffices to construct a solution q ∈ L∞ (R+ ; Lp ) of (4.2). Indeed, by the regularity results for the Stokes operator and recalling that Yi is smooth we deduce that Yiα is smooth too. From Proposition 3.1 we deduce that kT(q)kW 3,p 6 CkqkLp . Therefore u ∈ L∞ (R+ ; W 3,p ). From the PDE verified by q one can immediately see that ∂t q is bounded in the sense of distributions, so q must be continuous in time with values in D 0 . Since q ∈ L∞ (R+ ; Lp ) we infer by density of C0∞ in Lp that q ∈ Cb0,w (R+ ; Lp ). Then u is also weakly continuous in time: u ∈ Cb0,w (R+ ; W 3,p ). To solve (4.2) we will regularize it by introducing an artificial viscosity. More precisely, for ε > 0 let us consider the following PDE ε

ε

ε

ε

∂t q + u · ∇q − ε∆q = 0

ε

ε

with u = T(q ) +

N X

γi Yiα

(4.3)

i=1

with Dirichlet boundary conditions q ε ∂Ω = 0

14

(4.4)

and some smooth initial data q ε (0) ∈ C0∞

(4.5)

such that q ε (0) → q0 in Lp as ε → 0. The global existence of smooth solutions of (4.3)–(4.5) can be proved with classical methods, see for instance [25, Chapter 15]. Moreover, the Lp norms of the solutions decrease in time: kq ε (t)kLp 6 kq ε (0)kLp

∀t > 0.

Using also Proposition 3.1 we infer that q ε is bounded in L∞ (R+ ; Lp ) independently of ε and uε is bounded in L∞ (R+ ; W 3,p ) independently of ε. Then we can extract a subsequence of q ε that we denote again by q ε and some qˇ ∈ L∞ (R+ ; Lp ) and u ˇ ∈ L∞ (R+ ; W 3,p ) such that q ε * qˇ weak∗ in L∞ (R+ ; Lp ) and ˇ weak∗ in L∞ (R+ ; W 3,p ). uε * u

(4.6)

We now pass to the limit in (4.3) in the sense of distributions when ε → 0. Obviously ∂t q ε → ∂t q and ε∆q ε → 0 in the sense of distributions when ε → 0. It remains to show that uε q ε → u ˇ qˇ in the sense of distributions. To do that, we observe first from (4.3) that ∂t q ε is bounded in L∞ (R+ ; W −2,p ). Since the embedding W −2,p ,→ W −3,p is compact we deduce from the Arzel`a-Ascoli theorem that there exists a subsequence −3,p ). This of q ε , again denoted by q ε , such that q ε → qˇ strongly in L∞ loc ([0, ∞); W ε ε strong convergence combined with (4.6) implies that u q → u ˇ qˇ in the sense of distributions. Indeed, the product (u, q) 7→ uq is continuous from W 3,p × W −3,p into D 0 as can be seen from the following estimate: Z ∀ϕ ∈ C0∞ uqϕ 6 kqkW −3,p kuϕkW 3,p 6 CkqkW −3,p kukW 3,p kϕkW 3,∞ . Ω

We conclude that we can pass to the limit ε → 0 in (4.3) to deduce that ∂t qˇ + u ˇ · ∇ˇ q = 0. −3,p ) we From the uniform in time convergence: q ε → qˇ strongly in L∞ loc ([0, ∞); W infer that qˇ(0) = lim q ε (0) = q0 . ε→0

the proof of Theorem 4.1 it remains to prove that u ˇ = T(ˇ q) + PNTo complete α . We know that q ε = curl(uε − α∆uε ) so, after passing to the limit ε → 0 γ Y i=1 i i in the sense of distributions, we get that qˇ = curl(ˇ u − α∆ˇ u). On the other hand, from (4.6) we have that v ε = uε − α∆uε → vˇ = u ˇ − α∆ˇ u weak∗ in L∞ (R+ ; W 1,p ). In particular we have convergence of the trace of v ε on the boundary to the trace of vˇ on the boundary. So the circulations of v ε on each Γi converge towards the circulations of vˇ on each Γi . But the circulation of v ε on Γi is γi . Indeed, if we denote by γiε the

15

circulation of v ε on Γi then the construction performed at the beginning of Section 3 implies that N X uε = T(q ε ) + γiε Yiα (4.7) i=1

(see relation (3.4)). Comparing to the second part of (4.3) and observing that the vector fields Yiα are linearly independent (because Yi = (1 + αA)Yiα are linearly independent) we get that γiε = γi . We infer that the circulation of vˇ on Γi is γi . This information combined with the relation qˇ = curl(ˇ u − α∆ˇ u) implies that u ˇ = P α . This completes the proof of Theorem 4.1. T(ˇ q) + N γ Y i i i=1

5

Passing to the limit α → 0

In this section we show the convergence part of Theorem 1.2. Let uα the solution constructed in Theorem 4.1 and let us also denote v α = uα − α∆uα and q α = curl v α . Multiplying (1.1) by uα and integrating in space and time implies that the Hα1 norm of the velocity is conserved: kuα (t)kHα1 = kuα0 kHα1

∀t > 0.

By hypothesis, we know that kuα0 kHα1 is bounded uniformly in α hence uα bounded in L∞ (R+ ; Hα1 )

(5.1)

uniformly in α. Moreover, from the transport equation verified by q α we know that the Lp norm of q α is also conserved: kq α (t)kLp = kq0α kLp ∀t > 0 so q α bounded in L∞ (R+ ; Lp )

(5.2)

uniformly in α. Relation (5.1) implies that uα is bounded in L∞ (R+ ; L2 ). Using also relation (5.2) we deduce that there exists a subsequence uαk of uα , some vector field u and some scalar function ω such that uαk * u weak∗ in L∞ (R+ ; L2 )

(5.3)

weak∗ in L∞ (R+ ; Lp ).

(5.4)

and q αk * ω

Because uαk is divergence free and tangent to the boundary, the weak convergence stated in relation (5.3) implies that u is also divergence free and tangent to the boundary. Since αk curl ∆uαk → 0 in the sense of distributions we have that q αk = curl uαk − αk curl ∆uαk → curl u in the sense of distributions. By uniqueness of limits in the sense of distributions we infer that ω = curl u.

16

We need to prove that u verifies the Euler equation (1.3). In order to do that, we shall pass to the limit α → 0 in (1.1). A simple calculation shows that the α–Euler equations can be written under the following form ∂t (uα − α∆uα ) + div(uα ⊗ uα ) − α

X

∂j ∂i (uαj ∂i uα ) + α

j,i

X

∂j (∂i uαj ∂i uα )

j,i

−α

X

∂i (∂i uαj ∇uαj ) = −∇π α (5.5)

j,i

for some π α (see [19]). Because uαk → u in the sense of distributions we have that ∂t uαk → ∂t u in the sense of distributions and also ∂t ∆uαk → ∂t ∆u in the sense of distributions so αk ∂t ∆uαk → 0 in the sense of distributions. Recall also that the limit of a gradient is gradient. Let us now show that the last three terms on the left-hand side of (5.5) go to 0 in the sense of distributions. Let us consider for example the term αk ∂j (∂i uαj k ∂i uαk ). Thanks to Proposition 3.3 we know that there exists some η < 12 such that kuα kH 1 6 Cα−η (kuα kHα1 + kq α kLp ). We bound kαk ∂i uαj k ∂i uαk kL1 6 Cαk kuαk k2H 1 6 Cαk 1−2η (kuαk kHα1 + kq αk kLp )2 6 Cαk 1−2η −→ 0 when We infer that αk ∂i uαj k ∂i uαk → 0 in the sense of distributions so P αk → 0. αk j,i ∂j (∂i uαj k ∂i uαk ) → 0 in the sense of the distributions. One can show in a similar manner that the remaining two terms from (5.5) with coefficient α also go to 0 in the sense of distributions. It remains to pass to the limit in the term uαk ⊗ uαk . To do that we require compactness of the sequence uαk . This will be obtained via time-derivative estimates. To get these time-derivative estimates it is more practical to work in L2 based function spaces. We denoted by A2 be the Stokes operator seen as an unbounded operator in L2σ . s

s

For s > 0 we define X s to be domain of A22 with norm kf kX s = kA22 f kL2 . We also define X −s to be the dual space of X s . Estimates on the time derivative of uα − α∆uα are easy to obtain directly from the PDE (1.1) but we need estimates on ∂t uα and we must be careful about the dependence on α. Let us consider a test vector field ϕ ∈ X 4 and let us define ϕα = (1 + αA2 )−1 ϕ. One can use the classical results about the domain of As2 (see for example [10, Chapter 4]) to observe that ϕα ∈ D(A32 ). Expressing both ϕ and ϕα in terms of an orthonormal base of eigenfunctions of A2 as in [10, Chapter 4] and using the regularity results in that reference, one can easily see that we have kϕα kH 4 6 CkA22 ϕα kL2 6 CkA22 ϕkL2 = CkϕkX 4 .

17

(5.6)

Recall that since ϕα is divergence free and tangent to the boundary (even vanishing on the boundary) we have that Pϕα = ϕα . We multiply (5.5) by ϕα = Pϕα to obtain Z XZ α α α α α α uαj ∂i uα · ∂j ∂i ϕα u · ∇ϕ · u + α h∂t (u − α∆u ), Pϕ i = Ω

j,i

+α

XZ j,i

∂i uαj ∂i uα

Ω α

· ∂j ϕ − α

Ω

XZ j,i

∂i uαj ∇uαj · ∂i ϕα .

Ω

We now bound each of these terms. First, by the H¨older inequality and by Sobolev embeddings we have that Z uα · ∇ϕα · uα 6 Ckuα k2 2 k∇ϕα kL∞ 6 Ckuα k2 2 kϕα kH 3 6 CkϕkX 4 L L Ω

where we also used (5.1) and (5.6). Similarly, Z X α uαj ∂i uα · ∂j ∂i ϕα 6 Cαkuα kL2 kuα kH 1 kϕα kW 2,∞ j,i

Ω 1

6 Cα 2 kuα k2Hα1 kϕα kH 4 6 CkϕkX 4 and X α j,i

Z Ω

∂i uαj ∂i uα · ∂j ϕα − α

XZ j,i

Ω

∂i uαj ∇uαj · ∂i ϕα 6 Cαkuα k2H 1 k∇ϕα kL∞ 6 Ckuα k2Hα1 kϕα kH 3 6 CkϕkX 4 .

On the other hand, we have that h∂t (uα − α∆uα ), Pϕα i = hP∂t (uα − α∆uα ), ϕα i = h∂t (uα + αA2 uα ), ϕα i = h∂t uα , (1 + αA2 )ϕα i = h∂t uα , ϕi. We deduce from the above estimates the following bound: |h∂t uα , ϕi| 6 CkϕkX 4 . This implies that ∂t uα is bounded in L∞ (R+ ; X −4 ). In particular, the uα are equicontinuous in time with values in X −4 . The uα are also bounded in X −4 because by (5.1) they are bounded in L2 and L2σ = X 0 ,→ X −4 . Moreover, by compact Sobolev embeddings we know that the embedding X −4 ,→ X −5 is compact. Finally,

18

the Arzel` a-Ascoli theorem implies that there exists a subsequence of uαk , again denoted by uαk , such that uαk (t) → u(t) in X −5 uniformly in time: −5 uαk → u strongly in L∞ ). loc ([0, ∞); X

(5.7)

Thanks to Proposition 3.3 we know that there exists some s0 ∈ (0, 12 ) such that uαk is bounded in L∞ (R+ ; H s0 ). Therefore in L∞ (R+ ; X s0 ) too. Consequently u ∈ L∞ (R+ ; X s0 ). By interpolation and using (5.7) we deduce that 2 uαk → u strongly in L∞ loc ([0, ∞); L ).

(5.8)

We infer that uαk ⊗ uαk → u ⊗ u in the sense of distributions and therefore div(uαk ⊗ uαk ) → div(u ⊗ u) in the sense of distributions too. We proved that u verifies the incompressible Euler equation. Moreover, we recall (5.4) which says in particular that ω = curl u ∈ L∞ (R+ ; Lp ). To complete the proof s,p ) for all s < 1 . of Theorem 1.2 it suffices to show that uαk → u in L∞ loc ([0, ∞); W p We consider two cases, depending on p being larger or smaller than 2. If p 6 2 we have that L2 ⊂ Lp so from (5.8) we deduce that uαk → u in ∞ Lloc ([0, ∞); Lp ). But we know from Proposition 3.3 and from the boundedness of s,p ) for all s < 1 . By kuα kHα1 and of kq α kLp that uαk is bounded in L∞ loc ([0, ∞); W p s,p ) for all s < 1 . interpolation we conclude that uαk → u in L∞ ([0, ∞); W loc p 0

p 6 2. So we have the Sobolev embedding W01,p ,→ L2 . If p > 2 then p0 ≡ p−1 Passing to the dual we obtain that L2 ,→ W −1,p . Then we deduce from (5.8) that −1,p ). We conclude as above by interpolation that uαk → u uαk → u in L∞ loc ([0, ∞); W s,p ∞ in Lloc ([0, ∞); W ) for all s < p1 . This completes the proof of Theorem 1.2.

Remark 5.1. If p = ∞ we have that L∞ ⊂ Lr for any r so if q0α is bounded in L∞ it is also bounded in any Lr with r finite. Therefore one can still pass to the limit α → 0 using the case p < ∞. The limit solution u is a Yudovich solution, i.e. a solution of the incompressible Euler equation with bounded vorticity (see [27]). Indeed, on one hand we know from (5.4) that q α converges to ω and on the other hand q α is bounded in L∞ (R+ × Ω). So necessarily ω ∈ L∞ (R+ × Ω) which implies that u is a Yudovich solution. We conclude that Theorem 1.2 remains true in the case p = ∞ with the following modifications in the conclusion: • the solution uα belongs to the space L∞ (R+ ; W 3,r ) for all r < ∞ instead of L∞ (R+ ; W 3,∞ ); s,r ) for all s < • the convergence holds true in L∞ loc ([0, ∞); W

6

1 r

and r < ∞.

The case of second grade fluids

The equation of motion of second grade fluids read as follows: X ∂t (u−α∆u)−ν∆u+u·∇(u−α∆u)+ (u−α∆u)j ∇uj = −∇π, j

19

div u = 0. (6.1)

We endow this equation with the Dirichlet boundary conditions too: u ∂Ω = 0.

(6.2)

We observe that the α–Euler equations are a particular case of second grade fluids, more precisely they are the vanishing viscosity case ν = 0. We refer to the recent book [8] for an extensive discussion of various aspects of the second grade fluids. As for the α–Euler equations, we use the notation v = u − α∆u and q = curl v. Let us mention at this point that convergence towards a solution of the Euler equation when α, ν → 0 was proved in the case of the Navier boundary conditions without any condition on the relative sizes of ν and α in dimension two, see [4], and with the condition αν bounded in dimension three, see [5]. In the case of the Dirichlet boundary conditions, convergence towards a solution of the Navier-Stokes equations when α → 0 and ν > 0 is fixed was proved in [3], see also [19]. We would now like to know if the solutions of (6.1)–(6.2) converge to a solution of the incompressible Euler equation (1.3)–(1.4) when both α and ν converge to 0. The only result in that direction is given in [22] where convergence towards a H 3 solution of the Euler equation is proved under the assumption that αν is bounded. Theorem 6.1 ([22]). Let u0 ∈ H 3 be divergence free and tangent to the boundary. Assume that uα,ν verifies 0 α,ν • u ∈ H 3 , div uα,ν = 0 and uα,ν = 0; 0

• •

1 2

α k∇uα,ν 0 kL2 → α,ν u0 → u0 in L2

0

0 and α

3 2

0 ∂Ω α,ν ku0 kH 3 is

bounded as α, ν → 0;

as α, ν → 0.

Assume moreover that αν is bounded. Then the unique global H 3 solution of (6.1)– 2 converges in L∞ (6.2) with initial data uα,ν 0 loc ([0, ∞); L ) when α → 0 towards the 3 unique global H solution u of (1.3)–(1.4) with initial data u0 . We would now like to extend our result for the α–Euler equations to the second grade fluids, proving convergence of (6.1)–(6.2) towards solutions of (1.3)–(1.4) with Lp vorticity on multiply-connected domains. Our convergence result is based on Lp estimates uniform in α for the vorticity q. Let us remark right away that such estimates cannot hold true when α and ν are of the same size. More precisely, we have the following observation. Proposition R α,ν 6.2. Under the hypotheses of Theorem 6.1 assume in addition that both ν have non-zero limits when α → 0. Then for any r > 1 the vorticity α and Ω q0 q α,ν is unbounded in Lrloc ((0, ∞) × Ω). Proof. Let us apply the curl operator to (6.1). We find that q α,ν = curl(uα,ν − α∆uα,ν ) verifies the following PDE: ∂t q α,ν − ν curl ∆uα,ν + uα,ν · ∇q α,ν = 0. Integrating in space yields Z Z Z d q α,ν − ν curl ∆uα,ν + div(uα,ν q α,ν ) = 0. dt Ω Ω Ω

20

(6.3)

Because uα,ν vanishes on the boundary, the Stokes formula implies that R the last term on the left-hand side vanishes. For the same reason we have that Ω curl uα,ν = 0. We infer that Z Z 1 curl ∆uα,ν = − q α,ν . α Ω Ω We deduce that

d dt Z

so

Z q

α,ν

Ω

q

α,ν

ν + α

(t) = e

Z

ν −α t

Ω

q α,ν = 0

Ω

Z Ω

q0α,ν .

(6.4)

By hypothesis, there exist `1 , `2 6= 0 such that Z ν → `1 and q0α,ν → `2 . α Ω Relation (6.4) implies that Z

q α,ν (t) → e−t`1 `2 .

(6.5)

Ω

Now let us assume by absurd that q α,ν is bounded in Lrloc ((0, ∞) × Ω) for some α,ν r > 1. Then there is a subsequence of q α,ν ,Ralso denoted converges to R by q , which r α,ν some q weakly in Lloc ((0, ∞) × Ω). Then Ω q → Ω q weakly in Lrloc ((0, ∞)). In view of (6.5) we infer that Z q(t, x) dx = e−t`1 `2

(6.6)

Ω

almost everywhere in time. But uα,ν → u so α curl ∆uα,ν → 0 in the sense of distributions. Consequently α,ν q = curl uα,ν − α curl ∆uα,ν → curl u in the sense of distributions. By uniqueness of limits in the sense of distributions, we infer that q = curl u. This is a contradiction because for a solution of the Euler equation the integral of vorticity is conserved in time while (6.6) implies that the integral of q is not constant in time. This completes the proof. Proposition 6.2 shows that we cannot hope to adapt our approach to second grade fluids if ν and α are of the same size. But if ν is slightly smaller in size than α then we can prove convergence to the Euler equations. Theorem 6.3. Let Ω be a smooth bounded domain of R2 and 1 < p < ∞. Assume that ν 6 α1+ε for some ε > 0 independent of α. Let u0 ∈ W 1,p be divergence free and tangent to the boundary. Let uα,ν be such that 0 α,ν α,ν α,ν 3,p • u ∈ W , div u = 0 and u = 0; 0

• •

0

1 α,ν 2 kuα,ν 0 kL2 , α k∇u0 kL2 and 2 uα,ν 0 → u0 in L as α → 0.

0

∂Ω

kq0α,ν kLp

are bounded independently of α and ν;

21

Then there exists a global solution uα,ν ∈ L∞ (R+ ; W 3,p ) of (6.1)–(6.2). Moreover, there exists a subsequence of solutions uαk ,νk and a global solution u of the Euler 2 equations (1.3)–(1.4) with initial data u0 such that uαk ,νk → u in L∞ loc ([0, ∞); L ). In p addition, if ε > 12 then the limit solution has Lp vorticity: curl u ∈ L∞ loc ([0, ∞); L ). 1 r ∞ r If ε < 2 then the limit solution has L vorticity, curl u ∈ Lloc ([0, ∞); L ), for any r 1 verifying 1 < r 6 p and r < 1−2ε . Remark 6.4. The conclusion of Theorem 6.3 is slightly better than stated in the s,r ) for all s < 1 and, by sense that we actually obtain convergence in L∞ loc ([0, ∞); W r 0 s ) for all s0 < min(1 − 1 , 1 ). Here r is either Sobolev embeddings, in L∞ ([0, ∞); H loc r 2 1 p if ε > 12 , or any real number verifying 1 < r 6 p and r < 1−2ε if ε < 12 . A second remark is that Theorem 6.3 is somewhat weaker than Theorem 1.2 in the sense that the limit solution does not always have vorticity in Lp as we would expect from the initial vorticity belonging to Lp . The remainder of this section is devoted to the proof of Theorem 6.3. We will not give all the details as the proof is very similar to the proof of Theorem 1.2. We will only underline the differences. For clarity reasons we drop the superscript α,ν on the various quantities. If we analyze the proof we gave for the α–Euler equations, we realize that there are three main ingredients that need to be checked in the case of second grade fluids: • Hα1 estimates for the velocity u; • Estimates for the circulations of v on each Γi ; • Lp estimates for q. The Hα1 estimates for u go through easily. Indeed, if we multiply (6.1) by u, integrate in the x variable and do some integrations by parts using that u vanishes on the boundary we obtain that d kuk2Hα1 + 2νk∇uk2L2 = 0. dt So the Hα1 norm of u decreases. The circulations of v on each Γi are not conserved anymore but can nevertheless be computed and shown to be decreasing. More precisely, let Z γi (t) = v(t) · n⊥ . Γi

We have the following result. Lemma 6.5. Let u be a sufficiently smooth solution of (6.1). Then for every i ∈ {1, . . . , n} the circulation of v on Γi is given by ν

γi (t) = γi (0)e− α t . Proof. We proceed like in the proof of Lemma 2.3 by multiplying (6.1) by n⊥ and integrating on Γi . We get Z Z d ⊥ v·n −ν ∆u · n⊥ = 0. dt Γi Γi

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Since u vanishes on Γi we have that ∆u = − α1 (u − α∆u) = − αv on Γi . We infer that Z Z d ν v · n⊥ + v · n⊥ = 0. dt Γi α Γi Consequently Z

⊥

ν t −α

Z

v(t) · n = e

v(0) · n⊥ .

Γi

Γi

It remains to see if we can get Lp estimates for q and this is where the trouble lies. We can prove the following: Lemma 6.6. For any δ > 0 and 1 < r 6 p there exists a constant C depending solely on δ, r and Ω such that kq(t)kLr 6 (kq0 kLr + ku0 kHα1 )eCtνα

3 + 1 −δ −2 2r

.

(6.7)

Proof. We will make Lr estimates on equation (6.3). The rigorous way to proceed is r to multiply that PDE by q(q 2 + κ) 2 −1 , integrate in space, do the necessary estimates and let κ → 0 at the end. This kind of argument is well-known so, for the sake of the simplicity, we are taking the liberty of letting κ = 0 from the beginning and we are making a slightly formal argument. More precisely, we multiply (6.3) by q|q|r−2 and integrate in space to obtain that Z Z Z ∂t q q|q|r−2 − ν curl ∆u q|q|r−2 + u · ∇q q|q|r−2 = 0. Ω

Ω

Ω

We observe that ∂t q q|q|r−2 = 1r ∂t |q|r and ∇q q|q|r−2 = 1r ∇|q|r . Let ω = curl u. Making an integration by parts and recalling that curl ∆u = ω−q α we deduce that Z 1d ν ν ν kqkrLr + kqkrLr = ω q|q|r−2 6 kωkLr kqkr−1 Lr r dt α α Ω α so, after simplifying kqkr−1 Lr on both sides, d ν ν kqkLr + kqkLr 6 kωkLr . dt α α

(6.8)

Using (3.22) and recalling that the Hα1 norm of u is decreasing we bound 1

1

1

1

kωkLr 6 kukW 1,r 6 Cα− 2 + 2r −δ (kqkLr + kukHα1 ) 6 Cα− 2 + 2r −δ (kqkLr + ku0 kHα1 ). Using this in (6.8) implies that 3 1 d kqkLr 6 Cνα− 2 + 2r −δ (kqkLr + ku0 kHα1 ). dt

The Gronwall inequality completes the proof of the lemma.

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Recalling that we assumed ν 6 α1+ε we deduce from (6.7) the following bound − 1 + 1 +ε−δ kq(t)kLr 6 kq0 kLr + ku0 kHα1 eCtα 2 2r 6 kq0 kLr + ku0 kHα1 eCt

(6.9)

provided that ε+

1 1 > + δ. 2r 2

(6.10)

We consider two cases. 1 If ε > 12 then we choose r = p and δ = ε + 2p − 12 and we use (6.9) to deduce p that q is bounded in the space L∞ loc ([0, ∞); L ) independently of α and ν. Given p the boundedness of the L norm of q, the decay of the Hα1 norm of uα,ν and the explicit formula for the circulations γi (t) one can argue as in the case of the α–Euler equations and pass to the limit in (6.1) towards a solution of the Euler equation. Indeed, the additional term −ν∆uα,ν is linear and goes to 0 as ν goes to 0. The limit p α,ν → u holds true solution have vorticity in L∞ loc ([0, ∞); L ) and the convergence u 1 ∞ s,p in the space Lloc ([0, ∞); W ) for any s < p (with the strong topology). 1 If ε < 12 then we choose an r such that 1 < r 6 p and r < 1−2ε . The condition 1 1 (6.10) is verified for δ = ε + 2r − 2 > 0. We obtain then from (6.9) that q is r bounded in the space L∞ loc ([0, ∞); L ) independently of α and ν. In this case we s,r ) for any s < 1 and the limit solution have obtain convergence in L∞ loc ([0, ∞); W r r ∞ vorticity in Lloc ([0, ∞); L ). The proof of Theorem 6.3 is completed.

Acknowledgments D.I. has been partially funded by the ANR project Dyficolti ANR-13-BS010003-01 and by the LABEX MILYON (ANR-10-LABX-0070) of Universit´e de Lyon, within the program ”Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

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A. V. Busuioc: Universit´e de Lyon, Universit´e de Saint-Etienne – CNRS UMR 5208 Institut Camille Jordan – Facult´e des Sciences – 23 rue Docteur Paul Michelon – 42023 Saint-Etienne Cedex 2, France. Email: [email protected] D. Iftimie: Universit´e de Lyon, Universit´e Lyon 1 – CNRS UMR 5208 Institut Camille Jordan – 43 bd. du 11 Novembre 1918 – Villeurbanne Cedex F-69622, France. Email: [email protected]

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