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Inviscid limit for vortex patches in a bounded domain
Applied Mathematics Letters 25 (2012) 1367–1372
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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Inviscid limit for vortex patches in a bounded domain Quansen Jiu a,∗ , Yun Wang b a
School of Mathematical Sciences, Capital Normal University, Beijing 100048, PR China
b
The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
article
info
Article history: Received 23 April 2010 Accepted 2 December 2011 Keywords: Inviscid limit Navier boundary condition Vortex patches
abstract In this paper, we consider the inviscid limit of the incompressible Navier–Stokes equations in a smooth, bounded and simply connected domain Ω ⊂ Rd , d = 2, 3. We prove that for a vortex patch initial data, the weak Leray solutions of the incompressible Navier–Stokes equations with Navier boundary conditions will converge (locally in time for d = 3 and globally in time for d = 2) to a vortex patch solution of the incompressible Euler equation as the viscosity vanishes. In view of the results obtained in Abidi and Danchin (2004) [5] and Masmoudi (2007) [3] which dealt with the case of the whole space, we derive an almost 3
optimal convergence rate (ν t ) 4 −ε for any small ε > 0 in L2 . © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The incompressible Navier–Stokes equations read as
∂u − ν 1u + (u · ∇)u + ∇ p = 0, ∂t div u = 0,
(1.1)
where u = (u1 , . . . , ud )(d = 2 or 3) is the velocity fields, p is the pressure function and ν is the kinetic viscosity. Formally, when ν = 0, (1.1) becomes the following incompressible Euler equations:
∂u + (u · ∇)u + ∇ p = 0, ∂t div u = 0.
(1.2)
The inviscid limit for the incompressible Navier–Stokes equations in the whole space has been well understood (see [1–7] and references therein) for both smooth and non-smooth initial data. However, in the case of a bounded domain, the inviscid limit for the Navier–Stokes equations with Dirichlet boundary conditions is still a completely open problem. This is mainly due to the difference between the Dirichlet boundary conditions of the incompressible Navier–Stokes equations (1.1) and the tangential boundary conditions of the incompressible Euler equations (1.2) and a boundary layer will appear near the boundary of the domain. This paper is concerned with the inviscid limit problem of the incompressible Navier–Stokes equations (1.1) with the following Navier boundary conditions:
⃗ = 0, u·n ∗
[D(u)⃗n + α u]tan = 0,
on ∂ Ω × (0, +∞),
Corresponding author. E-mail addresses: [email protected] (Q. Jiu), [email protected] (Y. Wang).
0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.12.003
(1.3)
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Q. Jiu, Y. Wang / Applied Mathematics Letters 25 (2012) 1367–1372
⃗ is the unit exterior normal to the boundary ∂ Ω , D(u) = where Ω ⊂ Rd (d = 2 or 3) is a smooth bounded domain, n 1 [∇ u + (∇ u)T ] is the rate of strain tensor and [D(u)⃗n + α u]tan is the tangential component of the vector D(u)⃗n + α u. Here 2 α = α(x, t ) is a known function representing the friction coefficient of the material. The Navier boundary conditions, introduced by Navier in [8], say that the tangential component of the viscous stress at the boundary is proportional to the tangential velocity. They were rigorously justified as a homogenization of the no-slip condition on a rough boundary in [9] and widely used when studying the inviscid limit of the incompressible flows in a bounded domain (see [10–15]) in recent years. Of particular interest of this paper is the inviscid limit for vortex patches in a bounded domain. It is known that when the initial data are vortex patch ones (see Definition 2.1 for details), there exists a unique solution to the incompressible Euler equations which preserves the vortex-patch structures globally (in time) in the whole plane [16,17] and locally in three-dimensional whole space [18]. In a smooth, bounded and simply connected domain Ω ⊂ Rd (d = 2, 3), the vortex patch solutions of the incompressible Euler equations were derived in [19,20]. In this paper, we will show that the weak Leray solutions of the incompressible Navier–Stokes equations with Navier boundary conditions will tend to a vortex patch solution of the incompressible Euler equations as the viscosity vanishes if the initial data are vortex patch ones. Moreover, 3
we obtain that the convergence rate in L2 is (ν t ) 4 −ε for any small ε > 0. In the case of the whole plane, √ Constantin and Wu studied the inviscid limit for the 2D vortex patches in [6,7] and obtained the convergence rate in L2 is ν t. Abidi and Danchin 3
improved the convergence rate to be (ν t ) 4 in L2 which is optimal since the circular vortex patches provide a lower bound (see [5]). Later, Masmoudi extended the results to the case of three-dimensional whole space in [3]. Recently, Sueur [21] dealt with the vorticity internal transition layers for the Navier–Stokes equations and described how the smoothing effect is (micro-)localized in the case where vortex patches are prescribed as initial data, using the method of asymptotic expansion. In the case of the two-dimensional bounded domain, the inviscid limit for the incompressible Navier–Stokes equations with √ Navier-boundary conditions was discussed in [13] and the obtained convergence rate in L2 is ν t for initial vorticity in L∞ . Our results here apply to both 2D and 3D vortex patches in a bounded domain and the convergence rate obtained in this paper is almost optimal in view of the results in [5,3]. Since we consider the case of the bounded domain, estimates in Besov space in [5,3] cannot be used directly and we will use the interpolation space theory to deduce that the vorticity belongs to L∞ ([0, T ∗ ); H s (Ω )) for some T ∗ > 0 and s > 0. More subtle estimates will be given in this paper. Meanwhile, whether the 3
convergence rate can be improved to (ν t ) 4 is still open. The paper is organized as follows. In Section 2, we will give some preliminaries and the main results. Section 3 is devoted to the proof of the main result. 2. Preliminaries and main results Let Ω ⊂ Rd (d = 2, 3) be a smooth, bounded and simply connected domain. The initial-boundary problem to the incompressible Euler equations is written as
∂u + (u · ∇)u + ∇ p = 0, ∂t
(x, t ) ∈ Ω × [0, +∞)
div u = 0, (x, t ) ∈ Ω × [0, +∞), u · n⃗ = 0, (x, t ) ∈ ∂ Ω × [0, +∞), u(x, 0) = u0 (x), x ∈ Ω .
(2.1)
Denote by (uν , pν ) the solutions of the incompressible Navier–Stokes equations with corresponding kinetic viscosity ν . The initial-boundary problem to the incompressible Navier–Stokes equations with Navier boundary conditions is written as
ν ∂u − ν 1uν + (uν · ∇)uν + ∇ pν = 0, ∂t
(x, t ) ∈ Ω × (0, +∞)
ν
div u = 0, (x, t ) ∈ Ω × [0, +∞), ν ν ν u · n⃗ = 0, ν [D(u )⃗n + α u ]tan = 0, u(x, 0) = u0 (x), x ∈ Ω .
(2.2)
(x, t ) ∈ ∂ Ω × [0, +∞),
Let ω0 = curl u0 be the initial vorticity of u0 . In this paper, for any vector-valued function ϕ, D(ϕ) denotes the symmetric part of ∇ϕ , i.e., D(ϕ) =
∇ϕ + (∇ϕ)T 2
.
Denote by C r , C 1+r (0 < r < 1) the usual Hölder space. In particular, Ccr (Rd ) consists of functions in C r (Rd ) with compact support. Let Lp (Ω ), W s,p (Ω ) be the usual Sobolev spaces defined in Ω , where 1 ≤ p ≤ ∞ and s is permitted to be a real number. If p = 2, W s,2 (Ω ) is denoted by H s (Ω ). H0s (Ω ) is the closure of C0∞ (Ω ) in H s (Ω ). Define ∞ C0∞ ,σ (Ω ) = {f |f ∈ C0 (Ω ), div f = 0},
Cσ∞ (Ω ) = {f |f ∈ C ∞ (Ω ), div f = 0}. 2 1 ∞ 1 L2σ (Ω ) is the closure of C0∞ ,σ (Ω ) in L (Ω ), and Hσ (Ω ) is the closure of Cσ (Ω ) in H (Ω ).
Q. Jiu, Y. Wang / Applied Mathematics Letters 25 (2012) 1367–1372
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We first recall the definition of a vortex patch in a bounded domain (see [19,3]). Definition 2.1. Let 0 < r < 1. The vorticity ω = curl u of a vector field u is called a C r vortex patch of support P if the following decomposition holds:
ω = (ωi χP + ωe χΩ \P )|Ω , where P is an open set of class C 1+r , ωi , ωe ∈ Ccr (Rd )(d = 2, 3) and χP , χΩ \P are the characteristic functions of P and Ω \ P respectively.
⃗ = ωe · n⃗ on ∂ P. Notice that when d = 3, curl u is of divergence free, we need ωi · n If the initial data of the incompressible Euler equations is a C r vortex patch, the global existence of 2-d vortex patch solutions and the local existence of 3-d vortex patch solutions have been proved (see [19,20]). More precisely, one has Theorem 2.1. Let u0 be a divergence free vector field in Rd (d = 2, 3), tangent to ∂ Ω , whose vorticity ω0 is a C r vortex patch of support P, the boundary of ∂ P is a (d − 1)-dimensional compact submanifold of Rd . If P ⊂ Ω , then there exists a T ∗ > 0 such that the Euler equations (2.1) have a (unique) solution u ∈ L∞ ([0, T ∗ ); Lip(Ω )). Moreover, ω(t ) = curl u(t ) remains a vortex patch, whose support Ψ (t , P ) is of class C 1+r for any t ∈ [0, T ∗ ), Ψ denoting the flow of u. In addition, T ∗ > 0 can be arbitrarily large if d = 2. Remark 2.1. Under assumptions of Theorem 2.1, if P is tangent to ∂ Ω , a little regularity may be lost. However, local existence of 3-D vortex patch of C s (0 < s < r ) and global existence of 2-D vortex patch of C s (0 < s < r ) is proved in [19]. Now we give the definition of a Leray weak solution of the incompressible Navier–Stokes equations with Navier boundary conditions. Definition 2.2. We call a vector field uν (t , x) : [0, +∞) × Ω → Ω , denoted by u(t , x) here, a weak Leray solution of (2.2) if u verifies (1) u ∈ Cw ([0, ∞); L2σ (Ω )) Lloc ([0, ∞); Hσ1 (Ω )); ⃗ =0 (2) u verifies the system of Eqs. (2.2) under the following weak form: for every ϕ ∈ C0∞ ([0, ∞); Cσ∞ (Ω )) with ϕ · n on ∂ Ω ,
∞ ∞ α u · ϕ + 2ν D(u)D(ϕ) + (u · ∇)u · ϕ 0 Ω 0 Ω ∞∂Ω 0 = u · ∂t ϕ + u(0) · ϕ(0).
2ν
∞
0
Ω
Ω
(3) u verifies the energy inequality, for all t ≥ 0,
∥u(t )∥2L2 (Ω ) + 4ν
t 0
∂Ω
α|u|2 + 4ν
t 0
Ω
|D(u)|2 ≤ ∥u(0)∥2L2 (Ω ) .
We remark that the global existence of the Leray weak solution in the case of Dirichlet boundary conditions is well known for any u0 ∈ L2σ (Ω ). The extensions of this result to the case of Navier boundary conditions is straightforward by the Galerkin method. The main result of the paper is stated as Theorem 2.2. Suppose that the assumptions of Theorem 2.1 hold and u ∈ L∞ ([0, T ∗ ); Lip(Ω )) is the vortex patch solution of the incompressible Euler equations with initial data u0 . Suppose that uν are Leray weak solutions of the incompressible Navier–Stokes equations with Navier boundary conditions (2.2). The corresponding initial data uν (0) is uniformly bounded in L2 (Ω ), and α ∈ L∞ (∂ Ω ). Then for all 0 < T < T ∗ and any small ϵ > 0, one has
∥(uν − u)(t )∥L2 (Ω ) ≤ C ((ν t )
1+β−ϵ 2
+ ∥uν (0) − u0 ∥L2 (Ω ) ),
where β = min( 21 , r ) and C is a constant depending only on ϵ, u, T , ∥α∥L∞ (∂ Ω ) and M ≡ supν ∥uν (0)∥L2 (Ω ) . 3. Proof of main result Since we are concerned with the case of the bounded domain, the estimates in Besov space as in [5,3] cannot be used directly. However, we have Lemma 3.1. Suppose that ω = curl u is the vortex patch solution to the incompressible Euler system, derived in Theorem 2.1. Then, for any s < β = min(r , 12 ), one has ω ∈ L∞ ([0, T ∗ ); H s (Ω )).
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Proof. It is proved in [19] that the vortex patch solution has the following structures:
ω(x, t ) = ωi (x, t )χP (t ) (x) + ωe (x, t )χΩ \P (t ) (x),
t ∈ [0, T ∗ ),
where
ωi , ωe ∈ L∞ ([0, T ∗ ); C r˜ (Rd )),
P (t ) ∈ L∞ ([0, T ∗ ); C 1+˜r (Rd ))
for any r˜ < r, which means that for any t ∈ [0, T ∗ ), P (t ) is a C 1+˜r domain, and the C 1+˜r -norm of the boundary ∂ P (t ) is locally bounded. Hence H d−1 (∂ P (t )), the (d − 1)-dimensional Hausdorff measure of ∂ P (t ) is locally bounded which induces that
χP (t ) (x), χΩ \P (t ) (x) ∈ L∞ ([0, T ∗ ); L∞ (Rd ) ∩ BV (Rd )). Following Lemmas 4.2 and 4.3 in [3], after extending ω to the whole space by zero extension, we derive
ω(x, t ) ∈ L∞ ([0, T ∗ ); B˙ s2˜ ,∞ (Rd )), where s˜ = min(˜r , 21 ) and B˙ s2˜ ,∞ is the classical homogeneous Besov space (see [22] for definition). Using the fact that ω(x, t ) ∈ L∞ ([0, T ∗ ); L2 (Rd )), one has
ω(x, t ) ∈ L∞ ([0, T ∗ ); B˙ s2˜ ,∞ (Rd ) ∩ L2 (Rd )).
(3.3)
Moreover, for any s < β = min(r , ), there exists a s˜ > s such that (3.3) holds. Thus standard interpolation theory (see [22]) yields that 1 2
ω(x, t ) ∈ L∞ ([0, T ∗ ); H s (Ω )). The proof of Lemma 3.1 is completed.
The following are some known facts, of which the proofs are omitted here. Lemma 3.2 (See [23]). For any s ≥ 1 and 1 < p < ∞, there exists a positive constant C , depending only on Ω , s, p, such that for any vector-valued function w , one has
∥w∥W s,p (Ω ) ≤ C [∥div w∥W s−1,p (Ω ) + ∥curl w∥W s−1,p (Ω ) + ∥w · n⃗∥W s−1/p,p (∂ Ω ) + ∥w∥W s−1,p (Ω ) ]. Lemma 3.3 (Korn’s Inequality See [24]). Let ω ∈ H 1 (Ω ). Then there exists a constant C depending only on the domain Ω , such that
∥w∥H 1 (Ω ) ≤ C (∥D(w)∥L2 (Ω ) + ∥w∥L2 (Ω ) ). Now we are ready to prove our main result. Proof of Theorem 2.2. Let v ν = uν − u. For any fixed T < T ∗ , one has for every 0 < t ≤ T ,
∥uν (t )∥2L2 (Ω ) + 4ν
t ∂Ω
0
α|uν |2 + 4ν
t Ω
0
|D(uν )|2 ≤ ∥uν (0)∥2L2 (Ω ) ,
(3.4)
∥u(t )∥2L2 (Ω ) = ∥u0 ∥2L2 (Ω ) .
(3.5) ν
Here (3.4) is the energy inequality for the Leray weak solution u and (3.5) is the energy equality for the vortex patch solution u. Using u as a test function in the weak form satisfying by the Leray weak solution uν (see Definition 2.2), we obtain
ν
Ω
u · u(t )dx + 2ν
t 0
t + 0
Ω
ν
∂Ω
α u · udSdτ + 2ν
t Ω
0
(uν · ∇)uν · udxdτ =
Ω
D(uν ) : D(u)dxdτ
uν (0) · u0 dx.
(3.6)
Adding (3.4) and (3.5) and then subtracting (3.6), one deduces 1 2
ν
∥v (t )∥
2 L2 (Ω )
+ 2ν
t 0
ν
Ω
|D(v )| dxdτ ≤ 2
1 2
ν
∥v (0)∥
− 2ν
t 0
≡
1 2
2 L2 (Ω )
∂Ω
t − Ω
0
(v ν · ∇)u · v ν dxdτ
α uν · v ν dSdτ − 2ν
∥v ν (0)∥2L2 (Ω ) +
t 0
Ω
D(u) : D(v ν )dxdτ
(3.7)
t
(I2 + I3 + I4 )dτ . 0
(3.8)
Q. Jiu, Y. Wang / Applied Mathematics Letters 25 (2012) 1367–1372
1371
Now we estimate the terms on the right hand of (3.7). By Hölder’s inequality,
|I2 | = (v ν · ∇)u · v ν dx ≤ ∥∇ u∥L∞ (Ω ) ∥v ν ∥2L2 (Ω ) .
(3.9)
Ω
For the term I3 , since v ν = uν − u, one has
|I3 | ≤ 2ν
α u · v dS + 2ν ν
∂Ω
α|v | dS . ν 2
∂Ω
Making use of Lemmas 3.2 and 3.3 and the interpolation inequality, then
ν
α u · v dS ≤ ν∥α∥L∞ (∂ Ω ) ∥u∥L2 (∂ Ω ) ∥v ν ∥L2 (∂ Ω ) ν
∂Ω
≤ C ν∥α∥L∞ (∂ Ω ) ∥u∥H 1/2 (Ω ) ∥v ν ∥H 1−s (Ω ) 1−s ≤ C ν∥α∥L∞ (∂ Ω ) (∥ω∥H s (Ω ) + ∥u∥L2 (Ω ) )∥v ν ∥sL2 (Ω ) ∥v ν ∥L2 (Ω ) + ∥Dv ν ∥L2 (Ω ) 2s ν ν ν 1+s ∞ ≤ C ν∥α∥L (∂ Ω ) ∥v ∥L2 (Ω ) + ∥v ∥L2 (Ω ) + ∥D(v ν )∥2L2 (Ω ) , 4
and
ν
∂Ω
α|v ν |2 dS ≤ C ν∥α∥L∞ (∂ Ω ) ∥v ν ∥2 1
H 2 (Ω )
ν 2
≤ C ν∥α∥L∞ (∂ Ω ) ∥v ∥L2 (Ω ) +
ν 4
∥D(v ν )∥2L2 (Ω ) .
Hence the term |I3 | is estimated as
2s ν |I3 | ≤ C ν∥α∥L∞ (∂ Ω ) ∥v ν ∥2L2 (Ω ) + ∥v ν ∥L12+(sΩ ) + ∥D(v ν )∥2L2 (Ω ) .
(3.10)
2
Now we come to the term I4 . Since s < β, ω ∈ L∞ (0, T ; H s (Ω )). Using the duality between H s (Ω ) and H −s (Ω ) (note that s < 21 , H s (Ω ) = H0s (Ω )) and the interpolation inequality, one has
D(u) : D(v ν )dx ≤ C ∥D(u)∥H s (Ω ) ∥D(v ν )∥H −s (Ω ) Ω
≤ C [∥ω∥H s (Ω ) + ∥u∥L2 (Ω ) ] · ∥v ν ∥sL2 (Ω ) ∥v ν ∥H1−1 (sΩ ) ≤ C [∥ω∥H s (Ω ) + ∥u∥L2 (Ω ) ]∥v ν ∥sL2 (Ω ) (∥v ν ∥L2 (Ω ) + ∥D(v ν )∥L2 (Ω ) )1−s . By the way, from (3.4),
∥uν (t )∥2L2 (Ω ) + 4ν
t
∥D(uν )∥2L2 (Ω ) dτ t ν ν 2 ∞ ≤ ∥u (0)∥L2 (Ω ) + C ν∥α∥L (∂ Ω ) ∥u ∥L2 (Ω ) + ∥D(uν )∥L2 (Ω ) ∥uν ∥L2 (Ω ) dτ 0 t t ν 2 ν 2 ≤ ∥u (0)∥L2 (Ω ) + C ν ∥u ∥L2 (Ω ) dτ + 2ν ∥D(uν )∥2L2 (Ω ) dτ , 0
0
0
ν
ν
which implies that ∥u ∥L∞ (0,T ;L2 (Ω )) and ∥v ∥L∞ (0,T ;L2 (Ω )) are uniformly bounded by some constant C depending on M , T and ∥α∥L∞ (∂ Ω ) . Hence by the Young’s inequality,
ν |I4 | = ν D(u) : D(v )dx Ω
≤ C ν∥v ν ∥L2 (Ω ) + C ν∥v ν ∥sL2 (Ω ) ∥D(v ν )∥1L2−(sΩ ) 2s ν ν ν 1+s ≤ C ν ∥v ∥L2 (Ω ) + ∥v ∥L2 (Ω ) + ∥D(v ν )∥2L2 (Ω ) ,
(3.11)
4
where C is a constant depending on Ω , s, M , T , ∥α∥L∞ (∂ Ω ) . Putting (3.9)–(3.11) into (3.7), we get 1 2
∥v ν (t )∥2L2 (Ω ) ≤
1 2
∥v ν (0)∥2L2 (Ω ) +
t
0
(∥∇ u∥L∞ (Ω ) + C )∥v ν ∥2L2 (Ω ) dτ + C ν
t
0
2s
∥v ν ∥L12+(sΩ ) dτ .
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Q. Jiu, Y. Wang / Applied Mathematics Letters 25 (2012) 1367–1372
By Gronwall’s lemma, we deduce that 2
2
∥v ν (t )∥L12+(sΩ ) ≤ C ∥v ν (0)∥L12+(sΩ ) + C ν t , where C is a constant depending on ϵ, u, T , ∥α∥L∞ (∂ Ω ) and M. The proof of the theorem is completed.
Acknowledgments The authors would like to thank Professor Zhouping Xin for his interest in this topic and valuable discussions. The paper was started when the first author was visiting the Institute of Mathematical Sciences (IMS) of The Chinese University of Hong Kong. The first author’s research is partially supported by the National Natural Sciences Foundation of China (No. 11171229). References [1] H.S.G. Swann, The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in R3 , Trans. Amer. Math. Soc. 157 (1971) 373–397. [2] J.Y. Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids, Comm. Partial Differential Equations 21 (11–12) (1996) 1771–1779. [3] N. Masmoudi, Remarks about the inviscid limit of the Navier–Stokes system, Comm. Math. Phys. 270 (2007) 777–788. [4] P.L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Models, in: Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, New York, 1996. [5] H. Abidi, R. Danchin, Optimal bounds for the inviscid limit of Navier–Sokes equations, Asymptot. Anal. 38 (1) (2004) 35–46. [6] P. Constantin, J. Wu, Inviscid limit for vortex patches, Nonlinearity 8 (5) (1995) 735–742. [7] P. Constantin, J. Wu, The inviscid limit for non-smooth vorticity, Indiana Univ. Math. J. 45 (1) (1996) 67–81. [8] C.L.M.H. Navier, Sur les lois de l’équilibrie et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France 369 (1827). [9] W. Jäger, A. Mikelić, On the roughness-induced efective boundary conditions for an incompressible viscous flow, J. Differential Equations 170 (2001) 96–122. [10] T. Clopeau, A. Mikelić, R. Robert, On the vaninshing viscosity limit for the 2D incompressible Navier–Stokes equations with the friction type boundary conditions, Nonlinearity 11 (1998) 1625–1636. [11] D. Iftimie, G. Planas, Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions, Nonlinearity 19 (2006) 899–918. [12] Q.S. Jiu, D.J. Niu, Vanishing viscous limits for the 2D lake equations with Navier boundary conditions, J. Math. Anal. Appl. 338 (2008) 1070–1080. [13] J.P. Kelliher, Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane, SIAM J. Math. Anal. 38 (2006) 210–232. [14] M.C. Lopes Filho, H.J. Nussenzveig Lopes, G. Planas, On the inviscid limit for 2D incompressible flow with Navier friction condition, SIAM J. Math. Anal. 36 (2005) 1130–1141. [15] Y.L. Xiao, Z.P. Xin, On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition, Comm. Pure Appl. Math. 60 (7) (2007) 1027–1055. [16] J.Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. Éc. Norm. Supér. (4) 14 (2) (1993) 517–542. [17] A.L. Bertozzi, P. Constantin, Global regularity for vortex patches, Comm. Math. Phys. 152 (1) (1993) 19–28. [18] P. Gamblin, X. Saint Raymond, On three-dimensional vortex patches, Bull. Soc. Math. France 123 (3) (1995) 375–424. [19] A. Dutrifoy, On 3D vortex patches in bounded domains, Comm. Partial Differential Equations 28 (2003) 1237–1263. [20] N. Depauw, Poche de tourbillon pour Euler 2D dans un ouvert à bord, J. Math. Pures Appl. 78 (1995) 313–351. [21] F. Sueur, Vorticity internal transition layers for the Navier–Stokes equations, arXiv:0812.2145v1. [22] H. Triebel, Theory of Function Spaces, in: Monographs in Mathematics, vol. 100, Birkhäuser Verlag, Basel, 2006. [23] J.P. Bourguignon, H. Brezis, Remarks on the Euler equation, J. Funct. Anal. 15 (1974) 341–363. [24] C.O. Horgan, Korn’s inequalities and their applications in continuum mechanics, SIAM Rev. 37 (4) (1995) 491–511.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Inviscid limit for vortex patches in a bounded domain Quansen Jiu a,∗ , Yun Wang b a
School of Mathematical Sciences, Capital Normal University, Beijing 100048, PR China
b
The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
article
info
Article history: Received 23 April 2010 Accepted 2 December 2011 Keywords: Inviscid limit Navier boundary condition Vortex patches
abstract In this paper, we consider the inviscid limit of the incompressible Navier–Stokes equations in a smooth, bounded and simply connected domain Ω ⊂ Rd , d = 2, 3. We prove that for a vortex patch initial data, the weak Leray solutions of the incompressible Navier–Stokes equations with Navier boundary conditions will converge (locally in time for d = 3 and globally in time for d = 2) to a vortex patch solution of the incompressible Euler equation as the viscosity vanishes. In view of the results obtained in Abidi and Danchin (2004) [5] and Masmoudi (2007) [3] which dealt with the case of the whole space, we derive an almost 3
optimal convergence rate (ν t ) 4 −ε for any small ε > 0 in L2 . © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The incompressible Navier–Stokes equations read as
∂u − ν 1u + (u · ∇)u + ∇ p = 0, ∂t div u = 0,
(1.1)
where u = (u1 , . . . , ud )(d = 2 or 3) is the velocity fields, p is the pressure function and ν is the kinetic viscosity. Formally, when ν = 0, (1.1) becomes the following incompressible Euler equations:
∂u + (u · ∇)u + ∇ p = 0, ∂t div u = 0.
(1.2)
The inviscid limit for the incompressible Navier–Stokes equations in the whole space has been well understood (see [1–7] and references therein) for both smooth and non-smooth initial data. However, in the case of a bounded domain, the inviscid limit for the Navier–Stokes equations with Dirichlet boundary conditions is still a completely open problem. This is mainly due to the difference between the Dirichlet boundary conditions of the incompressible Navier–Stokes equations (1.1) and the tangential boundary conditions of the incompressible Euler equations (1.2) and a boundary layer will appear near the boundary of the domain. This paper is concerned with the inviscid limit problem of the incompressible Navier–Stokes equations (1.1) with the following Navier boundary conditions:
⃗ = 0, u·n ∗
[D(u)⃗n + α u]tan = 0,
on ∂ Ω × (0, +∞),
Corresponding author. E-mail addresses: [email protected] (Q. Jiu), [email protected] (Y. Wang).
0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.12.003
(1.3)
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Q. Jiu, Y. Wang / Applied Mathematics Letters 25 (2012) 1367–1372
⃗ is the unit exterior normal to the boundary ∂ Ω , D(u) = where Ω ⊂ Rd (d = 2 or 3) is a smooth bounded domain, n 1 [∇ u + (∇ u)T ] is the rate of strain tensor and [D(u)⃗n + α u]tan is the tangential component of the vector D(u)⃗n + α u. Here 2 α = α(x, t ) is a known function representing the friction coefficient of the material. The Navier boundary conditions, introduced by Navier in [8], say that the tangential component of the viscous stress at the boundary is proportional to the tangential velocity. They were rigorously justified as a homogenization of the no-slip condition on a rough boundary in [9] and widely used when studying the inviscid limit of the incompressible flows in a bounded domain (see [10–15]) in recent years. Of particular interest of this paper is the inviscid limit for vortex patches in a bounded domain. It is known that when the initial data are vortex patch ones (see Definition 2.1 for details), there exists a unique solution to the incompressible Euler equations which preserves the vortex-patch structures globally (in time) in the whole plane [16,17] and locally in three-dimensional whole space [18]. In a smooth, bounded and simply connected domain Ω ⊂ Rd (d = 2, 3), the vortex patch solutions of the incompressible Euler equations were derived in [19,20]. In this paper, we will show that the weak Leray solutions of the incompressible Navier–Stokes equations with Navier boundary conditions will tend to a vortex patch solution of the incompressible Euler equations as the viscosity vanishes if the initial data are vortex patch ones. Moreover, 3
we obtain that the convergence rate in L2 is (ν t ) 4 −ε for any small ε > 0. In the case of the whole plane, √ Constantin and Wu studied the inviscid limit for the 2D vortex patches in [6,7] and obtained the convergence rate in L2 is ν t. Abidi and Danchin 3
improved the convergence rate to be (ν t ) 4 in L2 which is optimal since the circular vortex patches provide a lower bound (see [5]). Later, Masmoudi extended the results to the case of three-dimensional whole space in [3]. Recently, Sueur [21] dealt with the vorticity internal transition layers for the Navier–Stokes equations and described how the smoothing effect is (micro-)localized in the case where vortex patches are prescribed as initial data, using the method of asymptotic expansion. In the case of the two-dimensional bounded domain, the inviscid limit for the incompressible Navier–Stokes equations with √ Navier-boundary conditions was discussed in [13] and the obtained convergence rate in L2 is ν t for initial vorticity in L∞ . Our results here apply to both 2D and 3D vortex patches in a bounded domain and the convergence rate obtained in this paper is almost optimal in view of the results in [5,3]. Since we consider the case of the bounded domain, estimates in Besov space in [5,3] cannot be used directly and we will use the interpolation space theory to deduce that the vorticity belongs to L∞ ([0, T ∗ ); H s (Ω )) for some T ∗ > 0 and s > 0. More subtle estimates will be given in this paper. Meanwhile, whether the 3
convergence rate can be improved to (ν t ) 4 is still open. The paper is organized as follows. In Section 2, we will give some preliminaries and the main results. Section 3 is devoted to the proof of the main result. 2. Preliminaries and main results Let Ω ⊂ Rd (d = 2, 3) be a smooth, bounded and simply connected domain. The initial-boundary problem to the incompressible Euler equations is written as
∂u + (u · ∇)u + ∇ p = 0, ∂t
(x, t ) ∈ Ω × [0, +∞)
div u = 0, (x, t ) ∈ Ω × [0, +∞), u · n⃗ = 0, (x, t ) ∈ ∂ Ω × [0, +∞), u(x, 0) = u0 (x), x ∈ Ω .
(2.1)
Denote by (uν , pν ) the solutions of the incompressible Navier–Stokes equations with corresponding kinetic viscosity ν . The initial-boundary problem to the incompressible Navier–Stokes equations with Navier boundary conditions is written as
ν ∂u − ν 1uν + (uν · ∇)uν + ∇ pν = 0, ∂t
(x, t ) ∈ Ω × (0, +∞)
ν
div u = 0, (x, t ) ∈ Ω × [0, +∞), ν ν ν u · n⃗ = 0, ν [D(u )⃗n + α u ]tan = 0, u(x, 0) = u0 (x), x ∈ Ω .
(2.2)
(x, t ) ∈ ∂ Ω × [0, +∞),
Let ω0 = curl u0 be the initial vorticity of u0 . In this paper, for any vector-valued function ϕ, D(ϕ) denotes the symmetric part of ∇ϕ , i.e., D(ϕ) =
∇ϕ + (∇ϕ)T 2
.
Denote by C r , C 1+r (0 < r < 1) the usual Hölder space. In particular, Ccr (Rd ) consists of functions in C r (Rd ) with compact support. Let Lp (Ω ), W s,p (Ω ) be the usual Sobolev spaces defined in Ω , where 1 ≤ p ≤ ∞ and s is permitted to be a real number. If p = 2, W s,2 (Ω ) is denoted by H s (Ω ). H0s (Ω ) is the closure of C0∞ (Ω ) in H s (Ω ). Define ∞ C0∞ ,σ (Ω ) = {f |f ∈ C0 (Ω ), div f = 0},
Cσ∞ (Ω ) = {f |f ∈ C ∞ (Ω ), div f = 0}. 2 1 ∞ 1 L2σ (Ω ) is the closure of C0∞ ,σ (Ω ) in L (Ω ), and Hσ (Ω ) is the closure of Cσ (Ω ) in H (Ω ).
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We first recall the definition of a vortex patch in a bounded domain (see [19,3]). Definition 2.1. Let 0 < r < 1. The vorticity ω = curl u of a vector field u is called a C r vortex patch of support P if the following decomposition holds:
ω = (ωi χP + ωe χΩ \P )|Ω , where P is an open set of class C 1+r , ωi , ωe ∈ Ccr (Rd )(d = 2, 3) and χP , χΩ \P are the characteristic functions of P and Ω \ P respectively.
⃗ = ωe · n⃗ on ∂ P. Notice that when d = 3, curl u is of divergence free, we need ωi · n If the initial data of the incompressible Euler equations is a C r vortex patch, the global existence of 2-d vortex patch solutions and the local existence of 3-d vortex patch solutions have been proved (see [19,20]). More precisely, one has Theorem 2.1. Let u0 be a divergence free vector field in Rd (d = 2, 3), tangent to ∂ Ω , whose vorticity ω0 is a C r vortex patch of support P, the boundary of ∂ P is a (d − 1)-dimensional compact submanifold of Rd . If P ⊂ Ω , then there exists a T ∗ > 0 such that the Euler equations (2.1) have a (unique) solution u ∈ L∞ ([0, T ∗ ); Lip(Ω )). Moreover, ω(t ) = curl u(t ) remains a vortex patch, whose support Ψ (t , P ) is of class C 1+r for any t ∈ [0, T ∗ ), Ψ denoting the flow of u. In addition, T ∗ > 0 can be arbitrarily large if d = 2. Remark 2.1. Under assumptions of Theorem 2.1, if P is tangent to ∂ Ω , a little regularity may be lost. However, local existence of 3-D vortex patch of C s (0 < s < r ) and global existence of 2-D vortex patch of C s (0 < s < r ) is proved in [19]. Now we give the definition of a Leray weak solution of the incompressible Navier–Stokes equations with Navier boundary conditions. Definition 2.2. We call a vector field uν (t , x) : [0, +∞) × Ω → Ω , denoted by u(t , x) here, a weak Leray solution of (2.2) if u verifies (1) u ∈ Cw ([0, ∞); L2σ (Ω )) Lloc ([0, ∞); Hσ1 (Ω )); ⃗ =0 (2) u verifies the system of Eqs. (2.2) under the following weak form: for every ϕ ∈ C0∞ ([0, ∞); Cσ∞ (Ω )) with ϕ · n on ∂ Ω ,
∞ ∞ α u · ϕ + 2ν D(u)D(ϕ) + (u · ∇)u · ϕ 0 Ω 0 Ω ∞∂Ω 0 = u · ∂t ϕ + u(0) · ϕ(0).
2ν
∞
0
Ω
Ω
(3) u verifies the energy inequality, for all t ≥ 0,
∥u(t )∥2L2 (Ω ) + 4ν
t 0
∂Ω
α|u|2 + 4ν
t 0
Ω
|D(u)|2 ≤ ∥u(0)∥2L2 (Ω ) .
We remark that the global existence of the Leray weak solution in the case of Dirichlet boundary conditions is well known for any u0 ∈ L2σ (Ω ). The extensions of this result to the case of Navier boundary conditions is straightforward by the Galerkin method. The main result of the paper is stated as Theorem 2.2. Suppose that the assumptions of Theorem 2.1 hold and u ∈ L∞ ([0, T ∗ ); Lip(Ω )) is the vortex patch solution of the incompressible Euler equations with initial data u0 . Suppose that uν are Leray weak solutions of the incompressible Navier–Stokes equations with Navier boundary conditions (2.2). The corresponding initial data uν (0) is uniformly bounded in L2 (Ω ), and α ∈ L∞ (∂ Ω ). Then for all 0 < T < T ∗ and any small ϵ > 0, one has
∥(uν − u)(t )∥L2 (Ω ) ≤ C ((ν t )
1+β−ϵ 2
+ ∥uν (0) − u0 ∥L2 (Ω ) ),
where β = min( 21 , r ) and C is a constant depending only on ϵ, u, T , ∥α∥L∞ (∂ Ω ) and M ≡ supν ∥uν (0)∥L2 (Ω ) . 3. Proof of main result Since we are concerned with the case of the bounded domain, the estimates in Besov space as in [5,3] cannot be used directly. However, we have Lemma 3.1. Suppose that ω = curl u is the vortex patch solution to the incompressible Euler system, derived in Theorem 2.1. Then, for any s < β = min(r , 12 ), one has ω ∈ L∞ ([0, T ∗ ); H s (Ω )).
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Proof. It is proved in [19] that the vortex patch solution has the following structures:
ω(x, t ) = ωi (x, t )χP (t ) (x) + ωe (x, t )χΩ \P (t ) (x),
t ∈ [0, T ∗ ),
where
ωi , ωe ∈ L∞ ([0, T ∗ ); C r˜ (Rd )),
P (t ) ∈ L∞ ([0, T ∗ ); C 1+˜r (Rd ))
for any r˜ < r, which means that for any t ∈ [0, T ∗ ), P (t ) is a C 1+˜r domain, and the C 1+˜r -norm of the boundary ∂ P (t ) is locally bounded. Hence H d−1 (∂ P (t )), the (d − 1)-dimensional Hausdorff measure of ∂ P (t ) is locally bounded which induces that
χP (t ) (x), χΩ \P (t ) (x) ∈ L∞ ([0, T ∗ ); L∞ (Rd ) ∩ BV (Rd )). Following Lemmas 4.2 and 4.3 in [3], after extending ω to the whole space by zero extension, we derive
ω(x, t ) ∈ L∞ ([0, T ∗ ); B˙ s2˜ ,∞ (Rd )), where s˜ = min(˜r , 21 ) and B˙ s2˜ ,∞ is the classical homogeneous Besov space (see [22] for definition). Using the fact that ω(x, t ) ∈ L∞ ([0, T ∗ ); L2 (Rd )), one has
ω(x, t ) ∈ L∞ ([0, T ∗ ); B˙ s2˜ ,∞ (Rd ) ∩ L2 (Rd )).
(3.3)
Moreover, for any s < β = min(r , ), there exists a s˜ > s such that (3.3) holds. Thus standard interpolation theory (see [22]) yields that 1 2
ω(x, t ) ∈ L∞ ([0, T ∗ ); H s (Ω )). The proof of Lemma 3.1 is completed.
The following are some known facts, of which the proofs are omitted here. Lemma 3.2 (See [23]). For any s ≥ 1 and 1 < p < ∞, there exists a positive constant C , depending only on Ω , s, p, such that for any vector-valued function w , one has
∥w∥W s,p (Ω ) ≤ C [∥div w∥W s−1,p (Ω ) + ∥curl w∥W s−1,p (Ω ) + ∥w · n⃗∥W s−1/p,p (∂ Ω ) + ∥w∥W s−1,p (Ω ) ]. Lemma 3.3 (Korn’s Inequality See [24]). Let ω ∈ H 1 (Ω ). Then there exists a constant C depending only on the domain Ω , such that
∥w∥H 1 (Ω ) ≤ C (∥D(w)∥L2 (Ω ) + ∥w∥L2 (Ω ) ). Now we are ready to prove our main result. Proof of Theorem 2.2. Let v ν = uν − u. For any fixed T < T ∗ , one has for every 0 < t ≤ T ,
∥uν (t )∥2L2 (Ω ) + 4ν
t ∂Ω
0
α|uν |2 + 4ν
t Ω
0
|D(uν )|2 ≤ ∥uν (0)∥2L2 (Ω ) ,
(3.4)
∥u(t )∥2L2 (Ω ) = ∥u0 ∥2L2 (Ω ) .
(3.5) ν
Here (3.4) is the energy inequality for the Leray weak solution u and (3.5) is the energy equality for the vortex patch solution u. Using u as a test function in the weak form satisfying by the Leray weak solution uν (see Definition 2.2), we obtain
ν
Ω
u · u(t )dx + 2ν
t 0
t + 0
Ω
ν
∂Ω
α u · udSdτ + 2ν
t Ω
0
(uν · ∇)uν · udxdτ =
Ω
D(uν ) : D(u)dxdτ
uν (0) · u0 dx.
(3.6)
Adding (3.4) and (3.5) and then subtracting (3.6), one deduces 1 2
ν
∥v (t )∥
2 L2 (Ω )
+ 2ν
t 0
ν
Ω
|D(v )| dxdτ ≤ 2
1 2
ν
∥v (0)∥
− 2ν
t 0
≡
1 2
2 L2 (Ω )
∂Ω
t − Ω
0
(v ν · ∇)u · v ν dxdτ
α uν · v ν dSdτ − 2ν
∥v ν (0)∥2L2 (Ω ) +
t 0
Ω
D(u) : D(v ν )dxdτ
(3.7)
t
(I2 + I3 + I4 )dτ . 0
(3.8)
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Now we estimate the terms on the right hand of (3.7). By Hölder’s inequality,
|I2 | = (v ν · ∇)u · v ν dx ≤ ∥∇ u∥L∞ (Ω ) ∥v ν ∥2L2 (Ω ) .
(3.9)
Ω
For the term I3 , since v ν = uν − u, one has
|I3 | ≤ 2ν
α u · v dS + 2ν ν
∂Ω
α|v | dS . ν 2
∂Ω
Making use of Lemmas 3.2 and 3.3 and the interpolation inequality, then
ν
α u · v dS ≤ ν∥α∥L∞ (∂ Ω ) ∥u∥L2 (∂ Ω ) ∥v ν ∥L2 (∂ Ω ) ν
∂Ω
≤ C ν∥α∥L∞ (∂ Ω ) ∥u∥H 1/2 (Ω ) ∥v ν ∥H 1−s (Ω ) 1−s ≤ C ν∥α∥L∞ (∂ Ω ) (∥ω∥H s (Ω ) + ∥u∥L2 (Ω ) )∥v ν ∥sL2 (Ω ) ∥v ν ∥L2 (Ω ) + ∥Dv ν ∥L2 (Ω ) 2s ν ν ν 1+s ∞ ≤ C ν∥α∥L (∂ Ω ) ∥v ∥L2 (Ω ) + ∥v ∥L2 (Ω ) + ∥D(v ν )∥2L2 (Ω ) , 4
and
ν
∂Ω
α|v ν |2 dS ≤ C ν∥α∥L∞ (∂ Ω ) ∥v ν ∥2 1
H 2 (Ω )
ν 2
≤ C ν∥α∥L∞ (∂ Ω ) ∥v ∥L2 (Ω ) +
ν 4
∥D(v ν )∥2L2 (Ω ) .
Hence the term |I3 | is estimated as
2s ν |I3 | ≤ C ν∥α∥L∞ (∂ Ω ) ∥v ν ∥2L2 (Ω ) + ∥v ν ∥L12+(sΩ ) + ∥D(v ν )∥2L2 (Ω ) .
(3.10)
2
Now we come to the term I4 . Since s < β, ω ∈ L∞ (0, T ; H s (Ω )). Using the duality between H s (Ω ) and H −s (Ω ) (note that s < 21 , H s (Ω ) = H0s (Ω )) and the interpolation inequality, one has
D(u) : D(v ν )dx ≤ C ∥D(u)∥H s (Ω ) ∥D(v ν )∥H −s (Ω ) Ω
≤ C [∥ω∥H s (Ω ) + ∥u∥L2 (Ω ) ] · ∥v ν ∥sL2 (Ω ) ∥v ν ∥H1−1 (sΩ ) ≤ C [∥ω∥H s (Ω ) + ∥u∥L2 (Ω ) ]∥v ν ∥sL2 (Ω ) (∥v ν ∥L2 (Ω ) + ∥D(v ν )∥L2 (Ω ) )1−s . By the way, from (3.4),
∥uν (t )∥2L2 (Ω ) + 4ν
t
∥D(uν )∥2L2 (Ω ) dτ t ν ν 2 ∞ ≤ ∥u (0)∥L2 (Ω ) + C ν∥α∥L (∂ Ω ) ∥u ∥L2 (Ω ) + ∥D(uν )∥L2 (Ω ) ∥uν ∥L2 (Ω ) dτ 0 t t ν 2 ν 2 ≤ ∥u (0)∥L2 (Ω ) + C ν ∥u ∥L2 (Ω ) dτ + 2ν ∥D(uν )∥2L2 (Ω ) dτ , 0
0
0
ν
ν
which implies that ∥u ∥L∞ (0,T ;L2 (Ω )) and ∥v ∥L∞ (0,T ;L2 (Ω )) are uniformly bounded by some constant C depending on M , T and ∥α∥L∞ (∂ Ω ) . Hence by the Young’s inequality,
ν |I4 | = ν D(u) : D(v )dx Ω
≤ C ν∥v ν ∥L2 (Ω ) + C ν∥v ν ∥sL2 (Ω ) ∥D(v ν )∥1L2−(sΩ ) 2s ν ν ν 1+s ≤ C ν ∥v ∥L2 (Ω ) + ∥v ∥L2 (Ω ) + ∥D(v ν )∥2L2 (Ω ) ,
(3.11)
4
where C is a constant depending on Ω , s, M , T , ∥α∥L∞ (∂ Ω ) . Putting (3.9)–(3.11) into (3.7), we get 1 2
∥v ν (t )∥2L2 (Ω ) ≤
1 2
∥v ν (0)∥2L2 (Ω ) +
t
0
(∥∇ u∥L∞ (Ω ) + C )∥v ν ∥2L2 (Ω ) dτ + C ν
t
0
2s
∥v ν ∥L12+(sΩ ) dτ .
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By Gronwall’s lemma, we deduce that 2
2
∥v ν (t )∥L12+(sΩ ) ≤ C ∥v ν (0)∥L12+(sΩ ) + C ν t , where C is a constant depending on ϵ, u, T , ∥α∥L∞ (∂ Ω ) and M. The proof of the theorem is completed.
Acknowledgments The authors would like to thank Professor Zhouping Xin for his interest in this topic and valuable discussions. The paper was started when the first author was visiting the Institute of Mathematical Sciences (IMS) of The Chinese University of Hong Kong. The first author’s research is partially supported by the National Natural Sciences Foundation of China (No. 11171229). References [1] H.S.G. Swann, The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in R3 , Trans. Amer. Math. Soc. 157 (1971) 373–397. [2] J.Y. Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids, Comm. Partial Differential Equations 21 (11–12) (1996) 1771–1779. [3] N. Masmoudi, Remarks about the inviscid limit of the Navier–Stokes system, Comm. Math. Phys. 270 (2007) 777–788. [4] P.L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Models, in: Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, New York, 1996. [5] H. Abidi, R. Danchin, Optimal bounds for the inviscid limit of Navier–Sokes equations, Asymptot. Anal. 38 (1) (2004) 35–46. [6] P. Constantin, J. Wu, Inviscid limit for vortex patches, Nonlinearity 8 (5) (1995) 735–742. [7] P. Constantin, J. Wu, The inviscid limit for non-smooth vorticity, Indiana Univ. Math. J. 45 (1) (1996) 67–81. [8] C.L.M.H. Navier, Sur les lois de l’équilibrie et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France 369 (1827). [9] W. Jäger, A. Mikelić, On the roughness-induced efective boundary conditions for an incompressible viscous flow, J. Differential Equations 170 (2001) 96–122. [10] T. Clopeau, A. Mikelić, R. Robert, On the vaninshing viscosity limit for the 2D incompressible Navier–Stokes equations with the friction type boundary conditions, Nonlinearity 11 (1998) 1625–1636. [11] D. Iftimie, G. Planas, Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions, Nonlinearity 19 (2006) 899–918. [12] Q.S. Jiu, D.J. Niu, Vanishing viscous limits for the 2D lake equations with Navier boundary conditions, J. Math. Anal. Appl. 338 (2008) 1070–1080. [13] J.P. Kelliher, Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane, SIAM J. Math. Anal. 38 (2006) 210–232. [14] M.C. Lopes Filho, H.J. Nussenzveig Lopes, G. Planas, On the inviscid limit for 2D incompressible flow with Navier friction condition, SIAM J. Math. Anal. 36 (2005) 1130–1141. [15] Y.L. Xiao, Z.P. Xin, On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition, Comm. Pure Appl. Math. 60 (7) (2007) 1027–1055. [16] J.Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. Éc. Norm. Supér. (4) 14 (2) (1993) 517–542. [17] A.L. Bertozzi, P. Constantin, Global regularity for vortex patches, Comm. Math. Phys. 152 (1) (1993) 19–28. [18] P. Gamblin, X. Saint Raymond, On three-dimensional vortex patches, Bull. Soc. Math. France 123 (3) (1995) 375–424. [19] A. Dutrifoy, On 3D vortex patches in bounded domains, Comm. Partial Differential Equations 28 (2003) 1237–1263. [20] N. Depauw, Poche de tourbillon pour Euler 2D dans un ouvert à bord, J. Math. Pures Appl. 78 (1995) 313–351. [21] F. Sueur, Vorticity internal transition layers for the Navier–Stokes equations, arXiv:0812.2145v1. [22] H. Triebel, Theory of Function Spaces, in: Monographs in Mathematics, vol. 100, Birkhäuser Verlag, Basel, 2006. [23] J.P. Bourguignon, H. Brezis, Remarks on the Euler equation, J. Funct. Anal. 15 (1974) 341–363. [24] C.O. Horgan, Korn’s inequalities and their applications in continuum mechanics, SIAM Rev. 37 (4) (1995) 491–511.