VORTICITY LAYERS OF THE 2D NAVIER-STOKES EQUATIONS WITH A SLIP TYPE BOUNDARY CONDITION

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1 VORTICITY LAYERS OF THE 2D NAVIER-STOKES EQUATIONS WITH A SLIP TYPE BOUNDARY CONDITION GUNG-MIN GIE 1 AND CHANG-YEOL JUNG 2 Abstract. We study the asymptotic behavior, at small viscosity ε, of the Navier- Stokes equations in a 2D curved domain. The Navier-Stokes equations are supplemented with the slip boundary condition, which is a special case of the Navier friction boundary condition where the friction coefficient is equal to two times the curvature on the boundary. We construct an artificial function, which is called a corrector, to balance the discrepancy on the boundary of the Navier-Stokes and Euler vorticities. Then, performing the error analysis on the corrected difference of the Navier Stokes and Euler vorticity equations, we prove convergence results in the L 2 norm in space uniformly in time, and in the norm of H 1 in space and L 2 in time with rates of order ε 3/4, and ε 1/4 respectively. In addition, using the smallness of the corrector, we obtain the convergence of the Navier-Stokes solution to the Euler solution in the H 1 norm uniformly in time with rate of order ε 1/4. Contents 1. Introduction 1 2. Vorticity formulation of the problem and main result 4 3. Curvilinear coordinate system 6 4. Asymptotic expansions of ω ε and ψ ε at order ε Construction of correctors Estimates on the correctors 9 5. Error Analysis at order ε 0 : Proof of Theorem An application 14 Acknowledgements 15 References Introduction We consider the motion of a fluid governed by the Navier-Stokes equations: u ε t ε uε +(u ε )u ε + p ε = F, in Ω (0,T), div u ε = 0, in Ω (0,T), u ε t=0 = u 0, in Ω. (1.1) Date: December 13, Mathematics Subject Classification. 35B25, 35C20, 76D05, 76D10. Key words and phrases. boundary layers, singular perturbations, Navier-Stokes equations, Euler equations, Navier friction boundary condition, Slip boundary condition. 1

2 2 G. GIE AND C. JUNG The domain Ω is a simply connected and bounded domain in R 2 with smooth boundary Γ, ε > 0 is the (small) viscosity parameter and T > 0 is fixed. The equations are to be solved for the velocity of the fluid u ε and pressure p ε given the forcing function F and initial velocity u 0. The Navier friction boundary condition is given in the form, where S(u ε ) := 1 2( u ε +( u ε ) ) = u ε n = 0, [S(u ε )n+αu ε ] tan = 0, on Γ, (1.2) 1 2 u ε 1 x ( u ε 1 y + uε 2 x ) 1 ( u ε ) 1 2 y + uε 2 x u ε. (1.3) 2 y In (1.2) and (1.3), u ε = (u ε 1, u ε 2), α is the friction coefficient, which is a smooth function on Γ, independent of ε, n is the outer unit normal vector on Γ, and the notation [ ] tan denotes the tangential component of a vector on Γ. Thanks to Lemma 4.1 in [12], the Navier friction boundary condition (1.2) can be written, { uε n = 0, on Γ, ω ε := curl u ε = (2γ α)[u ε (1.4) ] tan, on Γ, where curl u ε := u ε 2 / x uε 1 / y and γ is the curvature on the boundary Γ. As a special case, when α is equal to 2γ, we obtain the so-called slip (or Lions) boundary condition as u ε n = 0, ω ε := curl u ε = 0, on Γ. (1.5) In what follows, we supplement the Navier-Stokes equations(1.1) with the slip boundary condition (1.5). Setting ε = 0 in (1.1) with (1.5), we formally obtain the Euler equations, u 0 t +(u0 )u 0 + p 0 = F, in Ω (0,T), div u 0 = 0, in Ω (0,T), (1.6) u 0 n = 0, on Γ (0,T), u 0 t=0 = u 0, in Ω. We aim to study the asymptotic behavior of the Navier-Stokes solutions u ε at small viscosity. Our emphasis is not on the optimal regularity requirements on the data. Thus, in this article, we assume that where u 0 H C (Ω), F C loc ([0, );C (Ω)) and Γ is of class C, (1.7) H = { v L 2 (Ω) div v = 0 and v n = 0 on Γ }. Under the regularity assumption (1.7), for each fixed viscosity ε > 0 andany time T > 0, we have the existence of a strong solution to the Navier-Stokes equations; see, e.g., [15].

3 VORTICITY LAYERS OF THE 2D NSE WITH A SLIP BOUNDARY CONDITION 3 The Euler equations are well-posed (see, e.g., [10], [11], [19], or [18]) so that, e.g, for any time T > 0, u 0, curl u 0 L (0,T;C (Ω)). (1.8) Then the vorticity formulations of the Navier-Stokes equations with slip boundary condition and Euler equations, which appear in (2.1) and (2.5), are well-posed. The vanishing viscosity limit problem of the Navier-Stokes equations with the Navier friction(hence, as a special case, with the slip) boundary condition has been well-studied; see, e.g., [4], [6], [7], [13], [14], [15], [21], and [22], and other references therein. (See, e.g., [5] and [9] as well for some related problems.) In particular, in [4] or [7], using the boundary layer analysis, the authors prove the convergence of u ε to u 0 in the sense that u ε u 0 L (0,T;L 2 (Ω)) Cε 3 4, u ε u 0 L 2 (0,T;H 1 (Ω)) Cε 1 4, (1.9) for a constant C, depending on the data, but independent of the viscosity ε. Both works [4] and [7] are 3D results, but one can easily verify that their analysis are valid in 2D as well. (In [4], thanks to the regularity results in [15], using the anisotropic Agmon s inequality, the authors also prove the uniform convergence in time and space of u ε to u 0 at the rate of ε 3/8 δ for a small δ > 0 depending on the regularity of the data.) On the other hand, in [15], using the so-called conormal Sobolev spaces, the authors obtain very strong regularity results on the 3D Navier-Stokes equations with the Navier friction boundary condition. Then, using the compactness argument, they obtain the convergence results of u ε to u 0 as those in (1.9) without any rate. They obtain the uniform convergence in time and space as well. Concerning the slip boundary condition (1.5) (or its 3D version; see (1.11) below), in a 2 or 3D domain with flat boundaries, the convergence in higher order Sobolev norms is well-studied in, e.g., [1], [8], or [21]; for the case of a 3D curved domain, see, e.g., [2] or [20]. From the boundary layer analysis point of view, in [8], the authors study the boundary layers of the barotropic quasigeostrophic equations (6.1) in a 2D channel domain, which is essentially the vorticity formulation of the 2D Navier-Stokes equations with the slip boundary condition; see Section 6 for more details. More precisely, they obtain asymptotic expansions of the vorticity and stream function, and prove that u ε u 0 L (0,T;H 1 (Ω)) Cε 1 4, (1.10) for a constant C, depending on the data, but independent of the viscosity ε. In addition, they also obtain complete asymptotic expansions of the vorticity and stream function at any order ε j, j 0, with respect to the viscosity ε. In a recent work, [20], the authors considered the Navier-Stokes equations (1.1) in a 3D general smooth domain supplemented with the 3D version of (1.5): u ε n = 0, (curl u ε ) n = 0, on Γ. (1.11) (For the derivation of (1.11) from the generalized Navier friction boundary condition, see, e.g., [4].) Since the slip type boundary condition (1.11) is a special case of the Navier boundary condition, they first start with the asymptotic expansion of u ε in [7]: u ε u 0 + εv, (1.12)

4 4 G. GIE AND C. JUNG where εv is the boundary layer corrector introduced in [7]. By taking curl on the asymptotic expansion (1.12) and analyzing the vorticity formulations of the Navier- Stokes and Euler equations, they prove that curl u ε curl u 0 εcurl ( v v Γ ) L (0,T;L 2 (Ω)) Cε1 2 ; (1.13) see (6.30) in [20]. Then, from (1.13) and lemmas in [20], (1.10) follows in the end. In addition, the uniform convergence in time and space is also obtained with optimal rate ε 1/2, but only for (1.11). In this article, mainly aiming to prove (1.10) in a 2D curved domain (see (2.10) in Theorem 2.3), we study the vorticity layers of the 2D Navier-Stokes equations (1.1) supplemented with the slip boundary condition (1.5). Here, we follow and generalize the methodology in [8], which is simpler and more direct than that used in [20]. In fact, to manage the difference, caused by the slip boundary condition, of the Navier-Stokes and Euler vorticities on the boundary, we construct an asymptotic expansion of the Navier- Stokes vorticity. Then, using the asymptotic expansion of the Navier-Stokes vorticity, we obtain the convergence result (1.10) as well as the convergence of the corrected difference in (2.9). This convergence result (2.9) of the corrected difference, which is better than (1.13), is considered as optimal in a curved domain; see Remark 2.4 below. In this sense, at least for the 2D case, the boundary layer analysis performed in this article is more precise than that in [20]. The article is organized as follows: in Section 2, we derive the vorticity formulation of (1.1) and (1.5), and state our main result in Theorem 2.3 that is verified in the following sections. In Section 3, to manage the geometrical difficulty of the domain, we introduce an orthogonal curvilinear system adapted to the boundary so that we can treat the normal and tangential variables separately near the boundary. In Section 4, we construct a corrector function corresponding roughly to the Prandtl equation of the Navier-Stokes vorticity equation. This corrector balances the difference of the Navier- Stokes and Euler vorticities on the boundary, and we obtain an asymptotic expansion of the Navier-Stokes vorticity. Then, in Section 5, performing the error analysis, we verify the convergence results at order ε 0 as stated in (2.9) and (2.10). Additionally, in Section 6, we introduce an ocean circulation model as an application of our analysis in this article. 2. Vorticity formulation of the problem and main result For the Navier-Stokes equations (1.1), we introduce the stream function ψ ε and the scalar vorticity ω ε such that ψ ε = u ε, ω ε = curl u ε, where = ( / y, / x). (In addition, we have ψ ε = ω ε.) Then the Navier- Stokes equation (1.1) with the slip boundary condition (1.5) can be written in the vorticity from: ω ε t ε ωε +J(ψ ε, ω ε ) = f := curl F, in Ω (0,T), ω ε = 0, on Γ, (2.1) ω ε t=0 = ω 0 := curl u 0, in Ω,

5 with VORTICITY LAYERS OF THE 2D NSE WITH A SLIP BOUNDARY CONDITION 5 { ψ ε = ω ε, in Ω (0,T), ψ ε = 0, on Γ. The bilinear operator J(, ), appearing in (2.1), is defined as (2.2) J(v, w) := v w x y v w y x, (2.3) for smooth functions v and w. The impermeable boundary condition u ε n = 0 in (1.5) is written in terms of the stream function as ψ ε = 0, on Γ, (2.4) τ where τ denotes the tangential vector on Γ. However, since the domain considered in this article is assumed to be simply connected, without loss of generality, we instead impose the boundary condition (2.2) 2. Remark 2.1. The bilinear term J(ψ ε, ω ε ) in (2.1) is equal to (u ε )ω ε, which is more commonly used in the vorticity form of the 2D Navier-Stokes equations. However, to make the steam function useful in the analysis below, in this article, we rather use the expression J(, ); see, e.g., (5.20) and (5.21). Formally setting ε = 0 in (2.1), or taking the scalar curl on (1.6), we find the corresponding limit problem, with ω 0 t +J(ψ0,ω 0 ) = f, in Ω (0,T), ω 0 t=0 = ω 0, in Ω, { ψ 0 = ω 0, in Ω (0,T), ψ 0 = 0 on Γ. Here ψ 0 = u 0 and ω 0 = curl u 0 for the Euler solution u 0 of (1.6). (2.5) (2.6) Ourtasknowistostudytheasymptoticbehavior, atsmallviscosityε > 0, ofsolutions to (2.1), associated with its limit problem (2.5). As appearing in (2.1) and (2.5), since there is a discrepancy of ω ε and ω 0 on the boundary Γ, we expect the boundary layers of the vorticity to occur. Therefore, to prove the convergence, e.g., of ω ε to ω 0, we construct a corrector function which fixes the difference ω ε ω 0 on Γ, and then perform the error analysis in the following sections. For this purpose, it is useful to have the convergence result of u ε to u 0, e.g., L 2 -convergence of u ε to u 0 uniformly in time. Thus, borrowing from, e.g., [4], we use (1.9) 1 in our analysis below, which is equivalent to (ψ ε ψ 0 ) L (0,T;L 2 (Ω)) κε 3 4 ; (2.7) using (2.7) is to focus on the vorticity layer analysis, which is the main task in this article. However, as explained in Remark 5.1, one can verify (2.7) directly in the error analysis.

6 6 G. GIE AND C. JUNG Definition 2.2. In (2.7) and throughout this article, we use the notation convention, κ = κ(t, ω 0, f, Ω, σ C 2) is a constant depending on the data, but independent of ε, where σ is a smooth (of class C ) cut-off function, defined in (4.8). For the analysis below, we impose a consistency condition on the initial data ω 0 : ω 0 = 0, on Γ. (2.8) Now, using the corrector functions θ, defined in (4.11), we state our main result: Theorem 2.3. Under the regularity assumptions(1.7) and (2.8), the corrected difference ψ ε (ψ 0 +θ) of the Navier-Stokes and Euler stream functions converges to zero in the sense that { ψ ε (ψ 0 +θ) L (0,T;H 2 (Ω)) κε4, 3 (2.9) ψ ε (ψ 0 +θ) L 2 (0,T;H 3 (Ω)) κε4. 1 Moreover, we have the uniform H 1 convergence of the Navier-Stokes solution u ε to the Euler solution: u ε u 0 L (0,T;H 1 (Ω)) κε 1 4. (2.10) Remark 2.4. Concerning a curved domain, the estimate (2.9) is considered as optimal, because the curvature on the boundary cause some error of order ε 3/4 ; see, e.g., Remark 3.1. As studied in [3], this loss of ε 1/4 from the optimal rate ε can be recovered by adding additional corrector, which we omit in this article. 3. Curvilinear coordinate system We let x = (x,y) denotes the Cartesian coordinate system in R 2. We choose small but fixed δ > 0, depending only on the geometry of the domain Ω, so that the tubular neighborhoodω 3δ, withwidth3δ neartheboundaryγ, iswell-defined. Then, weconsider an orthogonal coordinate system ξ = (ξ 1,ξ 2 ) in Ω 3δ,ξ := (0,L) (0,3δ) for some L > 0, adapted to the boundary Γ, where the coordinate direction of ξ 1 (or ξ 2 ) is tangential (or normal) to Γ. (Ω 2δ and Ω 2δ,ξ are defined in the same manner.) In addition, Ω 3δ,ξ is periodic in ξ 1 with period L, and the covariant basis of ξ is given by g i = x ξ i, i = 1,2, for any x Ω 3δ. Without loss of generality, we assume that g 2 = 1. Then g 2 = n on the boundary Γ, and hence the ξ 2 variable measures the distance from the boundary Γ directed inward to Ω. We define h := g 1 g 1 > 0, in the closure of Ω 3δ,ξ, so that the function h(ξ) > 0 in Ω 3δ,ξ is the magnitude of Jacobian determinant from ξ to x. Moreover, since Ω 3δ is smooth (see (1.7)) and h is strictly positive, without loss of generality, we assume that h = (Jacobian determinant from ξ to x) > 0, in the closure of Ω 3δ,ξ. (3.1) We introduce the normalized covariant basis, e 1 = g 1 g 1, e 2 = g 2.

7 VORTICITY LAYERS OF THE 2D NSE WITH A SLIP BOUNDARY CONDITION 7 Then, for a scalar function F in Ω 3δ,ξ, we write the gradient of F in ξ variable, F = 1 F e 1 + F e 2. (3.2) h ξ 1 ξ 2 The Laplacian of F can be written in the form: F = SF +LF + 2 F, (3.3) ξ2 2 where SF := 1 ( 1 F ), LF := 1 h F. (3.4) h ξ 1 h ξ 1 h ξ 2 ξ 2 Thus, S is a linear combination of tangential differential operators with smooth coefficients, and L is proportional to the normal derivative / ξ 2. Remark 3.1. In our setting of the curvilinear system, one can write the explicit expression of the Jacobian determinant as h(ξ 1, ξ 2 ) = 1 γ(ξ 1 )ξ 2 where γ(ξ 1 ) is the curvature on the boundary Γ. Form this point, we infer from (3.4) that L γ(ξ 1 ) ξ 2 +O(ε 1 2 ) ξ 2, where O(ε 1/2 ) is of order ε 1/2. The term γ(ξ 1 ) / ξ 2, which is identically zero for any flat domain, cause some critical error of order ε 3/4 as appearing in (2.9). For two smooth functions F and G in Ω 3δ,ξ, the bilinear operator J(, ) appearing in (2.3) can be written in ξ as J(F, G) = 1 F G 1 F G. (3.5) h ξ 1 ξ 2 h ξ 2 ξ 1 4. Asymptotic expansions of ω ε and ψ ε at order ε 0 Aiming to prove the convergence results in Theorem 2.3, we propose an asymptotic expansion of ω ε and ψ ε, ω ε ω 0 +Θ, ψ ε ψ 0 +θ, (4.1) where, as constructed below, Θ is a vorticity corrector that will balance the difference of ω ε ω 0 on the boundary Γ, and θ is a corrector for the stream functions which will be derived from Θ Construction of correctors. To derive the equation of Θ, we set ω temp = ω ε ω 0, ψ temp = ψ ε ψ 0, (4.2) and insert into the difference of the equations (2.1) and (2.5): ω temp ε ω temp +J(ψ ε,ω ε ) J(ψ 0,ω 0 ) = ε ω 0, in Ω (0,T), t ω temp = ω 0, on Γ, ω temp t=0 = 0, in Ω, with { ψtemp = ω temp, in Ω (0,T), ψ temp = 0, on Γ. (4.3) (4.4)

8 8 G. GIE AND C. JUNG Here we see that ω temp 0 on Γ, and hence we expect boundary layers of the vorticity ω ε to appear near Γ. To extract the essential part of the equation (4.3) at small viscosity, using (4.2), we first write the difference of the nonlinear terms in (4.3) in the from, J(ψ ε, ω ε ) J(ψ 0, ω 0 ) = J(ψ temp, ω temp )+J(ψ temp, ω 0 )+J(ψ 0, ω temp ). Then, expecting that ψ temp is a lower order term than ω temp for small viscosity, and knowing that the data ω 0 and ψ 0 are smooth, we find that J(ψ ε, ω ε ) J(ψ 0, ω 0 ) J(ψ 0, ω temp ). (4.5) Now, in the tubular neighborhood Ω 3δ,ξ, using (3.3), (3.5) and (4.5), we write the equation (4.3) 1 in the curvilinear coordinates ξ. Then, in the resulting equation, using the ansatz, / ξ 2 ε 1/2 / ξ 1, we collect all terms of order ε 0 : ω temp t ε 2 ω temp ξ h ξ2 =0 ψ 0 ξ 1 ω temp ξ 2 1 h ξ2 =0 this is the main equation to define the corrector Θ. For the boundary condition of Θ, we use (4.3) 2 : ψ 0 ξ 2 ω temp ξ 1 = 0, in Ω 3δ,ξ ; (4.6) ω temp = ω 0, on Γ, i.e., ξ 2 = 0. (4.7) We define a smooth cut-off function, σ(ξ 2 ) C (R + ), with { 1, 0 ξ2 δ, σ(ξ 2 ) := 0, ξ 2 2δ. (4.8) Then, using (4.6), (4.7) and (4.8), we define a corrector function Θ = Θ(ξ; t) of ω ε, { σ(ξ2 )Θ(ξ; t), in Ω 2δ, Θ := (4.9) 0, in Ω\Ω 2δ. where Θ is a solution of Θ Θ t ε ξ2 2 h ξ2 =0 ψ 0 Θ 1 ξ 1 ξ 2 h ξ2 =0 ψ 0 ξ 2 Θ ξ 1 = 0, ξ (0,L) (0,2δ), Θ = ω 0, on (0,L) {ξ 2 = 0}, Θ = 0, on (0,L) {ξ 2 = 2δ}, Θ is periodic in ξ 1 with period L, Θ t=0 = 0. (4.10) Remark 4.1. The equation (4.10) is exactly the same as (3.12) in [8] with µ = 0, and 1/h ξ2 =0 and 2δ replaced by 1. Hence the well-posedness result of (4.10) follows from Section 4.3 in [8]. Thanks to (4.4) and (4.8), we define a corrector function θ = θ(ξ; t) of ψ ε in the form, { σ(ξ2 )θ(ξ; t), in Ω 2δ, θ := (4.11) 0, in Ω\Ω 2δ,

9 VORTICITY LAYERS OF THE 2D NSE WITH A SLIP BOUNDARY CONDITION 9 where θ is a solution of θ = Θ, ξ (0,L) (0,2δ), θ = 0, at ξ 2 = 0 or 2δ, θ is periodic in ξ 1 with period L. (4.12) From (4.9) and (4.11), we infer that m+n Θ ξ1 m ξ2 n = 0, ξ2 2δ m+n θ ξ1 m ξ2 n = 0, m,n 0. (4.13) ξ2 2δ 4.2. Estimates on the correctors. As mentioned in Remark 4.1, the system (4.10) is well-posed. Moreover, thanks to Lemma 4.3 in [8], the solution Θ of (4.10) satisfies the estimates below: Lemma 4.2. Under the assumptions (1.7) and (2.8), for p = 2,, we have ) r m+n Θ ε ξ1 m ξ2 n κε 1 2p n 2, r,m 0, 0 n 2. (4.14) L (0,T;L p (Ω 2δ )) ( ξ2 Using (4.9), (4.13) and Lemma 4.2, we see that, for p = 2,, Θ satisfies the following estimates: ) r m+n Θ ε ξ1 m κε 1 2p n 2, r,m 0, 0 n 2. (4.15) ξn 2 L (0,T;L p (Ω)) ( ξ2 We now state and prove the estimates on θ, solution of (4.12) below. Lemma 4.3. Under the assumptions (1.7) and (2.8), we have m+n θ ξ1 m κε 3 4 n 1 2, m 0, n = 1,2. (4.16) ξn 2 L (0,T;L 2 (Ω 2δ )) Proof. To verify (4.16) with n = 1, we differentiate equation (4.12) m times in ξ 1, and find m θ = m Θ, ξ (0,L) (0,2δ), ξ1 m ξ1 m m θ ξ1 m m θ ξ1 m = 0, at ξ 2 = 0 or 2δ, is periodic in ξ 1 with period L. (4.17)

10 10 G. GIE AND C. JUNG We multiply (4.17) by ( m θ/ ξ1 m )h(ξ) and integrate over Ω 2δ,ξ to find m θ ξ1 m 2 m Θ m θ dx L 2 (Ω 2δ ) Ω 2δ ξ1 m ξ1 m ε 1 2 ξ 2 m Θ 1 m θ ε ξ 2 Hence, we obtain ξ m 1 L 2 (Ω 2δ ) ξ m 1 L 2 (Ω 2δ ) (using Lemma 4.2 and Hardy s inequality for ξ2 1 m θ/ ξ1 m) κε 3 4 m θ L 2 (Ω 2δ ). ξ m 1 m θ κε 3 4, (4.18) L 2 (Ω 2δ ) ξ m 1 and then, (4.16) with n = 1 follows from (3.2) and (4.18). To prove (4.16) with n = 2, we infer from (4.17) and Lemma 4.2 that m θ κε 1 4. (4.19) L 2 (Ω 2δ ) ξ m 1 On the other hand, using (3.3) and (4.19), we notice that m+2 θ ξ1 m ξ2 2 κε S m θ L 2 (Ω 2δ ) ξ1 m + L m θ L 2 (Ω 2δ ) ξ1 m L 2 (Ω 2δ ) 2 κε 1 4 +κ m+i θ +κ m+1 θ L L 2 (Ω 2δ ) ξ1 m ξ 2 2 (Ω 2δ ) i=1 ξ1 m+i (using Poincaré inequality for m+i θ/ ξ1 m+i, i = 1,2) 2 κε 1 4 +κ m+i θ +κ m+1 θ L. L 2 (Ω 2δ ) ξ1 m ξ 2 2 (Ω 2δ ) i=1 ξ1 m+i Hence (4.16) with n = 2 follows from (4.16) with n = 1, (4.18) and (4.20). (4.20) Using (4.11), (4.13) and Lemma 4.3, we see that θ satisfies the following estimates, m+n θ ξ1 m κε 3 4 n 1 2, m 0, n = 1,2. ξn 2 L (4.21) (0,T;L 2 (Ω 2δ )) Here, thanks to Poincaré inequality for m θ/ ξ1 m and (4.18), we used the fact that m θ κε4, 3 m 0. (4.22) L (0,T;L 2 (Ω 2δ )) ξ m 1 5. Error Analysis at order ε 0 : Proof of Theorem 2.3 To perform the error analysis on the corrected difference of the Navier-Stokes and Euler vorticities, we set ω = ω ε ω 0 Θ, ψ = ψ ε ψ 0 θ. (5.1)

11 VORTICITY LAYERS OF THE 2D NSE WITH A SLIP BOUNDARY CONDITION 11 Using equations (2.1), (2.2), (2.5), (2.6), (3.3), (4.9), (4.10), (4.11) and (4.12), the equations of ω and ψ read: ω t ε ω +J(ψε,ω ε ) J(ψ 0,ω 0 ) = ε ω 0 Θ +ε Θ, in Ω (0,T), t ω = 0, on Γ, i.e., at ξ 2 = 0, (5.2) ω t=0 = 0, in Ω, with ψ = ω + 1 h σ θ +σ θ+2σ θ, in Ω (0,T), h ξ 2 ξ 2 ψ = 0, on Γ. Using (5.1), we write the difference of nonlinear terms, (5.3) J(ψ ε, ω ε ) J(ψ 0, ω 0 ) = J(ψ ε, ω)+j(ψ 0, Θ)+J(ψ ε ψ 0, ω 0 )+J(ψ ε ψ 0, Θ). (5.4) Then, thanks to (5.4), multiplying (5.2) by ω and integrating over Ω, we find where 1 d 4 2dt ω 2 L 2 (Ω) +ε ω 2 L 2 (Ω) κε2 + ω 2 L 2 (Ω) + J i, (5.5) J 1 = J 2 = J 3 = J 4 = One can easily verify that Ω Ω Ω Ω J(ψ ε, ω)ωdx, i=1 ( Θ ) t ε Θ+J(ψ0, Θ) ωdx, J(ψ ε ψ 0, ω 0 )ωdx, J(ψ ε ψ 0, Θ)ωdx, (5.6) J 1 = 0. (5.7) For the term J 2, using (4.13), we first notice that ( Θ ) J 2 = t ε Θ+J(ψ0, Θ) ωdx. (5.8) Ω 2δ Then, using (3.3), (3.5) and (4.9) in the tubular neighborhood, we find Θ t ε Θ+J(ψ0, Θ) = σ Θ ( ) t ε S +L+ 2 (σθ ) 1 ψ 0 (σθ) + σ 1 ψ 0 Θ ξ 2 h ξ 1 ξ 2 h ξ 2 ξ 1 = (using (4.10) and Taylor s expansion of 1/h in ξ 2 at ξ 2 = 0) = E 1 +E 2, (5.9)

12 12 G. GIE AND C. JUNG where E 1 = εσsθ ε ( Lσ ) Θ εσlθ εσ Θ 2εσ Θ 1 σ( ξ 2 h 1 h ( 1 E 2 = σ h h 1 ) ψ 0 Θ +σ 1 ψ 0 Θ. ξ2 =0 ξ 1 ξ 2 h ξ 1 Using (5.8) and (5.9), we write ξ2 =0 ) ψ 0 Θ, ξ 2 ξ 1 (5.10) J 2 E i ω L 1 (Ω 2δ ). (5.11) i=1,2 Near the boundary Γ, using the Taylor expansion in ξ 2 of the smooth function 1/h, we observe that 1 h 1 κε 1 ξ 2 2. (5.12) h ξ2 =0 ε Then, remembering thats isatangentialdifferential operatorinξ 1 andlisproportional to / ξ 2, and using (4.14) and (5.12), we find E 1 ω L 1 (Ω 2δ ) E 1 L 2 (Ω 2δ ) ω L 2 (Ω 2δ ) κε 3 4 ω L 2 (Ω) κε ω 2 L 2 (Ω). (5.13) Using (5.12), we estimate, E 2 ω L 1 (Ω 2δ ) κε 1 σ ψ0 L ( ) ξ2 2 Θ L ω L ξ 2 ξ 1 (Ω 2δ ) ε ξ (Ω 2δ ) (Ω 2δ ) +κε σ ψ0 L ξ 2 Θ L ξ 2 ξ 1 (Ω 2δ ) ε L 2 (Ω 2δ ) ω 2 (Ω 2δ ). (5.14) Then, since σ( ψ 0 / ξ 1 ) and σ ( ψ 0 / ξ 1 ) vanish at ξ 2 = 0, 2δ, using the regularity of ψ 0 and (4.14), we find E 2 ω L 1 (Ω 2δ ) κε 3 4 ω L 2 (Ω) κε ω 2 L 2 (Ω). (5.15) Combining (5.11), (5.13) and (5.15), we obtain Thanks to (2.7), the term J 3 is easy to estimate: J 2 κε ω 2 L 2 (Ω). (5.16) J 3 κ ω 0 L (Ω) (ψ ε ψ 0 ) L 2 (Ω) ω L 2 (Ω) κε 3 4 ω L 2 (Ω) κε ω 2 L 2 (Ω). (5.17) To estimate the term J 4, using (2.3) and (4.9), we write where J 4,1 = J 4,2 = J 4 J 4,1 + J 4,2, (5.18) 1 Ω 2δ h Ω 2δ 1 h (ψ ε ψ 0 ) ξ 1 (ψ ε ψ 0 ) ξ 2 (σθ) ω dx, ξ 2 (σθ) ω dx. ξ 1 (5.19)

13 VORTICITY LAYERS OF THE 2D NSE WITH A SLIP BOUNDARY CONDITION 13 Using (4.14) with p = and Hardy s inequality for (ψ ε ψ 0 )/ ξ 1, we estimate the term J 4,1, J 4,1 κε (ψ ε ψ 0 ) L ξ 2 (σθ) L ω L ξ 2 ξ 1 2 (Ω 2δ ) ε ξ 2 2 (Ω 2δ ) (Ω 2δ ) ( (ψ ε ψ 0 ) ) L κ ω L ξ (Ω) (Ω 2δ ) (5.20) (using (5.1) 2 and (4.21)) ( ( ψ ) ) L κ +ε 3 4 ω L ξ (Ω) (Ω 2δ ) κ ( ψ H 2 (Ω) +ε 3 4) ω L 2 (Ω). Then, since ψ = 0 on the boundary Γ, we find J 4,1 κ ( ψ L 2 (Ω) +ε 3 4) ω L 2 (Ω) (using (5.3), (4.16) and (4.22)) κ ( ω L 2 (Ω) +ε 3 4) ω L 2 (Ω) κ ω 2 L 2 (Ω) +κε3 2. (5.21) Using (2.7) and (4.15), the term J 4,2 is easy to estimate: J 4,2 κ (ψ ε ψ 0 ) L 2 (Ω) Θ L ω L ξ 2 1 (Ω) (Ω) (5.22) κε 3 4 ω L 2 (Ω) κε ω 2 L 2 (Ω). Combining (5.21) and (5.22), we find J 4 κε 3 2 +κ ω 2 L 2 (Ω). (5.23) Now, from (5.5), (5.7), (5.16), (5.17) and (5.23), we infer that 1 d 2dt ω 2 L 2 (Ω) +ε ω 2 L 2 (Ω) κε3 2 +κ ω 2 L 2 (Ω). (5.24) Applying the Gronwall inequality, we obtain ω L (0,T;L 2 (Ω)) κε 3 4, ω L 2 (0,T;H 1 (Ω)) κε 1 4, (5.25) and, using (4.16), (4.22) and (5.3) as well, we find ψ L (0,T;L 2 (Ω)) κε 3 4, ψ L 2 (0,T;H 1 (Ω)) κε 1 4. (5.26) Then, since ψ vanishes on Γ, (2.9) follows from (5.26). Finally, thanks to (4.21), (2.10) follows from (2.9), and hence the proof of Theorem 2.3 is now complete. Remark 5.1. Multiplying (5.2) by ψ(= ψ ε ψ 0 θ) in (5.1), and performing the error analysis on ψ, one can verify that ψ L (0,T;L 2 (Ω)) κε 3 4, ψ L (0,T;L 2 (Ω)) κε 1 4. (5.27)

14 14 G. GIE AND C. JUNG Then borrowing (2.7) from the earlier works, e.g., [4] or [7], becomes unnecessary. However, to focus on the vorticity layer analysis, we omit this process in this article. 6. An application Analysis in this article can be extended to a homogeneous model of ocean circulation (see, e.g., [16], [17] or [8]). More precisely, concerning the barotropic quasigeostrophic equations in a 2D general (curved) domain, we apply the β-plane approximation at the mid-latitudes, and use the stream function formulation of the model. Then one can write the non-dimensionalized barotropic quasigeostrophic equations in the form, ψ ε ε 2 ψε +µ ψ ε +J( ψ ε, ψ ε +βy) = g, in Ω (0,T), t ψ ε = 0, on Γ (0,T), ψ ε = 0, on Γ (0,T), ψ ε t=0 = ψ 0, in Ω, (6.1) where, ε is a small, but strictly positive viscosity parameter, g and ψ 0 are smooth data, T > 0 is a fixed time, and µ,β 0 are fixed constants. The bilinear operator J(, ) in (6.1) is the same as (2.3), and Ω R 2 is a simply connected bounded domain with smooth boundary Γ where the x and y axes are parallel to the directions of latitude and longitude of the earth respectively. The corresponding inviscid problem of (6.1) is ψ 0 t +µ ψ 0 +J( ψ 0, ψ 0 +βy) = g, in Ω (0,T), ψ 0 = 0, on Γ (0,T), ψ 0 t=0 = ψ 0, in Ω. (6.2) The model (6.1) (or (6.2)) with µ = β = 0 is exactly the same as the vorticity formulation of the Navier-Stokes equations(2.1)(or the Euler equations(2.5)). Moreover the terms involved in µ and β are lower order derivative terms than bi-laplacian 2, and hence they do not cause much difficulties in the analysis. Therefore, by following the modified, but almost the same, analysis in this article, one can prove the analogue of Theorem 2.3: For smooth and compatible data ψ 0, g and Γ, we have { ψε ( ψ 0 +σ θ) L (0,T;H 2 (Ω)) κε 3 4, ψ ε ( ψ 0 +σ θ) L 2 (0,T;H 3 (Ω)) κε 1 4, (6.3) and ψ ε ψ 0 L (0,T;H 2 (Ω)) κε 1 4. (6.4)

15 VORTICITY LAYERS OF THE 2D NSE WITH A SLIP BOUNDARY CONDITION 15 Here σ is the cut-off function, defined in (4.8), θ is a solution of (4.12) with θ and Θ respectively replaced by θ and Θ, and Θ is a solution of the system below: Θ ε 2 Θ +µ Θ+ 1 ψ 0 Θ t ξ2 2 1 ψ 0 Θ = 0, ξ (0,L) (0,2δ), h ξ2 =0 ξ 1 ξ 2 h ξ2 =0 ξ 2 ξ 1 Θ = ψ 0, on (0,L) {ξ 2 = 0}, Θ = 0, on (0,L) {ξ 2 = 2δ}, Θ is periodic in ξ 1 with period L, Θ t=0 = 0. (6.5) As shown in [8], Θ, solution of (6.5) enjoys the estimates in (4.14), and thus, by applying the same analysis in this article as for the 2D Navier-Stokes equations, one can verify that (6.3) and (6.4) hold true. Acknowledgements The authors would like to thank Professors Roger Temam and James P. Kelliher for the fruitful discussions and their helpful advice. The first author is supported by NSF grant DMS and AMS-Simons travel grants. The second author is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( ). References [1] H. Beirão da Veiga and F. Crispo. Sharp inviscid limit results under Navier type boundary conditions. An L p theory. J. Math. Fluid Mech., 12(3): , [2] H. Beirão da Veiga and F. Crispo. The 3-D inviscid limit result under slip boundary conditions. A negative answer. J. Math. Fluid Mech., 14(1):55 59, [3] Gung-Min Gie, Makram Hamouda, and Roger Temam. Boundary layers in smooth curvilinear domains: parabolic problems. Discrete Contin. Dyn. Syst., 26(4): , [4] Gung-Min Gie and James P. Kelliher. Boundary layer analysis of the Navier Stokes equations with generalized Navier boundary conditions. J. Differential Equations, 253(6): , , 5, 14 [5] Makram Hamouda, Chang-Yeol Jung, and Roger Temam. Boundary layers for the 2D linearized primitive equations. Commun. Pure Appl. Anal., 8(1): , [6] Dragoş Iftimie and Gabriela Planas. Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity, 19(4): , [7] Dragoş Iftimie and Franck Sueur. Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Anal., 199(1): , , 4, 14 [8] Chang-Yeol Jung, Madalina Petcu, and Roger Temam. Singular perturbation analysis on a homogeneous ocean circulation model. Anal. Appl. (Singap.), 9(3): , , 4, 8, 9, 14, 15 [9] Chang-Yeol Jung and Roger Temam. Convection-diffusion equations in a circle: the compatible case. J. Math. Pures Appl. (9), 96(1):88 107, [10] Tosio Kato. On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Rational Mech. Anal., 25: , [11] TosioKato.NonstationaryflowsofviscousandidealfluidsinR 3.J. Functional Analysis, 9: , [12] James P. Kelliher. Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM Math Analysis, 38(1): ,

16 16 G. GIE AND C. JUNG [13] M. C. Lopes Filho, H. J. Nussenzveig Lopes, and G. Planas. On the inviscid limit for 2d incompressible flow with Navier friction condition. SIAM Math Analysis, 36(4): , [14] Milton Lopes Filho. Boundary layers and the vanishing viscosity limit for incompressible 2d flow. arxiv: v1 [math.ap], [15] Nader Masmoudi and Frédéric Rousset. Uniform regularity for the Navier-Stokes equation with Navier boundary condition. Arch. Ration. Mech. Anal., 203(2): , , 3 [16] Joseph Pedlosky. Ocean circulation theory. Springer-Verlag, Berlin, [17] Rick Salmon. Lectures on geophysical fluid dynamics. Oxford University Press, New York, [18] R. Temam. Local existence of C solutions of the Euler equations of incompressible perfect fluids. In Turbulence and Navier-Stokes equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975), pages Lecture Notes in Math., Vol Springer, Berlin, [19] Roger Temam. On the Euler equations of incompressible perfect fluids. J. Functional Analysis, 20(1):32 43, [20] Lizhen Wang, Zhouping Xin, and Aibin Zang. Vanishing viscous limits for 3D Navier-Stokes equations with a Navier-slip boundary condition. Journal of Mathematical Fluid Mechanics, pages /s , 4 [21] Yuelong Xiao and Zhouping Xin. On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Comm. Pure Appl. Math., 60(7): , [22] Yuelong Xiao and Zhouping Xin. Remarks on vanishing viscosity limits for the 3D Navier-Stokes equations with a slip boundary condition. Chin. Ann. Math. Ser. B, 32(3): , Department of Mathematics, University of California, Riverside, 900 University Ave., Riverside, CA 92521, U.S.A. 2 Ulsan National Institute of Science and Technology, San 194, Banyeon-ri, Eonyangeup, Ulju-gun, Ulsan, Republic of Korea address: ggie@math.ucr.edu address: cjung@unist.ac.kr