A mathematical justification of a thin film approximation for the flow down an inclined plane

J. Math. Anal. Appl. 444 (2016) 804–824

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A mathematical justification of a thin film approximation for the flow down an inclined plane Hiroki Ueno ∗ , Tatsuo Iguchi Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

a r t i c l e

i n f o

Article history: Received 7 December 2015 Available online 5 July 2016 Submitted by C.E. Wayne Keywords: Navier–Stokes equations Free boundary problem Thin film approximation Inclined plane Burgers equation

a b s t r a c t We consider a two-dimensional motion of a thin film flowing down an inclined plane under the influence of the gravity and the surface tension. In order to investigate the stability of such flow, we often use a thin film approximation, which is an approximation obtained by the perturbation expansion with respect to the aspect ratio of the film. The famous examples of the approximate equations are the Burgers equation, Kuramoto–Sivashinsky equation, KdV–Burgers equation, KdV– Kuramoto–Sivashinsky equation, and so on. In this paper, we give a mathematically rigorous justification of a thin film approximation by establishing an error estimate between the solution of the Navier–Stokes equations and those of approximate equations. © 2016 Elsevier Inc. All rights reserved.

1. Introduction In this paper, we consider a two-dimensional motion of a liquid film of a viscous and incompressible fluid flowing down an inclined plane under the influence of the gravity and the surface tension on the interface. The motion can be mathematically formulated as a free boundary problem for the incompressible Navier–Stokes equations. We assume that the domain Ω(t) occupied by the liquid at time t ≥ 0, the liquid surface Γ(t), and the rigid plane Σ are of the form ⎧ 2 ⎪ ⎨ Ω(t) = {(x, y) ∈ R | 0 < y < h0 + η(x, t)}, Γ(t) = {(x, y) ∈ R2 | y = h0 + η(x, t)}, ⎪ ⎩ Σ = {(x, y) ∈ R2 | y = 0}, * Corresponding author. E-mail addresses: [email protected] (H. Ueno), [email protected] (T. Iguchi). http://dx.doi.org/10.1016/j.jmaa.2016.06.064 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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where h0 is the mean thickness of the liquid film and η(x, t) is the amplitude of the liquid surface. Here we choose a coordinate system (x, y) so that x axis is pointed to the streamwise direction and y axis is normal to the plane. We consider fluctuations of the Nusselt flat film solution, which is the stationary laminar flow given by η1 = 0,

u1 = (ρg sin α/2μ)(2h0 y − y 2 ),

v1 = 0,

p1 = p0 − ρg cos α(y − h0 ),

(1.1)

where ρ is a constant density of the liquid, g is the acceleration of the gravity, α is the angle of inclination, μ is the shear viscosity coefficient, and p0 is an atmospheric pressure. Throughout this paper, we assume that the flow is l0 -periodic in the streamwise direction x. Rescaling the independent and dependent variables by using h0 , l0 , the typical amplitude of the liquid surface a0 , U0 = ρgh20 sin α/2μ, and P0 = ρgh0 sin α, the equations are written in the non-dimensional form ⎧ ⎨

  2 1 δut + (¯ u + εu) · ∇δ u + (u · ∇δ )¯ u + ∇δ p − Δδ u = 0 R R ⎩∇ ·u = 0 δ ⎧  ⎪ ¯ ) − εpI n D δ (εu + u ⎪ ⎪  ⎪ ⎨ 1 δ2W εηxx n = − εη + tan α sin α (1 + (εδηx )2 ) 32 ⎪ ⎪ ⎪ ⎪   ⎩ ηt + 1 − (εη)2 + εu ηx − v = 0

in

Ωε (t), t > 0,

in

Ωε (t), t > 0,

on Γε (t), t > 0,

(1.2)

(1.3)

on Γε (t), t > 0,

u = 0 on Σ, t > 0.

(1.4)

Here, δ, ε, R, and W are non-dimensional parameters defined by δ=

h0 , l0

ε=

a0 , h0

R=

ρU0 h0 , μ

W=

σ , ρgh20

where σ is the surface tension coefficient. Note that δ is the aspect ratio of the film, ε represents the magnitude of nonlinearity, R is the Reynolds number, and W is the Weber number. Moreover, we used

¯ = (¯ u, 0)T , u notations u = (u, δv)T , u ¯ = 2y − y 2 , ∇δ = (δ∂x , ∂y )T , Δδ = ∇δ · ∇δ , D δ f = 12 ∇δ (f T ) + T   , and n = (−εδηx , 1)T . In this scaling, the liquid domain Ωε (t) and the liquid surface Γε (t) are ∇δ (f T ) of the form

Ωε (t) = {(x, y) ∈ R2 | 0 < y < 1 + εη(x, t)}, Γε (t) = {(x, y) ∈ R2 | y = 1 + εη(x, t)}.

Concerning a mathematical analysis of the problem in the case of δ = ε = 1, Teramoto [15] showed that the initial value problem to the Navier–Stokes equations (1.2)–(1.4) has a unique solution globally in time under the assumptions that the Reynolds number and the initial data are sufficiently small. Nishida, Teramoto, and Win [11] showed the exponential stability of the Nusselt flat film solution under the assumptions that the angle of inclination is sufficiently small and x ∈ T in addition to the assumptions in [15]. Furthermore, Uecker [16] studied the asymptotic behavior for t → ∞ of the solution in the case of x ∈ R and showed that the perturbations of the Nusselt flat film solution decay like the self-similar solution of the Burgers equation under the assumptions that the initial data are sufficiently small and R < Rc . Here, Rc = 45 tan1 α is the critical Reynolds number given by Benjamin [2]. On the other hand, Ueno, Shiraishi, and Iguchi [17] derived a uniform estimate for the solution of (1.2)–(1.4) with respect to δ when the Reynolds number, the angle of inclination, and the initial data are sufficiently small.

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Benney [3] derived the following single nonlinear evolution equation 8 (Rc − R)δηxx + C1 δ 2 ηxxx 15 2 W 3 δ ηxxxx = O(δ 3 + ε2 δ + εδ 2 ) + C2 εδ(ηηxx + ηx2 ) + 3 sin α

ηt + 2(1 + εη)2 ηx −

(1.5)

by using the method of a perturbation expansion of the solution (u, v, p) with respect to δ under the thin film regime δ  1. Here, C1 = C1 (R, α) and C2 = C2 (R, α) are constants independent of δ, ε, and W. Explicit forms of C1 and C2 will be given in Section 3 (see (3.14)). Many approximate equations are obtained from (1.5) by assuming that parameters ε, W, and R have appropriate orders in δ. In the following, we assume ε = δ and R < Rc and set η(x, t) = ζ(x − 2t, εt).

(1.6)

I. Burgers equation Assuming W1 ≤ W ≤ δ −1 W2 in (1.5), we have ηt + 2ηx + 4εηηx −

8 (Rc − R)δηxx = O(δ 2 ). 15

Plugging (1.6) in the above equation and passing to the limit ε = δ → 0, we obtain ζτ + 4ζζx −

8 (Rc − R)ζxx = 0. 15

(1.7)

II. Burgers equation with a fourth order dissipation term Assuming W = δ −2 W2 in (1.5), we have ηt + 2ηx + 4εηηx −

8 2 W2 (Rc − R)δηxx + δηxxxx = O(δ 2 ). 15 3 sin α

Plugging (1.6) in the above equation and passing to the limit ε = δ → 0, we obtain ζτ + 4ζζx −

8 2 W2 (Rc − R)ζxx + ζxxxx = 0. 15 3 sin α

(1.8)

III. Burgers equation with dispersion and nonlinear terms Assuming W1 ≤ W ≤ W2 in (1.5), we have ηt + 2ηx + 4εηηx −

8 (Rc − R)δηxx + C1 δ 2 ηxxx + C2 εδ(ηηxx + ηx2 ) + 2ε2 η 2 ηx = O(δ 3 ). 15

Plugging (1.6) in the above equation and neglecting the terms of O(δ 3 ), we obtain ζτ + 4ζζx −

   8 (Rc − R)ζxx + δ C1 ζxxx + C2 ζζxx + ζx2 + 2ζ 2 ζx = 0. 15

IV. Burgers equation with fourth order dissipation, dispersion, and nonlinear terms Assuming W = δ −1 W2 in (1.5), we have ηt + 2ηx + 4εηηx −

8 (Rc − R)δηxx 15

+ C1 δ 2 ηxxx + C2 εδ(ηηxx + ηx2 ) + 2ε2 η 2 ηx +

2 W2 2 δ ηxxxx = O(δ 3 ). 3 sin α

(1.9)

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Plugging (1.6) in the above equation and neglecting the terms of O(δ 3 ), we obtain 8 ζτ + 4ζζx − (Rc − R)ζxx 15 

  2 W2 2 2 ζxxxx = 0. + δ C1 ζxxx + C2 ζζxx + ζx + 2ζ ζx + 3 sin α

(1.10)

We remark that (1.9) and (1.10) are higher order approximate equations to the Burgers equation (1.7). In this paper, we assume R  Rc in order to use a uniform estimate in δ for the solution of the Navier–Stokes equations. Uniform estimates in δ for the solution play a most important role in the justification for these approximations. Here if we could assume R > Rc , then (1.8) would be the Kuramoto–Sivashinsky equation ˜ > 0, then we would obtain the δ-independent KdV–Burgers (see [10,13,14]). If we could assume Rc −R = δ R equation ζτ + 4ζζx −

˜ 8R ζxx + C1 ζxxx = 0 15

(1.11)

by plugging (1.6) in (1.5) and passing to the limit ε = δ 2 → 0 under the assumption W1 ≤ W ≤ W2 . More˜ < 0, we would obtain the δ-independent KdV–Kuramoto–Sivashinsky over if we could assume Rc −R = −δ R equation

ζτ + 4ζζx +

˜ 8R 2 W2 ζxx + C1 ζxxx + ζxxxx = 0 15 3 sin α

(1.12)

by plugging (1.6) in (1.5) and passing to the limit ε = δ 2 → 0 under the assumption W = δ −1 W2 . More details or a list of useful references about the thin film approximation can be found in [6,7,9,12,17]. In this paper, we will give a mathematically rigorous justification of these thin film approximations by establishing error estimates between the solution of the Navier–Stokes equations (1.2)–(1.4) and those of the approximate equations (1.7)–(1.10). To our knowledge, this is the first rigorous justification of a thin film approximation in the sense of comparing the solution of the Navier–Stokes equations with those of the approximate equations. We remark that Bresch and Noble [5] justified the shallow water model by proving that remainder terms converge to 0 as δ → 0 (see also [4]). Moreover, Giacomelli and Otto [8] justified a lubrication approximation in the sense that an equilibrium contact angle is preserved throughout the evolution for a Darcy flow. We also note that we cannot just yet justify the Kuramoto–Sivashinsky equation, the δ-independent KdV–Burgers equation (1.11), and the KdV–Kuramoto–Sivashinsky equation (1.12) because without the assumption R  Rc we have not yet obtain a uniform estimate in δ for the solution. The plan of this paper is as follows. In Section 2, we transform the problem in a time dependent domain to a problem in a time independent domain and then give our main theorem, that is, error estimates between the solution of the Navier–Stokes equations and those of the approximate equations. We remark that the error estimates are not only for the amplitude function η but also for the velocity field u and pressure p. In Section 3, we construct an approximate solution of the Navier–Stokes equations by using Benney’s method as follows. We first fix η = η(x, t) arbitrarily and let (u, v, p) be a solution of (1.2)–(1.4) except a kinematic boundary condition   ηt + 1 − (εη)2 + εu ηx − v = 0. Expanding the solution with respect to the small parameter δ as

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⎧ 2 ⎪ ⎪ ⎨u = u(0) + δu(1) + δ u(2) + · · · , v = v(0) + δv(1) + δ 2 v(2) + · · · , ⎪ ⎪ ⎩p = p + δp + δ 2 p + · · · (0)

(1)

(2)

and substituting these into the equations, we obtain ordinary differential equations in y together with boundary conditions for each order of δ. Solving the boundary value problems, we determine coefficients in the above expansion. Then, neglecting higher order terms in δ, we obtain an approximate solution of the Navier–Stokes equations for the arbitrary function η. We note that the approximate solution is just a polynomial in y whose coefficients depend on η and its derivatives. Substituting the approximate solution into the above kinematic boundary condition, we can recover the approximate equation for η given in Section 1. In Section 4, we derive an energy estimate for a difference between the solution of the Navier–Stokes equations and the approximate solution constructed in Section 3. Since the approximate solution satisfies the Navier–Stokes equations approximately, the difference satisfies linearized Navier–Stokes equations with non-homogeneous terms. Therefore, we apply the energy estimate for the solution of Navier–Stokes equations obtained in [17] to the difference. In [17], this energy estimate was the most important and essential step in order to derive the uniform estimate in δ for the solution of the Navier–Stokes equations. This energy structure allows us to derive the desired error estimates and hence Section 4 is the main part in this paper. Finally, in Section 5 we complete error estimates. That is, we specify the arbitrary function η as the solution of each approximate equation and estimate nonlinear terms appearing in the right-hand side of the energy inequality in terms of energy functions, where we use essentially the uniform estimate for the solution of the Navier–Stokes equations obtained in [17]. We remark that calculations performed in nonlinear estimates are technical because we need to carefully treat the dependence of δ in the estimates. Notation. We put Ω = T × (0, 1) and Γ = T × {y = 1}, where T is the flat torus T = R/Z. For a Banach space X, we denote by · X the norm in X. For 1 ≤ p ≤ ∞, we put u Lp = u Lp (Ω) , u = u L2 , |u|Lp = u(·, 1) Lp (T) , and |u|0 = |u|L2 . We denote by (·, ·)Ω and (·, ·)Γ the inner products of L2 (Ω) and L2 (Γ), respectively. For s ≥ 0, we denote by H s (Ω) and H s (Γ) the L2 Sobolev spaces of order s on Ω and Γ, respectively. The norms of these spaces are denoted by · s and | · |s . For a function u = u(x, y)  on Ω, a Fourier multiplier P (Dx ) (Dx = −i∂x ) is defined by (P (Dx )u)(x, y) = un (y)e2πinx , n∈Z P (n)ˆ 1 1 −2πinx −1 where u ˆn (y) = 0 u(x, y)e dx is the Fourier coefficient in x. We put ∂y f (x, y) = − y f (x, z)dz and k i j Dδ f = {(δ∂x ) ∂y f | i + j = k}. f  g means that there exists a non-essential positive constant C such that f ≤ Cg holds. 2. Main results We rewrite the system (1.2)–(1.4) according to [1,17]. Transforming the problem in the moving domain Ω(t) to a problem in the fixed domain Ω by using an appropriate diffeomorphism, and introducing new unknown function (u , v  , p ) to keep the solenoidal condition, we obtain ⎧ 2 1 ⎪ ¯ ux + u ¯y v) + δpx − (δ 2 uxx + uyy ) = δ 2 f1 ⎪ ⎪ δ(ut + u R R ⎪ ⎨ 2 1 δ 2 (vt + u ¯vx ) + py − δ(δ 2 vxx + vyy ) = δ 2 f2 ⎪ ⎪ R R ⎪ ⎪ ⎩ ux + vy = 0 ⎧ ⎪ δ 2 vx + uy − 2(1 + εη)2 η = δ 3 h1 ⎪ ⎪ ⎨ δ2W 1 η + ηxx = δ 2 h2 p − δv y − ⎪ tan α sin α ⎪ ⎪ ⎩ ηt + ηx − v = δ 2 η 2 ηx =: δ 2 h3

in Ω, t > 0, in Ω, t > 0,

(2.1)

in Ω, t > 0,

on Γ, t > 0, on Γ, t > 0, on Γ, t > 0,

(2.2)

H. Ueno, T. Iguchi / J. Math. Anal. Appl. 444 (2016) 804–824

u = v = 0 on Σ, t > 0,

809

(2.3)

where we dropped the prime sign in the notation and f1 , f2 , h1 , and h2 are collections of nonlinear terms. See [17] for more details on the explicit forms of these nonlinear terms. In the following, we will consider the initial value problem to (2.1)–(2.3) under the initial conditions η|t=0 = η0

on Γ,

(u, v)T |t=0 = (u0 , v0 )T

in Ω.

(2.4)

1 (0) Here, we assume 0 η0 (x)dx = 0 and denote h1 determined from initial data by h1 . We impose the following assumption on the non-dimensional parameters and initial data. Assumption 2.1. Let R0 , R1 , α0 , W1 , c0 , and M be positive constants and m ≥ 2 be an integer. (1) Conditions for parameters Parameters R, α, W, δ, and ε satisfy R1 ≤ R ≤ R0 ,

0 < α ≤ α0 ,

W1 ≤ W,

0 < ε = δ ≤ 1.

(2) Smallness of initial data Initial data (η0 , u0 , v0 ) and parameters W and δ satisfy |(1 + δ|Dx |)2 η0 |2 + (1 + |Dx |)2 (u0 , δv0 )T + (1 + |Dx |)2 Dδ (u0 , δv0 )T √ + (1 + |Dx |)2 Dδ2 (u0 , δv0 )T + δ 2 W|(1 + δ|Dx |)η0x |3 + δ 2 W (1 + |Dx |)2 δv0xy ≤ c0 . (3) Regularity of initial data Initial data (η0 , u0 , v0 ) satisfies (1 + |Dx |)m+1 (u0 , v0 )T H 2 (Ω) + |η0 |m+4 ≤ M. (4) Compatibility conditions Initial data (η0 , u0 , v0 ) and parameters δ and ε satisfy ⎧ ⎪ ⎨ u0x + v0y = 0 (0) u0y + δ 2 v0x − 2(1 + εη0 )2 η0 = δ 3 h1 ⎪ ⎩u = v = 0 0 0

in Ω, on Γ, on Σ.

Remark 2.1. Under the assumption that there exist small positive constants R0 , α0 , and c0 such that Assumption 2.1 is fulfilled, Ueno, Shiraishi, and Iguchi [17] proved the global in time uniform estimate with respect to δ for the solution of the Navier–Stokes equations (2.1)–(2.4). See also Proposition 4.2 in this paper. For later use, we define the norm of a difference between the solution (η δ , uδ , v δ , pδ ) of the Navier–Stokes equations (2.1)–(2.4) and the approximate solution (ζ app , uapp , v app , papp ) as D(t; ζ app , uapp , v app , papp ) := |η δ (t) − ζ app (· − 2t, εt)|20 + (1 + |Dx |)m (uδ − uapp )(t) 2

(2.5)

+ (1 + |Dx |)m−1 (v δ − v app )(t) 2 + (1 + |Dx |)m−1 (pδ − papp )(t) 2 . Let ζ I , ζ II , ζ III , and ζ IV be the solution of (1.7)–(1.10) under the initial condition ζ|τ =0 = η0 , respectively.

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Now we are ready to state our main results in this paper. Note that the definitions of the approximate solutions uI , v I , pI , uII , . . . appeared in the following statement will be given in Section 5 (see (5.1) and (5.25)–(5.27)). Theorem 2.2. There exist small positive constants R0 and α0 such that the following statement holds: Let m be an integer satisfying m ≥ 2, 0 < R1 ≤ R0 , 0 < W1 ≤ W2 , and 0 < α ≤ α0 . There exists small positive constant c0 such that if the initial data (η0 , u0 , v0 ) and the parameters δ, ε, R, and W satisfy Assumption 2.1, then we have the following estimates. I. Burgers equation If the parameters δ and W and the initial data η0 and u0 satisfy W1 ≤ W ≤ δ −1 W2 ,

|η0 |m+7 + δ −1 (1 + |D|x )m+1 u0yy ≤ M < ∞,

(2.6)

then the following error estimate holds. D(t; ζ I , uI , v I , pI ) ≤ Cδ 2 e−cεt .

(2.7)

II. Burgers equation with a fourth order dissipation term If the parameters δ and W and the initial data η0 and u0 satisfy W = δ −2 W2 ,

|η0 |m+12 + δ −1 (1 + |D|x )m+1 u0yy ≤ M < ∞,

(2.8)

then the following error estimate holds. D(t; ζ II , uII , v II , pII ) ≤ Cδ 2 e−cεt .

(2.9)

III. Burgers equation with dispersion and nonlinear terms If the parameters δ and W and the initial data η0 and u0 satisfy W1 ≤ W ≤ W2 ,

|η0 |m+13 + δ −2 (1 + |D|x )m+1 (u0yy − uIII yy |t=0 ) ≤ M < ∞,

(2.10)

then the following error estimate holds. D(t; ζ III , uIII , v III , pIII ) ≤ Cδ 4 e−cεt .

(2.11)

IV. Burgers equation with fourth order dissipation, dispersion, and nonlinear terms If the parameters δ and W and the initial data η0 and u0 satisfy W = δ −1 W2 ,

|η0 |m+17 + δ −2 (1 + |D|x )m+1 (u0yy − uIV yy |t=0 ) ≤ M < ∞,

(2.12)

then the following error estimate holds. D(t; ζ IV , uIV , v IV , pIV ) ≤ Cδ 4 e−cεt .

(2.13)

Here, positive constants C and c depend on R1 , W1 , W2 , α, and M but are independent of δ, ε, R, and W. Remark 2.2. The assumptions for u0yy in (2.6) and (2.8) represent the restriction on the initial profile of the velocity. Moreover, the assumptions for u0yy in (2.10) and (2.12) mean that the initial profile of the velocity have to be equal to that of the approximate solution up to O(δ 2 ).

H. Ueno, T. Iguchi / J. Math. Anal. Appl. 444 (2016) 804–824

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Remark 2.3. We see formally that the order of error terms in (1.7) is of O(δ), which implies that the error estimates (2.7) and (2.9) are natural. In a similar way, we see that the error estimates (2.11) and (2.13) are natural. Remark 2.4. By introducing the slow time scale τ = εt, the norm decays exponentially and uniformly in τ . 3. Approximate solution of the Navier–Stokes equations In this section, following Benney’s perturbation method [3] we will construct an approximate solution of the Navier–Stokes equations. Hereafter, we assume ε = δ. By a straightforward calculation, we can rewrite (2.1)–(2.3) as follows. ⎧ 2 1 2 (2) (3) ⎪ ¯ ux + u ¯y v) + δpx − (δ 2 uxx + uyy ) = −δ ηuyy + δ 2 f1 + δ 3 f1 δ(ut + u ⎪ ⎪ R R R ⎪ ⎨ 2 1 2 (2) (3) ¯vx ) + py − δ(δ 2 vxx + vyy ) = δ ηpy + δ 2 f2 + δ 3 f2 δ 2 (vt + u ⎪ ⎪ R R R ⎪ ⎪ ⎩ ux + vy = 0 ⎧ 2 ⎨ δ vx + uy − 2(1 + δη)2 η = δ 3 h1 2 ⎩ p − δvy − 1 η + δ W ηxx = δ 2 h(2) + δ 3 h(3) 2 2 tan α sin α

in Ω, t > 0, in Ω, t > 0, in

Ω, t > 0,

on Γ, t > 0, (3.2) on Γ, t > 0,

u = v = 0 on Σ, t > 0, ηt + ηx − v = δ 2 h3

(3.1)

(3.3)

on Γ, t > 0,

(3.4)

where ⎧  1 2 (2) ⎪ 3η uyy − 2ηpx + 2yηx py + ηt u + yηt uy f1 = ⎪ ⎪ R ⎪ ⎪ ⎪ ⎪ ⎪ + y 2 ηx u + 2y(y − 1)ηux − y 2 (y − 2)ηx uy − uux − vuy + 2(2y − 1)ηv, ⎨ (3.5)

 1 ⎪ (2) ⎪ − 2η 2 py + 2ηx uy + 2ηuxy , f2 = ⎪ ⎪ ⎪ R ⎪ ⎪ ⎪ ⎩ (2) h2 = 2ηηx + ηx u + ηux .

We proceed to construct an approximate solution of the Navier–Stokes equations following Benney [3]. Let η = η(x, t) be an arbitrary function. For any δ ∈ (0, 1], let (u, v, p) be a solution of (3.1)–(3.3) and we expand (u, v, p) as ⎧ 2 ⎪ ⎪ ⎨u = u(0) + δu(1) + δ u(2) + · · · , v = v(0) + δv(1) + δ 2 v(2) + · · · , ⎪ ⎪ ⎩p = p + δp + δ 2 p + · · · (0) (1) (2)

(3.6)

and substitute this into (3.1)–(3.3), we obtain a sequence of equations for each order of δ. By assuming W = O(1), the O(1), O(δ), and O(δ 2 ) problems are as follows. ⎧ ⎪ = 0, p(0)y = 0, u(0)x + v(0)y = 0 u ⎪ ⎨ (0)yy 1 u(0)y = 2η, p(0) = η ⎪ tan α ⎪ ⎩ u(0) = v(0) = 0

in

Ω,

on Γ, on Σ,

(3.7)

H. Ueno, T. Iguchi / J. Math. Anal. Appl. 444 (2016) 804–824

812

⎧ ⎪ = R(u(0)t + (2y − y 2 )u(0)x + 2(1 − y)v(0) ) + 2p(0)x + 2ηu(0)yy u ⎪ ⎪ (1)yy ⎪ ⎨2p u(1)x + v(1)y = 0 (1)y = v(0)yy + 2ηp(0)y , 2 ⎪ u(1)y = 4η , p(1) = −u(0)x ⎪ ⎪ ⎪ ⎩ u(1) = v(1) = 0

in Ω, in Ω, on Σ,

⎧ ⎪ u(2)yy = R(u(1)t + (2y − y 2 )u(1)x + 2(1 − y)v(1) ) ⎪ ⎪ ⎪ ⎪ (2) ⎪ ⎪ + 2p(1)x + 2ηu(1)yy − u(0)xx − Rf1 (η, u(0) , v(0) , p(0) ) ⎪ ⎪   ⎪ ⎪ ⎨2p(2)y = v(1)yy + 2ηp(1)y − R v(0)t + (2y − y 2 )v(0)x + Rf2(2) (η, u(0) , v(0) , p(0) ) ⎪ u(2)x + v(2)y = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u(2)y = −v(0)x + 2η 3 , ⎪ ⎪ ⎪ ⎪ ⎩u = v = 0 (2) (2)

(2)

p(2) = −u(1)x + h2 (η, u(0) ) −

(3.8)

on Γ,

W ηxx sin α

in

Ω,

in

Ω,

in

Ω,

(3.9)

on Γ, on Σ.

Solving the above boundary value problem for the ordinary differential equations, we have ⎧ ⎪ u = 2yη, ⎪ ⎨ (0) v(0) = −y 2 ηx , ⎪ ⎪ ⎩p = 1 η, (0) tan α ⎧ 1 3 

2    ⎪ 2y) tan1 α + 16 y 4 − 23 y R ηx + 4yη 2 , ⎪ ⎨u(1) =  3 y − y Rηt+ (y −    1 5 1 2  1 4 v(1) = − 12 y + 12 y 2 Rηxt + − 13 y 3 + y 2 tan1 α + − 30 y + 3 y R ηxx − 4y 2 ηηx , ⎪ ⎪ ⎩p = −(1 + y)η , (1)

(3.10)

(3.11)

x

⎧ 1 5 1 3  5 ⎪ u(2) = 60 y − 6 y + 12 y R2 ηtt ⎪ ⎪ ⎪

 1 4 1 3 2  R  1 7  2 ⎪ 1 6 1 4 1 3 101 ⎪ ⎪ + ηxt ⎪ 12 y − 3 y + 3 y tan α + − 252 y + 45 y − 12 y − 9 y + 180 y R ⎪

 2 3   1 6  R ⎪ ⎪ 1 5 1 4 2 2 ⎪ + − 3 y − y + 5y + − 90 y + 15 y − 6 y + 5 y tan α ⎪ ⎪ ⎪  1 8  2 ⎪ 2 7 1 4 ⎪ + − 560 y + 315 y − 18 y + 121 ηxx ⎪ ⎪ 630 y R ⎪ ⎪ ⎪ 4 3 3 4 ⎪ + 2yη + R( 3 y − 4y)ηηt + {R(y − 4y) + (3y 2 − 6y) tan1 α }ηηx , ⎪ ⎪ ⎪  1 6  2 ⎪ 1 4 5 2 ⎪ R ηxtt ⎪ 360 y + 24 y − 24 y ⎨v(2) = −   1 8  R  2 1 5 1 4 1 2 1 7 1 5 1 4 101 2 + − 60 y + 12 y − 3 y tan R ηxxt α + 2016 y − 315 y + 60 y + 36 y − 360 y ⎪

 1 4 1 3 5 2  1 7  R ⎪ ⎪ 1 6 1 5 1 2 ⎪ + y + y − 2 y + 630 y − 90 y + 30 y − 5 y tan α ⎪ ⎪ ⎪ 6 1 93   ⎪ 1 1 5 121 2 ⎪ + 5040 y − 1260 y 8 + 90 y − 1260 y R2 ηxxx ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ − 3y 2 η 2 ηx + R − 13 y 4 + 2y 2 (ηx ηt + ηηtx ) ⎪ ⎪

   ⎪ ⎪ ⎪ + R − 15 y 5 + 2y 2 + (−y 3 + 3y 2 ) tan1 α (ηx2 + ηηxx ), ⎪ ⎪ ⎪  

W  1 2  1  1 5 1 4 1   ⎪ 1 1 ⎪ p(2) = 12 y + 16 Rηxt + − sin ⎪ ⎪ α + − 2 y + y + 2 tan α + − 10 y + 6 y + 3 y + 10 R ηxx ⎪

 ⎪ ⎩ + R(4y − 4) − 5y + 3 ηηx .

(3.12)

Note that u(0) , v(0) , p(0) , . . . are just polynomials in y whose coefficients depend on η. Then, neglecting higher order terms in δ, we obtain the following approximate solution of the Navier–Stokes equations for an arbitrary function η. ⎧ app 2 ⎪ ⎪ ⎨u (y; η) = u(0) + δu(1) + δ u(2) , v app (y; η) = v(0) + δv(1) + δ 2 v(2) , ⎪ ⎪ ⎩papp (y; η) = p + δp + δ 2 p . (0)

(1)

(2)

(3.13)

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In order to make the approximate solution satisfy the kinematic boundary condition (3.4), η is required to satisfy the following equation. ηt + 2ηx + 4εηηx −

8 (Rc − R)δηxx + C1 δ 2 ηxxx + C2 εδ(ηηxx + ηx2 ) + 2ε2 η 2 ηx = O(δ 3 ), 15

where C1 = 2 +

32 2 40 R R − , 63 63 tan α

C2 =

2 16 R− 5 tan α

(3.14)

and the above equation is the approximate equation given in Section 1. Here, (3.14) is the explicit form of the coefficients appearing in (1.5). Thus far we have assumed W = O(1). Taking into account that W is contained only in the second equation in (3.2) and modifying the O(δ) problem under the assumption W ≤ O(δ −1 ), we see that (uI0 , v0I , pI0 ) and (uI1 , v1I , pI1 ), which are defined by

uI0 (y; η) := u(0) ,

v0I (y; η) := v(0) ,

pI0 (y; η) := p(0) ,

uI1 (y; η) := u(1) ,

v1I (y; η) := v(1) ,

pI1 (y; η) := p(1) −

(3.15)

δW sin α ηxx ,

are the solutions of the problem. Putting v = v0I + δv1I and substituting this into (3.4), we obtain the approximate equation ηt + 2ηx + 4εηηx −

8 (Rc − R)δηxx = O(δ 2 ). 15

Similarly, modifying the O(1) and O(δ) problems under the assumption W = O(δ −2 ) and putting

uII 0 := u(0) ,

v0II := v(0) ,

uII 1 := u(1) −

δ2 W 2 sin α (y

δ2 W sin α ηxx ,  δ2 W 1 3 v1II := v(1) + sin α 3y

pII 0 := p(0) −

− 2y)ηxxx ,

 − y 2 ηxxxx ,

pII 1 := p(1) ,

(3.16)

we obtain the approximate equation ηt + 2ηx + 4εηηx −

8 2 W2 (Rc − R)δηxx + δηxxxx = O(δ 2 ). 15 3 sin α

Moreover, putting ⎧ IV IV IV ⎪ ⎪ ⎨u0 := u(0) , v0 := v(0) , p0 := p(0) , δW v1IV := v(1) , pIV uIV 1 := u(1) , 1 := p(1) − sin α ηxx , ⎪ ⎪ ⎩uIV := u − δW (y 2 − 2y)η , v IV := v + δW  1 y 3 − y 2 η xxx xxxx , (2) (2) 2 2 sin α sin α 3

(3.17) pIV 2 := p(2) +

W sin α ηxx

and v = v0IV + δv1IV + δ 2 v2IV and substituting this into (3.4), we obtain the approximate equation ηt + 2ηx + 4εηηx −

8 (Rc − R)δηxx 15

+ C1 δ 2 ηxxx + C2 εδ(ηηxx + ηx2 ) + 2ε2 η 2 ηx + under the assumption W = O(δ −1 ).

2 W2 2 δ ηxxxx = O(δ 3 ) 3 sin α

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4. Energy estimate In this section, we will derive an energy estimate, which is most important step in order to obtain error estimates. Using the arbitrary function η and the approximate solution (uapp , v app , papp ) = (uapp (y; η), v app (y; η), papp (y; η)), we define ψ1 , ψ2 , φ1 , φ2 , and φ3 by the following equalities.

⎧ 1 2 ⎪ ⎪ +u ¯uapp +u ¯y v app ) + δpapp ψ1 (y; η) := 3 δ(uapp t x x ⎪ ⎪ δ R ⎪  ⎪ ⎪ 1 2 app ⎪ (1) app app app app ⎪ ⎪ − (δ uxx + uyy ) − δf1 (η, u , v , p ) , ⎪ ⎪ R ⎪ ⎪

 ⎪ ⎪ ⎪ 1 2 app 1 (1) ⎪ 2 app app 2 app app app app ⎪ ¯vx ) + py − δ(δ vxx + vyy ) − δf2 (η, u , p ) , ⎨ψ2 (y; η) := 3 δ (vt + u δ R R

 1 ⎪ ⎪ ⎪ φ1 (η) := 3 δ 2 vxapp + uapp − 2(1 + δη)2 η |y=1 , ⎪ y ⎪ δ ⎪ ⎪

 ⎪ ⎪  ⎪ δ2 ⎪ app  , ⎪φ2 (η) := 1 papp − δvyapp − 1 η + W ηxx − δ 2 h(2) (η, u ) ⎪ 2  3 ⎪ δ tan α sin α ⎪ y=1 ⎪ ⎪ ⎪ ⎪ ⎩φ (η) := 1 {η + η − v app − δ 2 h (η)}| , 3 t x 3 y=1 δ3

(4.1)

where (1)

f1

2 (2) = − ηuapp + δf1 , R yy

(1)

f2

=

2 app (2) ηp + δf2 . R y

(4.2)

Here, ψ1 , ψ2 , φ1 , φ2 , and φ3 measure how much (η, uapp , v app , papp ) fails to be the solution of the Navier– Stokes equations and in the next section we will give explicit forms of these functions (see (5.3)). Then, by (4.1) and the definition of the approximate solution constructed in Section 3, it satisfies the following equations. ⎧ 2 1 app app ⎪ ¯uapp +u ¯y v app ) + δpapp − (δ 2 uapp ⎪ x x xx + uyy ) ⎪ δ(ut + u ⎪ R R ⎪ ⎪ (1) ⎪ ⎪ = δf1 (η, uapp , v app , papp ) + δ 3 ψ1 (y; η) in Ω, t > 0, ⎪ ⎪ ⎪ ⎨ 2 1 app app ¯vxapp ) + papp − δ(δ 2 vxx + vyy ) δ 2 (vtapp + u y ⎪ ⎪ R R ⎪ ⎪ ⎪ (1) ⎪ = δf2 (η, uapp , papp ) + δ 3 ψ2 (y; η) in Ω, t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ app ux + vyapp = 0 in Ω, t > 0, ⎧ 2 app − 2(1 + δη)2 η = δ 3 φ1 (η) δ vx + uapp ⎪ y ⎪ ⎪ ⎪ ⎨ δ2W 1 (2) papp − δvyapp − η+ ηxx = δ 2 h2 (η, uapp ) + δ 3 φ2 (η) ⎪ ⎪ tan α sin α ⎪ ⎪ ⎩ ηt + ηx − v app = δ 2 h3 (η) + δ 3 φ3 (η) uapp = v app = 0

on

(4.3)

on Γ, t > 0, on

Γ, t > 0,

on

Γ, t > 0,

Σ, t > 0.

(4.4)

(4.5)

In other words, the approximate solution satisfies the Navier–Stokes equations approximately with reminder terms ψ1 , ψ2 , φ1 , φ2 , and φ3 . Let (η δ , uδ , v δ , pδ ) be the solution of (3.1)–(3.4) and we set H := η δ − η,

U := uδ − uapp ,

V := v δ − v app ,

P := pδ − papp .

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Taking the difference between (3.1)–(3.4) and (4.3)–(4.5), we have ⎧ 2 1 ⎪ ¯Ux + u ¯y V ) + δPx − (δ 2 Uxx + Uyy ) δ(Ut + u ⎪ ⎪ ⎪ R R ⎪ ⎪ ⎪ (3) ⎪ ⎪ = F1 + δ 3 f1 (η δ , uδ , v δ , pδ ) − δ 3 ψ1 (y; η) ⎨ 2 1 δ 2 (Vt + u ¯Vx ) + Py − δ(δ 2 Vxx + Vyy ) ⎪ ⎪ R R ⎪ ⎪ ⎪ ⎪ (3) ⎪ = F2 + δ 3 f2 (η δ , uδ , v δ , pδ ) − δ 3 ψ2 (y; η) ⎪ ⎪ ⎩ Ux + Vy = 0 ⎧   2 δ 3 δ δ δ 3 ⎪ ⎪ δ Vx + Uy − 2 + b(η , η) H = δ h1 (η , u , v ) − δ φ1 (η) ⎪ ⎨ δ2W 1 (3) H + Hxx = G2 + δ 3 h2 (η δ , uδ , v δ ) − δ 3 φ2 (η) P − δV − y ⎪ tan α sin α ⎪ ⎪ ⎩ Ht + Hx − V = G3 − δ 3 φ3 (η)

in

Ω, t > 0, (4.6)

in in

Ω, t > 0, Ω, t > 0,

on Γ, t > 0, on

Γ, t > 0,

(4.7)

on Γ, t > 0,

U = V = 0 on Σ, t > 0,

(4.8)

⎧  (1)  (1) ⎪ F1 = δ f1 (η δ , uδ , v δ , pδ ) − f1 (η, uapp , v app , papp ) , ⎪ ⎪ ⎪ ⎪   ⎪ ⎨F2 = δ f (1) (η δ , uδ , pδ ) − f (1) (η, uapp , papp ) , 2 2 ⎪b = 2δ δ(η δ )2 + (2 + δη)η δ + δη 2 + 2η , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩G = δ 2 h(2) (η δ , uδ , v δ ) − h(2) (η, uapp , v app ), G = δ 2 h (η δ ) − h (η). 2 3 3 3 2 2

(4.9)

where

Note that (4.6)–(4.8) are linearized Navier–Stokes equations with non-homogeneous terms. For convenience, we set U := (U, δV )T ,

F := (F1 , F2 )T ,

(3)

(3)

f := (f1 , f2 )T ,

ψ := (ψ1 , ψ2 )T .

We proceed to derive an energy estimate to (4.6)–(4.8) following [17]. In view of the energies obtained in [17] (see (3.6)–(3.8) and (3.24) in [17]), we put  1 2 δ2W |H|20 + |Hx |20 R tan α sin α

  2 δ2W 2 1 2 + β1 δ 2 U x 2 + δ |Hx |20 + δ |Hxx |20 R tan α sin α

  1 4 2 δ2W 4 4 2 2 2 + β2 δ U xx + δ |Hxx |0 + δ |Hxxx |0 R tan α sin α

  2 δ2W 2 1 2 + β3 δ 2 U t 2 + , δ |Ht |20 + δ |Htx |20 R tan α sin α

E0 (H, U ) := δ 2 V 2 +

F0 (H, U , P ) := δ U x 2 + δ ∂y−1 Px 2 + δ|Hx |20 + δ 3 W|Hxx |20 + δ 5 W2 |Hxxx |20

+ δ ∇δ U x 2 + δ 3 ∇δ U xx 2 + δ ∇δ U t 2 . Here, β1 , β2 , and β3 are appropriate positive constants (see (3.28) in [17]). Integrating by parts and using the third equation in (4.7) and Poincaré’s inequality, we see that for any  > 0 there exists a positive constant C such that

H. Ueno, T. Iguchi / J. Math. Anal. Appl. 444 (2016) 804–824

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δ 3 |({F + δ 3 f − δ 3 ψ}xx , U xx )Ω | ≤ δ 5 U xxx 2 + C δ( F x 2 + δ 6 f x 2 + δ 6 ψ x 2 ), |(H, (bH)x )Γ | ≤ δ|Hx |20 + C δ −1 |(bH)x |20 , δ 2 W|(Hxx , (bH)x )Γ | ≤ δ 3 W|Hxx |20 + C δW|(bH)x |20 , δ 2 W|(Hxx , G3 − δ 3 φ3 )Γ | ≤ δ 3 W|Hxx |20 + C δW(|G3 |20 + δ 6 |φ3 |20 ), δ 6 W|(Hxxxx , δ 3 φ3xx )Γ | ≤ δ 5 W2 |Hxxx |20 + C δ 13 |φ3xxx |20 , δ 4 W|(Hxxt , G3t − δ 3 φ3t )Γ | ≤ (δ 5 W2 |Hxxx |20 + δ 5 Uxxx 20 ) + C (1 + W2 )δ 3 (|G3t |20 + δ 6 |φ3t |20 ) + δ 5 (|G3xx |20 + δ 6 |φ3xx |20 ). Here, we used the inequality |V (·, 1)|0 = |V (·, 1) − V (·, 0)|0 ≤ Vy = Ux thanks to the third equation in (4.6) and the second equation in (4.8). In the following, we use frequently this type of inequality without any comment. Taking into account the above inequality and (3.27) in [17], we need to estimate the following quantities. N01 (Z1 ) := (δW + δ −1 )|(bH)x |20 + δ 3 |(bH)xx |20 + δ|(bH)t |20

(4.10)

+ δ −1 |G2 |20 + δ|G2x |20 + δ 2 ||Dx | 2 G2x |20 + δ|(G2t , δVt )Γ | 1

+ δW|G3 |20 + δ 3 |G3x |20 + δ 5 |G3xx |20 + δ 3 W2 |G3t |20 + δ 6 W|(Hxxxx , G3xx )Γ | + δ −1 F 2 + δ F x 2 + δ|(F t , U t )Ω |, 1

N02 (Z2 ) := δ 5 |h1 |20 + δ 7 |h1x |20 + δ 8 ||Dx | 2 h1x |20 + δ 4 |(h1t , Ut )Γ | (3)

1 2

(3)

(3)

(4.11)

(3)

+ δ 5 |h2 |20 + δ 7 |h2x |20 + δ 8 ||Dx | h2x |20 + δ 4 |(h2t , δVt )Γ | + δ 5 f 2 + δ 7 f x 2 + δ 4 |(f t , U t )Ω |, 1

N03 (Z3 ) := δ 5 |φ1 |20 + δ 7 |φ1x |20 + δ 8 ||Dx | 2 φ1x |20 + δ 7 |φ1t |20 + δ 5 |φ2 |20 + δ 7 |φ2x |20

(4.12)

1 2

+ δ 8 ||Dx | φ2x |20 + δ 7 |φ2t |20 + δ 7 W|φ3 |20 + δ 9 |φ3x |20 + δ 11 |φ3xx |20 + δ 13 |φ3xxx |20 + δ 9 W2 |φ3t |20 + δ 5 ψ 2 + δ 7 ψ x 2 + δ 7 ψ t 2 , where Z1 = (H, U , bH, G2 , G3 , F ),

(3)

Z2 = (U , h1 , h2 , h3 , f ),

Z3 = (φ1 , φ2 , φ3 , ψ).

For an integer m ≥ 2, we set Em (H, U ) :=

m 

E0 (∂xk H, ∂xk U ),

Fm (H, U , P ) :=

k=0

m 

F0 (∂xk H, ∂xk U , ∂xk P ),

(4.13)

k=0

Nm1 (H, U , P ; η) :=

m 

 N01 (∂xk Z1 ) + |(∂xk H, ∂xk G3 )Γ | ,

(4.14)

k=0

Nm2 (U )

:=

m 

N02 (∂xk Z2 ),

(4.15)

k=0

Nm3 (H; η) :=

m 

k=0

 N03 (∂xk Z3 ) + |(∂xk H, δ 3 ∂xk φ3 )Γ | .

(4.16)

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m m Here, the terms k=0 |(∂xk H, ∂xk G3 )Γ | and k=0 |(∂xk H, δ 3 ∂xk φ3 )Γ | come from (3.30) in [17]. Applying ∂xk to (4.6)–(4.8), using [17, Proposition 3.6], and adding the resulting inequalities for 0 ≤ k ≤ m, we obtain the following lemma. Lemma 4.1. There exist small positive constants R0 and α0 such that if 0 < R1 ≤ R ≤ R0 , W1 ≤ W, and 0 < α ≤ α0 , then the solution (H, U, V, P ) of (4.6)–(4.8) satisfies d Em + Fm ≤ C(Nm1 + Nm2 + Nm3 ), dt

(4.17)

where the constant C is independent of δ, R, and W. For later use, we modify the energy and dissipation functions Em and Fm as E˜m (H, U ) := Em (H, U ) + (1 + |Dx |)m U 2 + (1 + |Dx |)m Uy 2 , 5 2

(4.18) 7 2

F˜m (H, U , P ) := Fm (H, U , P ) + δ|(1 + δ|Dx |) Ht |2m + (δ 2 W)2 δ 2 ||Dx | H|2m

(4.19)

+ δ −1 (1 + |Dx |)m (1 + δ|Dx |)(∇δ P, Uyy ) 2 + δ (1 + |Dx |)m−1 ∇δ Pt 2 . We also introduce another energy function Dm by Dm (H, U ) := |(1 + δ|Dx |)2 H|2m + δ 2 (1 + |Dx |)m V 2 + δ 2 (1 + |Dx |)m U x 2 (4.20) √ + (1 + |Dx |)m Dδ2 U 2 + (δ 2 W)2 |(1 + δ|Dx |)Hx |2m+1 + δ 2 W (1 + |Dx |)m δVxy 2 ,

˜m = E˜m (η δ , uδ ) and F˜m = F˜m (η δ , uδ , pδ ) and using which does not include any time derivatives. Setting E [17, Theorem 2.2 and Proposition 6.1], the following uniform estimate holds. Proposition 4.2. There exist small positive constants R0 and α0 such that the following statement holds: Let m be an integer satisfying m ≥ 2, 0 < R1 ≤ R0 , 0 < W1 ≤ W2 , and 0 < α ≤ α0 . There exists small positive constant c0 such that if the initial data (η0 , u0 , v0 ) and the parameters δ, ε, R, and W satisfy Assumption 2.1 and W ≤ δ −2 W2 , then the solution (η δ , uδ , v δ , pδ ) of (2.1)–(2.4) satisfies ∞ ˜2 (t) ≤ c0 , E

˜m+1 (t) + sup E t≥0

F˜m+1 (t)dt ≤ C,

˜m+1 (t) ≤ Ce−cδt . E

0

Here, positive constants C and c depend on R1 , W1 , W2 , α, and M but are independent of δ, ε, R, and W. Moreover, we easily obtain the following lemma. Lemma 4.3. Let α > 0, 0 < R1 ≤ R < Rc . There exists small positive constant c1 such that if s ≥ 2 and |η0 |22 ≤ c1 , then the problems (1.7)–(1.10) under the initial condition ζ|τ =0 = η0 have unique solutions ζ I , ζ II , ζ III , and ζ IV , respectively, which satisfy ∞ sup |ζ τ ≥0

I

(τ )|2s

|ζxI (τ )|2s dτ ≤ C|η0 |2s ,

+ 0

∞ sup |ζ II (τ )|2s + τ ≥0

|ζ I (τ )|2s ≤ C|η0 |2s e−cδt ,

0



 II |ζxII (τ )|2s + |ζxx (τ )|2s dτ ≤ C|η0 |2s ,

|ζ II (τ )|2s ≤ C|η0 |2s e−cδt ,

H. Ueno, T. Iguchi / J. Math. Anal. Appl. 444 (2016) 804–824

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∞ sup |ζ

III

τ ≥0

(τ )|2s

|ζxIII (τ )|2s dτ ≤ C|η0 |2s ,

+ 0

∞ sup |ζ IV (τ )|2s + τ ≥0

Here, Rc = and R.

5 1 4 tan α

|ζ III (τ )|2s ≤ C|η0 |2s e−cδt ,

 IV  IV |ζx (τ )|2s + δ|ζxx (τ )|2s dτ ≤ C|η0 |2s ,

|ζ IV (τ )|2s ≤ C|η0 |2s e−cδt .

0

is the critical Reynolds number and positive constants C and c are independent of δ

5. Error estimate We will show (2.11) under Assumption 2.1 and (2.10) by combining the energy estimate obtained in Section 4 and nonlinear estimates which will be performed in this section. We can show the other claims in Theorem 2.2 in the same way as the proof of (2.11) and we will comment about the discrepancy at the end of this section. Now, we specify the arbitrary function η as the solution of the approximate equation. Let ζ III be the solution of (1.9) under the initial condition ζ III |τ =0 = η0 and we put η III (x, t) := ζ III (x −2t, εt) and ⎧ III app III ⎪ ⎪ ⎨u (x, y, t) := u (y; η (x, t)), (5.1) v III (x, y, t) := v app (y; η III (x, t)), ⎪ ⎪ ⎩pIII (x, y, t) := papp (y; η III (x, t)), where (uapp , v app , papp ) was defined by (3.13). Then, we have 8 III III (Rc − R)δηxx − C1 δ 2 ηxxx 15

   III − 4δη III ηxIII − δ 2 C2 η III ηxx + (ηxIII )2 + 2(η III )2 ηxIII .

ηtIII = −2ηxIII +

(5.2)

Using the approximate solution (5.1), we define ψ1 , ψ2 , φ1 , φ2 , and φ3 by (4.1). By using the equality (5.2) to eliminate the t derivatives of η III , we can rewrite these terms as follows. ⎧ ⎪ ψ1 (y; η III ) = C1 (y)∂x3 η III + C2 (y)δ∂x4 η III + · · · + C7 (y)δ 6 ∂x9 η III + N1III , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ψ (y; η III ) = C8 (y)∂x3 η III + C9 (y)δ∂x4 η III + · · · + C15 (y)δ 7 ∂x10 η III + N2III , ⎪ ⎨ 2 φ1 (η III ) = C16 ∂x3 η III + C17 δ∂x4 η III + · · · + C21 δ 5 ∂x8 η III + N3III , ⎪ ⎪ ⎪ ⎪ ⎪ φ2 (η III ) = C22 ∂x3 η III + C23 δ∂x4 η III + · · · + C26 δ 4 ∂x7 η III + N4III , ⎪ ⎪ ⎪ ⎩ φ3 (η III ) = C27 ∂x4 η III + C28 δ∂x5 η III + · · · + C30 δ 3 ∂x7 η III + N5III ,

(5.3)

where C1 , . . . , C15 are polynomials in y, C16 , . . . , C30 are constants, and N1III , . . . , N5III are collections of the nonlinear terms of the form 1 Φ0 (δη III , δ 2 ∂x η III , . . . , δ 5 ∂x4 η III ; y)Φ0 (δ 2 ∂x η III , . . . , δ 10 ∂x9 η III ; y). δ3

(5.4)

Here we generally denote polynomials of f by the same symbol Φ = Φ(f ) and Φ0 is such a function satisfying Φ0 (0) = 0. We also use such a function Φ0 depending also on y ∈ [0, 1] and denote it by Φ0 (f ; y), that is, Φ0 (0; y) ≡ 0. Let (η δ , uδ , v δ , pδ ) be the solution of (2.1)–(2.3) and we set H III := η δ − η III , III U III := (uδ − uIII , δ(v δ − v III ))T , E˜m := E˜m (H III , U III ), and so on. We prepare several lemmas to proceed the error estimate. In particular, we estimate nonlinear terms defined by (4.14)–(4.16) in terms of energy functions.

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Lemma 5.1. Under the same assumption as Proposition 4.2, for any  > 0 there exists a positive constant C such that we have ˜m (t)F˜m+1 (t), Nm2 (U III )(t) ≤ F˜m (t) + C δ 4 E

(5.5)

where Nm2 is the collection of nonlinear terms defined by (4.15). (3)

Proof. By the explicit forms of f , h1 , and h2 (see (3.5) and Section 2), we can obtain the desired estimate in the same but more easier way as proving [17, Lemmas 5.11 and 5.12]. 2 Lemma 5.2. Under the same assumption as Proposition 4.2, for any  > 0 there exists a positive constant C such that we have Nm3 (H; η III )(t) ≤ F˜m (t) + C δ 5 |ηxIII (t)|2m+12 ,

where Nm3 is the collection of nonlinear terms defined by (4.16). Proof. By the well-known inequalities ∂xk (f g)  f L∞ ∂xk g + g L∞ ∂xk f and ∂xk Φ0 (f ; y) ≤   m 3 k C( f L∞ ) ∂xk f , (5.2)–(5.4) lead to 1 + |η III |2m+12 δ 5 |ηxIII |2m+12 . Moreover, by k=0 N0 (∂x Z3 )  Poincaré’s inequality and (5.4), we see that |(∂xk H, δ 3 ∂xk φ3 )Γ | ≤ δ|∂xk Hx |20 + C δ 5 |∂xk φ3 |20 ≤ F˜m +   C 1 + |η III |2m+12 δ 5 |ηxIII |2m+12 . These together with Lemma 4.3 imply the desired inequality. 2 Lemma 5.3. Under the same assumption as Proposition 4.2, for any  > 0 there exists a positive constant C such that we have

˜2 (t) + )F˜ III (t) + C E ˜m (t)F˜m+1 (t) ˜m (t)F˜ III (t) + δ 4 E Nm1 (H III , U III , P III ; η III )(t) ≤ (C E m 2  III (t) , + δ 5 |ηxIII (t)|2m+12 + (F˜m (t) + δ|ηxIII (t)|2m+12 )E˜m

(5.6)

where Nm1 is the collection of nonlinear terms defined by (4.14). Proof. In this proof, we omit the symbol III appeared in a superscript of solutions for simplicity. By (3.5), (4.2), and (4.9), we see that F is consist of terms of the form ⎧ ⎪ δΦ0 (η δ , δηxδ ; y)(∇δ Uy , ∇δ P ) + δ 2 (η δ )2 (Uyy , Py ), ⎪ ⎪ ⎪ ⎪ δ δ δ ⎪ ⎪ ⎨δΦ0 (η , δηx , u ; y)(δV, δUx ), δΦ0 (δηxδ , δηtδ , δv δ ; y)(U, Uy ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪δΦ0 (η, u, ∇δ u, ∇δ uy , ∇δ p; y)(δHx , δHt , U, δV ), ⎪ ⎩ 2 δ δ η (uyy + py )H and that G2 = δ 2 {η δ (2Hx + Ux ) + ηxδ U + (2ηx + ux )H + uHx }, G3 = δ 2 {(η δ )2 Hx + (η δ + η)ηx H}, and   bH = 2δ δ(η δ )2 + (2 + δη)η δ + δη 2 + 2η H. Note that using (5.1) and (5.2), we can express the approximate solutions u, ∇δ u, uyy , and ∇δ p in terms of η and its x derivatives. In view of these, by putting ⎧ ⎪ Φ1 ⎪ ⎪ ⎪ ⎨Φ2 ⎪Φ3 ⎪ ⎪ ⎪ ⎩ 4 Φ

δ δ = Φ(η δ , δηxδ , δηtδ , δ 2 ηxx , δ 2 ηtx , uδ ; y), δ δ δ = Φ(δηxδ , δηtδ , δ 2 ηxx , δ 2 ηtx , δ 2 ηtt , δv δ , δuδx , δuδt ; y),

= Φ(η δ , δηxδ ; y), = Φ(η, δηx , . . . , δ 10 ∂x10 η; y),

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⎧ 2 2 3 ⎪ ⎪ ⎨W := (δHx , δHt , δ Hxx , δ Htx , δ Hxxx , δV, δU x , δU t , δ∇δ Ux , δ∇δ Ut , ∇δ Uy , ∇δ Uxy , ∇δ P, ∇δ Px , δUx |Γ , δUt |Γ , δ 2 Uxx |Γ , δ 5/2 |Dx |5/2 U |Γ ), ⎪ ⎪ ⎩Q := (H, δH , δH , δ 2 H , δ 2 H , δ 3 H , U , ∇ U , δU , U | ), x

t

xx

tx

xxx

δ

t

Γ

it suffices to estimate ⎧ ⎪ I1 ⎪ ⎪ ⎪ ⎪ ⎪ I2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪I3 ⎪ ⎪ ⎪ ⎨I 4 ⎪ I5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ I6 ⎪ ⎪ ⎪ ⎪ ⎪ I7 ⎪ ⎪ ⎪ ⎩ I8

= δ ∂xk (Φ10 W ) 2 , = δ ∂xk (Φ20 Q) 2 , = δ 3 |(∂xk (η δ U tx ), ∂xk Vt )Γ |, = δ 2 |(∂xk (Φ30 ∇δ Uty ), ∂xk U t )Ω |, = δ 2 |(∂xk (Φ30 ∇δ Pt ), ∂xk U t )Ω |, = δ ∂xk (Φ1 Φ40 Q) 2 , = δ 4 |(∂xk (Φ40 Htt ), ∂xk Vt )Γ |, = δ 6 W|(∂xk Hxxxx , ∂xk G3xx )Γ |

for 0 ≤ k ≤ m. By Proposition 4.2 and (u, v) L∞  (uy , vy ) + (uxy , vxy ) thanks to the boundary condition u|y=0 = v|y=0 = 0, we obtain ˜2 , Φ10 2L∞  E

˜m , ∂xk Φ10 2 + ∂xk Φ10y 2  E

(5.7)

Φ20 2L∞  F˜2 ,

∂xk Φ20 2 + ∂xk Φ20y 2  F˜m .

(5.8)

In the same way as the proof of Lemma 5.2, we have δ Φ40 2L∞  δ|ηx |2m+12 ,

δ( ∂xk Φ40 2 + ∂xk Φ40y 2 )  δ|ηx |2m+12 ,

|Φ40 |2m− 1  |η|2m+12 .

(5.9)

2

On the other hand, it is easy to see that W 2 + Wx 2  F˜2 , Q 2 + Qx 2  E˜2 ,

∂xk W 2  F˜m , ∂xk Q 2  E˜m ,

1

(5.10) (5.11) 5

where we used the trace theorem |f |20 +δ||Dx | 2 f |20  f 2 +δ 2 fx 2 + fy 2 to estimate the term δ 5 ||Dx | 2 U |20 . In the following, we often use the inequality ∂xk (af )  a L∞ ∂xk f + ( ∂xk a + ∂xk ay )( f + fx ),

(5.12)

which have been shown in [17, Lemma 5.2]. ˜2 F˜m + E ˜m F˜2 . As for I2 , by (5.8), (5.11), and As for I1 , by (5.7), (5.10), and (5.12), we have I1  E (5.12), we have I2  F˜m E˜m . As for I3 , by integration by parts, we have I3  C δ 3 |η δ Utx |2m− 1 +δ 3 |Vt |2m+ 1 ≤ 2 2 ˜2 F˜m + E ˜m F˜2 ) + F˜m . As for I4 , by integration by parts in y, we have C (E   I4 ≤ C δ 2 ∂xk (Φ30 ∇δ Ut ) 2 + ∂xk (Φ30y ∇δ Ut ) 2 + δ 3 |(∂xk (Φ30 Utx ), ∂xk U t )Γ | + δ 2 |(∂xk (Φ30 Uty ), ∂xk U t )Γ | + δ ∂xk U ty 2 ≤ I4,1 + I4,2 + I4,3 + F˜m ,

H. Ueno, T. Iguchi / J. Math. Anal. Appl. 444 (2016) 804–824

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  where I4,1 = C δ 2 ∂xk (Φ30 ∇δ Ut ) 2 + ∂xk (Φ30y ∇δ Ut ) 2 , I4,2 = δ 3 |(∂xk (Φ30 Utx ), ∂xk U t )Γ |, and I4,3 = δ 2 |(∂xk (Φ30 Uty ), ∂xk U t )Γ |. The estimates for I4,1 and I4,2 are reduced to the estimates for I1 and I3 , respectively. Thus, taking into account that we can eliminate the term Uy |Γ in I4,3 by the first equation in (4.7),

˜2 F˜m + E ˜m (F˜2 +δ 4 F˜m+1 + this together with the estimates for I2 , I3 , δ 3 h1 , and δ 3 φ1 yields I4 ≤ F˜m +C E  2 5 2 |η|m+12 δ |ηx |m+12 ) . As for I5 , it suffices to show the case of k ≥ 1 because we can treat easily the case of k = 0. Integrating by parts in x, (5.7), and (5.12), we have I5 ≤ δ 3 ∂xk U tx 2 + C δ ∂xk−1 (Φ30 ∇δ Pt ) 2 ≤   ˜2 F˜m + E ˜m F˜2 . As for I6 , by (5.7), (5.9), (5.11), and (5.12), we have F˜m + C E

I6  δ Φ40 2L∞ ( ∂xk Φ1 2 + ∂xk Φ1y 2 )( Q 2 + Qx 2 ) + Φ1 2L∞ ( ∂xk Φ40 2 + ∂xk Φ40y 2 )( Q 2 + Qx 2 ) + Φ1 2L∞ Φ40 2L∞ ∂xk Q 2



2 ˜ ˜m + |η|2  (E m+12 )δ|ηx |m+12 Em .

As for I7 , it suffices to show the case of k ≥ 1 because we can treat easily the case of k = 0. By the third equation in (4.7), integration by parts, and the trace theorem, we have 1

I7 ≤ C δ 4 ||Dx | 2 ∂xk−1 (Φ40 Vt )|20 + C δ 5 |∂xk (Φ40 Hxt + Φ40 G3t )|20 + C δ 5 |δ 3 ∂xk φ3t |20   1 +  δ 4 ||Dx | 2 ∂xk Vt |20 + δ 3 |∂xk Vt |20 ≤ I7,1 + I7,2 + I7,3 + F˜m , 1

where I7,1 = C δ 4 ||Dx | 2 ∂xk−1 (Φ40 Vt )|20 , I7,2 = C δ 5 |∂xk (Φ40 Hxt + Φ40 G3t )|20 , and I7,3 = C δ 5 |δ 3 ∂xk φ3t |20 . By the trace theorem, the second equation in (4.6), and (5.9), we have 1

I7,1  |Φ40 |2m− 1 δ 3 |Vt |2L∞ + δ|Φ40 |2L∞ δ 3 ||Dx | 2 ∂xk−1 Vt |20 2

 |Φ40 |2m− 1 δ 3 Utxx 2 + δ|Φ40 |2L∞ (δ 2 ∂xk Ut 2 + δ 4 ∂xk Vt 2 ) 2



|η|2m+12 F˜2

+ δ|ηx |2m+12 E˜m .

Recalling the explicit form of G3 , we see that the estimate of I7,2 is reduced to I6 . Taking into account

˜m + that we have already estimated I7,3 in the proof of Lemma 5.2, we obtain I7 ≤ C |η|2m+12 F˜2 + (E  2 2 2 5 2 |η|m+12 )δ|ηx |m+12 E˜m + |η|m+12 δ |ηx |m+12 + F˜m . As for I8 , integration by parts, (5.7), and (5.9) lead to 7

5

δ 6 W|(∂xk Hxxxx , ∂xk G3xx )Γ | ≤ (δ 2 W)2 δ 2 ||Dx | 2 H|2m + C δ 6 ||Dx | 2 G3 |2m

 ˜2 F˜m . ≤ F˜m + C δ 2 (F˜m + δ|ηx |2m+12 )E˜2 + E ˜m and |η|2 Therefore, by the boundedness of the terms E m+12 which comes from Proposition 4.2 and Lemma 4.3, the proof is complete. 2 Lemma 5.4. Under the same assumption as Proposition 4.2, we have III III ˜m+1 (t) + |η III (t)|2 E˜m (t)  Em (t) + δ 4 (E m+12 ), III III III F˜m (t)  Fm (t) + (F˜m (t) + δ|ηxIII (t)|2m+12 )E˜m (t)

(5.13) (5.14)

˜m (t)F˜m+1 (t) + δ 5 |η III (t)|2 + δ4 E x m+12 , III III Em (t)  Dm (t) + δ 4 .

(5.15)

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Proof. In view of the discrepancy of non-homogeneous terms in the equations, modifying the proof of (6.2) III in [17, Lemma 6.2], we obtain (5.13). Taking into account that we can eliminate Uyy in Fm by using the first equation in (4.6), modifying the proof of (6.3) in [17, Lemma 6.2], it is not difficult to check that (5.14) holds. Moreover, modifying the proof of (6.10) in [17], we obtain (5.15). 2 Lemma 5.5. Under the same assumption as Proposition 4.2, we have III Dm (0)  δ 4 .

Remark 5.1. This lemma together with (5.15) yields III Em (0)  δ 4 .

(5.16)

Proof. By the second and third equations in the compatibility conditions in Assumption 2.1, we see that y 1 u0 (x, y) = yu0y (x, 1) −

u0yy (x, w)dwdz 0



= 2yη0 +

4yδη02

(5.17)

z

+

2yδ 2 η03





+ δy − δv0x +

(0)  δ 2 h1

y 1 −

u0yy (x, w)dwdz. 0

z

III It follows from (2.10) and (1 + |Dx |)m+1 uIII , that is, (3.10)–(3.13) yy |t=0  δ (see the explicit form of u m+1 III and (5.1)) that (1 + |Dx |) u0yy  δ. Thus, by (5.17), the explicit form of u , (2.10), and the uniform (0) estimate for δ 2 |h1 |m+1 (see the proof of Lemma 5.1), we obtain (1 + |Dx |)m+1 U |t=0  δ. Combining this and the first equation in the compatibility conditions leads to (1 + |Dx |)m V |t=0  δ. Therefore, in view of the definition of Dm (see (4.20)), using these and H|t=0 = 0, we obtain the desired estimate. 2

Proof of (2.11) in Theorem 2.2. By Proposition 4.2, Lemmas 4.1, 5.1–5.3, and (5.13) and (5.14) in Lemma 5.4, if c0 and  are sufficiently small, then we have   d III III III ˜m (t)F˜ III (t) + δ 4 ϕ2 (t) , E (t) + F˜m (t) ≤ C1 ϕ1 (t)Em (t) + E 2 dt m

(5.18)

where ϕ1 (t) = F˜m (t) + δ|ηxIII (t)|2m+12 ,

˜m (t)F˜m+1 (t) + δ|η III (t)|2 ϕ2 (t) = E x m+12 .

(5.19)

By considering the case of m = 2 in (5.18) and using Gronwall’s inequality and Proposition 4.2, if c0 is t sufficiently small, then we have E2III (t) + 0 F˜2III (s)ds ≤ ϕ3 (t), where  ϕ3 (t) =

E2III (0) exp

t C1

 t t 4 ϕ1 (s)ds + C1 δ ϕ2 (s) exp C1 ϕ1 (σ)dσ ds,

0

0

(5.20)

s

which leads to t F˜2III (s)ds ≤ ϕ3 (t). 0

(5.21)

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˜m+1 (t) and Note that by Proposition 4.2 and Lemma 4.3, we have the exponential decay estimate for E III 2 III III ˜ |η (t)|m+13 . This together with (5.18), Gronwall’s inequality, and δ Em  Fm which comes from |H|0  |Hx |0 and V  Vy = Ux (see (4.13) and (4.19)) yields III Em (t)

 t  III ≤ Em (0) exp C1 ϕ1 (s)ds + ϕ4 (t) e−cδt , 0

where t ϕ4 (t) = C1



F˜2III (s)

 t + δ F˜m+1 (s) exp C1 ϕ1 (σ)dσ ds. 4



(5.22)

s

0

Combining the above inequality and (5.13) and (5.15) in Lemma 5.4, we obtain   III III E˜m (t) ≤ C2 δ 4 + Dm (0) + ϕ4 (t) e−cδt .

(5.23)

Here, recalling the definition η III (x, t) = ζ III (x − 2t, εt) and the assumption ε = δ and using Lemma 4.3, ∞ ∞ we have 0 δ|ηxIII (t)|2s dt = 1ε 0 δ|ζxIII (τ )|2s dτ  |η0 |s . By this, the integrability of F˜m+1 which comes from Proposition 4.2, and (5.16), we have ϕ3 (t)  δ 4 (see (5.19) and (5.20)). This together with (5.21) leads to ϕ4 (t)  δ 4 (see (5.22)). Combining this, (5.23), and Lemma 5.5, we have III E˜m (t) ≤ C3 δ 4 e−cεt ,

(5.24)

which implies D(t; ζ III , uIII , v III , pIII )  δ 4 e−cεt (see (2.5) and (4.18)). Here, we used V  Vy = 1 Ux . Moreover, by taking into account the equality P (x, y, t) = P (x, 1, t) − y Py (x, z, t)dz and using the second equation in (4.6), the second equation in (4.7), and the uniform estimate (5.24), we easily obtain (1 + |Dx |)m (pδ − pIII )(t) 2  δ 4 e−cεt . Note that in the case of O(δ −1 ) ≤ W ≤ O(δ −2 ) we can estimate the δ2 W m ˜III term sin α ∂x Hxx which comes from the second equation in (4.7) by Em+1 . Therefore, the proof of (2.11) in Theorem 2.2 is complete. 2 We proceed to prove (2.7), (2.9), and (2.13). Let ζ I , ζ II , and ζ IV be the solutions for (1.7), (1.8), and (1.10), respectively under the initial condition ζ I |τ =0 = ζ II |τ =0 = ζ IV |τ =0 = η0 . We put η I (x, t) := ζ I (x − 2t, εt), η II (x, t) := ζ II (x − 2t, εt), η IV (x, t) := ζ IV (x − 2t, εt) and ⎧ ⎪ uI (x, y, t) := uI0 (y; η I (x, t)) + δuI1 (y; η I (x, t)), ⎪ ⎨ v I (x, y, t) := uI0 (y; η I (x, t)) + δv1I (y; η I (x, t)), ⎪ ⎪ ⎩ I p (x, y, t) := pI0 (y; η I (x, t)) + δpI1 (y; η I (x, t)),

(5.25)

⎧ II II II ⎪ uII (x, y, t) := uII 0 (y; η (x, t)) + δu1 (y; η (x, t)), ⎪ ⎨ II II II v II (x, y, t) := uII 0 (y; η (x, t)) + δv1 (y; η (x, t)), ⎪ ⎪ ⎩ II II II II p (x, y, t) := pII 0 (y; η (x, t)) + δp1 (y; η (x, t)),

⎧ IV IV IV ⎪ (x, t)) + δuIV (x, t)) + δ 2 uIV (x, t)), uIV (x, y, t) := uIV 0 (y; η 1 (y; η 2 (y; η ⎪ ⎨ IV v IV (x, y, t) := uIV (x, t)) + δv1IV (y; η IV (x, t)) + δ 2 v2IV (y; η IV (x, t)), 0 (y; η ⎪ ⎪ ⎩ IV IV IV IV (x, t)) + δpIV (x, t)) + δ 2 pIV (x, t)), p (x, y, t) := pIV 0 (y; η 1 (y; η 2 (y; η

(5.26)

(5.27)

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where uI0 , v0I , pI0 , . . . were defined by (3.15)–(3.17). In view of this, by applying the same argument as showing (2.11), it is not difficult to check that (2.7), (2.9), and (2.13) hold. Therefore, the proof of Theorem 2.2 is complete. 2 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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