First order decoupled method of the primitive equations of the ocean I: Time discretization

J. Math. Anal. Appl. 412 (2014) 895–921

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First order decoupled method of the primitive equations of the ocean I: Time discretization ✩ Yinnian He ∗ Center for Computational Geosciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China

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Article history: Received 2 May 2013 Available online 12 November 2013 Submitted by W. Layton

In this article, we study the time discretization of the first order decoupled semi-implicit scheme of the 3D primitive equations of the ocean in the case of the non-slip and non-heat flux boundary conditions on the side. The almost unconditional stability of the scheme and the optimal error estimates of the time discrete velocity, pressure and density are provided. © 2013 Elsevier Inc. All rights reserved.

Keywords: Primitive equations of the ocean Stability Optimal error estimate Decoupled semi-implicit scheme

1. Introduction Given a smooth bounded domain ω ⊂ R 2 and the cylindrical domain Ω = ω × (−d, 0) ⊂ R 3 , consider in Ω the following 3D viscous primitive equations of the ocean:

ut + L 1 u + (u · ∇)u + w ∂z u + ∇ P + f k × u = F 1 ,

(1.1)

θt + L 2 θ + (u · ∇)θ + w ∂z θ − σ w = F 2 ,

(1.2)

∇ · u + ∂ z w = 0,

(1.3)

∂ z P + γ θ = 0.

(1.4)

The unknowns for the 3D viscous PEs are the fluid velocity field (u , w ) = (u 1 , u 2 , w ) ∈ R with u = (u 1 , u 2 ) being horizontal, the density θ and the pressure P . Here f = f 0 (β + y ) is the given Coriolis rotation frequency with β -plane approximation, F 1 and F 2 are two given functions, k is vertical unit vector, σ > 0 is the stratification constant of the ocean and γ > 0 is the gravitational constant. The elliptic operators L 1 and L 2 are given respectively as the following: 3

L i = −νi − μi ∂z2 ,

i = 1, 2.

Here the positive constants ν1 , μ1 are the horizontal and vertical viscosity coefficients; while the positive constants ν2 , μ2 are the horizontal and vertical thermal diffusivity coefficients and

ut =

∂u , ∂t

θt =

∂θ , ∂t

∇ = (∂x , ∂ y ),

= ∂xx + ∂ y y ,

with i = 1, 2, 3 and (x1 , x2 , x3 ) = (x, y , z). ✩

*

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0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.11.013

∂ xi =

∂ , ∂ xi

∂xi xi = ∂x2i ,

896

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

We partition the boundary of Ω into the following three parts:

  ¯ z=0 , Γu = (x, y , z) ∈ Ω;   ¯ z = −d , Γb = (x, y , z) ∈ Ω;   ¯ (x, y ) ∈ ∂ ω, −d  z  0 . Γs = (x, y , z) ∈ Ω; Next, we provide the system (1.1)–(1.4) with the following boundary conditions-with the wind-driven on the top surface and non-slip and non-heat flux on the side walls and the bottom (see, e.g., page 246 in [3], page 160 in [11] and page 1037 in [17]):

on Γu ,

∂ z u = dτ ∗ ,

on Γb ,

∂ z u = 0,

on Γs ,

u · n = 0,

on Γs ,

u = 0,

  ∂z θ = −α θ − θ ∗ ;

w = 0,

w = 0, ∂ z θ = 0; ∂u ∂θ × n = 0, = 0, ∂n ∂n

or

∂θ = 0, ∂n

where τ ∗ = τ ∗ (x, y ) is the wind stress on ocean surface, α is a positive constant, n is the normal vector of Γs and θ ∗ = θ ∗ (x, y ) is the typical density distribution of the top surface of the ocean. Based on the above boundary condition, it is natural to assume that τ ∗ (x, y ) and θ ∗ (x, y ) satisfy

∂τ ∗ × n = 0, ∂n

τ ∗ · n = 0,

or

τ ∗ = 0, and

∂θ ∗ = 0 on ∂ ω. ∂n

Due to these boundary conditions, we can convert the previous boundary conditions into the homogeneous by replacing (u , θ) by (u + 12 [(z + d)2 − 13 h3 ]τ ∗ , θ + θ ∗ ) (refer to page 248 in [3]). Hence, we consider the following boundary conditions for the 3D viscous PEs:

w |Γu ∪Γb = 0,

(1.5)

∂z u |Γu ∪Γb = 0, or

u · n|Γs = 0,

∂z u |Γu ∪Γb = 0,

∂u × n|Γs = 0, ∂n

(1.6-1)

u |Γs = 0,

(1.6-2)

∂θ |Γ = 0 , ∂n s

∂z θ|Γb = (∂z θ + α θ)|Γu = 0,

(1.7)

refer to (28)–(29) on page 248 in [3] and (1.3)–(1.4) on page 160 in [11] for the boundary condition (1.5), (1.6-1) and (1.7), and Remark 2.1 on page 1038 in [17] for the boundary condition (1.5), (1.6-2) and (1.7). Also, the initial conditions of u (x, y , z, t ) and θ(x, y , z, t ) should be given by

u (x, y , z, 0) = u 0 (x, y , z),

θ(x, y , z, 0) = θ0 (x, y , z).

(1.8)

Using the Dirichlet boundary condition (1.5) of w on Γu ∩ Γb and (1.3)–(1.4), we have

z w (x, y , z, t ) = −

0 ∇ · u (x, y , ξ, t ) dξ,

−d

∇ · u (x, y , ξ, t ) dξ = 0, −d

z P (x, y , z, t ) = p (x, y , t ) − γ

θ(x, y , ξ, t ) dξ. −d

With the above statements, one obtains the initial boundary value problem of the 3D viscous PEs:

z ut + L 1 u + ∇ p (x, y , t ) − γ

∇θ(x, y , ξ, t ) dξ + f k × u + (u · ∇)u −

−d

z θt + L 2 θ + σ

∇ · u (x, y , ξ, t ) dξ + (u · ∇)θ − −d

∇ · u¯ = 0,

 z

 z

 ∇ · u (x, y , ξ, t ) dξ ∂z u = F 1 ,

(1.9)

−d



∇ · u (x, y , ξ, t ) dξ ∂z θ = F 2 ,

(1.10)

−d

(1.11)

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

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together with the boundary condition (1.6)–(1.7) and the initial condition (1.8), where

¯ x, y ) = φ(

1 d

0

¯ φ˜ = φ − φ,

φ(x, y , z) dz, −d

for any function φ(x, y , z) in Ω . Remark. Recall [3,11], F 1 = 0 and γ = 1 in (1.1) and (1.4) and σ = 0 in (1.2) and (1.10). The 3D viscous PEs are very important research subjects in the field of geophysical fluid, at both the theoretical and numerical levels. There are some well-known difficulties associated with this fundamental equation for 3D oceanic model since their strong nonlinearity. A mathematical study of the PEs originated in a series of articles by Lions, Temam and Wang in the early 1990s [14–16], where a mathematical formulation for the PEs, which resembles that of the Navier–Stokes equations, was established. Also, an asymptotic analysis and the finite-dimensional behavior of the 3D viscous PEs in a thin domain as the depth of the domain goes to zero were studied in [9,10]. For a more extensive discussion and review on this subject, the reader is referred to the recent articles [2,3,7,11,19,20]. A numerical simulation study of the first order coupled method based on a two-grid finite difference method for the PEs of the ocean was proposed by Medjo and Temam in [17]. However, the error analysis was not given in [17]. To the best of our knowledge, it is the first time that the first order decoupled method for the 3D viscous PEs of the ocean is represented. The derivation of error estimates for the numerical solutions of these equations has been challenging. In this paper, the time discretization of the first order decoupled method will be represented and analyzed in detail. This paper provides sharp error estimates in the first order decoupled temporal discretization. Next, the spatial discretization of the first order decoupled method can be made by the finite difference, finite element, finite volume element and spectral methods. Here, the finite element discretization (spatial discretization) part of the first order decoupled method are discussed in the further work [6] and the numerical computation part of the first order decoupled method will be done in the further work. In this paper we analyze only the decoupled time discretization of problem (1.9)–(1.11) in the finite time interval [0, T ] with the boundary conditions (1.5)–(1.7) and the initial condition (1.8), the spatial variable remaining continuous. Setting τ be time step size and tn = nτ and T = N τ , we consider the time discrete approximation (un , pn , θ n ) of 0 (u (tn ), p (tn ), θ(tn )). We write u (tn ) = u¯ (tn ) + u˜ (tn ) with u¯ (tn ) = d−1 −d u (x, y , z, tn ) dz and (u¯ n , u˜ n ) is a time discrete approximation of (u¯ (tn ), u˜ (tn )). In our decoupled scheme, we first find (u¯ n , pn ) by the Stokes problem in 2D domain ω , then find u˜ n by the linearized equations in 3D domain without θ n . Based on (un , pn ) with un = u¯ n + u˜ n , we finally find θ n by the linearized equation in 3D domain. The advantage of using the proposed time decoupled discretization over the other time coupled discretization is that (u¯ n , pn ), u˜ n and θ n can be solved in relative small systems, respectively, while the unknowns (un , pn ) and θ n for the time coupled discretization need to be solved by a whole large system. Thus, the time decoupled discretization is efficiency in computational aspect. Also, we have provided the available optimal error estimates with the first order convergence rate with time decoupled discretization. In order to obtain some suitable the error estimates for the numerical solution of the time discretization for PEs of ocean, we need to recall the H 2 -regularity results of the solution and some of its time derivatives for the 3D viscous PEs of ocean provided in recent paper [7]. Theorem 1.1. Assume that the initial data (u 0 , θ0 ) has the H 2 -regularity, and

(∂z F 1 , ∂z F 2 ),

     ∂ti F 1 , ∂ti F 2 ∈ L ∞ R + ; L 2 (Ω)2 × L ∞ R + ; L 2 (Ω) ,



i = 0, 1 , . . . , m .

Then, the solution (u , p , θ) of the 3D viscous PEs satisfies the following bounds:









2 2 σ 2m−2 (t ) ∂tm u (t ) L 2 + ∂tm θ(t )L 2

2  2  2  2  2   + σ (t ) ∂tm u (t ) H 1 + ∂tm θ(t ) H 1 + ∂tm−1 u (t ) H 2 + ∂tm−1 θ(t ) H 2 + ∂tm−1 p (t ) H 1  κ ,

t





















2 2 2 2 2 e α2 (s−t ) σ 2m−2 (s) ∂tm u  H 1 + ∂tm θ  H 1 + σ (s) ∂tm u  H 2 + ∂tm θ  H 2 + ∂tm p  H 1



ds  κ ,

(1.12) (1.13)

0

t







 

2 2 e α2 (s−t ) σ 2m−1 (s) ∂tm+1 u  L 2 + ∂tm+1 θ  L 2 ds  κ ,

(1.14)

0

for all t  0 and m = 1, 2, 3, where α2 > 0 is a fixed constant and κ is a general positive constant depending on the data (ω, α , σ , γ , d, ν1 , μ1 , ν2 , μ2 , f , F 1 , F 2 , u 0 , θ0 ), which can take the different value at its different occurrences.

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Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

Here, for any function(or vector function) φ(x, y , z, t ) (or φ(x, y , t )), for convenience, we write it into φ(t ) and φ n is an approximation of φ(tn ). Also, we will discuss the stability of the first order decoupled semi-implicit scheme and the optimal error estimates of the time discrete solution (un , pn , θ n ) with n = 1, . . . , N. This paper is organized as follows. In Section 2, some basic mathematical setting and some important inequalities are recalled and some basic lemmas and some estimates of the nonlinear terms are provided. In Section 3, the stability with respect to t of the numerical schemes is concluded. In Section 4, the optimal error estimates of the time discrete solution (un , pn , θ n ) based on the decoupled semi-implicit scheme are obtained. 2. Preliminaries For the 3D domain Ω and 2D domain ω and m  0, p  1, we introduce the standard Sobolev spaces H m (Ω) and H m (ω) or H m (Ω)2 and H m (ω)2 with the norms · H m and · H m (ω) and semi-norms | · | H m and | · | H m (ω) , respectively. For some detail case of the Sobolev spaces, the reader can refer to Adams [1]. Set

H 2 = L 2 (Ω),

X 2 = V 2 = H 1 (Ω),





H 1 = v ∈ L 2 (Ω)2 ; div v¯ = 0, v · n|Γs = 0 ,





X 1 = v ∈ H 1 (Ω)2 ; v · n|Γs = 0 , in the case of (1.6-1) and





X 1 = v ∈ H 1 (Ω)2 ; v |Γs = 0 , in the case of (1.6-2), and V 1 = X 1 ∩ H 1 . And we define the Sobolev space X 0 in the 2D domain by

 0 1 2 −1 ¯ ¯ ¯ X 0 = X 1 = v ∈ H (ω) ; v = d v (x, y , z) dz, v ∈ X 1 ,

V 0 = { v¯ ∈ X 0 ; ∇ · v¯ = 0}.

−d

We also use (·, ·)Ω and (·, ·)ω to denote the inner product in L 2 (Ω) and L 2 (ω) or L 2 (Ω)2 and L 2 (ω)2 or L 2 (Ω)4 and L 2 (ω)4 , respectively. We denote by A 1 the Stokes-type operator associated with the primitive equations (see [3,11,12]), that is A 1 = P L 1 , where P is the L 2 -orthogonal projection from L 2 (Ω)2 to H 1 . Also, we write A 2 = L 2 . Therefore, we define the bilinear forms ai : X i × X i → R, i = 1, 2 as follows:

a1 (u , v ) = ν1 (∇ u , ∇ v )Ω + μ1 (u z , v z )Ω ,







1

1



a2 (θ, η) = ν2 (∇θ, ∇ η)Ω + μ2 (θz , ηz )Ω + μ2 α θ( z = 0), η( z = 0) ω = A 22 θ, A 22 η Ω , where



1

1



a1 (u , v ) = A 12 u , A 12 v Ω

∀u , v ∈ V 1 .

Define

D ( A i ) = {φ ∈ H i ; A i φ ∈ H i },

i = 1, 2,

with the norm A i · L 2 . We have the following Poincaré inequalities[1,3]:

4γ0 u 2L 2  ν1 ∇ u 2L 2



4γ0 θ 2L 2  μ2 ∂z θ 2L 2

∀u ∈ X 1 ;  2  + α θ(z = 0) L 2 (ω) ∀θ ∈ X 2 ,

(2.1) (2.2)

for some positive constant γ0 depending on Ω or (ω, d). Here and after, we shall use the letters c and C (with or without subscripts) to denote a general positive constant the data (ω, α , σ , γ , d, ν1 , μ1 , ν2 , μ2 ), respectively, which can take the different value at their different occurrences. For the regularity results of the Navier–Stokes equations, Heywood and Rannacher in [8] made an assumption on the regularity of the solution of the Stokes system when the domain is sufficient smooth. Also, we need a further assumption on the regularity results of the solution of the Stokes-type system associated with the primitive equations of ocean and the modified Poisson equation when the domain ω is sufficient smooth. (A1). For a given g 1 ∈ H k (Ω)2 , the steady modified Stokes-type system

−ν1 v − μ1 ∂zz v + ∇ q(x, y ) = g 1 in Ω,

div v¯ (x, y ) = 0 in ω

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

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admits a unique solution ( v , q) ∈ H 2+k (Ω)2 × L 20 (ω) ∩ H 1+k (ω) for the boundary conditions ∂z v |Γu ∪Γb = 0 and v · n|Γs = 0, ∂v ∂n

× n|Γs = 0 or v |Γs = 0 such that

v 2H 2+k + q 2H 1+k (ω)  c g 1 2H k ,

(2.3)

for k = 0, 1 and for a given g 2 ∈ H k (Ω), the elliptic equation

−ν2 φ − μ2 ∂zz φ = g 2 in Ω admits a unique solution φ ∈ H 2+k (Ω) for the boundary conditions ∂z φ|Γb = (∂z φ + α φ)|Γu = 0, and ∂ n |Γs = 0 such that ∂φ

φ H 2+k  c g 2 2H k ,

(2.4)

for k = 0, 1. The second part in the assumption (A1) is the classical results. Some details of the first part in the assumption (A1) can be found on pages 2740–2741 in [12], pages 56–57 in [21] and pages 308 and 311 in [22] in the case of the boundary conditions (1.5), (1.6-2) and (1.7). In the case of the boundary conditions (1.5), (1.6-1) and (1.7), some results in the assumption (A1) can be proved by the similar manner as in [12,21,22]. We usually make the following assumption about the prescribed data for problem (1.5)–(1.11): (A2) The initial data (u 0 , θ0 ) ∈ D ( A 1 ) × D ( A 2 ) and ( F 1 , F 2 ) ∈ L ∞ ([0, T ]; L 2 (Ω)2 ) × L ∞ ([0, T ]; L 2 (Ω)) such that for some positive constant C 0 ,

              A 1 u 0 22 +  A 2 θ0 22 + sup  F 1 (t )22 +  F 2 (t )22 +  F 1 (t )2 1 +  F 2 (t )2 1 L L L L H H 0t  T

2  2  2  2   + sup  F 1t (t ) L 2 +  F 2t (t ) L 2 +  F 1tt (t ) L 2 +  F 2tt (t ) L 2  C 0 .

(2.5)

0t  T

Also, we recall the following important inequality(see [3,7])

c 1 φ 2H 2  A 1 φ 2L 2  c 0 φ 2H 2 ,

(2.6)

c 1 ψ 2H 2

(2.7)



A 2 ψ 2L 2

 c 0 ψ 2H 2 ,

for φ ∈ D ( A 1 ), ψ ∈ D ( A 2 ) and

 0

  ∇ u (x, y , ξ ) dξ

ω −d

0

1

1

1

1

|φz || w | dz dx dy  c 0 ∇ u L22

u L22 φz L22 ∇φz L22 w L 2 ,

(2.8)

−d

for u ∈ D ( A 1 ), (φ, w ) ∈ D ( A 1 ) × L 2 (Ω)2 or (φ, w ) ∈ D ( A 2 ) × L 2 (Ω) and the following Sobolev and Ladyzhenskaya inequalities [1,3–5,13]: 1

1

φ L 3 (Ω)  c 0 φ L22 |φ| H2 1 ,

φ L 6  c 0 |φ| H 1 ,

(2.9)

for all φ ∈ X 1 or X 2 , where the norm · L q denotes · L q (Ω)2 or · L q (Ω) and 1  1 1

φ L ∞ (Ω) + ∇φ L 3 (Ω) + ∂z φ L 3 (Ω)  c 0  A i2 φ  L22 A i φ L22 ,

(2.10)

for all φ ∈ H 2 (Ω)2 ∩ X 1 or H 2 (Ω)2 ∩ X 2 with i = 1, 2. It is easy to see [3] that



 z

 z

∇ · v (x, y , ξ ) dξ, φ −d

− Ω

 0

 ∇φ(x, y , ξ ) dξ, v

−d

= Ω

∇ · v (x, y , ξ ) dξ, −d

−d

∀v ∈ X1, φ ∈ X2, f k × v · v | v |4k = 0 ∀ v ∈ L 2 (Ω)2 , k = 0, 1. By the definitions, we have

 12 2  A u  2 = ν1 ∇ u 22 + μ1 ∂z u 22 , 1 L L L    12 2 2 2  A θ  2 = ν2 ∇θ 2 + μ2 θz 2 + μ2 α θ(z = 0)22 , 2 L L (ω ) L L   2 2 2 2 2  2 2

A 1 u L 2 = ν1

u L 2 + 2ν1 μ1 ∇∂z u L 2 + μ1 ∂z u L 2 ,



0

 2

A 2 θ 2L 2 = ν22

θ 2L 2 + 2ν2 μ2 ∇θz 2L 2 + μ22 θzz 2L 2 + 2ν2 μ2 α ∇θ(z = 0) L 2 (ω) ,

φ(x, y , ξ ) dξ ω

(2.11) (2.12)

900

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

and

 12 2  1   1   A v  2 =  A 2 v¯ 22 +  A 2 v˜ 22 1 1 1 L L L

A 1 v 2L 2

=

A 1 v¯ 2L 2

+ A 1 v˜ 2L 2

∀v ∈ V 1, ∀ v ∈ H 2 (Ω)2 ∩ V 1 .

Moreover, we define the trilinear forms:



  z



b( v , φ, ψ) = ( v · ∇)φ, ψ Ω −





∇ · v dξ ∂z φ, ψ

, Ω

−d

for all v ∈ X 1 , (φ, ψ) ∈ H 1 (Ω)2 × H 1 (Ω)2 or (φ, ψ) ∈ H 1 (Ω) × H 1 (Ω). It is easy to see that



4k

b v , φ, φ|φ|



=−

1 2 + 4k

 0

 2+4k ∇ · v dξ, φ( z = 0)

 (2.13)

, ω

−d

for all v ∈ X 1 , φ ∈ H 1 (Ω) or H 1 (Ω)2 and k = 0, 1. With the above statements and the boundary conditions (1.5)–(1.7), we deduce the weak form: find (u , p , θ)(t ) ∈ X 1 × M 0 × X 2 with t  0 such that ( v , q, η) ∈ X 1 × M 0 × X 2

(ut , v )Ω + a1 (u , v ) + b(u , u , v ) + ( f k × u , v )Ω − γ



 z ∇θ(x, y , ξ, t ) dξ, φ

= ( F 1 , v )Ω ,

 z

(θt , η)Ω + a2 (θ, η) + b(u , θ, η) + σ

− (∇ · v , p )Ω + (∇ · u , q)Ω Ω

−d

(2.14)

 ∇ · u (x, y , ξ, t ) dξ, η

= ( F 2 , η)Ω ,

(2.15)

Ω

−d



together with the initial condition (1.8), where M 0 = L 20 (ω) = {q ∈ L 2 (Ω); ω q(x, y ) dx dy = 0}. For the aim of some numerical analysis, we make the following simple calculations:

       ∇ φ|φ|2k 2 = |∇φ|2 |φ|4k + 1 + 1 ∇|φ|2k 2 |φ|2 , k        ∂z φ|φ|2k 2 = |∇φ|2 |φ|4k + 1 + 1 ∇|φ|2k 2 |φ|2 , k



 2 1 1  L i φ, φ|φ|4 Ω   L i2 φ|φ|2  L 2 ,

(2.16)

2

for k = 1, 2 and φ ∈ X 1 or φ ∈ X 2 , and

 1   (φ − ψ) · φ|φ|4 = |φ|2 − |ψ|2 + |φ − ψ|2 |φ|4 2    2 1  1 1 2 1 = |φ|6 − |ψ|6 + |φ| + |ψ|2 |φ|2 − |ψ|2 + |φ − ψ|2 |φ|4 , 6

3

6

for all φ, ψ ∈ X 1 or φ, ψ ∈ X 2 . Also, we need the following Sobolev and Ladyzhenskaya’s inequalities in

φ L 4 (ω)  c 0 φ

1 2 L 2 (ω )

φ L 8 (ω)  c 0 φ

3 4 L 6 (ω )

∇φ

1 2 L 2 (ω )

L



ω

ω (see [1,13,3]):

,

for all φ ∈ X 0 . It follows from (2.18) that

= (ω )

(2.17)

1 4 L 2 (ω )

∇φ

,    2

φ 12  c 0 φ 6L 6 (ω) ∇ φ|φ|2  L 2 (ω) , L 12 (ω)

  φ|φ|2 66

2

2  4  |φ|6 |φ|12 dx dy  φ|φ|2  L 4 (ω) φ|φ|2  L 8 (ω)

     3 2  c 0 φ|φ|2 L 2 (ω) ∇ φ|φ|2  L 2 (ω) φ|φ|2  L 6 (ω) ,

(2.18)

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

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which yields

  φ|φ|2 66

L (ω )

 2   4  c 0 φ|φ|2  L 2 (ω) ∇ φ|φ|2  L 2 (ω)   4 = c 0 φ 6L 6 (ω) ∇ φ|φ|2 L 2 (ω) .

(2.19)

In order to analyze the stability and convergence of the numerical solution, we need the following uniform Gronwall lemma [18]. Lemma 2.1. Let C 0 is a positive constant and an , bn , dn be three positive series satisfying m 

am + τ

m −1 

bn  τ

n =1

dn an + C 0 ,

Then m 

am + τ

m  1,

(2.20)

n =0

 bn  C 0 exp

τ

n =1

m −1 

 ∀m  1.

dn

(2.21)

n =0

Lemma 2.2. Let Ω1 ⊂ R m1 and Ω2 ⊂ R m2 be two measurable sets, where m1 and m2 are positive integers. Suppose that f (ξ, η) is measurable over Ω1 × Ω2 . Then

   Ω1

   f (ξ, η) dη

p

 1p dξ

Ω2

 

   f (ξ, η) p dξ

 Ω2

 1p dη .

(2.22)

Ω1

3. First order decoupled semi-implicit scheme: stability Setting τ be time step size, tn = nτ , T = N τ and u 0 = u 0 , θ 0 = θ0 , then we can define the first order time discrete schemes of (1.9)–(1.11) as follows. First order decoupled semi-implicit scheme. For each n and find (un , pn , θ n ) ∈ X 1 × M 0 × X 2 satisfying





n





n



n





 z



n

dt u , v Ω + a 1 u , v − ∇ · v , p Ω + ∇ · u , q Ω − γ



n −1

n −1





n −1





 ∇θ

n −1

Ω

−d



  + f k × un , v Ω

(x, y , ξ ) dξ, v

, v¯ + b u , u , v˜ =   z  n   n      n −1 dt θ , φ Ω + a 2 θ , φ + σ ∇ · u (x, y , ξ ) dξ, φ + b un−1 , θ n , φ = F 2n , φ Ω , +b u

,u

n

F 1n , v Ω ,

(3.1) (3.2)

Ω

−d

for all ( v , q, φ) ∈ X 1 × M 0 × X 2 , v˜ = v − v¯ and n

dt u =

1

τ

n

u −u

n −1



n

dt θ =

,

1

τ

n

θ −θ

n −1



F 1n

,

=

1

tn F 1 (t ) dt ,

τ

F 2n

=

t n −1

1

tn F 2 (t ) dt .

τ t n −1

For the convenience of numerical analysis, we define dt u ∈ H 1 and dt θ ∈ H 2 such that 0



0





0

 z



 ∇θ (x, y , ξ ) dξ, v Ω

−d

0





0



 z



      + f k × u 0 , v Ω + b u 0 , u 0 , v = F 10 , v Ω ,

0

dt u , v Ω +a1 u , v − γ



0

    + b u 0 , θ 0 , φ = F 20 , φ Ω ,

0

dt θ , φ Ω +a2 θ , φ + σ

∇ · u (x, y , ξ ) dξ, φ −d

Ω

for all ( v , φ) ∈ V 1 × V 2 . Hence, we deduce from the assumption (A2) that

 0 dt u 

L2

and

 0 dt θ 

L2

are bounded.

902

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

Here (un , pn ) can be solved by (3.1) without θ n and θ n can be solved by (3.2) without (un , pn ), respectively. Set

u¯ n = d−1

un = u¯ n + u˜ n ,

0 u˜ n = un − u¯ n ,

un (x, y , z) dz,

−d

then (3.1) can be rewritten as follows: for each n and (un−1 , θ n−1 ), find (u¯ n , pn , u˜ n ) ∈ X 1 × M 0 × X 2 such that











dt u¯ , v¯ Ω + a1 u¯ , v¯ − ∇ · v¯ , p n

=



n

n





+ ∇ · u¯ , q n

 Ω

 z −γ

∇θ





= F˜ 1n , v

Ω

(3.3)

 z



dt u˜ n , v Ω + a1 u˜ n , v − γ



 Ω

    + f k × u¯ n , v¯ Ω + b un−1 , un−1 , v¯

dξ, v¯

−d



F¯ 1n , v¯ Ω ,



 n −1

∇θ n−1 dξ − d−1

−d



0  z

    + f k × u˜ n , v Ω + b un−1 , un , v˜

∇θ n−1 dξ dz, v Ω

−d −d

(3.4)

,

for all v ∈ X 1 and q ∈ M 0 , together with the initial condition:

u¯ 0 = u¯ 0 (x, y ),

u˜ 0 = u˜ 0 (x, y , z).

Here (u¯ n , pn ) can be solved by the Stokes equations (3.3) in the 2D domain ω without u˜ n and u˜ n can be solved by the linearized equations (3.4) in the 3D domain Ω . Hence, the scheme (3.2)–(3.4) is an Euler decoupled semi-implicit scheme with respect to θ n , (u¯ n , pn ) and u˜ n . Theorem 3.1. Suppose that the assumptions (A1)–(A2) hold and 0 < τ < 1 satisfies the following stability condition:

8σ γ d 2







ν1−1 + ν2−1 τ  1,

1 c 3 1 + σ −1 κ0 + μ− 1 κ1



8ν1−1 σ −1 κ2 τ  1,

2c 2 κ12



ν1−1 κ1 + (2ν1 μ1 )−1 κ2 τ  1,





ν1−2 + σ −2 τ  1,

c 4 σ − 1 κ2 τ  1 ,



 2 2 −1 −1 4ν1−1 σ −1 κ2 + c 5 κ12 + c 5 μ− + μ− κ1 κ2 τ  1 , 1 (2ν1 ) 2 (2ν2 )



(3.5)

for some positive constants c 2 , c 3 , c 4 , c 5 . Then there hold



2



2

σ um L 2 + γ θ m L 2 + τ

m   1 2

 12 n 2 σ  A 1 u L 2 + γ  A 22 θ n L 2 n =1

 2  2  κ0 = σ u 0  L 2 + γ θ 0  L 2 + γ0−1

T

 2 2

 σ  F 1 (t )L 2 + γ  F 2 (t )L 2 dt ,

(3.6)

0 m   1 2  1 2  2  2   2 σ u˜ m L 6 + γ θ m L 6 +  A 12 um L 2 +  A 22 θ m L 2 + ν1 ∇ u¯ n − u¯ n−1 L 2 n =1

m            A 1 un 22 +  A 2 θ n 22 + σ dt un 22 + γ dt θ n 22  κ1 , +τ L L L L

(3.7)

n =1



2



2



2



2

σ dt um L 2 + γ dt θ m L 2 + σ ∇ um L 3 + σ um L ∞ + τ  m 2 p  1

H (ω )

for m  0. Here u 0 , θ0 , T ).

m     

 n 2  u  3 +  θ n 2 3 +  p n 2 2 +τ n =1

H

H

H (ω )



m     1

 12  A dt un 22 +  A 2 dt θ n 22  κ2 , 1 2 L L n =1

 κ3 ,

(3.8)

κi with i = 0, 1, 2, 3 is a general positive constant depending on the data (Ω, α , σ , γ , d, ν1 , μ1 , ν2 , μ2 , f , F 1 , F 2 ,

Before proving Theorem 3.1, we need the following 8 lemmas. L 2 -estimates. Now, we shall provide the following L 2 -stability of the numerical solution (un , pn , θ n ).

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

903

Lemma 3.1. Under the assumptions of Theorem 3.1, there hold



2



2

σ um L 2 + γ θ m L 2 +

m σ 

2

n =1









2 2 1 − 8ν1−1 un−1  L ∞ τ un − un−1  L 2 + τ

T

 2  2  σ u 0 L 2 + γ θ 0 L 2 + γ0−1

m   1 2

 12 n 2 σ  A 1 u L 2 + γ  A 22 θ n L 2 n =1

 2 2

 σ  F 1 (t )L 2 + γ  F 2 (t )L 2 dt .

(3.9)

0

Proof. We take ( v , q) = σ (un , pn )τ in (3.1), φ = γ θ n τ in (3.2) and adding these two relations and using (2.11)–(2.13) with div u¯ n = 0, we obtain

    2 2 γ  n 2   θ  2 − θ n−1 22 + θ n − θ n−1 22 −  u n −1  L 2 +  u n − u n −1  L 2 + L L L

σ  n 2 u

2

L2

2

 1 2  1 2  σ  + σ  A 12 un  L 2 τ + γ  A 22 θ n  L 2 τ + b un−1 , u¯ n , un − un−1 τ  z n

− σγ

2

 n

∇θ dξ, u − u

n −1

 n

τ + σγ Ω

−d

 z n

∇ · u dξ, θ − θ

n −1

τ Ω

−d

    = σ F 1n , un Ω τ + γ F 2n , θ n Ω τ .

(3.10)

Using (2.1)–(2.2) and the Hölder inequality, we obtain the following estimates:

 









 



σ b un−1 , u¯ n , un − un−1   σ un−1 L ∞ ∇ u¯ n L 2 un − un−1 L 2  2  2 σ ν1  n 2  ∇ u¯ L 2 + 2ν1−1 σ un−1  L ∞ un − un−1  L 2 ,   z    n n n −1 σγ ∇θ dξ, u − u  −d

8

Ω

         σ γ dun − un−1 L 2 ∇θ n  L 2  

  z    n n n −1 σγ ∇ · u dξ, θ − θ  −d

Ω

γ ν2  8

2  2 ∇θ n  L 2 + 2ν2−1 γ σ 2 d2 un − un−1  L 2 ,

       τ  σ γ d∇ un  L 2 θ n − θ n−1  L 2  

σ ν1  8

2  2 ∇ un  L 2 + 2γ 2 σ ν1−1 d2 θ n − θ n−1  L 2 ,

      2   σ ν1  n 2 1 σ  F 1n , un Ω   σ  F 1n L 2 un L 2  ∇ u L 2 + σ γ0−1  F 1n L 2 , 8

2

   2    γ μ2  n 2  1 ∂z θ  2 + α θ n ( z = 0)22 + γ γ0−1  F 2n  L 2 . γ  F 2n , θ n Ω   L L (ω ) 8

2

Combining these estimates with (3.10) and using the stability condition yields









 







1





1



2 2 2 2 2 2 σ un L 2 − un−1 L 2 + γ θ n L 2 − θ n−1 L 2 + σ  A 12 un L 2 τ + γ  A 22 θ n L 2 τ  2  2 σ + 1 − 8ν1−1 un−1  L ∞ τ un − un−1  L 2

2

 σ γ0−1

tn

t n −1

   F 1 (t )22 dt + γ γ −1 L

tn

0

   F 2 (t )22 dt . L

t n −1

Summing (3.11) from n = 1 to n = m, we deduce (3.9).

2

L 6 -estimates. Now, we shall provide the following L 6 -stability of the numerical solution (u˜ n , θ n ).

(3.11)

904

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

Lemma 3.2. Under the assumptions of Theorem 3.1, (u˜ n , θ n ) satisfies



2



2

σ u˜ m L 6 + γ θ m L 6  τ

m −1 

dn

n =0

T +c

 2  2  2

 n 2 σ u˜ L 6 + γ θ n L 6 + σ u˜ 0 L 6 + γ θ 0 L 6

m −1   2 2   1      12 n 2  A u  2 +  A 2 θ n 22 , σ  F 1 (t ) 2 + γ  F 2 (t ) 2 dt + c τ L

1

L

n =0

0

τ

L

2

(3.12)

L

m   1   2 2

 12  n  n 2 2 σ  L 1 u˜ u˜  L 2 + γ  L 22 θ n θ n  L 2 n =1

τ

m  n =1

+ cτ



 





 

2 4 4 dn−1 u˜ n−1  L 6 u˜ n  L 6 + u˜ n−1  L 6 +

3 

2

 2

σ u˜ 0 L 6 + γ θ 0 L 6

2

m m          

 n 2  n 4

 n−1 2  n 4  F  2 u˜  6 +  F n 22 θ n 46 + c τ ∇ u  2 θ  6 + ∇θ n−1 22 u˜ n 46 , 1 L 2 L L L L L L L n =1

(3.13)

n =1

where





 



 

2 2 dn = c ∇ u˜ n+1  L 2 + u˜ n+1  L 2 ∇ u˜ n+1  L 2 .

Proof. Taking φ = γ θ n |θ n |4 ∈ X 2 in (3.2) and v˜ = σ u˜ n |u˜ n |4 ∈ X 1 in (3.4) and using (2.11)–(2.13) with k = 1 and (2.16)–(2.17), we obtain

 0   n−1 n n  n 4   n  n 4  σ  n 6 σ  12  n  n 2 2 − 1 n − 1 n L 1 u˜ u˜ + σ b u˜ , u¯ , u˜ u˜  + d σ u˜ ⊗ u˜ dξ, ∇ u˜ u˜  dt u˜ L 6 + L2 6

2

 z − σγ

∇θ

n −1

ξ −d

−1

−d

 6

γ 6

dt θ n  L 6 +

γ  2

1

∇θ

n −1

  n  n 4



Ω

dξ dz, u˜ u˜

Ω

−d −d

 4   σ F˜ 1n , u˜ n u˜ n  Ω , 

0  z

−d

(3.14)

   

2 2 L 22 θ n θ n   L 2 + σ γ

 z

 4 ∇ · un−1 dξ, θ n θ n 



 4    γ F 2n , θ n θ n  Ω . Ω

−d

Using (2.1)–(2.2), (2.9)–(2.10), (2.16)–(2.19) and Lemma 2.2, we have



 4   γ  12  n  n 2 2        L θ θ   2 + γ c  F n 22 θ n 46 , 2 L 2 L L Ω

γ  F 2n , θ n θ n 

8

  z      n −1 n  n 4 σγ ∇ ·u dξ, θ θ  

−d



Ω

  γ  1      2  4 2 2     L 22 θ n θ n   L 2 + γ c ∇ un−1  L 2 θ n  L 6 ,  8

   σ   n  n 2 2       L u˜ u˜   2 + σ c  F n 22 u˜ n 46 , 1 L 1 L L Ω 1 2

4 σ  F˜ 1n , u˜ n u˜ n 

16

  z  0  z     n −1 −1 n −1 n  n 4 σγ ∇θ dξ − d ∇θ dξ, u˜ u˜  −d

−d −d

   4   σ b u˜ n−1 , u¯ n , u˜ n u˜ n   Ω

   σ ∇ u¯ n 

0   L 2 (ω )

−d

ω

 n−1 2  n 10 u˜  u˜  dx dy

 12 dz

Ω

  σ  1         L 2 u˜ n u˜ n 2 22 + σ c ∇θ n−1 22 u˜ n 46 ,  L L L  16 1

(3.15)

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

   σ ∇ u¯ n  L 2 (ω)

0  

−d

   σ ∇ u¯ n 

L 2 (ω )

 n−1 6 u˜  dx dy

ω

 n −1  u˜ 

L 2 (ω )

 n −1  u˜ 

 n 15 u˜  dx dy

 13 dz

ω

 0  

 n 15 u˜  dx dy

L 6 (Ω)

−d

   σ ∇ u¯ n 

 16  

 0 L 6 (Ω)

 56

 25 dz

ω

 n  125 u˜  12 L

(ω )

 56

 n  n 2  65 u˜ u˜   6

L (ω )

dz

−d

     σ ∇ u¯ n  L 2 (ω) u˜ n−1  L 6 (Ω)

 0

 n  125 u˜  6

L (ω )

  n  n 2  65 ∇ u˜ u˜   2

   σ ∇ u¯ n 

L 2 (ω )

 n −1  u˜ 

L 2 (ω )

 n −1  u˜ 

 0 L 6 (Ω)

L (ω )

−d

L 6 (Ω)

 13  0

 n 6 u˜  6

 56

L (ω )

−d

   σ ∇ u¯ n 

905

dz

dz

 12

  n  n 2 2 ∇ u˜ u˜   2

L (ω )

dz

−d

 n 2 u˜  6

L (Ω)

  n  n 2  ∇ u˜ u˜  

L 2 (Ω)

 2  2  4 σ  1   2 2   L 12 u˜ n u˜ n  L 2 + σ c ∇ u¯ n  L 2 u˜ n−1  L 6 u˜ n  L 6 , 8

σd

 0    n  n 4  n −1 n   u˜ ⊗ u˜ dξ, ∇ u˜ u˜  

−1 

−d

Ω

    

2  0   0  12       n  n 2  4   n − 1 n n ⊗ u˜ dξ  u˜  dx dy dz  c ∇ u˜ u˜   L 2 (Ω)  u˜   −d ω −d

 σd

  n  n 2  ∇ u˜ u˜   L 2 (Ω)

−1 

3   0  13  0    13  12    n 12   n −1 n   ⊗ u˜ dξ  dx dy u˜ dx dy dξ  u˜   ω −d

   2   σ c ∇ u˜ n u˜ n  

 0   L 2 (Ω)

−d

   2   σ c ∇ u˜ n u˜ n  

 0 L 2 (Ω)

−d

 n −1 3 u˜ ⊗ u˜ n  dx dy

 13 dξ

 n −1  u˜ 

L 2 (Ω)

 n −1  u˜ 

 n 4 u˜  12 L

−d

ω

L 6 (ω )

 n  13 u˜  2 L

 n  23 ∇ u˜  2 dξ (ω ) L (ω )

−d

   2   σ c ∇ u˜ n u˜ n  

ω

 0

L 6 (Ω)

 n  13 u˜  2

L (Ω)

 0

 n  23 ∇ u˜  2

 0

(ω )

dz

 n 2 u˜  6 L

  n  n 2  23 ∇ u˜ u˜   2 dξ (ω ) L (ω )

−d

 n 2 u˜  6 L

L (Ω)

 12

  n  n 2  23 ∇ u˜ u˜   2 dz (ω ) L (ω )

 12

 12

−d

   2     1  2      2  1  σ c ∇ u˜ n u˜ n   L 2 (Ω) u˜ n−1 L 6 (Ω) u˜ n  L32 (Ω) ∇ u˜ n  L32 (Ω) u˜ n  L 6 (Ω) ∇ u˜ n u˜ n   L32 (Ω) 

σ 

16

1





 











 







2 2 4 4 2 2 L 12 u˜ n u˜ n   L 2 + σ c 02 un−1  L 6 (Ω) + u˜ n  L 6 (Ω) u˜ n  L 2 ∇ u˜ n  L 2 (Ω) u˜ n−1  L 6 .

Combining these estimates with (3.14)–(3.15) yields

σ  n 6 σ  1   2 2 γ  6 γ  1   2 2 dt u˜ L 6 +  L 12 u˜ n u˜ n   L 2 + dt θ n  L 6 +  L 22 θ n θ n   L 2 6

4

6

4

2  4  2  4 4     2 σ  σ c  F 1n  L 2 + ∇θ n−1  L 2 u˜ n  L 6 + dn−1 u˜ n−1  L 6 u˜ n  L 6 + u˜ n−1  L 6 6  n−1 2  n 4  n 2       + γ c F 2 L2 + ∇ u θ L6 . L2

Thus, (3.12)–(3.13) follow from (3.16) and Lemma 2.1.

2

(3.16)

906

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

H 1 -estimates. First, we shall provide the following stability of the numerical solution ∇ u¯ n . Lemma 3.3. Under the assumptions of Theorem 3.1, (um , pm , θ m ) satisfies the following stability:



2

ν1 ∇ u¯ m L 2 + τ τ

m −1  n =0

where



m m      2    2  A 1 u¯ n 22 + ν1 − 4u¯ n−1 L ∞ τ ∇ u¯ n − u¯ n−1 L 2 L n =1

n =1

 2  2 dn ν1 ∇ u¯ n  L 2 + ν1 ∇ u¯ 0  L 2 + c

T

m −1 m     12  n  n 2 2  12 n 2   F 1 (t )22 dt + c τ  L u˜ u˜   2 + c τ A u  2, 1 1 L L L n =0

0

2 

(3.17)

n =1

2

dn = c u¯ n  L 2 ∇ u¯ n+1  L 2 . Proof. Taking v = A 1 un τ ∈ H 1 and q = 0 in (3.3), and noting A 1 un = A 1 u¯ n , we obtain

   2 2 2 2      ∇ u¯ n  L 2 − ∇ u¯ n−1  L 2 + ∇ u¯ n − u¯ n−1  L 2 +  A 1 u¯ n  L 2 τ + f k × u¯ n , A 1 u¯ n Ω τ + b un−1 , un−1 , A 1 u¯ n τ 2   = F¯ 1n , A 1 u¯ n Ω τ . (3.18)

ν1 

Using (2.1)–(2.2) and (2.9)–(2.10), we have

  n −1 n −1           b u , u , A 1 u¯ n    u¯ n−1 · ∇ u¯ n−1 , A 1 u¯ n  +  ∇ · u˜ n−1 ⊗ u˜ n−1 , A 1 u¯ n  Ω Ω  n−1  12  n−1  12  n  12  3   n −1    n   n 2            c u¯ ∇ u¯ ∇ u¯ L 2 A 1 u¯ L 2 + u¯ ∇ u¯ − u¯ n−1  L 2  A 1 u¯ n  L 2 L∞ L2 L2  1  1   2  1   + c ∇ u˜ n−1  22  L 2 u˜ n−1 u˜ n−1   22  A 1 u¯ n  2 1

1

L



L

L

 2  2  2  2  2 2 1 1   + c u¯ n−1  L 2 ∇ u¯ n−1  L 2 ∇ u¯ n  L 2 + c ∇ u˜ n−1  L 2 +  L 12 u˜ n−1 u˜ n−1  L 2 8  n −1  2   n  n−1 2    + 2 u¯ , ∇ u¯ − u¯ L∞ L2            1 2  f k × u¯ n , A 1 u¯ n   c u¯ n  2  A 1 u¯ n  2   A 1 u¯ n  2 + c ∇ u¯ n 22 , L L Ω L L  A 1 u¯ n 22 L 4

8

     n    F¯ , A 1 u¯ n   1  A 1 u¯ n 22 + 2 F n 22 . 1 1 L L Ω 8

Combining these estimates with (3.18) yields















 







2 2 2 2 2 ν1 ∇ u¯ n L 2 − ∇ u¯ n−1 L 2 + ν1 − 4un−1 L ∞ τ ∇ u¯ n − u¯ n−1 L 2 +  A 1 u¯ n L 2 τ

 2  2  2  2  dn−1 ν1 ∇ u¯ n−1  L 2 τ + c ∇ u˜ n−1  L 2 τ + c ∇ u¯ n  L 2 τ + c  F 1n  L 2 τ .

Summing the above inequality from n = 1 to n = m, we deduce (3.17).

2

Next, we shall provide the following stability of the numerical solution ∂z un . Lemma 3.4. Under the assumptions of Theorem 3.1, (un , pn , θ n ) satisfies the following stability: m   m 2 2  2

 ∂ z u  2 + τ ν1 ∇∂z un L 2 + μ1 ∂zz un L 2 L n =1

+τ

m 

n =1

τ

m −1  n =0









 2 dn ∂z un  L 2 + c

T





    n 2 τ  ∂ z u − u n −1  2

2

L

0

for some positive constant c 2 , where





L

m −1      12 n 2   F 1 (t )22 dt + τ  A θ  2 + ∂z u 0 22

2  2

 + ν1 ∇∂z u 0  L 2 + μ1 ∂zz u 0  L 2 τ , 



4 4 4 4 1 − c 2 ∇ u¯ n−1  L 2 + u˜ n−1  L 6 + ∇ u¯ n  L 2 + u˜ n  L 6







n =0

L

L

(3.19)



 

4 4 4 4 dn = c ∇ u¯ n+1  L 2 + u˜ n+1  L 6 + ∇ u¯ n  L 2 + u˜ n  L 6 .

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

907

Proof. Taking v = −∂zz un τ ∈ H 1 and q = 0 in (3.1), we obtain

            ∂z un 22 − ∂z un−1 22 + ∂z un − un−1 22 + ν1 ∇∂z un 22 τ + μ1 ∂zz un 22 τ + γ θ n−1 , ∇ · ∂z un τ L L L L L Ω 2     n−1  n  + ∂z u · ∇ u , ∂ z u n Ω τ + 2 u n −1 · ∇ ∂ z u n , ∂ z u n Ω τ   n = F 1 , −∂zz un Ω τ . (3.20) 1 

Using (2.1)–(2.2) and (2.9)–(2.10), we have

 n−1  n           √     ∂z u · ∇ u , ∂z un Ω   2un  L 6 ∂z un−1  L 3 ∇∂z un  L 2 + ∇∂z un−1  L 2 ∂z un  L 3 2 2 2 2 ν1  μ1  ν1  μ1   ∇∂z un  L 2 + ∂zz un  L 2 + ∇∂z un−1  L 2 + ∂zz un−1  L 2 8 8 8 8 4 2 2   4    + c ∇ u¯ n  L 2 + u˜ n  L 6 ∂z un−1  L 2 + ∂z un  L 2 ,           2 un−1 · ∇ ∂z un , ∂z un Ω   2un−1  L 6 ∇∂z un  L 2 ∂z un  L 3 2 2 4 4  2   ν1  μ1   ∇∂z un  2 + ∂zz un  2 + c ∇ u¯ n−1  2 + u˜ n−1  6 ∂z un  2 , L

8

L

8

L

L

L

√      2  2   ν1  γ  θ n−1 , ∇∂z un Ω   γ 2θ n−1 L 2 ∇∂z un L 2  ∇∂z un L 2 + 2γ 2 ν1−1 θ n−1 L 2 , 8

  n       F , −∂zz un    F n  2 ∂zz un  2  μ1 ∂zz un 22 + 2μ−1 F n 22 . 1 1 L 1 L 1 Ω L L 8

Combining these estimates with (3.20) yields

 n 2        



 ∂z u  2 − ∂z un−1 22 + 5 ν1 ∇∂z un 22 + μ1 ∂zz un 22 τ − 1 ν1 ∇∂z un−1 22 + μ1 ∂zz un−1 22 τ L L L L L L 4

4

Summing the above inequality from n = 1 to n = m, we obtain (3.19).

2

   4    4 4 4  2 + 1 − c 2 ∇ u¯ n−1 L 2 + u˜ n−1  L 6 + ∇ u¯ n  L 2 + u˜ n  L 6 τ ∂z un − un−1  L 2  2  1 2  2  c  F 1n  L 2 τ + c  A 22 θ n−1  L 2 τ + dn−1 ∂z un−1 L 2 τ .

Lemma 3.5. Under the assumptions of Theorem 3.1, (un , pn , θ n ) satisfies the following stability: m   m 2  2  2   2

 ∂z θ  2 + α θ m ( z = 0)22 + τ ν2 ∇∂z θ n L 2 + μ2 ∂zz θ n L 2 + μ2 α ∇θ n (z = 0)L 2 (ω) L L (ω ) n =1

+

m 

n =1

τ

m −1  n =0







 







  



2 2 2 2 2 α 1 − c3 1 + un−1 L 2 + ∂z un−1 L 2 ∇ un−1 L 2 + ∇∂z un−1 L 2 τ θ n (z = 0) − θ n−1 (z = 0)L 2 (ω)



2

dn α θ n ( z = 0) L 2 (ω) + c

T

       F 2 (t )22 dt + ∂z θ 0 22 + α θ 0 ( z = 0)22 L L L (ω )

0

m     

 12 n−1 2  A u  2 + ν1 ∇∂z un−1 22 τ + μ1 ∂zz un−1 22 + cτ 1 L L L n =1

+ cτ

m      

 12 n−1 2   A u  2 + ∇ u¯ n−1 42 + u˜ n−1 46 ∂z θ n 22 1 L L L L n =1

+ cτ

m   n 4  n−1 2    θ  6 ∂z u  2 + ∂z θ n 22 , L L L

(3.21)

n =1

for some positive constant c 3 , where





2



2  n 2    ∇ u  2 + ∇∂z un 22 .

dn = c 1 + un  L 2 + ∂z un  L 2

L

L

908

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

Proof. Taking φ = −∂zz θ n τ ∈ H 2 in (3.2), we obtain

      ∂z θ n 22 − ∂z θ n−1 22 + ∂z θ n − θ n−1 22 L L L

1  2

2  2  2 θ n ( z = 0) L 2 (ω) − θ n−1 ( z = 0) L 2 (ω) + θ n ( z = 0) − θ n−1 ( z = 0) L 2 (ω)  2  2  2   + ν2 ∇∂z θ n  L 2 τ + μ2 ∂zz θ n L 2 τ + μ2 α ∇θ n ( z = 0) L 2 (ω) − σ un−1 , ∇∂z θ n Ω τ           + α u n −1 · ∇ θ n , θ n ω  z =0 τ + ∂ z u n −1 · ∇ θ n , ∂ z θ n Ω τ + 2 u n −1 · ∇ ∂ z θ n , ∂ z θ n Ω τ   = F 2n , −∂zz θ n Ω τ . +

α  2

(3.22)

Using (2.1)–(2.2) and (2.9)–(2.10), we have



 

















α  un−1 · ∇ θ n , θ n ω z=0   α un−1 (z = 0)L 4 (ω) θ n (z = 0)L 4 (ω) ∇θ n (z = 0)L 2 (ω)   2 2 2 μ2 α  n 1  n −1  u ( z = 0) L 4 (ω) θ n ( z = 0) L 4 (ω) ∇θ ( z = 0) L 2 (ω) + 4αμ− 2 16 2 μ2 α  n  

∇θ ( z = 0) L 2 (ω) 2 2  2 2     2  + c 1 + un−1 L 2 + ∂z un−1  L 2 ∇ un−1  L 2 + ∇∂z un−1  L 2 θ n ( z = 0) L 2 (ω) ,  n−1  n           √     ∂z u · ∇ θ , ∂z θ n Ω   2θ n  L 6 ∂z un−1  L 3 ∇∂z θ n  L 2 + ∇∂z un−1  L 2 ∂z θ n  L 3 2 2 2 2 ν2  μ2  ν1  μ1   ∇∂z θ n  L 2 + ∂zz θ n  L 2 + ∇∂z un−1  L 2 + ∂zz un−1 L 2 8 8 8 8   4  2 2  + c  θ n  L 6  ∂ z u n −1  L 2 +  ∂ z θ n  L 2 ,           2 un−1 · ∇ ∂z θ n , ∂z θ n Ω   2un−1  L 6 ∇∂z θ n  L 2 ∂z θ n  L 3 2 2 4 4  2   ν2  μ2   ∇∂z θ n  L 2 + ∂zz θ n L 2 + c ∇ u¯ n−1  L 2 + u˜ n−1  L 6 ∂z θ n  L 2 , 8

8

8

     2  2   ν2  σ  un−1 , ∇∂z θ n Ω   σ un−1 L 2 ∇∂z θ n L 2  ∇∂z θ n L 2 + 2σ 2 ν2−1 un−1 L 2 , 8

      n       F , −∂zz θ n    F n  2 ∂zz θ n  2  μ2 ∂zz θ n 22 + 2μ−1  F n 22 . 2 2 L 2 L 2 L Ω L 8

Combining these estimates with (3.22) and using (2.1)–(2.2) yields

   

 n 2  n−1 2  2 + α θ n−1 ( z = 0)22 ∂z θ  2 + α θ n ( z = 0)22 − ∂ z θ L L (ω ) L L (ω )  2  2  2 + ν2 ∇∂z θ n  2 τ + μ2 ∂zz θ n  2 τ + μ2 α ∇θ n ( z = 0) 2 τ L

L (ω )

L

2 2  2 2      2 + α 1 − c 3 1 + un−1 L 2 + ∂z un−1  L 2 ∇ un−1  L 2 + ∇∂z un−1  L 2 τ θ n ( z = 0) − θ n−1 ( z = 0) L 2 (ω)



 2  1 2  2  2  c  F 2n L 2 τ + c  A 12 un−1  L 2 τ + ν1 ∇∂z un−1  L 2 τ + μ1 ∂zz un−1  L 2 τ 4 4  2  4  2 2     + c ∇ u¯ n−1  L 2 + u˜ n−1  L 6 ∂z θ n  L 2 τ + c θ n  L 6 ∂z un−1 L 2 + ∂z θ n  L 2 τ  2 + dn−1 α θ n−1 ( z = 0) L 2 (ω) τ . Summing (3.23) from n = 1 to n = m, we obtain (3.21).

2

Furthermore, we will give the estimates of ∇ un and ∇θ n . Lemma 3.6. Under the assumptions of Theorem 3.1, (un , pn , θ n ) satisfies the following bound:



1

2



1

2

σ  A 12 um L 2 + γ  A 22 θ m L 2 + τ + σ ν1 τ

m 



n =1





m  2  2

 σ  A 1 un  L 2 + γ  A 2 θ n  L 2 n =1



    n 2 τ ∇ u − un−1 L 2

2 2 1 − c 4 un−1  L ∞ + ∇ un−1  L 3

(3.23)

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

m −1 

τ

dn

 1 2

 12 n 2 σ  A u  2 + γ  A 2 θn 2 + c 1

n =0

+τ

2

L

T

L

909

2 2 

 σ  F 1 (t ) 2 + γ  F 2 (t ) 2 dt L

L

0

m  4 4   1 2   1 2   c ∇ u¯ n−1  L 2 + u˜ n−1  L 6 σ  A 12 un  L 2 +  A 22 θ n  L 2 τ n =1

 1 2  1 2  2 + σ  A 12 u 0 L 2 + γ  A 22 θ 0  L 2 + σ  A 1 u 0  L 2 τ ,

(3.24)

for some positive constant c 4 , where

 





 



 

2 2 2 2 dn−1 = c + c ∂z un  L 2 ∇∂z un  L 2 + ∂z θ n  L 2 ∇∂z θ n  L 2 .

Proof. Taking v = σ A 1 un ∈ H 1 and q = 0 in (3.1), and φ = γ A 2 θ n ∈ H 2 in (3.2) and adding them together, we obtain



 2  2   + γ dt θ n , A 2 θ n Ω + σ  A 1 u n  L 2 + γ  A 2 θ n  L 2       + σ b u , un , A 1 un − σ b un−1 , un − un−1 , A 1 u¯ n + γ b un−1 , θ n , A 2 θ n  z   z  n −1 n n −1 n − σγ ∇θ dξ, A 1 u + σγ ∇ ·u dξ, A 2 θ

σ dt u n , A 1 u n



Ω n −1

=σ



−d

F 1n ,

n



A1u Ω + γ



Ω

F 2n ,

A2θ

n

Ω

−d

 Ω

(3.25)

.

Using (2.1)–(2.2) and (2.9)–(2.10), we have

 





   Ω √  n−1    n         σ 2 u  L ∞ ∇ u − un−1  L 2 + ∇ un−1  L 3 un − un−1  L 6  A 1 u¯ n  L 2 2 2 2     2 σ   A 1 un  L 2 + σ c un−1  L ∞ + ∇ un−1  L 3 ∇ un − un−1  L 2 ,





σ b un−1 , un − un−1 , A 1 u¯ n  = σ  ∇ · un−1 ⊗ un − un−1 , A 1 u¯ n

8

  0

√          σ b un−1 , un , A 1 un   σ un−1 L 6 ∇ un L 3  A 1 un L 2 + σ 2

ω

 n −1  ∇ u  dz

−d

0

  n   ∂z u  A 1 un  dz dx dy

−d

       1  1  1  1    σ un−1 L 6 ∇ un  L 3  A 1 un  L 2 + σ c ∇ un−1  L22  un−1  L22 ∂z un  L22 ∇∂z un  L22  A 1 un  L 2  2 2 4 4  2  σ σ   A 1 un  2 +  A 1 un−1  2 + σ c ∇ u¯ n−1  2 + u˜ n−1  6 ∇ un  2 L

8

L

24

L

 2  2  2 + σ c ∂z un  2 ∇∂z un  2 ∇ un−1  2 , L

L

L

L

L

and

√          γ b un−1 , θ n , A 2 θ n   γ un−1 L 6 ∇θ n L 3  A 2 θ n L 2 + γ 2

  0 ω

−d

 n −1  ∇ u  dz

0

  n   ∂z θ  A 2 θ n  dz dx dy

−d

       1  1  1  1    γ un−1  L 6 ∇θ n  L 3  A 2 θ n  L 2 + σ c ∇ un−1  L22  un−1  L22 ∂z θ n  L22 ∇∂z θ n  L22  A 2 θ n L 2  2 2 4 4  2  γ σ   A 2 θ n  L 2 +  A 1 un−1  L 2 + σ c ∇ u¯ n−1  L 2 + u˜ n−1  L 6 ∇θ n  L 2 8 24  2  2  2 + σ c ∂z θ n  2 ∇∂z θ n  2 ∇ un−1  2 ,   z    n −1 n σγ ∇θ dξ, A 1 u  −d

L

Ω

  z    n −1 n σγ ∇ ·u dξ, A 2 θ  −d

L

L

      2  2 d σ    √ σ γ  A 1 un  L 2 ∇θ n  L 2   A 1 un  L 2 + σ γ 2 d2 ∇θ n−1  L 2 ,  8 2

Ω

       2  2 3 γ  σ γ d  ∇ u n −1  L 2  A 2 θ n  L 2   A 2 θ n  L 2 + 3γ σ 2 d 2  ∇ u n −1  L 2 , τ   2 8

910

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921



            σ  F n  2  A 1 un  2  σ  A 1 un 22 + 2σ  F n 22 , 1 L 1 L L Ω L 8  n       γ 2 2 γ  F 2 , A 2 θ n Ω    A 2 θ n L 2 + 2γ  F 2n L 2 .

σ  F 1n , A 1 un

8

Combining these estimates with (3.25) and noting









σ dt u n , A 1 u n Ω + γ dt θ n , A 2 θ n Ω 2 σ  12 n 2  12 n−1 2  12  n = A 1 u L2 − A 1 u + A 1 u − u n −1  L 2 L2 2τ 2 γ  12 n 2  12 n−1 2  12  n + A 2 θ L2 − A 2 θ + A 2 θ − θ n −1  L 2 , L2 2τ     σ dt u n , A 1 u n Ω + γ dt θ n , A 2 θ n Ω 2 2 2  2    σ γ   A 1 un  L 2 +  A 2 θ n  L 2 + 2 σ dt un  L 2 + γ dt θ n  L 2 , 8

8

we obtain

5  4

2

5 

2

1 

2

1 

2

σ  A 1 u n  L 2 + γ  A 2 θ n  L 2 − σ  A 1 u n −1  L 2 − γ  A 2 θ n −1  L 2 4

4

4

 1 2   2  2 2  2    4  c  A 12 un−1  L 2 ∇ un − un−1  L 2 + ∂z θ n  L 2 ∇∂z θ n  L 2 + ∂z un  L 2 ∇∂z un  L 2  4  2 2   2  2  + c ∇ un−1  L 2 ∇ un  L 2 + ∇θ n  L 2 + c  F 1n  L 2 + c  F 2n  L 2  2  2  2  2 + c dt un  L 2 + c dt θ n  L 2 + c ∇ un−1  L 2 + c ∇θ n−1  L 2

(3.26)

and

 1 2  1 2  1 2

 12 n 2 σ  A 1 u L 2 + γ  A 22 θ n L 2 − σ  A 12 un−1 L 2 + γ  A 22 θ n−1 L 2 2  2 2 5  1  + σ  A 1 u n  2 τ + γ  A 2 θ n  2 τ − σ  A 1 u n −1  2 τ 4

L

L

L

4

 2      2 + σ ν1 1 − c 4 un−1  L ∞ + ∇ un−1  L 3 τ ∇ un − un−1  L 2 2  1 2  2  2

 1  dn−1 σ  A 12 un−1  L 2 + γ  A 22 θ n−1  L 2 τ + 4σ  F 1n  L 2 τ + 4γ  F 2n  L 2 τ   1 2  4 4   1 2  + c ∇ u¯ n−1  L 2 + u˜ n−1  L 6 σ  A 12 un  L 2 +  A 22 θ n  L 2 τ ,

Summing (3.27) from n = 1 to n = m, we obtain (3.24).

(3.27)

2

H 2 -estimates. Now, we shall provide the following H 2 -stability of the numerical solution (un , pn , θ n ). Lemma 3.7. Under the assumptions of Theorem 3.1, for each 1  m  N there holds



2



2

σ dt um L 2 + γ dt θ m L 2 + τ +σ

m   n =1

m  2  1 2

 12 σ  A 1 dt un L 2 + γ  A 22 dt θ n L 2 n =1





 



2 4 1 − 4ν1−1 H (tn − t 2 )un−2  L ∞ τ − c 5 ∇ u¯ n − u¯ n−1  L 2 τ

 2  2  2 − c 5 ∂z un−1  L 2 ∇∂z un−1  L 2 τ dt un − dt un−1  L 2 +γ

m   n =1

τ

m −1  n =0

dn



 







2 2 2 1 − c 5 ∂z θ n−1  L 2 ∇∂z θ n−1  L 2 τ dt θ n − dt θ n−1  L 2

 2    n 2 σ dt u L 2 + γ dt θ n L 2 + c

T

 2 2

 σ  F 1t (t )L 2 + γ  F 2t (t )L 2 dt

0

 2  2 2 2  1

 1 + σ dt u 1  L 2 + γ dt θ 1  L 2 + 2 σ  A 12 dt u 1  L 2 +  A 22 dt θ 1  L 2 τ , for some positive constant c 5 , where

(3.28)

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

 





 

1





 

1

911





 

1



4 2 2 2 2 dn = c ∇ u¯ n+1 − u¯ n  L 2 + c  A 12 un  L 2  A 1 un  L 2 + c  A 12 un  L 2  A 1 un  L 2 +  A 22 θ n  L 2  A 2 θ n  L 2 , 1

1

and the bound of σ dt u 1 2L 2 + γ dt θ 1 2L 2 + 2[σ A 12 dt u 1 2L 2 + A 22 dt θ 1 2L 2 ]τ can be estimated by (3.1)–(3.2). Proof. It follows from (3.1)–(3.2) that for each n there hold





n





n



n



dtt u , v Ω + a1 dt u , v Ω − ∇ · v , dt p Ω







  + ∇ · dt u n , q − γ 

 z

 ∇ dt θ

n −1



(x, y , ξ ) dξ, v Ω

−d



+ f k × dt u , v − b dt u , u − u , v¯ − b u , dt u − dt u , v¯       + b dt un−1 , un , v + b un−2 , dt un , v = dt F 1n , v Ω ,   z   n       n n −1 dtt θ , φ Ω + a2 dt θ , φ Ω + σ ∇ · dt u (x, y , ξ ) dξ, φ + b dt u n − 1 , θ n , φ Ω + b u n − 2 , dt θ n , φ Ω n



= dt F 2n , φ

n −1

n

n −1

n −2

Ω

(3.29)

Ω

−d



n −1

n

(3.30)

,

for all ( v , q, φ) ∈ X 1 × M × X 2 . We take ( v , q) = σ (dt un , dt pn )τ in (3.29), φ = γ dt θ n τ in (3.30) and adding these two relations and using (2.11)–(2.13) with div dt u¯ n = 0, we obtain



σ  2





 







1





1

2 2 2 2 2 dt un  L 2 − dt un−1  L 2 + dt un − un−1  L 2 + σ  A 12 dt un  L 2 τ + γ  A 22 dt θ n  L 2 τ

+



γ 





 



2 2 2 dt θ n  L 2 − dt θ n−1  L 2 + dt θ n − θ n−1  L 2

2

 z

− σγ 



 z

∇ dt θ n−1 dξ, dt un

τ + σγ Ω

−d n −1

n −1





 ∇ · dt un−1 dξ, dt θ n Ω

−d n −2

− σ b dt u , u − u , dt u¯ τ − σ b u , dt u − dt u     + σ b dt u n − 1 , u n − 1 , dt u n τ + γ b dt u n − 1 , θ n − 1 , dt θ n τ     = σ dt F 1n , dt un Ω τ + γ dt F 2n , dt θ n Ω τ . n

n

τ

n

n −1

 , dt u¯ n τ

(3.31)

By some simple calculations, we have

 





 

 



σ b dt un−1 , un − un−1 , dt u¯ n   σ dt un−1 L 3 ∇ dt un L 2 u¯ n − u¯ n−1 L 6

√       + σ 2∇ dt un−1  L 2 dt un  L 3 u¯ n − u¯ n−1 L 6 2 2 σ ν1  σ ν1   ∇ dt u n  L 2 + ∇ dt u n − 1  L 2 16 16   2 2   4  + σ c ∇ u¯ n − u¯ n−1  2 dt un−1  2 + dt un  2 , L

     σ b un−2 , dt un − dt un−1 , dt u¯ n   σ un−2 

L

L

    ∇ dt u¯ n  2 dt un − dt un−1  2 L∞ L L     2 σ ν1  2 2  ∇ dt u¯ n L 2 + 2σ ν1−1 un−2  L ∞ dt un − dt un−1  L 2 , 8

   σ b dt un−1 , un−1 , dt un 

 0    n −1   n −1   n       n −1        d t u L 6 + σ  ∇ · dt u dξ   σ dt u ∇u L2 L3      0 × ω

−d

 n−1 2 ∂z u  dz

−d

2

 14    0 dx dy ω

−d

 n 2 dt u  dz

L 2 (ω )

2

 14 dx dy

         1  1  1  1  σ dt un−1  L 2 ∇ un−1  L 3 dt un L 6 + σ c ∇ dt un−1 L 2 ∂z un−1  L22 ∇∂z un−1  L22 dt un  L22 ∇ dt un  L22

912

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

2 2      2 σ ν1  ∇ dt u n  L 2 + ∇ dt un−1  L 2 + σ c ∇ un−1  L 2  A 1 un−1  L 2 dt un−1  L 2 16 16  2  2  2 + σ c ∂z un−1  L 2 ∇∂z un−1  L 2 dt un  L 2 ,    0    n −1 n −1  n −1   n −1   n       n n − 1 γ b dt u , θ , dt θ   γ dt u L 2 ∇θ L 3 dt θ L 6 + γ  ∇ · dt u  dξ    

σ ν1 

   0 × ω



γ 

−d

 n−1 2 ∂z θ  dz

−d

2

    0 1 4

dx dy ω

A 22 dt θ n  L 2 +

σ ν1 

2

 14 dx dy

−d

2      2 ∇ dt un−1 L 2 + γ c ∇θ n−1  L 2  A 2 θ n−1  L 2 dt un−1  L 2 8 8  2  2  2 + γ c ∂z θ n−1  2 ∇∂z θ n−1  2 dt θ n  2 , 1

2

 n 2 dt θ  dz

L 2 (ω )

L

L

L

and

  z    n −1 n σγ ∇ dt θ dξ, dt u  −d

Ω

  z    n −1 n σγ ∇ · dt u dξ, dt θ  −d

      2  2 σ ν1   ∇ dt un  L 2 + 2ν1−1 γ 2 σ d2 dt θ n−1  L 2 ,   dσ γ ∇ dt un  L 2 dt θ n−1  L 2   8

Ω

      2  2 γ ν2   ∇ dt θ n  L 2 + 2γ σ 2 ν2−1 d2 dt un−1  L 2 , τ  σ γ d∇ dt θ n  L 2 dt un−1  L 2   8

      σ ν1  ∇ dt un 22 + σ c dt F n 22 , σ  dt F 1n , dt un Ω   1 L L 8

 2  2   γ 1 γ  dt F 2n , dt θ n Ω    A 22 dt θ n L 2 + γ c dt F 2n L 2 . 8

Combining these estimates with (3.31) and using (2.1)–(2.2) and the stability condition yields



2



2



2



2

σ dt un L 2 + γ dt θ n L 2 − σ dt un−1 L 2 + γ dt θ n−1 L 2

 1 2 2  1 2 1  1 + σ  A 12 dt un  L 2 τ − σ  A 12 dt un−1  L 2 τ + γ  A 22 dt θ n L 2 τ 4  2   2   σ  n 2  + ∇ dt u L 2 − ∇ dt un−1  L 2 τ + σ 1 − 4ν1−1 un−2  L ∞ τ 4    2  2  2 4 − c 5 ∇ u¯ n − u¯ n−1 L 2 τ − c 5 ∂z un−1  L 2 ∇∂z un−1  L 2 τ dt un − dt un−1  L 2  2  2  2  + γ 1 − c 5 ∂z θ n−1  L 2 ∇∂z θ n−1  L 2 τ dt θ n − dt θ n−1  L 2 2  2  2  2       dn−1 σ dt un−1  L 2 + γ dt θ n−1  L 2 τ + c σ dt F 1n  L 2 + γ dt F 2n  L 2 τ .

Summing the above inequality from n = 2 to n = m, we obtain (3.28).

2

Lemma 3.8. Under the assumptions of Theorem 3.1, for each 1  m  N, there holds

5

4



2



2

ν1 ∇∂z un L 2 + μ1 ∂zz un L 2 −

1

4



2



2

ν1 ∇∂z un−1 L 2 + μ1 ∂zz un−1 L 2

 2  2  2  2  4  2  4  2  c dt un  L 2 + c  F 1n  L 2 + c un  L 2 + c θ n−1  L 2 + c ∇ un  L 2 ∂z un−1  L 2 + c ∇ un−1  L 2 ∂z un  L 2 ,  2   2 2 ν2 ∇∂z θ n L 2 + ν2 α ∇θ n (z = 0)L 2 + μ2 ∂zz θ n L 2    2  2  2 2 2    c dt θ n  L 2 + c  F 2n L 2 + c un−1  L 2 + c 1 + un−1  L 2 + ∂z un−1  L 2   2 2 2  1 2  × ∇ un−1  L 2 + ∇∂z un−1  L 2 + ∂zz un−1 L 2  A 22 θ n  L 2 ,  n 2          1        p  1  c dt un 22 + c un 22 + c ∇θ n−1 22 + c  F n 22 + c  A 2 un−1 22  A 1 un−1 22 +  A 1 un 22 , 1 L 1 H (ω ) L L L L L L 1  n 2  12       n 2 1  n−1 2 2 2 2  u  3 +  p  2   u  3 + c  A dt u n  2 + c  A 2 u n  2 + c  A 2 θ n − 1  2 H

H (ω )

4

H

1

L

1

L

 2  2  2 2   + c  F 1n  H 1 + c  A 1 un−1  L 2  A 1 un−1  L 2 +  A 1 un  L 2 ,

L

(3.32)

(3.33)

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

 n 2  1          θ  3  c  A 2 dt θ n 22 + c  A 1 un−1 22 + c  F n 2 1 + c  A 1 un−1 22  A 2 θ n 22 . 2 H 2 H L L L L

913

(3.34)

Proof. Taking ( v , q) = (−∂zz un , 0) in (3.1) and φ = −∂zz θ n in (3.2), we obtain



2



2

  + f k × un , −∂zz un Ω         + γ θ n−1 , ∇∂z un Ω + ∂z un−1 · ∇ un , ∂z un Ω + 2 un−1 · ∇ ∂z un , ∂z un Ω     = F 1n , −∂zz un Ω + F 2n , −∂zz θ n Ω , 2  2   2 ν2 ∇∂z θ n L 2 + ν2 α ∇θ n (z = 0)L 2 + μ2 ∂zz θ n L 2         + dt θ n , −∂zz θ n Ω − σ un−1 , ∇∂z θ n Ω + un−1 · ∇ θ n , θ n ω z=0       + ∂ z u n −1 · ∇ θ n , ∂ z θ n Ω + 2 u n −1 · ∇ ∂ z θ n , ∂ z θ n Ω   = F 2n , −∂zz θ n Ω . 

ν1 ∇∂z un L 2 + μ1 ∂zz un L 2 + dt un , −∂zz un



Ω

(3.35)

(3.36)

Using (2.1)–(2.2) and (2.9)–(2.10), we have

 n−1  n 2 2   ν1  μ1   ∂z u · ∇ u , ∂z un Ω   ∇∂z un  L 2 + ∂zz un  L 2 8 8  2  4  2 ν1  μ1  n −1  2 + + ∇∂z u ∂zz un−1  L 2 + c ∇ un  L 2 ∂z un−1 L 2 , L2 8

8

          ν1   ∇∂z un 22 + μ1 ∂zz un 22 + c ∇ un−1 42 ∂z un 22 , 2 u n −1 · ∇ ∂ z u n , ∂ z u n Ω   L L L L 8

8

 2  2   ν1  γ  θ n−1 , ∇∂z un Ω   ∇∂z un L 2 + 2γ 2 ν1−1 θ n−1 L 2 , 8

        f k × un , −∂zz un   μ1 ∂zz un 22 + c un 22 , L L Ω 24

        dt , −∂zz un   μ1 ∂zz un 22 + 6μ−1 dt un 22 , 1 L L Ω 24

 n            F , −∂zz un    F n  2 ∂zz un  2  μ1 ∂zz un 22 + 6μ−1  F n 22 , 1 1 L 1 L 1 L Ω L 24  n−1  n n     n −1  n            α u · ∇ θ , θ ω z=0  α u ( z = 0) L 4 (ω) θ ( z = 0) L 4 (ω) ∇θ n ( z = 0) L 2   2 2 2 μ2 α  n 1  n −1  u ( z = 0) L 4 (ω) θ n ( z = 0) L 4 (ω) ∇θ ( z = 0) L 2 (ω) + 4αμ− 2 16 2   2 2   μ2 α  n  ∇θ ( z = 0) L 2 (ω) + c 1 + un−1  L 2 + ∂z un−1  L 2 8

2 2  1 2   × ∇ un−1  L 2 + ∇∂z un−1  L 2  A 22 θ n  L 2 ,       n−1  n     ∂z u · ∇ θ , ∂z θ n Ω   ∇θ n  L 2 ∂z un−1 L 3 ∂z θ n  L 6 2 2 ν2  μ2   ∇∂z θ n  2 + ∂zz θ n  2 8

L

L

8

      2   + c 1 + ∂z un−1  L 2 ∇∂z un−1  L 2 + ∂zz un−1  L 2 ∇θ n  L 2 ,           2 un−1 · ∇ ∂z θ n , ∂z θ n Ω   2un−1  L 6 ∇∂z θ n  L 2 ∂z θ n  L 3 2 2  4  2 ν2  μ2   ∇∂z θ n  L 2 + ∂zz θ n  L 2 + c ∇ un−1  L 2 ∂z θ n  L 2 ,

8

8

and



σ  un−1 , ∇∂z θ n

            σ un−1  2 ∇∂z θ n  2  ν2 ∇∂z θ n 22 + 2σ 2 ν −1 un−1 22 , 2 L L Ω L L 8

     n    dt θ , −∂zz θ n   μ2 ∂zz θ n 22 + 2μ−1 dt θ n 22 , 2 L L Ω 8

     n    F , ∂zz θ n   μ2 ∂zz θ n 22 + 2μ−1  F n 22 . 2 2 L 2 L Ω 8

Combining these estimates with (3.35) and (3.36), respectively, we obtain (3.32) and (3.33). Furthermore, we deduce (3.34) from assumption (A1) and (3.3). 2

914

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

Proof of Theorem 3.1. Now, we will use the induction method to prove Theorem 3.1. It follows from assumption (A2) and (3.1)–(3.2) that (3.6)–(3.8) hold for m = 0, 1. Assuming that (3.6)–(3.8) hold for m = 0, 1, . . . , J , we need to prove (3.6)–(3.8) hold for m = J + 1. Set m = J + 1 in Lemmas 3.1–3.8. First, using the stability condition (3.5) and the induction assumption in Lemma 3.1, we have showed that (3.6) holds for m = J + 1. Second, it follows from (3.6), Lemma 3.2 and Lemma 2.1 that



2



J +1   1   2 2

 12  n  n 2 2 σ  L 1 u˜ u˜  L 2 + γ  L 22 θ n θ n  L 2  C 2 .

2

σ u˜ J +1 L 6 + γ θ J +1 L 6 + τ

(3.37)

n =1

Next, by using (3.5)–(3.6), the induction assumption, Lemma 3.3 and Lemma 2.1, we obtain



2

ν1 ∇ u¯ J +1 L 2 + τ

J +1 J +1       n   A 1 u¯ n 22 + ν1 ∇ u¯ − u¯ n−1 22  C 3 . L L n =1

(3.38)

n =1

Moreover, we deduce from (3.5)–(3.6), (3.37)–(3.38), the induction assumption, Lemma 3.4 and Lemma 2.1 that J +1    2  2

 ∂z u J +1 22 + τ ν1 ∇∂z un L 2 + μ1 ∂zz un L 2  C 4 . L

(3.39)

n =1

Furthermore, using (3.5)–(3.6), (3.37)–(3.39), the induction assumption, Lemma 3.5 and Lemma 2.1, we deduce

    ∂z θ J +1 22 + α θ J +1 ( z = 0)22 L

L

+τ (ω )

J +1  2  2  2

 ν2 ∇∂z θ n L 2 + μ2 ∂zz θ n L 2 + μ2 α ∇θ n (z = 0)L 2 (ω)  C 5 . (3.40) n =1

Again, using (3.5)–(3.6), (3.37)–(3.40), the induction assumption, Lemma 3.6 and Lemma 2.1, we conclude that (3.7) holds for m = J + 1. Finally, we deduce from (3.5)–(3.7), the induction assumption, Lemmas 3.7–3.8 and Lemma 2.1,



2



2

σ dt u J +1 L 2 + γ dt θ J +1 L 2 + τ 

2



2

J +1  2  1 2

 12 σ  A 1 dt un L 2 + γ  A 22 dt θ n L 2  C 6 ,

(3.41)

n =1

ν1 ∇∂z u J +1 L 2 + μ1 ∂zz u J +1 L 2  C 7 ,

(3.42)

 2   2 2 ν2 ∇∂z θ J +1 L 2 + ν2 α ∇θ J +1 (z = 0)L 2 (ω) + μ2 ∂zz θ J +1 L 2  C 8 .

(3.43)

Combining (3.41)–(3.43) with (3.27) and using (3.6)–(3.7), we obtain



2



2

σ  A 1 u J +1  L 2 + γ  A 2 θ J +1  L 2  C 9 .

(3.44)

Thus, (3.41)–(3.44), (2.10) and (3.34) show that (3.8) holds for m = J + 1. The proof ends.

2

4. First order decoupled semi-implicit scheme: error estimates In this section, we shall provide the error estimates of the numerical solution (un , pn , θ n ). Lemma 4.1. Under the assumptions of Theorem 3.1, (um , pm , θ m ) satisfies the following error estimate:



2



2

σ u (tm ) − um L 2 + γ θ(tm ) − θ m L 2 + τ Proof. Applying the integrate operator τ1









tn t n −1

m   1

 12  2 2 σ  A u (tn ) − un  2 + γ  A 2 θ(tn ) − θ n  2  κτ 2 . 1

2

L

n =1

L

(4.1)

·dt to (3.1) and (3.2), we obtain













dt u (tn ), v Ω + a1 u (tn ), v + b u (tn−1 ), u (tn−1 ), v¯ + b u (tn−1 ), u (tn ), v˜ + f k × u (tn ), v Ω

 z

−γ 



∇θ(tn−1 ) dξ, v −d

= F 1n , v



Ω



+ E n1 , v

 Ω

− ∇ · v, Ω

,



1



tn p (t ) dt

τ t n −1

  + ∇ · u (tn ), q Ω

Ω

(4.2)

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921











=



F 2n , φ Ω

+





 z



dt θ(tn ), φ Ω + a2 θ(tn ), φ + b u (tn−1 ), θ(tn ), φ + σ



915

∇ · u (tn−1 ) dξ, φ Ω

−d



E n2 , φ Ω ,

(4.3)

for all ( v , q, φ) ∈ X 1 × M 0 × X 2 , where





E n1 , v Ω

=

tn

1





(t − tn−1 )a1 ut (t ), v dt +

τ

  

 (t − tn ) b ut (t ), u (t ), v + b u (t ), ut (t ), v dt

τ

t n −1

+

tn

1

t n −1

tn

1

τ

  1 (t − tn−1 ) f k × ut (t ), v Ω dt − γ

(t − tn )

τ

t n −1

tn

 z

tn

∇θt (t ) dξ, v

dt Ω

−d

t n −1







b u (tn−1 ), ut (t ), v˜ dt ,

+

(4.4)

t n −1



E n2 ,

η

 Ω

=

1

tn





(t − tn−1 )a2 θt (t ), η dt +

τ t n −1

σ + τ

τ

(t − tn )

  

 (t − tn−1 ) b ut (t ), θ(t ), η + b u (t ), θt (t ), η dt

t n −1

 z

tn

tn

1



tn

∇ · ut (t ) dξ, η

dt − Ω

−d

t n −1





b ut (t ), θ(tn ), η dt .

(4.5)

t n −1

By some simple computations, we deduce







tn tn  12  12      A ut (t )22 dt + c τ 2  A θt (t )22 dt , 1 2 L L



2 2 τ  E n1  X   c τ 2 1 + max  A 1 u (t )L 2 0t  T

1



2

τ  E n2  X 

2

t n −1

(4.6)

t n −1

tn    12 2   2   A θt (t )22 dt   c τ 1 + max A 1 u (t ) L 2 2 L 0t  T

+ cτ

2



t n −1

 2  1 + max  A 2 θ(t ) L 2 0t  T

tn

 12   A ut (t )22 dt , 1

(4.7)

L

t n −1

and

   2 2  τ σ (tn ) E n1 L 2  c τ 2 1 + max  A 1 u (t )L 2 0t  T

tn + cτ 2

tn



2

σ (t ) A 1 ut (t )L 2 dt

t n −1









2 4 σ (t ) A 2 θt (t )L 2 dt + c τ 2 H (tn − t 1 ) max  A 1 u (t ) L 2 , 0t t 1

t n −1

    2  τ σ (tn ) E n2 L 2  c τ 2 1 + max  A 1 u (t )L 2 0t  T

tn



(4.8)

2

σ (t ) A 2 θt (t )L 2 dt

t n −1

tn   2   2 + c τ 2 1 + max  A 2 θ(t ) L 2 σ (t ) A 1 ut (t )L 2 dt 0t  T

t n −1

 2  2 + c τ H (tn − t 1 ) max  A 1 u (t ) L 2  A 2 θ(t ) L 2 , 2

0t t 1

where H (t ) = 0 if t < 0 and H (t ) = 1 if t  0.

(4.9)

916

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

Now, setting (en , r n , εn ) = (u (tn ) − un , 1t









tn tn−1



p (t ) dt − pn , θ(tn ) − θ n ), we deduce from (4.2)–(4.3) and (3.1)–(3.2) that













dt en , v Ω + a1 en , v + b en−1 , u (tn−1 ), v¯ + b un−1 , en−1 , v¯ + b en−1 , u (tn ), v˜ + b un−1 , en , v˜



+ f k × en , v

 z

 Ω

−γ



∇ε

n −1



n

dt ε , η Ω + a 2





n

      − ∇ · v , rn Ω + ∇ · en , q Ω = E n1 , v Ω ,

dξ, v

(4.10)

Ω

−d







ε ,η + b e

n −1





, θ(tn ), η + b u

n −1

 z



n

,ε ,η + σ

 ∇ ·e

n −1

  = E n2 , η Ω ,

dξ, η

(4.11)

Ω

−d

together with the initial condition e = 0 and ε = 0. Taking ( v , q) = σ (en , r n ) in (4.10) and η = γ εn in (4.11), adding these two relations and using (2.10)–(2.12), we find 0

0

2 γ  n 2     σ  n 2  n−1 2  n ε  2 − εn−1 22 + εn − εn−1 22 e L2 − e + e − e n −1  L 2 + L2 L L L 2τ

2τ

 1 2  1 2     + σ  A 12 en L 2 + γ  A 22 εn  L 2 + σ b en−1 , u (tn−1 ), e¯ n + σ b en−1 , u (tn ), e˜ n     + σ b un−1 , e¯ n , en − en−1 + γ b en−1 , θ(tn ), εn  z   z  n n n −1 n n n −1 + σγ ∇ · e dξ, ε − ε − σγ ∇ ε dξ, e − e

=σ



−d



E n1 , en Ω

+γ



ε

Due to the following estimates:

 

Ω



Ω

−d

E n2 , n Ω .



(4.12)



 

 



σ b en−1 , u (tn−1 ), e¯ n   σ en−1 L 2 ∇ e¯ n l2 ∇ u (tn−1 ) L ∞  2  n−1 2 σ  12 n 2     

16

A 1 e¯

L2

+ c A 1 u (tn−1 )

L2

e

L2

,

  2 2 2  4   σ  1 2  σ 1 σ b en−1 , u (tn ), e˜ n    A 12 e˜ n L 2 +  A 12 en−1 L 2 + c  A 1 u (tn )L 2 +  A 1 u (tn )L 2 en−1 L 2 , 16

 





16





 



σ b un−1 , e¯ n , en − en−1   σ un−1 L ∞ ∇ e¯ n L 2 en − en−1 L 2  2  2 σ  n 2  ν1 ∇ e¯ L 2 + 4σ ν1−1 un−1 L ∞ en − en−1 L 2 , 16

   γ b en−1 , θ(tn ), εn 

   z  z        n −1      n        n − 1 n − 1 dξ  ∇∂z θ(tn ) L 2 ε L 6 + c  e dξ  ∂z θ(tn ) L 6 ∇ εn  L 2  c e  L 2 ∇θ(tn ) L 3 ε  L 6 + c  e 3 3   L

−d

L

−d

2 2  4   γ  12 n 2 σ  12 n−1 2  A 2 ε L2 + A1 e + c  A 2 θ(tn ) L 2 +  A 2 θ(tn ) L 2 en−1  L 2 , L2 16

16

and

  z    n n n −1 σγ ∇ · e dξ, ε − ε  −d

Ω

   z      n n n −1 ∇ ε dξ, e − e  + σγ   −d

Ω

    

2 2  2  2 σ  γ   ν1 ∇ en  L 2 + ν2 ∇ εn L 2 + 2σ γ 2 d2 ν1−1 εn − εn−1 L 2 + 2γ σ 2 d2 ν2−1 en − en−1  L 2 , 8

8

   2  2     γ  1 2 σ  1 2 σ  E n1 , en Ω  + γ  E n2 , εn Ω    A 22 εn L 2 +  A 12 en L 2 + c  E n1  X  +  E n2  X  , 16

16

1

2

we deduce from (4.12) and the stability condition (3.5) that

 





 















2 2 2 2 2 2 σ en L 2 − en−1 L 2 + γ εn L 2 − εn−1 L 2 + σ 1 − 4σ γ d2 ν2−1 τ − 4ν1−1 un−1 L ∞ τ en − en−1 L 2

2 2  1 2   5  1 2 1  1 + γ 1 − 4σ γ d2 ν1−1 τ εn − εn−1  L 2 + σ  A 12 en  L 2 τ − σ  A 12 en−1  L 2 τ + γ  A 22 εn  L 2 τ 4 4  n−1 2   n 2   n −1  2

 n 2          dn−1 σ e E  + E  , 2 +γ ε 2 τ + c t L

L

1 X 1

2 X 2

(4.13)

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

917

where







 







 







 

2 4 2 4 2 4 dn−1 = c  A 2 θ(tn ) L 2 +  A 2 θ(tn ) L 2 + c  A 1 u (tn ) L 2 +  A 1 u (tn ) L 2 + c  A 1 u (tn−1 ) L 2 +  A 1 u (tn−1 ) L 2 .

Summing (4.13) from n = 1 to m and noting e 0 = 0 and



2



2

σ em L 2 + γ εm L 2 + τ τ

m −1 

dn

n =0

ε0 = 0, we deduce

m   1 2

 12 n 2 σ  A e  2 + γ  A 2 εn  2 1

n =1

2

L

L

m     2    n 2

 n 2  E   +  E n 2  . σ e L 2 + γ εn L 2 + c τ 1 X 2 X 1

n =1

(4.14)

2

2

Applying Lemma 2.1 to (4.14) and using (3.7)–(3.8), we have completed the proof of Lemma 4.1. Theorem 4.1. Under the assumption of Theorem 3.1, (um , pm , θ m ) satisfies the following error estimate:



1







1





2 2 σ σ (tm ) A 12 u (tm ) − um L 2 + γ σ (tm ) A 22 θ(tm ) − θ m L 2

+τ

m  n =1

+τ

m  n =1













2 2 σ (tn ) σ  A 1 u (tn ) − un L 2 + γ  A 2 θ(tn ) − θ n L 2

 

 





2 2 σ (tn ) σ dt u (tn ) − un L 2 + γ dt θ(tn ) − θ n L 2  κτ 2 .

Proof. Taking ( v , q) = σ ( A 1 en + dt en , 0) in (4.10) and

(4.15)

η = γ A 2 εn + γ dt εn in(4.11), adding these two relations, we find

 2  2 2 σ  12 n 2  12 n−1 2  12  n A 1 e L2 − A 1 e A 1 e − en−1  L 2 + σ dt en  L 2 + γ dt εn  L 2 2 + L τ  2  2 2 γ  12 n 2  12 n−1 2  12  n + A 2 ε L2 − A 2 ε A 2 ε − ε n −1  L 2 + σ  A 1 e n  L 2 + γ  A 2 ε n  L 2 2 + L τ     + σ b en−1 , u (tn−1 ) − u (tn ), A 1 e¯ n + dt e¯ n + σ b un−1 , en−1 − en , A 1 e¯ n + dt e¯ n

    + σ b en−1 , u (tn ), A 1 en + dt en + σ b un−1 , en , A 1 en + dt en     + γ b en−1 , θ(tn ), A 2 εn + dt εn + γ b un−1 , εn , A 2 εn + dt εn  z   z    n n n n −1 n n n −1 n n  ∇ε dξ, A 1 e + dt e + σγ ∇ ·e dξ, A 2 ε + dt ε + σ f k × e , A 1 e + dt e Ω − σ γ

=σ



E n1 ,

n

n



A 1 e + dt e Ω + γ



−d

E n2 ,

n

n

Ω



Ω

−d

A 2 ε + dt ε Ω .

(4.16)

Due to (2.8)–(2.10), we have the following estimates:

 



σ b en−1 , u (tn−1 ), A 1 e¯ n + dt e¯ n 

       en−1 L 6 ∇ u (tn−1 ) L 3  A 1 e¯ n + dt e¯ n  L 2    1   1  1  + c ∇ en−1  L 2  en−1  L22 ∂z u (tn−1 ) L22 ∇∂z u (tn−1 ) L22  A 1 e¯ n + dt e¯ n L 2 2  1 2  2  1  σ  n 2 σ  n−1 2 σ  n 2  A 1 e¯ L 2 + A1e dt e¯ L 2 + c 1 +  A 12 u (tn−1 ) L 2  A 1 u (tn−1 ) L 2  A 12 en−1  L 2 , 2 + L 16 16 16    σ b un−1 , en−1 , A 1 e¯ n + dt e¯ n 

         1  1  1    σ un−1  L ∞ ∇ en−1  L 2  A 1 e¯ n + dt e¯ n  L 2 + c ∇ un−1  L 2  un−1  L22 ∂z en−1  L22 ∇∂z en−1  L22  A 1 e¯ n + dt e¯ n  L 2 2  2  1 2  1  σ  n 2 σ  n−1 2 σ  n 2  A 1 e¯ L 2 + A1e dt e¯ L 2 + c 1 +  A 12 un−1  L 2  A 1 un−1  L 2  A 12 en−1  L 2 , 2 + L 16 16 16    σ b en−1 , u (tn ), A 1 e˜ n + dt e˜ n    1  1        1     en−1 L 6 ∇ u (tn ) L 3  A 1 e˜ n + dt e˜ n  L 2 + c ∇ en−1  L 2  en−1  L22 ∂z u (tn ) L22 ∇∂z u (tn ) L22  A 1 e˜ n + dt e˜ n  L 2   12     1  σ  n 2 σ  n−1 2 σ  n 2  A u (tn )22  A 1 u (tn )22  A 2 en−1 22 ,  A 1 e˜ A1e dt e˜ 2 + 2 + 2 +c 1+ 16

L

16

L

16

L

1

L

L

1

L

918

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

 



σ b un−1 , en , A 1 e˜ n + dt e˜ n 

         1  1  1    σ un−1  L ∞ ∇ en  L 2  A 1 e˜ n + dt e˜ n  L 2 + c ∇ un−1  L 2  un−1 L22 ∂z en  L22 ∇∂z en  L22  A 1 e˜ n + dt e˜ n  L 2 2  2  1 2  1  σ  n 2 σ  n 2 σ  n 2  A 1 e˜ L 2 + A 1e L2 + dt e˜ L 2 + c 1 +  A 12 un−1  L 2  A 1 un−1  L 2  A 12 en  L 2 , 16 16 16    γ b en−1 , θ(tn ), A 2 εn + dt εn         1     1  1   γ en−1  L 6 ∇θ(tn ) L 3  A 2 εn + dt εn  L 2 + c ∇ en−1  L 2  en−1  L22 ∂z θ(tn ) L22 ∇∂z θ(tn ) L22  A 2 εn + dt εn L 2 2  1 2  2  1  γ  n 2 σ  n−1 2 γ  n 2  A 2 ε L2 + A1e dt ε L 2 + c 1 +  A 22 θ(tn ) L 2  A 2 θ(tn ) L 2  A 12 en−1  L 2 , 2 + L 16 16 16    γ b un−1 , εn , A 2 εn + dt εn               γ un−1  ∞ ∇ εn  2  A 2 εn + dt εn  2 + c ∇ un−1  6 ∂z εn  3  A 2 εn + dt e˜ n  2  

γ 

16

L

L



2 A 2 n L2

ε

+

γ 



2 dt n  L 2

16

σ  f k × en , A 1 en + dt en

ε

L

L

L

L

2  2  1 2  + c 1 +  A 1 un−1  L 2  A 1 un−1 L 2  A 22 εn L 2 , 

    1    σ    A 1 en 22 + σ dt en 22 + c  A 2 en 22 , 1 L L L Ω 16

  z    n −1 n n σγ ∇ε dξ, A 1 e + dt e  −d

Ω

   z  n −1 n n γ ∇ ·e dξ, A 2 ε + dt ε  −d

Ω

16

  σ      1    A 1 en 22 + σ dt en 22 + c  A 2 εn−1 22 ,  2 L L L  16 16   γ      1    A 2 εn 22 + γ dt εn 22 + c  A 2 en−1 22 ,  1 L L L  16 16

  n     σ E , A 1 en + dt en + γ E n , A 2 εn + dt εn  1 2 Ω Ω  2    2 γ  n 2 σ  n 2 σ  n 2 σ  n 2 A 2 ε L2 + A 1e L2 + dt e L 2 + dt ε L 2 + c σ  E n1  L 2 + γ  E n2  L 2 .  16

16

16

16

Combining (4.16) with these inequalities and using Theorem 1.1 and Theorem 3.1, we obtain





1





1



1







1









2 2 2 2 2 2 σ  A 12 en L 2 −  A 12 en−1 L 2 + γ  A 22 εn L 2 −  A 22 εn−1 L 2 + σ dt en L 2 + γ dt εn L 2

2  1 2 5  1 2 1  1 + σ  A 12 en  L 2 τ − σ  A 12 en−1  L 2 τ + γ  A 22 εn  L 2 τ 4

4

2  1 2  1 2   2   1

 2  τ C σ  A 12 en−1  L 2 + σ  A 12 en  L 2 + γ  A 22 εn  L 2 + c τ σ  E n1  L 2 + γ  E n2  L 2 .

(4.17)

σ (tn ), we obtain  1 2 2  1 2

 1 2

 1 σ (tn ) σ  A 12 en L 2 + γ  A 22 εn L 2 − σ (tn−1 ) σ  A 12 en−1 L 2 + γ  A 22 εn−1 L 2 2  1 2  1 5 1 + σ σ (tn ) A 12 en  L 2 τ − σ σ (tn−1 ) A 12 en−1 L 2 τ

Multiplying (4.17) by

4

2

2  2  1 2

 + γ σ (tn ) A 22 εn  τ + σ (tn ) σ dt en  L 2 + γ dt εn  L 2 τ 2  1 2  2  1 2  1 2

 1

 2  C τ σ  A 12 en−1  L 2 + γ  A 22 εn−1  L 2 +  A 12 en  L 2 +  A 22 θ n  L 2 + c τ σ (tn ) σ  E n1  L 2 + γ  E n2  L 2 . Summing (4.18) from n = 1 to m and using Lemma 4.1 and (4.8)–(4.9), we have completed the proof of Theorem 4.1.

(4.18)

2

Theorem 4.2. Under the assumption of Theorem 3.1, (um , pm , θ m ) satisfies the following error estimate:



2

σ 2 (tm ) p (tm ) − pm L 2 (ω)  κτ 2 .

(4.19)

Proof. It follows (4.10) that















dt e¯ n , v¯ Ω + ν1 ∇ e¯ n , ∇ v¯ Ω + b en−1 , u (tn−1 ), v¯ + b un−1 , en−1 , v¯



+ f k × e¯ n , v

 Ω

 z

−γ



∇ε −d

n −1



      − ∇ · v¯ , rn Ω + ∇ · e¯ n , q Ω = E¯ n1 , v¯ Ω ,

dξ, v¯ Ω

(4.20)

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

919

for all ( v¯ , q) ∈ X 0 × M 0 , where



E¯ n1 , v¯



tn

ν1 = Ω τ

  1 (t − tn−1 ) ∇ u¯ t (t ), v¯ Ω dt +

  

 (t − tn ) b ut (t ), u (t ), v¯ + b u (t ), ut (t ), v¯ dt

τ

t n −1

+

tn t n −1

tn

1

τ

 z

tn

  1 (t − tn−1 ) f k × u¯ t (t ), v¯ Ω dt − γ

t n −1

∇θt (t ) dξ, v¯

(t − tn )

τ

 dt .

−d

t n −1

(4.21)

Ω

It follows from (4.20)–(4.21) that















dtt e¯ n , v¯ Ω + ν1 ∇ dt e¯ n , ∇ v¯ Ω + b dt en−1 , u (tn−1 ), v¯ + b en−2 , dt u (tn−1 ), v¯



      + b dt un−1 , en−1 , v¯ + b un−2 , dt en−1 , v¯ + f k × dt e¯ n , v¯ Ω  z      n −1 −γ ∇ dt ε dξ, v¯ − ∇ · v¯ , dt rn Ω + ∇ · dt e¯ n , q Ω Ω

−d



= dt E¯ n1 , v¯

 Ω

(4.22)

,

for all ( v¯ , q) ∈ X 0 × M 0 , where





dt E¯ n1 , v¯ Ω

=

ν1

tn 



t − t n −1

1−

2

2 



∇ u¯ tt (t ), v¯

τ



dt + Ω

2

τ



∇ u¯ tt (t ), v¯

 Ω

dt

t n −2

tn 

1

t − t n −2

2

t n −1

−

tn−1 

ν1

t − tn

2











b utt (t ), u (t ), v¯ + 2b ut (t ), ut (t ), v¯ + b u (t ), utt (t ), v¯

τ

2



dt

t n −1

−

tn−1 

1

 1−

2

2 



t − t n −1









b utt (t ), u (t ), v¯ + 2b ut (t ), ut (t ), v¯ + b u (t ), utt (t ), v¯

τ



dt

t n −2

+

tn 

1

 1−

2

t − t n −1

+

γ



t − tn

+

γ

1−

2

t − t n −1

t − t n −2

τ

2





f k × u¯ tt (t ), v¯ Ω dt

dt Ω

−d



tn  t n −2

 ∇θtt (t ) dξ, v¯

t n −1 tn−1 

 1 f k × u¯ tt (t ), v¯ Ω dt + 2

2   z

τ

2



τ

t n −1

tn

2 

2   z

 ∇θtt (t ) dξ, v¯

τ

−d

t n −2

dt .

(4.23)

Ω

Thus, we deduce from Theorem 1.1 and (4.21) and (4.23) that



2

σ 2 (tn ) E¯ n1 L 2  κτ 2 ,

τ

N  n =1



2

σ 2 (tn )dt E¯ n1  X   κτ 2 .

(4.24)

0

Now, taking ( v¯ , q) = (dt e¯ n , dt r n ) in (4.22) and using (2.11)–(2.13), we have

        1  n 2 dt e¯  2 − dt e¯ n−1 22 + dt e¯ n − dt e¯ n−1 22 + ν1 ∇ dt e¯ n 22 + b dt en−1 , u (tn−1 ), dt e¯ n L L L L 2τ       + b en−2 , dt u (tn−1 ), dt e¯ n + b dt un−1 , en−1 , dt e¯ n + b un−2 , dt en−1 , dt e¯ n  z    n −1 n ∇ dt ε dξ, dt e¯ = dt E¯ n1 , dt e¯ n Ω . −γ −d

Ω

(4.25)

920

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

By some simple calculations, we have

  n −1        b dt e , u (tn−1 ), dt e¯ n   dt en−1  2 ∇ dt e¯ n  2 u (tn−1 ) 2 L L L    2  n −1  2 ν1  n 2   ¯  ∇ dt e L 2 + c A 1 u (tn−1 ) L 2 dt e  L 2 , 16        n −2   b e , dt u (tn−1 ), dt e¯ n   en−2  6 ∇ dt e¯ n  2 dt u (tn−1 ) 3 L L L   12 2  12 n−2 2 ν1  n 2    ∇ dt e¯ L 2 + c A 1 dt u (tn−1 ) L 2  A 1 e  L 2 , 16

   1   1    n −1 n −1  b dt u , e , dt e¯ n   ν1 ∇ dt e¯ n 22 + c  A 2 dt un−1 22  A 2 en−1 22 , 1 1 L L L 16

        n −2  b u , dt en−1 , dt e¯ n   ν1 ∇ dt e¯ n 22 + c  A 1 un−1 22 dt en−1 22 , L L L    z  n −1 n γ ∇ dt ε dξ, dt e¯  −d

16

Ω

  ν  2  2 1  ∇ dt e¯ n  L 2 + c dt εn−1  L 2 ,   16

 n        dt E¯ , dt e¯ n   ν1 ∇ dt e¯ n 22 + c dt E¯ n 2  . 1 1 X L Ω 16

0

Combining these inequalities with (4.25) yields

 n 2     dt e¯  2 − dt e¯ n−1 22 + ν1 ∇ dt e¯ n 22 τ L L L 2  n−1 2  1 2 2  2  1  n − 1  c  A 1 u (tn−1 ) L 2 +  A 1 u L 2 dt e  L 2 τ + c  A 12 dt u (tn−1 ) L 2  A 12 en−2  L 2 τ  1 2  1 2  2 2  + c  A 2 dt un−1  2  A 2 en−1  2 τ + c dt εn−1  2 τ + dt E¯ n   τ .

(4.26)

σ 2 (tn ) and noting σ 2 (tn )  σ 2 (tn−1 ) + 3τ σ (tn−1 ), we obtain 2 2 2    σ 2 (tn )dt e¯ n L 2 − σ 2 (tn−1 )dt e¯ n−1 L 2 + ν1 σ 2 (tn )∇ dt e¯ n L 2 τ 2   2  2   2    c σ (tn−1 ) 1 +  A 1 u (tn−1 ) L 2 +  A 1 un−1  L 2 σ dt en−1  L 2 + γ dt εn−1  L 2 τ 2 2  1 2  1 2  1  1 + c σ (tn−2 ) A 12 dt u (tn−1 ) L 2  A 12 en−2  L 2 τ + c σ (tn−1 ) A 12 dt un−1  L 2  A 12 en−1  L 2 τ 2  + c σ 2 (tn )dt E¯ n1  X  τ .

(4.27)

1

1

L

L

L

1 X 0

Multiplying (4.26) by

0

Summing (4.27) from n = 2 to n = m and using Theorem 4.1, Theorem 3.1, we obtain

2



σ 2 (tm )dt e¯ m L 2  κτ 2 .

(4.28)

Now, by (4.20) and the inf-sup condition of the 2D Stokes equations:

q L 2 (ω)  c sup

v¯ ∈ X 0

(∇ · v¯ , q)Ω ,

∇ v¯ L 2 (ω)

∀q ∈ M 0 ,

(4.29)

we deduce

 n r 

L 2 (ω )

  1      1   1    c dt e¯ n  L 2 + c ∇ e¯ n  L 2 + c  A 12 en−1 L 2  A 12 u (tn−1 ) L 2 +  A 12 un−1 L 2        + c e¯ n  L 2 + εn−1  L 2 + c  E¯ n1  L 2 ,

or

 









 



 





2 2 2 2 2 2 σ 2 (tn )rn L 2 (ω)  c σ 2 (tn )dt e¯ n L 2 + c σ (tn )∇ e¯ n L 2 + c e¯ n L 2 + εn−1 L 2 + c σ 2 (tn ) E¯ n1 L 2

 1 2  1 2  1 2  + c σ 2 (tn−1 ) A 12 en−1  L 2  A 12 u (tn−1 ) L 2 +  A 12 un−1 L 2 .

(4.30)

Using (4.24), (4.28), Lemma 4.1, Theorem 4.1, Theorem 1.1 and Theorem 3.1 in (4.30) with n = m, we obtain



2

σ 2 (tm )rm L 2 (ω)  κτ 2 .

(4.31)

Y. He / J. Math. Anal. Appl. 412 (2014) 895–921

921

Noting

   p (tm ) − pm 22 L

 2 tm   1    2 p (tm ) − p (t ) dt  (ω )  2 τ

L (ω )

t m −1

 tm 2 1     2 (t − tm−1 ) p (t ) dt  τ 2  2τ −1

 2 + 2rm  L 2 (ω)

L (ω )

t m −1

tm

 2 + 2rm  L 2 (ω)

2   2 (t − tm−1 )2  pt (t ) L 2 (ω) dt + 2rm  L 2 (ω) ,

t m −1

we have



2

σ 2 (tm ) p (tm ) − pm L 2 (ω)  2τ −1

tm



2



2

σ 2 (tm )(t − tm−1 )2  pt (t )L 2 (ω) dt + 2σ 2 (tm )rm L 2 (ω)

t m −1

tm  8τ



2



2

σ 2 (t ) pt (t )L 2 (ω) dt + 2σ 2 (tm )rm L 2 (ω) .

(4.32)

t m −1

Using (4.31) and (1.12) of Theorem 1.1 in (4.32), we obtain (4.19). The proof ends.

2

Acknowledgments The author would like to thank the Editor and the referees for their helpful comments and suggestions which helped to improve the presentation of the present paper. References [1] R.A. Adams, Sobolev Space, Academic Press, New York, 1975. [2] Chongsheng Cao, Edriss S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure Appl. Math. 56 (2003) 198–233. [3] Chongsheng Cao, Edriss S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. 166 (2007) 245–267. [4] P. Constantin, C. Foias, Navier–Stokes Equations, The University of Chicago Press, Chicago, IL, 1988. [5] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vols. I & II, Springer-Verlag, New York, 1994. [6] Yinnian He, First order decoupled method of the 3D primitive equations of the ocean II: finite element spatial discretization, IMA J. Numer. Anal. (2013), submitted and revised. [7] Yinnian He, Jianhua Wu, Global H 2 -regularity of the 3D primitive equations of the ocean, submitted for publication. [8] J.G. Heywood, R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem, I, Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982) 275–311. [9] C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear Anal. 61 (2005) 425–460. [10] C. Hu, Finite dimensional behaviors of the primitive equations under small depth assumption, Numer. Funct. Anal. Optim. 28 (2007) 853–882. [11] Ning Ju, The global attractor for the solutions to the 3D viscous primitive equations, Discrete Contin. Dyn. Syst. 17 (2007) 159–179. [12] I. Kukavica, M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity 20 (2007) 2739–2753. [13] O.A. Ladyženskaja, The Boundary Value Problem of Mathematical Physics, Springer-Verlag, New York, 1985. [14] J.L. Lions, R. Temam, S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity 5 (1992) 237–288. [15] J.L. Lions, R. Temam, S. Wang, On the equations of large-scale ocean, Nonlinearity 5 (1992) 1007–1053. [16] J.L. Lions, R. Temam, S. Wang, Mathematical theory for the coupled atmospere–ocean models, J. Math. Pures Appl. 74 (1995) 105–163. [17] T.T. Medjo, R. Temam, The two-grid finite difference method for the primitive equations of the ocean, Nonlinear Anal. 69 (2008) 1034–1056. [18] J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal. 38 (1990) 201–229. [19] R. Temam, Some mathematical aspects of geophysical fluid dynamics equations, Milan J. Math. 71 (2003) 175–198. [20] R. Temam, M. Ziane, Some mathematical problems in geophysical fluid dynamics, in: Handbook of Mathematical Fluid Dynamics, vol. 3, 2004. [21] M. Ziane, Regularity results for Stokes type systems related to climatology, Appl. Math. Lett. 8 (1995) 53–58. [22] M. Ziane, Regularity results for the stationary primitive equations of atmosphere and the ocean, Nonlinear Anal. 28 (1997) 289–313.