Decay estimates for steady magnetohydrodynamics equations in a semi-infinite cylinder

DECAY ESTIMATES FOR STEADY MAGNETOHYDRODYNAMICS EQUATIONS

IN A SEMI-INFINITE CYLINDER

Jong Chul Song

Abstract. This paper establishes exponential decay estimates for the solution of a stationary magnetohydrodynamics equations in a semi- innite pipe ow when homogeneous lateral surface boundary condi- tions are applied.

1. Introduction

The model equations of magnetohydrodynamics are discussed in the book of Ames and Straughan [1, p.154]. In this paper we investigate ex- ponential decay results for a solution of the model equations of magneto- hydrodynamics. Under appropriate homogeneous boundary conditions on the lateral surface of the cylinder we establish Saint-Venant type decay of solutions as the distance from the nite end of the cylinder tends to innity.

Several papers in the literature have dealt with spatial exponential decay in the Navier-Stokes equations (see Ames and Payne [3], Ames et al. [2], and Horgan and Wheeler [10]). For a survey of Saint-Venant type spatial decay results see Horgan and Knowles [8] and Horgan [6, 7]. More recent work on spatial decay in porous medium problems has been carried out by Payne and Song [12, 13], and Chadam and Qin [5].

We consider that a stationary magnetohydrodynamics ow occupies the interior of a semi-innite cylindrical pipe of arbitrary cross section and generators parallel to the x

₃

axis. Denoting the cross section of the pipe by D and the interior half cylinder by R, we introduce the notation:

R

z

= {(x

1

, x

2

, x

3

)|(x

1

, x

2

) ∈ D, x

3

> z ≥ 0}, D

z

= {(x

1

, x

2

, x

3

) |(x

1

, x

2

) ∈ D, x

3

= z },

Received June 7, 2000.

2000 Mathematics Subject Classication: 35B40, 35Q35, 76D99, 76E25.

Key words and phrases: magnetohydrodynamics equations, decay estimates, a priori bounds.

This work was supported by Hanyang University, Korea, made in the program year of 2000.

where z is a running variable along the x

₃

axis. Clearly, R

₀

≡ R and D

0

≡ D.

Throughout this paper we adopt the summation convention of summing in any term over a repeated index, Latin subscripts ranging from 1 to 3 and Greek subscripts from 1 to 2. A comma is used to denote partial dierentiation.

Let v

i

, h

i

, and p denote the velocity, magnetic eld, and pressure in the semi-innite pipe, respectively. The governing equations for the stationary magnetohydrodynamics are

(1.1)

v

_j

v

_i,j

− h

j

h

_i,j

= −p

,i

+ v

_i

in R, v

j

h

i,j

− h

j

v

i,j

= h

i

in R,

v

_j,j

= 0, h

_j,j

= 0 in R.

The symbol denotes the Laplace operator. The boundary conditions are given by

(1.2)

v

i

= 0, h

i

= 0 on ∂R\D

0

, v

i

= f

i

(x

1

, x

2

) on D

0

, h

_i

= h

_i

(x

₁

, x

₂

) on D

₀

.

The prescribed entrance proles f

_i

and h

_i

are assumed to be zero on ∂D

₀

. As x

3

→ ∞, v

i

is assumed to tend to the fully developed velocity eld ˆvδ

3i

, i.e., (0, 0, ˆ v(x

₁

, x

₂

)), corresponding to the net ow

(1.3)

∫

D0

f

3

dA ≡ F,

with the magnetic eld corresponding to the fully developed velocity eld vanishing. Thus ˆ v(x

₁

, x

₂

) may be characterized as the solution of the boundary value problem

(1.4)

ˆ

v

,αα

= ˆ p

,3

in D

z

, ˆ

v = 0 on ∂D

_z

,

∫

Dz

ˆ

vdA = F.

The gradient of the pressure ˆ p in (1.4)

₁

has the form ˆ p

_,i

= −P δ

3i

where P is a positive constant and δ

ij

is the Kronecker delta symbol.

In order to render our appropriate inequalities more explicit, we point out that the solution of (1.4) is nonnegative (by the maximum principle) and is just a multiple of the well studied torsion problem (where P is re- placed by 2), and F is just an appropriate multiple of the torsional rigidity.

Also bounds for the maximum value of the solution can be found in the

literature (see [3]).

We dene

(1.5) u

_i

= v

_i

− ˆvδ

3i

, q = p − ˆp.

Then the boundary value problem for u

i

and h

i

is

(1.6)

u

_i

= q

_,i

+ (u

_j

+ ˆ vδ

_3j

)u

_i,j

+ u

_α

v ˆ

_,α

δ

_3i

− h

j

h

_i,j

in R, h

_i

= (u

_j

+ ˆ vδ

_3j

)h

_i,j

− h

j

u

_i,j

− h

α

v ˆ

_,α

δ

_3i

in R, u

i,i

= 0, h

i,i

= 0 in R,

u

i

= 0, h

i

= 0 on ∂R \D

0

, u

_i

= f

_i

− ˆvδ

3i

on D

₀

, h

_i

= g

_i

on D

₀

,

(1.7) u

i

, h

i

, u

i,j

, h

i,j

, q = o(x

⁻¹₃

) uniformly in (x

1

, x

2

) as x

3

→ ∞.

In view of (1.3) and (1.4)

₃

and from the divergence theorem, it follows that

(1.8)

∫

Dz

u

₃

dA = 0 for all z ∈ [0, ∞).

Our goal in this paper is to derive an integro-dierential inequality for an energy integral E(z) dened in (3.1) of the form

(1.9) dE

dz + M

∫

_∞

z

E(ξ)dξ ≤ KE(z),

where for positive constants M and K are given in (3.27) and (3.28), re- spectively. We shall show that E decays to zero exponentially as z → ∞ provided (3.29) is met.

In the next section we record some auxiliary inequalities that we shall use. In section 3 we derive the dierential inequality for E, which integrates to yield exponential decay and in the nal section we establish a bound for E(0).

2. Auxiliary inequalities

We shall make frequent use of Schwarz’s inequality and the arithmetic mean-geometric mean inequality in our derivation of the rst-order integro- dierential inequality we seek. In addition, we need the following Babu ska- Aziz inequality (2.1) and inequalities (2.2), (2.4).

Lemma. Let R be a bounded three-dimensional region with Lipschitz

boundary and let χ be a bounded function in R of mean value zero. Then

there exists a vector function ω

_i

and a constant C depending only on the shape R such that

ω

i,i

= χ in R, ω

i

= 0 on ∂R, and

(2.1)

∫

R

ω

_i,j

ω

_i,j

dx ≤ C

∫

R

χ

²

dx.

A proof of this result is given in Ladyzhenskaya and Solonnikov [11] (see also Velte [15], and Horgan and Payne [9]).

We now record here the following Poincar e inequality. Let D be a plane domain with suciently smooth boundary ∂D, and let v be a suciently smooth function dened on the closure D of D. If v = 0 on ∂D, then

(2.2) λ

1

∫

D

v

²

dA ≤

∫

D

v

,α

v

,α

dA, where λ is the smallest positive eigenvalue of

w

_,αα

+ λw = 0 in D, w = 0 on ∂D.

Second, if ∂v/∂n == 0 on ∂D and ∫

D

v dA = 0, then

(2.3) µ

2

∫

D

v

²

dA ≤

∫

D

v

,α

v

,α

dA, where µ

₂

is the smallest positive eigenvalue of

w + µw = 0 in D, ∂w

∂n = 0 on ∂D,

∫

D

w dA = 0.

The inequality (2.2) has been well studied (see, e.g., Bandle [4]), whereas the inequality (2.1) was extensively used for energy decay estimates for Darcy and Brinkman ows investigated by Payne and Song [12].

Finally, we have the following Sobolev inequality [10, see Eq.(A.11)]

(2.4)

∫

Rz

v

⁴

dx ≤ 1

√ λ

1

(∫

Rz

v

_,j

v

_,j

dx )

2

, for suciently smooth functions v vanishing on ∂R

_z

\D

z

.

3. Decay estimates

In this section in order to derive an inequality which will imply expo- nential decay, we rst consider the energy integral

(3.1) E(z) =

∫

Rz

(u

i,j

u

i,j

+ h

i,j

h

i,j

)dx.

We integrate by parts and use boundary conditions for u

_i

and h

_i

to obtain (3.2) E(z) = −

∫

Dz

(u

i,3

u

i

+ h

i,3

h

i

)dA −

∫

Rz

(u

i

u

i

+ h

i

h

i

)dx.

From the denition of E(z), it follows that

(3.3) E

^′

(z) = −

∫

Dz

(u

_i,j

u

_i,j

+ h

_i,j

h

_i,j

)dA,

where the prime denotes dierentiation with respect to z. To establish the desired integro-dierential inequality (1.9), we rst form

(3.4) E

^′

(z) + γ

∫

_∞

z

E(ξ)dξ = −I

1

+ I

₂

, where we have used the dierential equations for u

i

and h

i

, (3.5) I

₁

=

∫

Dz

(u

_i,j

u

_i,j

+ h

_i,j

h

_i,j

)dA − γ 2

∫

Dz

(u

_i

u

_i

+ h

_i

h

_i

)dA,

I

2

= −γ

∫

_∞

z

∫

R_ξ

u

i

[q

,i

+ (u

j

+ ˆ vδ

3j

)u

i,j

+ u

α

ˆ v

,α

δ

3i

− h

j

h

i,j

] dxdξ

− γ

∫

_∞

z

∫

Rξ

h

_i

[(u

_j

+ ˆ vδ

_3j

)h

_i,j

− h

j

u

_i,j

− h

α

ˆ v

_,α

δ

_3i

] dxdξ, (3.6)

and γ is a positive parameter to be chosen. If we choose γ = 2λ

₁

, we drop the term I

1

which is nonnegative in view of (2.2).

In estimating the term I

₂

we integrate by parts and note that the term

∫

_∞

z

∫

Rξ

u

i,j

h

i

h

j

dxdξ cancels. Then we see I

₂

γ =

∫

Rz

u

₃

qdx + 1 2

∫

Rz

u

_i

u

_i

(u

₃

+ ˆ v)dx +

∫

_∞

z

∫

Rξ

ˆ

vu

_3,j

u

_j

dxdξ +

∫

Rz

ˆ

vu

²₃

dx −

∫

Rz

u

i

h

3

h

i

dx + 1 2

∫

Rz

(u

3

+ ˆ v)h

i

h

i

dx

−

∫

_∞

z

∫

Rξ

ˆ

vh

3,j

h

j

dxdξ −

∫

Rz

ˆ vh

²₃

dx

=

∑

8 n=1

J

n

. (3.7)

We have thus written I

₂

/γ as the sum of eight integrals J

_n

’s each of which

we must now bound in terms of E(z). Let us consider the sum of two

integrals J

₂

and J

₆

. Using Schwarz’s inequality repeatedly and inequalities

(2.2), (2.4) successively, we nd J

₂

+ J

₆

≤ 1 2

(∫

Rz

u

²₃

dx

)

_1/2

[{∫

Rz

(u

i

u

i

)

²

dx }

_1/2

+ {∫

Rz

(h

_i

h

_i

)

²

dx }

1/2

]

+ 1 2

ˆ v

_s

λ

₁

∫

Rz

(u

_i,j

u

_i,j

+ h

_i,j

h

_i,j

)dx

≤ λ

^−3/4₁

2

(∫

Rz

u

3,α

u

3,α

dx

)

_1/2

(∫

Rz

u

i,j

u

i,j

dx+

∫

Rz

h

i,j

h

i,j

dx )

+ 1 2

ˆ v

s

λ

1

∫

Rz

(u

i,j

u

i,j

+ h

i,j

h

i,j

)dx

≤ λ

^−3/4₁

2

√ E(0)E(z) + 1 2

ˆ v

s

λ

₁

E(z), (3.8)

where ˆ v

_s

= max

_Dz

v(x

₁

, x

₂

) and the last step following from the monotonic- ity of E(z). To make our inequalities more explicit, we mention here the bound for ˆ v

_s

obtained by Ames and Payne [3, see eq.(7.8)].

We apply Schwarz’s inequality and the inequality (2.2) to the sum of two integrals J

₃

and J

₇

to obtain

(3.9)

J

3

+ J

7

≤ √ v ˆ

s

λ

₁

{∫

_∞

z

∫

Rξ

(u

i,j

u

i,j

+ h

i,j

h

i,j

)dxdξ }

= v ˆ

_s

√ λ

₁

∫

_∞

z

E(ξ)dξ.

As for J

5

, an application of Schwarz’s inequality, inequalities (2.2), (2.4), and the arithmetic mean-geometric mean inequality yields the bound

J

5

≤ (∫

Rz

u

i

u

i

dx

)

_1/2

{∫

Rz

(h

i

h

i

)

²

dx

∫

Rz

h

⁴₃

dx }

_1/4

≤ 1 λ

^1/2₁

(∫

Rz

u

_i,α

u

_i,α

dx )

1/2

× 1 λ

^1/8₁

{∫

Rz

h

_i,j

h

_i,j

dx }

1/2

1 λ

^1/8₁

{∫

Rz

h

_3,j

h

_3,j

dx }

1/2

≤ λ

^−3/4₁

2

(∫

Rz

u

i,j

u

i,j

dx+

∫

Rz

h

i,j

h

i,j

dx )(∫

Rz

h

3,j

h

3,j

dx )

_1/2

≤ λ

^−3/4₁

2

√ E(0)E(z).

(3.10)

An application of the inequality (2.2) to the sum of two integrals J

₄

and J

8

leads to the bound

J

4

+ J

8

≤

∫

Rz

|ˆv|(u

²₃

+ h

²₃

)dx

≤ v ˆ

s

λ

1

∫

Rz

(u

3,α

u

3,α

+ h

3,α

h

3,α

)dx

≤ v ˆ

s

λ

₁

E(z).

(3.11)

Finally, to complete the derivation of the integro-dierential inequality (1.9), we need to eliminate the pressure term q in the J

₁

integral. This can be accomplished with the aid of (2.1) by using the arguments of Horgan and Payne [9], and Payne and Song [12]. Thus we rewrite the integral J

1

as (3.12)

∫

Rz

u

3

q dx =

∑

∞ n=0

∫

_z+(n+1)L

z+nL

∫

Dξ

u

3

q dAdξ,

for some positive constant L at our disposal. We introduce the further notation

(3.13) R

_zⁿ

= D × {z + nL < ξ < z + (n + 1)L}, so that we may write

(3.14)

∫

Rz

u

3

q dx =

∑

∞ n=0

∫

R_zⁿ

u

3

q dx.

We may proceed to bound each term on the right separately. In fact, we show how to bound an arbitrary term in terms of quantities which do not depend on z so that the same type of bound holds for each term. Recall that for each value of z( ≥ 0)

(3.15)

∫

Dz

u

₃

dA = 0.

This allows us to dene a vector eld ω

_i

satisfying

(3.16) ω

i,i

= u

3

in R

ⁿ_z

,

ω

i

= 0 on ∂R

ⁿ_z

.

Using the vector function ω

_i

, it follows that

∫

Rⁿ_z

u

3

qdx =

∫

Rⁿ_z

ω

i,i

qdx = −

∫

Rⁿ_z

ω

i

q

,i

dx

=

∫

Rⁿ_z

ω

_i

[ − u

i

+(u

_j

+ ˆ vδ

_3j

)u

_i,j

+u

_α

ˆ v

_,α

δ

_3i

−h

j

h

_i,j

]dx

=

∫

Rⁿ_z

ω

i,j

u

i,j

dx −

∫

Rⁿ_z

ω

i,j

(u

j

+ ˆ vδ

3j

)u

i

dx +

∫

Rⁿ_z

ω

₃

u

_α

v ˆ

_,α

dx +

∫

Rⁿ_z

ω

_i,j

h

_j

h

_i

dx

≤

∫

Rⁿ_z

ω

i,j

u

i,j

dx +

∫

Rⁿ_z

|ω

i,j

|(u

i

u

j

+ h

i

h

j

)dx

−

∫

Rⁿ_z

ˆ

vω

i,3

u

i

dx +

∫

Rⁿ_z

ω

3

u

α

ˆ v

,α

dx

= P

_n⁽¹⁾

+ P

_n⁽²⁾

+ P

_n⁽³⁾

+ P

_n⁽⁴⁾

. (3.17)

Now by means of Schwarz’s inequality and inequalities (2.1), (2.2), we have

(3.18)

P

_n⁽¹⁾

≤ (∫

Rⁿ_z

ω

i,j

ω

i,j

dx

)

1/2

(∫

Rⁿ_z

u

i,j

u

i,j

dx )

1/2

≤ (

C

∫

R_zⁿ

u

²₃

dx

)

_1/2

(∫

Rⁿ_z

u

i,j

u

i,j

dx )

_1/2

≤ (

C λ

₁

∫

Rⁿ_z

u

_3,α

u

_3,α

dx

)

1/2

(∫

Rⁿ_z

u

_i,j

u

_i,j

dx )

1/2

≤

√ C λ

₁

e

_n

(z),

where e

_n

(z) = ∫

Rⁿ_z

(u

_i,j

u

_i,j

+ h

_i,j

h

_i,j

)dx. Using Schwarz’s inequality repeat-

edly, inequalities (2.1), (2.4), and the monotonicity of e

_n

(z), for P

n⁽²⁾

we

obtain

(3.19)

P

_n⁽²⁾

≤ (∫

Rⁿ_z

ω

_i,j

ω

_i,j

dx )

1/2

×



 {∫

Rⁿ_z

(u

i

u

i

)

²

dx }

_1/2

+ {∫

Rⁿ_z

(h

i

h

i

)

²

dx }

_1/2





≤ 1 λ

^1/4₁

( C

∫

Rⁿ_z

u

²₃

dx )

1/2

× (∫

Rⁿ_z

u

i,j

u

i,j

dx +

∫

Rⁿ_z

h

i,j

h

i,j

dx )

≤ C

^1/2

λ

^3/4₁

√ e

_n

(0)e

_n

(z).

A bound for P

n⁽³⁾

is established through the use of Schwarz’s inequality and inequalities (2.1), (2.2). We have

(3.20)

P

_n⁽³⁾

≤ ˆv

s

(∫

Rⁿ_z

ω

_i,3

ω

_i,3

dx

)

1/2

(∫

Rⁿ_z

u

_i

u

_i

dx )

1/2

≤ ˆv

s

( C

∫

Rⁿ_z

u

²₃

dx )

1/2

(

1 λ

₁

∫

Rⁿ_z

u

i,α

u

i,α

dx )

1/2

≤ v ˆ

s

C

^1/2

λ

1

e

n

(z).

We turn now to the bound for P

n⁽⁴⁾

. On integrating by parts we use Schwarz’s inequality since u

α,α

= −u

3,3

and nd

(3.21)

P

_n⁽⁴⁾

= −

∫

Rⁿ_z

ˆ

vω

3,α

u

α

dx −

∫

Rⁿ_z

ˆ

vω

3

u

α,α

dx

≤ ˆv

s

(∫

Rⁿ_z

ω

3,α

ω

3,α

dx

)

1/2

(∫

Rⁿ_z

u

α

u

α

dx )

1/2

+ ˆ v

s

(∫

Rⁿ_z

ω

²₃

dx

)

_1/2

(∫

Rⁿ_z

u

²_3,3

dx )

.

^1/2

It then follows by applying inequalities (2.1) and (2.2) that

(3.22)

P

_n⁽⁴⁾

≤ ˆ v

_s

√ λ

1

(∫

Rⁿ_z

ω

_3,α

ω

_3,α

dx

)

1/2

(∫

Rⁿ_z

u

_α,β

u

_α,β

dx )

1/2

+ √ v ˆ

s

λ

₁

(∫

Rⁿ_z

ω

_3,α

ω

_3,α

dx

)

1/2

(∫

Rⁿ_z

u

²_3,3

dx )

1/2

≤ √ ˆ v

s

λ

1

(∫

Rⁿ_z

ω

i,j

ω

i,j

dx

)

1/2

(∫

R_zⁿ

u

α,β

u

α,β

dx )

1/2

+ v ˆ

_s

√ λ

1

(∫

Rⁿ_z

ω

_i,j

ω

_i,j

dx

)

_1/2

(∫

Rⁿ_z

u

²_3,3

dx )

_1/2

≤ ˆ v

s

√ C

√ λ

₁

(∫

Rⁿ_z

u

²₃

dx )

1/2

×



 (∫

Rⁿ_z

u

_α,β

u

_α,β

dx )

_1/2

+ (∫

R_zⁿ

u

²_3,3

dx )

_1/2



 .

Using the inequality (2.2), we have

(3.23)

P

_n⁽⁴⁾

≤ ˆ v

_s

√ C λ

1

(∫

Rⁿ_z

u

_3,α

u

_3,α

dx

)

_1/2

(∫

Rⁿ_z

u

_i,j

u

_i,j

dx )

_1/2

≤ ˆ v

_s

√ C λ

1

e

_n

(z).

Since C and λ

1

are independent of z and (3.24)

∑

∞ n=0

e

_n

(z) = E(z),

we may sum over n to bound J

1

in terms of E(z). Then we have (3.25) J

1

≤

{√ C λ

1

+ C

^1/2

λ

^3/4₁

√ E(0) + 2ˆ v

s

C

^1/2

λ

1

} E(z).

Substituting the bounds for the J

_n

’s into (3.7), inserting the bound for I

2

, and dropping the nonnegative term I

1

in (3.4), we obtain the desired integro-dierential inequality for the energy integral

(3.26) E

^′

(z) + M

∫

_∞

z

E(ξ)dξ ≤ KE(z),

where

(3.27) M = 2λ

₁

(

1 − √ v ˆ

s

λ

₁

)

,

(3.28) K = 2λ

₁

[

3 2

ˆ v

s

λ

₁

+

√ C

λ

₁

+ 2ˆ v

_s

C

^1/2

λ

₁

+ λ

^−3/4₁

(1 + √ C) √

E(0) ]

. To ensure decay, we require M > 0, or

(3.29) v ˆ

s

< √

λ

1

.

This condition yields a restriction on the ow. We shall subsequently re- quire somewhat stronger restrictions (4.26) in bounding the total energy E(0).

To investigate (3.26) we refer to Horgan and Wheeler [10, see eq.(4.15)].

It follows then that

(3.30) E(z) ≤ aE(0)e

^−bz

for all z ∈ [0, ∞), where

(3.31) a =

( K

²

+ 4M )

1/2

b , b = 1

2

{( K

²

+ 4M )

1/2

− K } .

To make (3.30) explicit we need a bound for E(0) in terms of data, which we derive in the next section.

4. A bound for E(0)

To bound E(0), we introduce auxiliary functions which are

(4.1)

ζ

i

= ˆ q

,i

in R, H

_i

= s

_,i

in R,

ζ

_i,i

= 0, H

_i,i

= 0 in R, ζ

i

= 0, H

i

= 0 on ∂R\D

0

, ζ

i

= f

i

− ˆvδ

3i

on D

0

, H

_i

= g

_i

on D

₀

,

where ˆ q and s are functions that are not prescribed a priori but are deter-

mined up to constants by the fact that all of the equations (4.1) are to be

satised.

We dene

(4.2)

X =

∫

R0

u

_i,j

u

_i,j

dx, Y =

∫

R0

h

_i,j

h

_i,j

dx, A =

∫

R0

ζ

_i,j

ζ

_i,j

dx, B =

∫

R0

H

_i,j

H

_i,j

dx.

From the triangle inequality, it follows that

(4.3)

E(0) = X + Y ≤ 2 {∫

R0

(u

_i

− ζ

i

)

_,j

(u

_i

− ζ

i

)

_,j

dx }

+ 2A + 2

{∫

R0

(h

i

− H

i

)

,j

(h

i

− H

i

)

,j

dx }

+ 2B.

First, we consider the leading term on the right side of (4.3). From the divergence theorem we see

∫

R0

(u

_i

− ζ

i

)

_,j

(u

_i

− ζ

i

)

_,j

dx = −

∫

R0

(u

_i

− ζ

i

)( u

_i

− ζ

i

)dx

= −

∫

R0

(u

i

− ζ

i

)[(q−ˆq)

,i

+(u

j

+ ˆ vδ

3j

)u

i,j

+u

α

v ˆ

,α

δ

3i

−h

j

h

i,j

]dx.

(4.4)

We integrate by parts in the rst and second term and obtain

−

∫

R0

(u

_i

− ζ

i

)[(q − ˆq)

,i

+ (u

_j

+ ˆ vδ

_3j

)u

_i,j

]dx

= 1 2

∫

D0

(f

₃

− ˆv)(f

i

− ˆvδ

3i

)(f

_i

− ˆvδ

3i

)dA + 1

2

∫

D0

ˆ

v(f

_i

− ˆvδ

3i

)(f

_i

− ˆvδ

3i

)dA

−

∫

R0

ζ

_i,j

u

_j

u

_i

dx +

∫

R0

ˆ

vζ

_i

u

_i,3

dx.

(4.5)

We bound the last two integrals by the appropriate integral inequalities

(4.6) −

∫

R0

ζ

i,j

u

j

u

i

dx ≤

√ A λ

^1/4₁

∫

R0

u

i,j

u

i,j

dx,

(4.7)

∫

R0

ˆ

vζ

i

u

i,3

dx ≤ ϵ

1

2 ˆ v

s

√ λ

1

∫

R0

u

i,j

u

i,j

dx + ϵ

⁻¹₁

2

ˆ v

s

√ λ

1

A.

An integration by parts in the third term of (4.4) results in

(4.8)

−

∫

R0

(u

₃

− ζ

3

)u

_α

v ˆ

_,α

dx =

∫

R0

ˆ

vu

_3,α

u

_α

dx +

∫

R0

ˆ

vu

₃

u

_α,α

dx

−

∫

R0

ˆ

vζ

_3,α

u

_α

dx −

∫

R0

ˆ

vζ

₃

u

_α,α

dx.

For the second integral on the right side of (4.8), integrating by parts since u

α,α

= −u

3,3

, we have

(4.9)

∫

R0

ˆ

vu

₃

u

_α,α

dx = −

∫

R0

ˆ

vu

₃

u

_3,3

dx

= 1 2

∫

D0

ˆ

v(f

3

− ˆv)

²

dA.

As for the remaining integrals on the right side of (4.8), applications of the appropriate inequalities enable us to conclude

(4.10)

∫

R0

ˆ

vu

_3,α

u

_α

dx ≤ √ v ˆ

s

λ

₁

∫

R0

u

_i,j

u

_i,j

dx,

(4.11) −

∫

R0

ˆ

vζ

_3,α

u

_α

dx ≤ ϵ

₂

2

ˆ v

_s

√ λ

1

∫

R0

u

_i,j

u

_i,j

dx + ϵ

⁻¹₂

2

ˆ v

_s

√ λ

1

A,

(4.12)

−

∫

R0

ˆ

vζ

3

u

α,α

dx ≤ ˆv

s

(∫

R0

ζ

₃²

dx

)

1/2

{∫

R0

(u

α,α

)

²

dx }

1/2

≤ √ v ˆ

s

λ

₁

(∫

R0

ζ

3,α

ζ

3,α

dx

∫

R0

u

i,j

u

i,j

dx )

1/2

≤ ϵ

₃

2

ˆ v

_s

√ λ

1

∫

R0

u

_i,j

u

_i,j

dx + ϵ

⁻¹₃

2

ˆ v

_s

√ λ

1

A.

Integrating by parts in the last term of (4.4), we apply Schwarz’s inequality and the inequality (2.4) to obtain

(4.13)

∫

R0

(u

i

− ζ

i

)h

j

h

i,j

dx = −

∫

R0

u

i,j

h

j

h

i

dx + +

∫

R0

ζ

i,j

h

j

h

i

dx

≤ −

∫

R0

u

i,j

h

j

h

i

dx + +

√ A λ

^1/4₁

∫

R0

h

i,j

h

i,j

dx.

Consider the third term on the right side of (4.3). On integrating by parts, we observe

∫

R0

(h

_i

− H

i

)

_,j

(h

_i

− H

i

)

_,j

dx = −

∫

R0

(h

_i

− H

i

)( h

_i

− H

i

)dx

= −

∫

R0

(h

_i

− H

i

)[(u

_j

+ ˆ vδ

_3j

)h

_i,j

− h

j

u

_i,j

− h

α

v ˆ

_,α

δ

_3i

− s

,i

]dx.

(4.14)

A further integration by parts in the leading term on the right side of (4.14) yields

(4.15) −

∫

R0

h

i,j

u

j

h

i

dx = 1 2

∫

D0

g

i

g

i

(f

3

− ˆv)dA,

(4.16) −

∫

R0

h

_i

ˆ vδ

_3j

h

_i,j

dx = 1 2

∫

D0

ˆ

vg

_i

g

_i

dA.

As for some of the remaining integral on the right side of (4.14), a straight- forward application of Schwarz’s inequality, the inequalities (2.2), (2.4) and the arithmetic mean-geometric mean inequality gives

(4.17)

∫

R0

H

i

u

j

h

i,j

dx ≤ B

^1/2

λ

^−1/4₁

2 (X + Y ),

(4.18)

∫

R0

(h

i

− H

i

)h

j

u

i,j

dx ≤

∫

R0

h

i

h

j

u

i,j

dx + + B

^1/2

λ

^−1/4₁

2 (X + Y ).

On integrating by parts in the penultimate term in (4.14), we are led to (4.19)

∫

R0

(h

3

− H

3

)h

α

v ˆ

,α

dx

= −

∫

R0

ˆ

v(h

3,α

− H

3,α

)h

α

dx −

∫

R0

ˆ

v(h

3

− H

3

)h

α,α

dx.

In a manner similar to the computation of each of integrals on the right side of (4.8), we arrive at

(4.20)

−

∫

R0

ˆ

vh

_3,α

h

_α

dx ≤ ˆv

s

(∫

R0

h

_3,α

h

_3,α

dx

∫

R0

h

_α

h

_α

dx )

1/2

≤ √ v ˆ

s

λ

1

∫

R0

h

i,j

h

i,j

dx,