A Phragmén–Lindelöf alternative result for the Navier–Stokes equations for steady compressible viscous flow

A Phragmén–Lindelöf alternative result for the Navier–Stokes equations for steady compressible viscous flow

J. Math. Anal. Appl. 340 (2008) 1480–1492 www.elsevier.com/locate/jmaa A Phragmén–Lindelöf alternative result for the Navier–Stokes equations for ste...

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J. Math. Anal. Appl. 340 (2008) 1480–1492 www.elsevier.com/locate/jmaa

A Phragmén–Lindelöf alternative result for the Navier–Stokes equations for steady compressible viscous flow ✩ Changhao Lin ∗ , Haobin Li School of Mathematical Sciences, South China Normal University, Guangzhou 510631, PR China Received 12 August 2007 Available online 29 September 2007 Submitted by B. Straughan

Abstract In this paper, the authors consider the Navier–Stokes equations for steady compressible viscous flow in three-dimensional cylindrical domain. A differential inequality for appropriate energy associated with the solutions of the Navier–Stokes isentropic flow in semi-infinite pipe is derived, from which the authors show a Phragmén–Lindelöf alternative result, i.e. the solutions for steady compressible viscous N–S flow problem either grow or decay exponentially as the distance from the entry section tends to infinity. In the decay case, the authors indicate how to bound explicitly the total energy in terms of data. © 2007 Elsevier Inc. All rights reserved. Keywords: Navier–Stokes equations; Steady compressible viscous flow; Phragmén–Lindelöf alternative; Decay estimates

1. Introduction In 1856, de Saint-Venant [24] formulated and conjectured a famous mathematical and mechanical principle which came to be known in subsequent literature as Saint-Venant’s principle and led to an extensive investigation in the framework of applied mathematics. Early work on Saint-Venant’s principle primarily focused on the boundary-value problems involving elliptic equations. It is Boley [5], who, in the 1950’s, first pointed out the validity of Saint-Venant’s principle for the heat equations. Since then an extensive attention has been paid to the parabolic equations (e.g. see [2, 14,18], and [10]). These studies are motivated by a desire to establish, for parabolic equations, spatial decay estimates analogous to those obtained for elliptic equations in the investigation of Saint-Venant’s principle in elasticity theory. The history and development of this question was explained and summarized in the work of Horgan and Knowles [10] and had been periodically updated by Horgan [8,9]. We also recall the monographs of Ames and Straughan [3], Flavin and Rionero [6] and Antonsev, Díaz and Shamarev [4], where the energy method is widely used to study the spatial behavior of solutions of partial differential equations. ✩ The work was supported by the National Natural Science Foundation of China (Grant #10471050), Guangdong Province Natural Science Foundation of China (Grants #031495, #7005795) and University Special Research Fund for Ph.D. Program (Grant #20060574002). * Corresponding author. E-mail address: [email protected] (C. Lin).

0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.09.037

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A number of papers in the literature have dealt with the flow of incompressible viscous fluid which is governed by steady or transient Navier–Stokes equations in a semi-infinite channel or pipe (see, e.g., [1,2,13,14,17,18]). These pipe and channel flow results may be regarded as Saint-Venant type decay results. In fact, the first paper to point out this connection was that of Horgan and Wheeler [13]. For other results of Saint-Venant type, see [1,2,17,18]. Of interest also are the paper [8,9] and [10] therein. However, it seems that, up to now, no analogous result concerning compressible viscous Navier–Stokes flow equations has been seen in the literature. To establish spatial decay estimate of Saint-Venant type, one always needs to assume that solution must satisfy some a priori decay criterions at infinity. Clearly, such assumptions seem slightly unnatural and artificial. The classical Phragmén–Lindelöf theorem, without some a priori asymptotical decay assumptions for solution, states that harmonic function which vanishes on the cylindrical surface must either grow or decay exponentially with distance from the finite end of the cylinder. Phragmén–Lindelöf alternative results were obtained by Horgan and Payne [12] for harmonic function with nonlinear boundary conditions on the lateral surface of a semi-infinite cylinder. Payne and Schaefer [23] established the Phragmén–Lindelöf type results in three type special domains in R 2 . Additional references for Phragmén–Lindelöf type results may be found in [15,16] and [19]. In the present paper, we are concerned with the flow of a compressible viscous fluid which is governed by steady Navier–Stokes flow equations in a semi-infinite pipe and establish a Phragmén–Lindelöf type growth-decay estimate for the problem. In Section 2, we formulate the basic boundary-value problem which provides the framework for our investigation. Section 3, we derive a basic differential inequality that leads directly to our main results of this paper. We then obtain growth-decay estimate for the energy expressions in Section 4. We finally show how the total energy in the decay result is bounded explicitly in terms of prescribed data in Section 5. 2. Formulation of the problem In this paper, we consider the steady state Navier–Stokes system for compressible viscous fluids in R 3 : div(ρu) = 0,

(2.1)

−μu − (μ + λ)∇ div u + ρu · ∇u = −∇P (ρ) + ρf + g,

(2.2)

where u = (u1 , u2 , u3 ) is the fluid’s velocity, ρ is its density and P (ρ) is pressure. Functions f = (f1 , f2 , f3 ) and g = (g1 , g2 , g3 ), which are prescribed, model outer force and force density. The constant viscosity coefficients μ and λ are assumed to satisfy 2 μ + λ  0, 3 for physical requests. We are interested only in the isentropic case, where the pressure is given by μ > 0,

(2.3)

P (ρ) = aρ r

(2.4)

with a positive constant a and r > 1. This is important from the physical point of view since air has the adiabatic constant r = 75 . For simplicity, we assume that there is potential function F of f , i.e. f = ∇F and there is a potential function G for g/ρ, i.e. g/ρ = ∇G. We also assume f, g ∈ L∞

(2.5)

F, G ∈ H 1 .

(2.6)

and

In recent years, the solvability to the various initial-boundary value problem for the full Navier–Stokes equations for both stationary and instationary compressible viscous fluids has been established. A standard result (cf. [7,20,21] and therein) is that there exists a solution u ∈ H 1,

ρ ∈ Lq ∩ L∞

where q > 0 depends on r.

(2.7)

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Let us formulate more precisely the problem. Our problem is defined on a semi-infinite cylinder R with arbitrary cross section D. The boundary ∂D of D is assumed to be smooth enough to admit application of the divergence theorem. To make the geometry precise we assume that the generators of the cylinder are parallel to the x3 -axis and the finite end of the cylinder is in the plane x3 = 0. Thus    (2.8) R = (x1 , x2 , x3 )  (x1 , x2 ) ∈ D, x3 > 0 . We shall also use the notations    Rz = (x1 , x2 , x3 )  (x1 , x2 ) ∈ D, x3 > z  0 ,    Dz = (x1 , x2 , x3 )  (x1 , x2 ) ∈ D, x3 = z .

(2.9) (2.10)

We rewrite the compressible Navier–Stokes flow equations in the following form: (ρui ),i = 0, −μui − (μ + λ)uj,j i + ρuj ui,j

  + a ρ r ,i = ρfi + gi ,

(2.11) in R

(2.12)

subject to the boundary conditions: ui = 0

on ∂D,

(2.13)

ui (x1 , x2 , 0) = hi (x1 , x2 )

in D,

(2.14)

where hi (i = 1, 2, 3) are assumed to satisfy the compatibility hi |∂D = 0. We will not make any a priori assumptions regarding the behavior of solution as x3 → ∞. In this paper, we assume (u, ρ), with u ∈ C 2 (R) ∩ C 1 (R) and ρ ∈ C 1 (R) ∩ L∞ (R), is the classical solution of the stationary compressible viscous flow problem (2.11)–(2.14). According to the embedding theorem of Sobolev space, our assumption is reasonable. We also assume ρ > 0, satisfying (2.15)

0 < ρ < ρ1 ,

ρ1 is a constant. In fact, the boundedness of density ρ is natural from the physical viewpoint. ∂u , and have adopted Throughout this paper, we have used a comma to indicate partial differentiation, i.e. u,i = ∂x i the summation of summing over repeated Latin indices from 1 to 3, over repeated Greek from 1 to 2. We note that, for any z > 0, by (2.11) and (2.13) 

 ρu3 dA = Dz



z  ρh3 dA +

D0

div(ρu) dA dη = 0 Dη

ρh3 dA.

(2.16)

D0

(2.16) displays that the area mean value of ρu3 is the same over each cross section. In what follows, we shall use the Poincaré inequality for Dirichlet integrable function ϕ which vanishes on ∂D, namely   λ1 ϕ 2 dA  ϕ,α ϕ,α dA, (2.17) D

D

where λ1 is the first eigenvalue of Membrane problem:  ψ,αα + λψ = 0 in D, ψ =0 on ∂D. Lower bounds for λ1 are well-known, see, e.g. [11]. In addition to inequality (2.17), we also apply the following Sobolev inequality which holds for ϕ ∈ C01 (D)    1 ϕ 4 dA  ϕ 2 dA ϕ,α ϕ,α dA. (2.18) 2 D

D

D

A derivation of (2.18) is given by Serrin [25] and Payne [22].

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3. Basic inequality For arbitrary z > z0 > 0, for the solution (u, ρ), making use of the divergence theorem, we find z 



  −μui,jj − (μ + λ)uj,j i + ρuj ui,j + a ρ r ,i − ρfi − gi ui dA dη

0= z0 Dη

 = −μ

 ui ui,3 dA + μ

Dz

ui ui,3 dA + μ



+ (μ + λ)

(ui,i )2 dA dη + z0 Dη

z

1 2

z

ui,i u3 dA + (μ + λ)

 ρu3 ui ui dA −

1 2

Dz0

z



ui,i u3 dA



ρu3 ui ui dA + a

 r ρ ,i ui dA dη

z0 Dη

Dz0



ρfi ui dA dη − z0 Dη



Dz

Dz





ui,j ui,j dA dη − (μ + λ) z0 Dη

Dz0

z



z 

(3.1)

gi ui dA dη. z0 Dη

We note that z  a

 r ρ ,i ui dA dη = r

z0 Dη

z  ρ r−1 ρ,i ui dA dη z0 Dη

=

ar r −1

ar = r −1

z 

 r−1  ρ ρui dA dη ,i

z0 Dη



ar ρ u3 dA − r −1



r

Dz

ρ r u3 dA.

(3.2)

Dz0

Since by assumption there exists a potential function F of f , we obtain z  −



z  ρfi ui dA dη = −

z0 Dη

F,i ρui dA dη = − z0 Dη

 Fρu3 dA +

Dz

Fρu3 dA.

(3.3)

Gρu3 dA.

(3.4)

Dz0

In a similar manner, we obtain z  −



z  gi ui dA dη = −

z0 Dη

G,i ρui dA dη = − z0 Dη

Dz

 Gρu3 dA + Dz0

A combination of (3.1)–(3.4) leads to       1 ar −μ ui ui,3 dA − (μ + λ) ui,i u3 dA + ρu3 ui ui dA + ρ r u3 dA − Fρu3 dA − Gρu3 dA 2 r −1 Dz

Dz

z



z0 Dη

Dz

Dz

Dz



ui,j ui,j dA dη − (μ + λ)

= −μ where

Dz

z

(ui,i )2 dA dη + w(z0 ), z0 Dη

(3.5)

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w(z0 ) = −μ

ui ui,3 dA − (μ + λ)

Dz0

 −

Dz0

 Fρu3 dA −

Dz0



1 ui,i u3 dA + 2

Dz0

ar ρu3 ui ui dA + r −1

 ρ r u3 dA Dz0

(3.6)

Gρu3 dA.

Dz0

We further define z 

z  ui,j ui,j dA dη − (μ + λ)

Φ(z) = w(z) = −μ z0 Dη

(ui,i )2 dA dη + w(z0 ).

(3.7)

z0 Dη

Differentiating (3.7) with respect to z yields   dΦ(z) = −μ ui,j ui,j dA − (μ + λ) (ui,i )2 dA < 0. dz Dz

(3.8)

Dz

Next section, we will proceed to derive a basic inequality  



Φ(z)  c1 −Φ  (z) 3/2 + c2 −Φ  (z) + c3 −Φ  (z) 1/2

(3.9)

for computable constants c1 , c2 and c3 . (3.9) will be utilized to lead to our Phragmén–Lindelöf alternative results. 4. Asymptotic behavior of solution This section is devoted to the derivation of asymptotic behavior of solution for the problem. By virtue of the definition of Φ(z), we have       Φ(z)  μ |ui ui,3 | dA + (μ + λ) |ui,i u3 | dA + 1 ρ|u3 ui ui | dA + ar ρ r |u3 | dA 2 r −1 Dz



+

 |Fρu3 | dA +

Dz

Dz

Dz



|Gρu3 | dA.

(4.1)

Dz

We next treat each term on the right of (4.1) with the aim of deriving appropriate upper bounds for those qualities. Making use of the Schwarz’s inequality and inequality (2.17), we obtain  1/2    μ μ |ui ui,3 | dA  μ ui,3 ui,3 dA ui ui dA  √ ui,j ui,j dA. (4.2) 2 λ1 Dz

Dz

Dz

Dz

Similarly, we obtain    ε 1 μ+λ 2 · μ ui,j ui,j dA + · (μ + λ) (ui,i ) dA (μ + λ) |ui,i u3 | dA  √ ε 2 λ1 μ Dz

Dz

Dz

√   μ+λ μ ui,j ui,j dA + (μ + λ) (ui,i )2 dA ; = √ 2 λ1 μ Dz

Dz

√ √ in (4.3), we set ε = μ + λ/ μ. Making use of the Sobolev inequality (2.18) and Poincaré’s inequality (2.17), we find  1/2   1 ρ1 2 2 |ρu3 ui ui | dA  u3 dA (ui ui ) dA 2 2 Dz

Dz

Dz

(4.3)

C. Lin, H. Li / J. Math. Anal. Appl. 340 (2008) 1480–1492

ρ1  √ 2 2



 u23 dA

Dz

ρ1  √ 2 2λ1



 ui ui dA

Dz

1485

1/2 ui,α ui,α dA

Dz

3/2 ui,j ui,j dA .

(4.4)

Dz

In a manner similar to (4.2), we obtain ar r −1

 ρ r |u3 | dA  Dη

arρ1r r −1

 Dz

1/2 1/2  arρ1r |D|1/2 u23 dA |D|1/2  ui,j ui,j dA √ (r − 1) λ1

(4.5)

Dz

where |D| denotes the area of D. Similarly, we have 

 |Fρu3 | dA  ρ1 Dz



Dz

Dz

 1/2  1/2 1/2 ρ1 F 2 dA u23 dA  √ F L2 (D) ui,j ui,j dA , λ1

ρ1 |Gρu3 | dA  √ G L2 (D) λ1



Dz

1/2 ui,j ui,j dA .

(4.6)

Dz

(4.7)

Dz

Substituting (4.2)–(4.7) into (4.1) leads to inequality  



Φ(z)  c1 −Φ  (z) 3/2 + c2 −Φ  (z) + c3 −Φ  (z) 1/2

(4.8)

with ρ1 ; c1 = √ 2 2λ1 μ3/2 √ μ+λ 1 +1 ; c2 = √ 2λ1 μ r

arρ1 |D|1/2 c3 = √ + ρ1 F L2 (D) + ρ1 G L2 (D) / λ1 μ. (r − 1) λ1 From (4.8), we get the following two inequalities:

3/2

1/2 Φ(z)  c1 −Φ  (z) + c2 −Φ  (z) + c3 −Φ  (z) ,



3/2 1/2 + c2 −Φ  (z) + c3 −Φ  (z) . −Φ(z)  c1 −Φ  (z)

(4.9) (4.10)

We recall that Φ  (z) < 0 for all z > 0. Thus if for some value of z, for instance z = z0 , Φ(z0 ) < 0, then Φ(z) < 0 for all z > z0 . In this case, applying A–G inequality to −Φ  (z) in (4.10), we find that

3/2

1/2 c2 c2  −Φ (z) −Φ  (z) + c3 + −Φ(z)  c1 + 2 2

3/2

1/2   =: M1 −Φ (z) + M2 −Φ (z)   3/2

1/2 M M2 1/2 3   M1 −Φ (z) + − √2 . (4.11) 3M1 3 3M1 Solving (4.11) for −Φ  (z), we obtain    Φ(z) M2 3/2 1/3 M2 1/2 2 − + −  −Φ  (z). M1 3M1 3M1

(4.12)

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Setting Φ(z) M2 3/2 + . Ψ (z) = − M1 3M1 3

(4.13)

Clearly, we have −Φ  (z) = 3M1 Ψ 2 Ψ  (z). In terms of Ψ , we now rewrite (4.12) as   M2 1/2 2 3M1 Ψ 2 Ψ   Ψ − . 3M1 We note that   Φ  (z) M2 3/2 1/3 M2 1/2 M2 1/2 = − + − > 0. Ψ− 3M1 M1 3M1 3M1

(4.14)

(4.15)

(4.16)

From (4.15), we obtain

 

M2 1/2 −1  M2 Ψ  Ψ +  1. 3M1 Ψ  + 2 3M1 M2 Ψ − M2 1/2 2 3M1 [Ψ − ( 3M ) ] 1

An integration of (4.17) from z0 to z yields  z

 M2 M2 1/2   z − z0 . 3M1 Ψ + 2 3M1 M2 ln Ψ − − M2 1/2  3M1 Ψ − ( 3M ) z0 1 We observe that Ψ  (z) 1 d = − < 0. M2 1/2 2 dz Ψ − ( M2 )1/2 [Ψ − ( 3M ) ] 3M1 1 Thus, for z > z0 , we have z  1  > 0. − M2 1/2  Ψ − ( 3M1 ) z0 Applying (4.20) to (4.19) leads to 1/2 2√3M1 M2   2√3M1 M2   M M 2 2 e3M1 Ψ (z0 ) Ψ (z0 ) − ln e3M1 Ψ (z) Ψ (z) − 3M1 3M1   M2 .  ln exp z − z0 − M2 1/2 Ψ (z0 ) − ( 3M ) 1 (4.21) deduces further to √ M2 1/2 2 3M1 M2 3M1 Ψ (z) Ψ (z) − e  Q(z0 )ez−z0 , 3M1 with constant Q(z0 ) being given by √   M2 1/2 2 3M1 M2 M2 . Q(z0 ) = Ψ (z0 ) − exp 3M1 Ψ (z0 ) − M2 1/2 3M1 Ψ (z0 ) − ( 3M ) 1 (4.22) implies 3M1 Ψ (z) z − z0 M2 1/2 exp √ Ψ (z) −  Q(z0 ) exp √ , 3M1 2 3M1 M2 2 3M1 M2

(4.17)

(4.18)

(4.19)

(4.20)

(4.21)

(4.22)

(4.23)

(4.24)

C. Lin, H. Li / J. Math. Anal. Appl. 340 (2008) 1480–1492

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where √

Q(z0 ) = Q(z0 ) 2

1 3M1 M2

.

(4.25)

Now, we use the primary inequality x < 1 + x < ex

(x > 0).

to (4.22) to obtain √   √ M2 1/2 3M1 + 2 M2 z − z0 exp Q(z0 ) exp √  exp · Ψ (z) . √ 3M1 2 M2 2 3M1 M2

(4.26)

(4.27)

It is readily to verify for large enough z0 Q(z0 )  1. Thus, from (4.27), we obtain

z − z0  ( 3M1 + 2 M2 )Ψ (z). √ 3M1 Recalling the definition of Ψ (z), (4.28), in fact, is equivalent to

Φ(z) M2 3/2  (z − z0 )3 . (3M1 + 2 3M1 M2 )3 − + M1 3M1 (4.29) further implies 

lim z−3 μui,j ui,j + (μ + λ)(ui,i )2 dx  K0 , z→+∞

(4.28)

(4.29)

(4.30)

Rz0 \Rz

where K0 is a computable constant depending on z0 . We have shown that if there exists a z0 > 0, such that Φ(z0 ) < 0, then Φ(z) cannot remain bounded for z > z0 . We next consider the case in which no such a z0 exists, i.e. Φ(z)  0 for all z  0. We now discuss (4.9) instead of (4.10). Proceeding as in (4.11), we obtain  

1/2 M2 1/2 3 M2 3/2 + − M1 . (4.31) Φ(z)  M1 −Φ  (z) 3M1 3M1 Solving (4.31) for −Φ  (z), we get    Φ(z) M2 3/2 1/3 M2 1/2 2 −Φ  (z)  + − . M1 3M1 3M1 Setting Ψ13 (z) =

Φ(z) M2 3/2 + , M1 3M1

substituting (4.33) into (4.32) leads to   M2 1/2 2 2  −3M1 Ψ1 Ψ1 (z)  Ψ1 (z) − . 3M1 We note that M2 1/2 Ψ1 (z) − > 0. 3M1 Upon integrating (4.34) from 0 to z yields  z

 M2 M2 1/2   −z. 3M1 Ψ1 + 2 3M1 M2 ln Ψ1 − −  M 2 1/2 3M1 Ψ1 − ( 3M1 ) 0

(4.32)

(4.33)

(4.34)

(4.35)

(4.36)

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From (4.34), we see Ψ1 (z) −1 d = < 0. M M2 1/2 2 dz Ψ1 (z) − ( 2 )1/2 [Ψ1 (z) − ( 3M ) ] 3M1 1

(4.37)

Henceforth, we have −

z    0. M2 1/2  Ψ1 − ( 3M1 ) 0 M2

(4.38)

For the left-hand side of (4.36), we neglect the first term evaluated at z and the third term evaluated at zero to obtain 

M2 1/2 z M2 2 3M1 M2 ln Ψ1 (z) −  −z + + 3M1 Ψ1 (0). (4.39)  M2 1/2 3M1 Ψ1 − ( 3M ) 0 1 M2 1/2 Next we show that M2 /[Ψ1 − ( 3M ) ] has a priori upper bound. 1

M2 1/2 From (4.37), we see that M2 /[Ψ1 − ( 3M ) ] is a monotonously increasing function. By virtue of the definition of 1

M2 1/2 Ψ1 (z), we observer that Ψ1 (z) is a monotonously decreasing function and has lower bound ( 3M ) . Consequently, 1 there exists a constant, denoted as q, such that

lim Ψ1 (z) = q.

(4.40)

z→∞

Thus, we obtain M2 M2 1/2 Ψ1 (z) − ( 3M ) 1

 lim

z→∞

M2 M2 1/2 Ψ1 (z) − ( 3M ) 1

=

M2 M2 1/2 q − ( 3M ) 1

.

Inserting (4.41) into (4.39) obtain √ M2 M2 1/2 z · exp √ Ψ1 (z) −  exp − √ M2 1/2 3M1 2 3M1 M2 2 3M1 (q − ( 3M ) ) 1    1 3M1 M2 1/2 . · exp · Ψ1 (0) · Ψ1 (0) − 2 M2 3M1 (4.42) can be rewritten as M2 1/2 Ψ1 (z) −  Q2 e−σ z 3M1 with

(4.41)

(4.42)

(4.43)

√     1 3M1 M2 1/2 M2 Ψ1 (0) + √ , Ψ1 (0) − Q2 = exp M2 1/2 2 M2 3M1 2 3M1 (q − ( 3M ) ) 1 1 . σ= √ 2 3M1 M2

Recalling the definition of Ψ1 (z), by (4.43), we find   Φ(z) M2 3/2 1/3 M2 1/2 + −  Q2 e−σ z . M1 3M1 3M1 A simplification of (4.44) yields

Φ(z)  M1 Q32 e−3σ z + 3M1 M2 Q22 e−2σ z + M2 Q2 e−σ z . Furthermore, according to the definition of Φ(z), it follows that

(4.44)

(4.45)

C. Lin, H. Li / J. Math. Anal. Appl. 340 (2008) 1480–1492



 ui,j ui,j dx + (μ + λ)

μ Rz

(ui,i )2 dx  M1 Q32 e−3σ z +

1489

3M1 M2 Q22 e−2σ z + M2 Q2 e−σ z ,

(4.46)

Rz

We have thus proved Theorem 4.1. Let (u, ρ) be the solution of problem (2.11)–(2.14) with density ρ satisfying (2.15). In addition, assume that the outer face f and force density g satisfy (2.5) and (2.6), then the following Phragmén–Lindelöf alternative result holds, i.e. either 

(4.47) μui,j ui,j + (μ + λ)(ui,i )2 dx  K1 , lim z−3 z→+∞

Rz0 \Rz

or



 μ

ui,j ui,j dx + (μ + λ)

Rz

(ui,i )2 dx  M1 Q32 e−3σ z +

3M1 M2 Q22 e−2σ z + M2 Q2 e−σ z ,

(4.48)

Rz

where K1 , Q2 , M1 , M2 and σ are computable constants. 5. Total energy bound of Φ(0) In order to make our decay estimates in Section 4 explicit, we need to compute a bound for Q2 . We see that Q2 depends on Φ(0) (via Ψ1 (0)). So we need to seek a bound for Φ(0) in terms of end data. We first introduce an auxiliary function ϕi (x1 , x2 , x3 ) = hi (x1 ,2 )e−δx3 ,

i = 1, 2, 3,

(5.1)

where constant δ > 0 will be determined later. Clearly, integrating by parts, we find      1 ar r Φ(0) = −μ ui ui,3 dA − (μ + λ) ui,i u3 dA + ρu3 ui ui dA + ρ u3 dA − Fρu3 dA 2 r −1 D0

D0

 −

D0

D0





D0

Gρu3 dA

D0

 =μ

 ui,j ϕi,j dx + (μ + λ)

R

ar + r −1 

R



 ρ r h3 dA − D0

ui,i ϕj,j dx + μ

Fρh3 dA −

D0

ui,j ϕi,j dx + (μ + λ)

=μ R





gi ϕi dx + R

ar r −1





ui,i ϕj,j dx + R

ρ r h3 dA + D0

1 2

R



 ρuj ui,j ϕi dx + a

R



ρh3 hi hi dA − D0

R

ρh3 hi hi dA D0

 r ρ ,i ϕi dx −

R



Fρh3 dA −

D0

After a computation, (5.2) deduces directly to that    Φ(0)  μ ui,j ϕi,j dx + (μ + λ) ui,i ϕj,j dx + ρuj ui,j ϕi dx + Data. R



Gρh3 dA

D0



ui,jj ϕi dx + (μ + λ) R



1 uj,j i ϕi dx + 2

 ρfi ϕi dx R

Gρh3 dA.

(5.2)

D0

(5.3)

R

In (5.3), we have used the upper bound ρ1 instead of ρ. Data denotes the integrals in terms of prescribed functions. Making use of the Schwarz’s inequality, we have

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 ui,j ϕi,j dx + (μ + λ)

μ R

μ  8

ui,i ϕj,j dx R





ui,j ui,j dx + 2μ R

ϕi,j ϕi,j dx +

μ+λ 8

R



 (ui,i )2 dx + 2(μ + λ) R

(ϕj,j )2 dx R

1 = Φ(0) + Data. 8 By using inequalities (2.17) and (2.18), we obtain  ρuj ui,j ϕi dx R



 ρh3 hi hi dA −

=−

(5.4)

D0

ρuj ui ϕi,j dx R

∞   Data + ρ1

1/2  1/2 (ui ui ) dA ϕi,j ϕi,j dA dη 2



0



1/2  ∞

 ∞  Data + ρ1

1/2

2

(ui ui ) dA dη

ϕi,j ϕi,j dA dη

0 Dη

0 Dη

 1/2 ρ1  Data + √ max ui ui dA λ1 z Dz

1/2  ∞

 ∞

ui,α ui,α dA dη 0 Dη

1/2 ϕi,j ϕi,j dA dη

(5.5)

0 Dη

Assume that   ui ui dA = max ui ui dA,

(5.6)

z

Dz1

.

Dz

then z1 

 max z

Dz

ui ui dA = 0

 ui ui dA dη + ui ui dA ,η



D0



z1  ui ui,η dη +

=2 0 Dη

 z1  2

ui ui dA

D0

1/2  z1 

ui ui dA dη 0 Dη

1 √ λ1



1/2 ui,η ui,η dA dη

+ Data

0 Dη

ui,j ui,j dx + Data.

(5.7)

R

Inserting (5.7) into (5.5), and using the inequality √ √ √ a + b  a + b, we obtain  1/2   2ρ1 ρuj ui,j ϕi dx   ϕi,j ϕi,j dx ui,j ui,j dx + Data = BΦ(0) + Data, 4 3 λ 1 R R R

(5.8)

C. Lin, H. Li / J. Math. Anal. Appl. 340 (2008) 1480–1492

1491

where 2ρ1 B=  μ 4 λ31 2ρ1 =  μ 4 λ31

1/2

 ∞ ϕi,j ϕi,j dx 0 Dη

 ∞

e

−2δη

0

2ρ1 1 =  ·√ 2δ μ 4 λ31



 dη

1/2 hi,α hi,α dA

D

1/2 hi,α hi,α dA .

(5.9)

D

Obviously, we can choose δ large enough so that B is small enough, and satisfies 1 0
(5.10)

(5.11)

We have finally established an explicit upper bound in terms of prescribed data for the total energy Φ(0). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

K.A. Ames, L.E. Payne, Decay estimates in steady pipe flow, SIAM J. Math. Anal. 20 (1989) 789–815. K.A. Ames, L.E. Payne, P.W. Schaefer, Spatial decay estimates in time-dependent Stokes flow, SIAM J. Math. Anal. 24 (1993) 1395–1413. K.A. Ames, B. Straughan, Non-Standard and Improperly Posed Problems, Academic Press, San Diego, 1997. S.N. Antonsev, J.I. Díaz, S. Shamarev, Energy Methods for Free Boundary Problems, Applications to Nonliear PDEs and Fluid Mechanics, Birkhäuser, Boston, 2002. B.A. Boley, Upper bounds and Saint-Venant’s principle for transient heat conduction, Quant. Appl. Math. 18 (1960) 205–207. J.N. Flavin, S. Rionero, Qualitative Estimate for Partial Differential Equations: An Introduction, CRC Press, Boca Raton, 1996. H. Frehse, S. Goj, M. Steinhauer, LP -estimates for the Navier–Stokes equations for steady compressible flow, Manuscripta Math. 116 (2005) 265–275. C.O. Horgan, Recent developments concerning Saint-Venant’s principle: An update, Appl. Mech. Rev. 42 (1989) 295–302. C.O. Horgan, Recent developments concerning Saint-Venant’s principle: A second update, Appl. Mech. Rev. 49 (1996) 101–111. C.O. Horgan, J.K. Knowles, Recent developments concerning Saint-Venant’s principle, in: I.Y. Wu, J.W. Hulchinson (Eds.), Adv. Appl. Mech., vol. 23, Academic Press, San Diego, 1983, pp. 179–264. C.O. Horgan, L.E. Payne, Inequalities of Korn, Friedrichs and Babuska-Aziz, Arch. Ration. Mech. Anal. 82 (1983) 165–179. C.O. Horgan, L.E. Payne, Phragmén–Lindelöf type alternative results for harmonic functions with nonlinear boundary conditions, Arch. Ration. Mech. Anal. 122 (1993) 123–144. C.O. Horgan, L.T. Wheeler, Spatial decay estimates for the Navier–Stokes equations with application to the problem of entry flow, SIAM J. Appl. Math. 35 (1978) 97–116. Changhao Lin, Spatial decay estimates and energy bounds for the Stokes flow equation, Stability Appl. Anal. Contin. Media 2 (1992) 249–264. Changhao Lin, L.E. Payne, A Phragmén–Lindelöf type results for second order quasilinear parabolic equation in R 2 , Z. Angew. Math. Phys. 45 (1994) 294–311. Changhao Lin, L.E. Payne, Phragmén–Lindelöf alternative for a class of quasilinear second order parabolic problems, Differential Integral Equations 8 (1995) 539–551. Changhao Lin, L.E. Payne, Spatial decay bounds in the channel flow of an incompressible viscous fluid, Math. Models Methods Appl. Sci. 14 (2004) 795–818. Changhao Lin, L.E. Payne, Spatial decay bounds in time dependent pipe flow of an incompressible viscous fluid, SIAM J. Appl. Math. 65 (2005) 458–474. Yan Lin, Changhao Lin, Phragmén–Lindelöf type alternative results for the Stokes flow equation, Math. Inequal. Appl. 9 (2006) 671–694. A. Matsumura, N. Yamagata, Global weak solutions of the Navier–Stokes equations for multidimensional compressible flow subject to large external potential forces, Osaka J. Math. 38 (2001) 399–418. S. Novo, A. Novotný, On the existence of weak solutions to steady compressible Navier–Stokes equations in domains with conical outlets, J. Math. Fluid Mech. 8 (2006) 187–210. L.E. Payne, Uniqueness criteria for steady state solutions of the Navier–Stokes equations, in: Simpos. Internaz. Appl. Anal. Fis. Mat., CagliariSarrari, 1964, Edizioni Cremonse, Rome, 1965, pp. 130–153.

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[23] L.E. Payne, P.W. Schaefer, Some Phragmén–Lindelöf type alternative results for the biharmonic equation, Z. Angew. Math. Phys. 45 (1994) 414–432. [24] A.J.C.B. de Saint-Venant, Mémoire sur la flexion des prismes, J. Math. Pures Appl. 1 (2) (1856) 89–189. [25] J. Serrin, The initial-value problem for the Navier–Stokes equations, in: Nonlinear Problems, University of Wisconsin Press, Madison, WI, 1963, pp. 69–98.