Electronic Journal of Differential Equations, Vol. 2007(2007), No. 02, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
STRUCTURAL STABILITY FOR BRINKMAN-FORCHHEIMER
EQUATIONS
YAN LIU, CHANGHAO LIN
Abstract. In this paper, we obtain the continuous dependence and convergence results for the Brinkman and Forchheimer coefficients of a differential
equation that models the flow of fluid in a saturated porous medium.
1. Introduction
The concept of structural stability in which the study of continuous dependence
(or stability) is on changes in the model itself rather than the initial data, has been
the subject of much recent study. Many references to the work of the nature are
given in the monograph of Ames and Straughan [1], which stress that continuous
dependence on the model itself, or structural stability, is every bit as important
as stability with respect to perturbations of the initial data. In particularly, the
stability of flow in porous media has attracted much more attention in the literature;
see [3, 4, 5, 6, 7, 8] and their references.
In this paper, we are interested in the Brinkman-Forchheimer equations governing the flow of fluid in a saturated porous medium,
∂ui
= λ∆ui − aui − b|u|ui − p,i
∂t
(1.1)
∂ui
=0
∂xi
where ui is the average fluid velocity in the porous medium, a is the Darcy coefficient, λ is the Brinkman coefficient, b is the Forchheimer coefficient, and p is the
pressure. λ, a and b are positive constants. Here also ∆ is the laplace operator,
and k · k denotes the norm of L2 .
We assume that Ω is a bounded, simply connected domain with boundary ∂Ω in
R3 . Associated with (1.1), we imposed the boundary condition
ui = 0
on ∂Ω × {t > 0}
(1.2)
and the initial condition
ui (x, 0) = fi (x).
2000 Mathematics Subject Classification. 35B40, 35K50,35K45.
Key words and phrases. Continuous dependence; Brinkman-Forchheimer equations;
Brinkman coefficient; Forcheimer coefficient; structural stability.
c
2007
Texas State University - San Marcos.
Submitted December 7, 2006. Published January 2, 2007.
1
(1.3)
2
Y. LIU, C. LIN
EJDE-2007/02
In this paper, the usual summation convection is employed with repeated Latin subscripts summed from 1 to 3. The comma is used to indicated partial differentiation
and the differentiation with respect to the direction xk is denoted as “, k”.
2. Continuous dependence for the Brinkman coefficient
To study the continuous dependence on λ, we let (ui , p) and (vi , q) solve the
following boundary initial-value problems for different Brinkman coefficients λ1
and λ2 ,
∂ui
= λ1 ∆ui − aui − b|u|ui − p,i in Ω × {t > 0}
∂t
∂ui
= 0 in Ω × {t > 0}
(2.1)
∂xi
ui = 0 on ∂Ω × {t > 0}
ui (x, 0) = fi (x),
x∈Ω
and
∂vi
= λ2 ∆vi − avi − b|v|vi − q,i in Ω × {t > 0}
∂t
∂vi
= 0 in Ω × {t > 0}
∂xi
vi = 0 on ∂Ω × {t > 0}
vi (x, 0) = fi (x),
(2.2)
x∈Ω
We define the difference variables wi , π and λ by
wi = ui − vi , π = p − q, λ = λ1 − λ2
(2.3)
and then (wi , π) satisfies the boundary initial-value problem
∂wi
= λ1 ∆ui − λ2 ∆vi − awi − b(|u|ui − |v|vi ) − π,i
∂t
∂wi
= 0 in Ω × {t > 0}
∂xi
wi = 0 on ∂Ω × {t > 0}
wi (x, 0) = 0,
in Ω × {t > 0}
(2.4)
x∈Ω
Multiplying (2.4)1 by wi and integrating over Ω, we get
1 d
kwk2
2 dt Z
Z
Z
λ1 ∇ui ∇wi dx +
λ2 ∇vi ∇wi dx − akwk2 − b (|u|ui − |v|vi )wi dx (2.5)
ZΩ
Z Ω
ZΩ
=−
λ∇ui ∇wi dx +
λ2 ∇wi ∇wi dx − akwk2 − b (|u|ui − |v|vi )wi dx
=−
Ω
Ω
Ω
Since
(|u|ui − |v|vi )wi
1
1
1
1
= |u|(ui − vi + vi )wi − |v|vi wi + |u|ui wi + |v|wi (ui − vi − ui )
2
2
2
2
1
1
= (|u| + |v|)wi wi dx + (|u| − |v|)2 (|u| + |v|)
2
2
(2.6)
EJDE-2007/02
STRUCTURAL STABILITY
3
Combining (2.5) and (2.6), using the Cauchy-Schwarz inequality and dropping some
negative items, we obtain
d
λ2
kwk2 ≤
k∇uk2
dt
2λ2
Integrating from 0 to t, we obtain
Z t
λ2
2
k∇uk2 dη
(2.7)
kwk ≤
2λ2 0
Rt
Our next step is to bound 0 k∇uk2 dη. Multiplying (2.1)1 by ui and integrating
over Ω, we see that
d
kuk2 + 2λ1 k∇uk2 ≤ 0
dt
Integrating from 0 to t, we obtain
Z t
2
kuk + 2λ1
k∇uk2 dη ≤ kf k2
(2.8)
0
thus
Z
t
k∇uk2 dη ≤
0
kf k2
2λ1
(2.9)
Combining (2.7) and (2.9), we obtain
λ2
kf k2
(2.10)
4λ1 λ2
Inequality (2.10) shows the continuous dependence on λ. However, the convergence
result can’t follow from (2.10) as λ1 → 0, λ2 = 0.
kwk2 ≤
3. Convergence as λ1 → 0, λ2 = 0
Let (ui , p) be a solution of (2.1) with λ1 → 0, (vi , p) be a solution of (2.1) with
λ1 → 0, wi , π are defined the same as in section 2.
∂wi
= λ1 ∆ui − awi − b(|u|ui − |v|vi ) − π,i in Ω × {t > 0}
∂t
∂wi
= 0 in Ω × {t > 0}
(3.1)
∂xi
wi = 0 on ∂Ω × {t > 0}
wi (x, 0) = 0,
x∈Ω
Multiplying (3.1)1 by wi and integrating over Ω, we find
1 d
kwk2
2 dt
Z
Z
= −λ1
∇ui ∇wi dx − akwk2 − b (|u|ui − |v|vi )wi dx
Ω
ZΩ
Z
Z
(3.2)
2
= −λ1
∇ui ∇ui dx + λ1
∇ui ∇vi dx − akwk − b (|u|ui − |v|vi )wi dx
Ω
Ω
Z Ω
λ1
≤
∇vi ∇vi dx − akwk2 .
4 Ω
R
The next step is to bound Ω vi,j vi,j dx. We know
Z
Z
vi,j vi,j dx =
vi,j (vi,j − vj,i )dx .
(3.3)
Ω
Ω
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Y. LIU, C. LIN
EJDE-2007/02
Using (2.2)1 with λ2 = 0, we get
Z tZ
vi,j (vi,j − vj,i )ds dη
0
Ω
Z Z
1 t
=
(vi,j − vj,i )(−vi,jt − b(|v|vi ),j − q,ij )dx dη
a 0 Ω
Z Z
Z Z
1 t
b t
=−
(vi,j − vj,i )vi,jt ds dη −
(vi,j − vj,i )
a 0 Ω
a 0 Ω
Z Z
vk vk,j
1 t
vi,ij q,j ds dη
×(
vi + |v|vi,j )ds dη +
|v|
a 0 Ω
Z Z
Z Z
Z Z
1 t
1 t
1 t
−
vj,ji q,i ds dη −
vi,j q,j ni dsdη +
vj,i q,i nj dsdη
a 0 Ω
a 0 ∂Ω
a 0 ∂Ω
Z
Z
1
1
=−
vi,j vi,j dx|η=t +
fi,j fi,j dx
2a Ω
2a Ω
Z Z
Z Z
b t
b t
vk vk,j vi vi,j
−
ds dη −
vi,j |v|vi,j ds dη
a 0 Ω
|v|
a 0 Ω
(3.4)
thus
Z tZ
Z
1
vi,j vi,j ds dη ≤
fi,j fi,j dx
(3.5)
2a Ω
0
Ω
Combining (3.2), (3.3) and (3.5), we get
Z
λ1
fi,j fi,j ds dη
kwk2 ≤
8a Ω
This inequality demonstrates the convergence u → v when λ1 → 0, λ2 = 0.
4. Continuous dependence for the Forchheimer coefficient b
To study continuous dependence on b, we let (ui , p) and (vi , q) solve the following
boundary initial-value problem for different coefficients b1 and b2 .
∂ui
= λ∆ui − aui − b1 |u|ui − p,i in Ω × {t > 0}
∂t
∂ui
= 0 in Ω × {t > 0}
∂xi
ui = 0 on ∂Ω × {t > 0}
ui (x, 0) = fi (x),
(4.1)
x∈Ω
and
∂vi
= λ∆vi − avi − b2 |v|vi − q,i in Ω × {t > 0}
∂t
∂vi
= 0 in Ω × {t > 0}
∂xi
vi = 0 on ∂Ω × {t > 0}
vi (x, 0) = fi (x),
(4.2)
x∈Ω
We define the difference variables
wi = ui − vi ,
π = p − q,
b = b 1 − b2 .
(4.3)
EJDE-2007/02
STRUCTURAL STABILITY
Then (wi , π) satisfy the boundary initial-value problem
∂wi
= λ∆wi − awi − (b1 |u|ui − b2 |v|vi ) − π,i
∂t
∂wi
= 0 in Ω × {t > 0}
∂xi
wi = 0 on ∂Ω × {t > 0}
wi (x, 0) = 0,
in Ω × {t > 0}
= −λ
|∇w|2 dx − akwk2 −
Ω
b
≤ −akwk −
2
2
Z
b
2
(4.4)
x∈Ω
Multiplying (4.4)1 by wi and integrating over Ω, we get
Z
Z
1 d
kwk2 = −λ
|∇w|2 dx − akwk2 − (b1 |u|ui − b2 |v|vi )wi dx
2 dt
Ω
Ω
For we have
b
b1 |u|ui − b2 |v|vi = (|u|ui + |v|vi ) + b̃(|u|ui − |v|vi )
2
2
where b̃ = b1 +b
2 . Combining (4.5), (4.6) and (2.6), we obtain
1 d
kwk2
2 dt Z
5
(4.5)
(4.6)
Z
(|u|ui + |v|vi )wi dx − b̃(|u|ui − |v|vi )wi dx
Ω
b̃
(|u|ui + |v|vi )wi dx −
2
Ω
Z
(|u| + |v|)wi wi dx
Ω
(4.7)
We then use the Cauchy-Schwarz and arithmetic geometric mean inequalities as
follows
Z
Z
Z
b
b2
1
| (|u|ui + |v|vi )wi dx| ≤
(|u|3 + |v|3 )dx + b̃ (|u| + |v|)wi wi dx (4.8)
2 Ω
2 Ω
8b̃ Ω
We now employ (4.8) in (4.7), after an integration, that
Z Z
Z t
1
b2 t
(|u|3 + |v|3 )ds dη
kwk2 + a
kwk2 dη ≤
2
8
b̃
0
Ω
0
(4.9)
From (4.1)1 , one deduce that
Z t
Z tZ
Z tZ
1
1
2
2
3
kuk + a
kuk dη + b1
|u| ds dη + λ
ui,j ui,j ds dη = kf k2 (4.10)
2
2
0
0
Ω
0
Ω
and so
Z tZ
1
|u|3 ds dη ≤
kf k2
(4.11)
2b
1
0
Ω
Similarly, from (4.2)1 , we can also get
Z tZ
1
|v|3 ds dη ≤
kf k2
(4.12)
2b
1
0
Ω
Inserting (4.11), (4.12) in (4.10), we find
Z t
2
kwk + 2a
kwk2 dη ≤
b2
kf k2
(4.13)
4b1 b2
0
This inequality establish continuous dependence on b, we note, however, that convergence as b1 → 0, b2 = 0 is not established from (4.13).
6
Y. LIU, C. LIN
EJDE-2007/02
5. Convergence as the Forchheimer coefficient b1 → 0 and b2 = 0
Now, let (ui , p) be the solution of (4.1), and (vi , q) be the solution of (4.2) with
b2 = 0. The object of this section is to demonstrate convergence of the solution ui
to the solution vi as b1 → 0. We also define the variables wi and π by
wi = ui − vi ,
π =p−q
(5.1)
and then (wi , π) satisfy the boundary initial-value problems
∂wi
= λ∆wi − awi − b1 |u|ui − π,i in Ω × {t > 0}
∂t
∂wi
= 0 in Ω × {t > 0}
∂xi
wi = 0 on ∂Ω × {t > 0}
wi (x, 0) = 0,
(5.2)
x∈Ω
Multiplying by wi and integrating over Ω, we obtain
1 d
kwk2 = −λk∇wk2 − akwk2 − b1
2 dt
Z
|u|ui wi dx
(5.3)
Ω
Using the Hölder inequality, we get
d
kwk2 ≤ 2b1
dt
Z
2/3 Z
1/3
|u|3 dx
|w|3 dx
− 2λk∇wk2 − 2akwk2
Ω
(5.4)
Ω
For a function F such that F = 0 on ∂Ω (see for example [2]), we have the Sobolev
inequality
Z
4
|F | dx ≤ c1
Z
Ω
1/2 Z
3/2
|F | dx
Fi,j Fi,j dx
2
Ω
Ω
Then, we use the Cauchy-Schwarz inequality, to get
Z
1/2 Z
1/2
|w| dx ≤
|w| dx
|w|4 dx
Ω
Ω
Ω
Z
3/4 Z
3/4
≤ c1
wi wi dx
wi,j wi,j dx
3
Z
2
Ω
(5.5)
Ω
Similarly,
Z
3
|u| dx ≤ c1
Ω
Z
Ω
3/4 Z
3/4
ui ui dx
ui,j ui,j dx
Ω
(5.6)
EJDE-2007/02
STRUCTURAL STABILITY
7
In view of (5.4) and (5.5), (5.3) can be rewritten as
Z t h Z
1/4 Z
1/4 Z
1/2
kwk2 ≤ 2b1 c1
wi wi dx
wi,j wi,j dx
ui ui dx
0
Ω
Ω
Ω
Z t
Z t
Z
1/2 i
w2 ds dη
×
k∇wk2 dη − 2a
ui,j ui,j dx
dη − 2λ
0
0
Ω
Z t
Z t h Z
Z t
1/4
kwk2 dη − 2b1 c1
ε1
k∇wk2 dη − 2a
wi wi dx
≤ 2λ
0
0
Ω
Z
Z0
Z
1/4 1/2 i
−1/2
× ε2
wi,j wi,j dx
max
ui ui dx ·
ε1 ε2
ui,j ui,j dx
dη
Ω
Ω
Ω
Z tZ
Z tZ
1
1
wi wi ds dη + ε2
wi,j wi,j ds dη
≤ ε1
4
4
0
Ω
0
Ω
Z tZ
4
+ (b1 c1 )2 · (ε1 ε2 )−1/2
ui,j ui,j ds dη
2
0
Ω
Z
Z tZ
Z t
Z t
− 2a max
ui ui dx
w2 ds dη − 2λ
|∇w|2 dη − 2a
kwk2 dη
Ω
0
Ω
0
0
If we choose ε1 = 8a, ε2 = 8λ, the above expression can be rewritten as
Z
Z tZ
b21 c21
2
kwk ≤
max
ui ui dx ·
ui,j ui,j ds dη
4(aλ)1/2
Ω
0
Ω
From (4.10), we have
Z
max
ui ui dx ≤ kf k2 ,
Ω
Z tZ
ui,j ui,j ds dη ≤
0
Ω
(5.7)
1
kf k2 ;
2λ
therefore, from (5.7), we have
kwk2 ≤
(b1 c1 )2
kf k4 ,
8a1/2 λ3/2
which shows the desired result.
Acknowledgments. The authors would like to express their gratitude to Professor
Hongliang Tu for his advice when in writing parts of this article.
References
[1] K. A. Ames and B. Straughan, Non-standard and improperly posed problems, Academic. Press.
[2] J. N. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equtions:An Introduction, CRC Press, Boca Raton (1996).
[3] J. Guo and P. N. Kaloni, Double diffusive convection in a porous medium, nonlinear stability
and the Brinkman effect, Stud. Appl. Math. 94 (1995), 341-358.
[4] L. E. Payne and B. Straughan, Convergence and continuous dependence for the BrinkmanForchheimer equations, Stud. Appl. Math. 102 (1999), 419-439.
[5] L. E. Payne and B. Straughan, Structural stability for the Darcy equations of flow in porous
media, Proc, Roy, Soc, London A 454 (1998), 1691-1698.
[6] L. E. Payne, J. C. Song and B. Straughan, Continuous dependence and convergence results for
Brinkman and Forchheimer models with variable viscosity, Proc. R. Soc. Lond. A (1999)45S,
2173-2190.
[7] B. Straughan, Effect of property variation and modeling on convection in a fluid overlying a
porous layer,Int. J. num. Anal. Meth. Geomech, 26 (2002)75-97.
[8] B. Straughan and K. Hutter, A priori bounds and structure stability for double-diffusive convection in corporating the soret effect, Proc. R. Soc. lond. A(1999)455, 767-777.
8
Y. LIU, C. LIN
EJDE-2007/02
Yan Liu
Department of Applied Mathematics, Guangdong University of Finance, Guangzhou,
510520, China
E-mail address: liuyan99021324@tom.com
Changhao Lin
School of Mathematical Sciences, South China Normal University, 510631, China
E-mail address: linchh@scnu.edu.cn