Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 584680, 16 pages
doi:10.1155/2012/584680
Research Article
Asymptotic Behavior of the 3D
Compressible Euler Equations with Nonlinear
Damping and Slip Boundary Condition
Huimin Yu
Department of Mathematics, Shandong Normal University, Jinan 250014, China
Correspondence should be addressed to Huimin Yu, hmyu@amss.ac.cn
Received 7 January 2012; Accepted 1 March 2012
Academic Editor: Laurent Gosse
Copyright q 2012 Huimin Yu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The asymptotic behavior as well as the global existence of classical solutions to the 3D
compressible Euler equations are considered. For polytropic perfect gas P ρ P0 ργ , time
asymptotically, it has been proved by Pan and Zhao 2009 that linear damping and slip boundary
effect make the density satisfying the porous medium equation and the momentum obeying the
classical Darcy’s law. In this paper, we use a more general method and extend this result to the
3D compressible Euler equations with nonlinear damping and a more general pressure term.
Comparing with linear damping, nonlinear damping can be ignored under small initial data.
1. Introduction and Main Results
We study the 3D compressible Euler equations with nonlinear damping:
ρt ∇ · ρ
u 0,
u⊗u
∇P ρ −αρ
u − βρ|
u|q−1 u
.
ρ
u t ∇ · ρ
1.1
This model represents the compressible flow through porous media with nonlinear external
force field. Here ρ, u,
and M ρ
u denote density, velocity, and momentum, respectively.
The pressure P is a smooth function of ρ such that P ρ > 0, P ρ > 0 for any ρ > 0.
Obviously, for polytropic perfect gas, the pressure term P ρ P0 ργ P0 > 0, γ > 1 satisfies
this condition. ∇· denotes the divergence in R3 and the symbol ⊗ denotes the Kronecker
appears in the momentum equation,
tensor product. The external term −αρ
u − βρ|
u|q−1 u
where α is a positive constant, β is another constant but can be either negative or positive.
The term −αρ
u is called the linear damping and throughout this paper we assume α ≡ 1.
2
Journal of Applied Mathematics
with q >1 is regarded as a nonlinear source to the linear damping,
The term −βρ|
u|q−1 u
where the symbol |
u| denotes u21 u22 u23 if we assume u
u1 , u2 , u3 . When β > 0, the
is nonlinear damping, while, when β < 0 this term is regarded as nonlinear
term −βρ|
u|q−1 u
accumulating. For convenience, we call both the two cases nonlinear damping.
System 1.1 is supplemented by the following initial and boundary conditions:
0 x,
ρ, u
x, 0 ρ0 , u
x
x1 , x2 , x3 ∈ Ω,
u
·n
|∂Ω 0, t ≥ 0,
ρ0 d
x ρ∗ > 0,
1.2
Ω
where Ω ∈ R3 is a bounded domain with smooth boundary ∂Ω, n
is the unit outward normal
vector of ∂Ω, and the last condition is imposed to avoid the trivial case, ρ ≡ 0.
For 1D case, system 1.1 and its corresponding p-system in Lagrangian coordinates
have been studied intensively during the past decades. When β 0, for various initial
and initial-boundary conditions, both the existence and large time behaviors of solutions
including classical and weak solutions were investigated, see 1–10 and the references
therein. 11–13 studied the p-system with nonlinear damping β / 0. The existence as well
as approximate behavior of smooth solutions to the initial boundary condition in half line
and Cauchy problem are considered.
From physical point of view, the 3D model 1.1 describes more realistic phenomena.
Also, the 3D compressible Euler equations carry some unique features, such as the effect of
vorticity, which are totally absent in the 1D case and make the problem more mathematically
challenging. Thus, due to strong physical background and significant mathematical
challenge, system 1.1 and its time-asymptotic behavior are of great importance and are
much less understood than its 1D companion. When β 0, investigations were carried
out among small smooth solutions and we refer the readers to 14–16. In the direction of
nonlinear damping β / 0, even the global existence of classical solutions is still open, no less
than the asymptotic behavior. In this paper, we consider the global existence and asymptotic
behavior of classical solutions to the 3D problem 1.11.2 with nonlinear damping and slip
boundary condition.
When β 0, it has been proved that see 15 the dissipation in the momentum
equations and the boundary effect make system 1.1 be approximated by the decoupled
system
ρt ΔP ρ ,
1.3
ρu
M
−∇P ρ ,
where the first equation is the famous porous medium equation with P ρ ργ /γ, while the
second one states Darcy’s law. The corresponding initial-boundary conditions change into
ρ
x, 0 ρ0 x,
∇P ρ · n
|∂Ω 0, t ≥ 0,
ρ0 d
x ρ∗ > 0.
Ω
1.4
Journal of Applied Mathematics
3
When β /
0, we will prove in this paper that the effect of nonlinear damping β < 0 or
accumulating β > 0 can be ignored comparing with the linear damping when the initial
perturbation around equilibrium state is small.
Before stating our main results, we give some notations. Throughout this paper, · and · s denote the norms of L2 Ω and H s Ω, respectively. For any vector valued function
F f1 , f2 , f3 : Ω → R3 , we denote
2 2 2
F2s f1 s f2 s f3 s ,
FL∞ f1 L∞ f2 L∞ f3 L∞ ,
1.5
furthermore, for any functional matrix M Mij n×n : Ω → Rn×n , we denote
M2s n Mij 2 ,
s
ML∞ i,j1
n Mij ∞ .
L
1.6
i,j1
The energy space under consideration is
X3 Ω, 0, T F : Ω × 0, T −→ R3 or R
1.7
equipped with norm
F3,T
3 2
l
ess sup |F| ess sup
∂t F·, t
0≤t≤T
0≤t≤T
l0
1/2
3−l
,
1.8
for any F ∈ X3 Ω, 0, T . The notation |Ω| denotes the measure of the R3 domain Ω. In this
paper, unless specified, C will denote a generic constant which is independent of time. The
followings are the main results of this paper.
0 · n
|∂Ω 0 for
Theorem 1.1. Suppose the initial data satisfy the compatibility condition ∂lt u
0 ∈ H 3 Ω and
0 ≤ l ≤ 2. Then there exists a constant δ > 0 such that if ρ0 − ρ∗ /|Ω| , u
0 3 ≤ δ, the initial boundary condition 1.1 and 1.2 exists a unique global
ρ0 − ρ∗ /|Ω|, u
solution ρ, u
in C1 Ω × 0, ∞ ∩ X3 Ω, 0, ∞. Moreover, the global solution satisfies
∗ ρ∗
−η1 t
ρ − ρ ·, t |
u·, t| ≤ C1 ρ0 −
,u
0 exp ,
|Ω|
|Ω|
3
1.9
for some certain positive constants C1 and η1 .
Theorem 1.2. Let ρ, u
be the unique global smooth solution of 1.1 and 1.2, ρ,
u
be the global
solution of 1.31.4. Then there exist constants C2 and η2 such that
·, t
ρ − ρ ·, t u
−
u
≤ C2 exp−η2 t
1
satisfies for big enough t > 0.
1.10
4
Journal of Applied Mathematics
Theorem 1.1 states that the global solution of 1.1 and 1.2 converges to the steady
state ρ∗ /|Ω|, 0 exponentially fast in time. To prove this theorem, we first change problem
1.1 into
ρt ∇ · ρ
u 0,
∇P ρ
u
t u
−β|
u|q−1 u
· ∇
uu
,
ρ
1.11
and then we consider the existence and large time behavior of perturbation solution. It is worth
pointing out that in 15 the pressure P is assumed to be P ρ ργ /γ, then the authors
can introduce a nonlinear transformation σ ρθ /θθ γ − 1/2 to reformulate the
perturbation system as a symmetric hyperbolic system. In this paper, the pressure is more
general and the transformation in 15 do not work any more. To overcome this difficulty,
we use a symmetrizer to reduce the system to a symmetric hyperbolic one in the sense
of Friedrichs. Due to the slip boundary condition, the basic energy estimates can not be
applied directly to spatial derivatives. Inspired by 15, 17, we use time-derivatives, which
still preserve the boundary conditions, to estimate the spatial derivatives.
Theorem 1.2 is our target result. To prove this theorem, we first claim system 1.3
and 1.4 decays to the steady state ρ∗ /|Ω|, 0 exponentially, too. Then using the triangular
inequality we get Theorem 1.2.
Now, we recall some inequalities which will be used in the following.
Lemma 1.3. Let Ω be any bounded domain in R3 with smooth boundary. Then
f ∞ Ω ≤ Cf 2 Ω,
L
H
f p Ω ≤ Cf 1 ,
L
H Ω
2≤p≤6
1.12
for some constant C > 0 depending only on Ω.
Lemma 1.4. Let u
∈ H s Ω be a vector-valued function satisfying u
·n
|∂Ω 0, where n
is the unit
outer normal of ∂Ω. Then
s−1 ∇ · u
s−1 us−1 ,
us ≤ C∇ × u
1.13
for s ≥ 1, and the constant C depends only on s and Ω.
2. Global Existence and Asymptotic Behavior
In this section, we will consider the global existence and the asymptotic behavior of system
1.11 and 1.2. Due to the boundary effect 1.22 and the dissipation in the velocity
equations, the kinetic energy is conjectured to vanish and the potential energy will converge
to a constant as time goes to infinity. Furthermore, since the conservation of mass and the
initial condition 1.23 , we expect
ρ, u
−→
ρ∗
,0
|Ω|
as t −→ ∞,
2.1
Journal of Applied Mathematics
5
where ρ, u
is the solution of problem 1.11 and 1.2, and |Ω| is the measure of Ω in R3 .
Without loss of generality, we assume |Ω| 1.
For this purpose, we consider the perturbation system:
0,
ϕ ρ∗ u
∇P ϕ ρ∗
u
t u
· ∇
uu
,
−β|
u|q−1 u
ϕ ρ∗
ϕt ∇ ·
2.2
where ϕ ρ − ρ∗ . Let gρ Q ρ P ρ/ρ, then Q ρ∇ρ ∇P ρ/ρ and system 2.2
changes into
ϕt ϕ ρ∗ ∇ · u
∇ϕ · u
0,
∗
· ∇
uu
g ϕ ρ ∇ϕ −β|
,
u|q−1 u
u
t u
2.3
i
i 0,
ϕt ϕ ρ∗ u
,i ϕ,i u
u
it u
j u
i,j u
i g ϕ ρ∗ ϕ,i −β|
u|q−1 u
i.
2.4
that is,
In matrix notation, we have
⎡ ⎤ ⎡
ϕt
1 ϕ ρ∗
u
1⎥
∗
⎢u
⎢
u
1
⎢ t ⎥ ⎢g ϕ ρ
2⎦
⎣u
⎣
t
0
0
u
3t
0
0
⎡
3
u
⎢
0
⎢
⎣
0
g ϕ ρ∗
Denoting w ϕ
u
0
u
3
0
0
0
0
u
1
0
⎡ ⎤
⎡
⎤ ϕ
u
2
0 ⎢ ,1 ⎥
1
u
⎢
⎥
⎥
⎢
0 ⎥⎢ ,1 ⎥ ⎢ 0 ⎣
0 ⎦⎢
g ϕ ρ∗
2,1 ⎥
⎦
⎣u
1
u
0
u
3
0 ϕ ρ∗
0
u
2
0
u
2
0
0
,1
⎡ ⎤
⎤ ϕ
0 ⎢ ,2 ⎥
1 ⎥
⎢u
0⎥
⎥⎢ ,2 ⎥
0 ⎦⎢
2,2 ⎥
⎦
⎣u
2
u
3
u
⎤
⎡ ⎤ ⎡
⎤ ϕ
0
0 ϕ ρ∗ ⎢ ,3 ⎥ ⎢
1 β|
u|q−1 u
1⎥
1 ⎥ ⎢u
⎥
⎢u
0
0 ⎥
⎥
⎥⎢ ,3 ⎥ ⎢
q−1 2 ⎥ 0.
3
2
⎢
⎥
⎢
2
⎦
0
u
β|
u| u
⎦
,3 ⎦ ⎣u
⎣u
0
u
3
u
3 β|
u|q−1 u
3
u
3,3
,2
2.5
and multiplying 2.5 on the left by the symmetrizer
D diag g ϕ ρ∗ , ϕ ρ∗ , ϕ ρ∗ , ϕ ρ∗ ,
2.6
we can rewrite the result to be
Dwt A1 w,1 A2 w,2 A3 w,3 f 0,
2.7
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Journal of Applied Mathematics
where
1
⎤
g ϕ ρ∗ u
ϕ ρ∗ g ϕ ρ∗
0
0
1
⎥
⎢
⎥
⎢ ϕ ρ∗ g ϕ ρ∗
0
0
ϕ ρ∗ u
⎥,
A1 ⎢
⎥
⎢
∗
1
0
0
0
ϕρ u
⎦
⎣
0
0
0
ϕ ρ∗ u
1
2
⎤
⎡
g ϕ ρ∗ u
0
0
ϕ ρ∗ g ϕ ρ∗
⎥
⎢
⎥
⎢
0
ϕ ρ∗ u
0
0
2
⎥,
A2 ⎢
⎥
⎢ ϕ ρ∗ g ϕ ρ∗
∗
2
0
0
ϕρ u
⎦
⎣
0
0
0
ϕ ρ∗ u
2
3
⎤
⎡
g ϕ ρ∗ u
0
0
ϕ ρ∗ g ϕ ρ∗
3
⎥
⎢
⎥
⎢
0
ϕ ρ∗ u
0
0
3
⎥,
⎢
A ⎢
3
⎥
∗
0
0
0
ϕρ u
⎦
⎣
∗
∗
∗
3
ϕρ g ϕρ
0
0
ϕρ u
⎡
⎤
0
⎢u
1
β|
u|q−1 u
1⎥
⎢
⎥
⎥
ϕ ρ∗ ⎢
q−1 2 ⎥.
2
⎢u
u| u
⎦
⎣ β|
⎡
2.8
u
3 β|
u|q−1 u
3
The existence of global solution to 2.7 can be proved by local existence result and
a priori estimates. The local existence is classical and we omit it here. As for the a priori
estimates, we first have the following estimate about the temporal derivatives of w.
Lemma 2.1. Let |w| ≤ δ be sufficiently small. Then there exists a positive constant C3 > 0 such
that
d
dt
Ω
∂kt w · D∂kt wd
x ρ∗
2
k
x, t d
x ≤ C3 |w|3 ,
∂t u
Ω
2.9
for k 0, 1, 2, 3.
Proof. (1) Zero Order Estimate.
Multiplying 2.7 by w and integrating over Ω, we have
Ω
w · Dwt d
x
3 i1
Ω
w · Ai w,i d
x
Ω
w · fd
x 0.
2.10
Journal of Applied Mathematics
7
Using Lemma 1.3 and Cauchy-Schwarz inequality, we have
Ω
1 d
1
w · Dwd
x−
w · Dt wd
x
2 dt Ω
2 Ω
1 d
≤
w · Dwd
x CDt L∞
x
|w|2 d
2 dt Ω
Ω
1 d
≤
w · Dwd
x C|w|3 .
2 dt Ω
w · Dwt d
x
2.11
For the second term coming from the left side of 2.10, we have
3 Ω
i1
3
1
w · Ai w d
x−
w Ai,i wd
x
i
2 Ω i1
Ω i1
3
1
≤
w · Ai n
i wds CAi,i ∞
x,
|w|2 d
L
2 ∂Ω
Ω
i1
w · Ai w,i d
x
1
2
3 2.12
where we have used the Divergence Theorem. Since
w·
3
·u
0.
Ai n
i w gϕ2 2ρgϕ ρ|
u|2 n
2.13
i1
Here g gρ, ρ ϕ ρ∗ , then 2.12 turns into
3 i1
Ω
w · Ai w,i d
x ≤ C|w|3 .
2.14
While the term
Ω
w · fd
x ρ∗ ϕ
ρ∗
1 β|
u|q−1 |
x≥
x,
u|2 d
u|2 d
|
2 Ω
Ω
2.15
for q > 1 and the smallness of δ. Thus, we have
d
dt
from 2.10 to 2.15.
Ω
w · Dwd
xρ
∗
Ω
x ≤ C|w|3
u|2 d
|
2.16
8
Journal of Applied Mathematics
(2) Higher Derivatives Estimate.
For k 1, 2, 3, taking ∂kt of system 2.7, we get
k! l k−l
∂t D∂t wt ∂lt Ai ∂k−l
.
w
,i
t
l!k − l!
l1
3
k
D ∂kt w Ai ∂kt w,i ∂kt f −
t
i1
2.17
Multiply 2.17 by ∂kt and integrate over Ω to have
Ω
3
∂kt w · D ∂kt w d
x
∂kt w · Ai ∂kt w,i d
x
∂kt w · ∂kt fd
x
t
Ω
k
k!
−
l!k
− l!
l1
Ω
i1
k
l i k−l
∂kt w · ∂lt D∂k−l
x.
t wt ∂t w · ∂t A ∂t w,i d
2.18
Ω
Just like the estimates from 2.11 and 2.15, we estimate the three terms on the left side of
2.18 as follows:
1 d
1
∂kt w · D ∂kt w d
x
∂kt w · D∂kt wd
x−
∂k w · Dt ∂kt wd
x
t
2 dt Ω
2 Ω t
Ω
1 d
k 2
≤
∂kt w · D∂kt wd
x CDt L∞
x
2.19
∂t w d
2 dt Ω
Ω
1 d
≤
∂k w · D∂kt wd
x C|w|3 ,
2 dt Ω t
3 3
3 1
1
∂kt w · Ai ∂kt w d
∂kt w · Ai ∂kt w,i d
x
x−
∂kt w · Ai,i ∂kt wd
x
i
2 Ω i1
2 Ω
i1 Ω
i1
3
1
k 2
k
i
k
≤
∂t w · A n
i ∂t wds CAi,i ∞
x
∂t w d
L
2 ∂Ω
Ω
i1
2.20
2 2
1
u
ρg∂kt ϕ ∂kt u
·n
ds
g ∂kt ϕ ρ∂kt u
·n
ds 2
2 ∂Ω
Ω
k 2
CAi,i ∞
x ≤ C|w|3 ,
∂t w d
L
Ω
∂kt w · ∂kt fd
x
Ω
∂kt u
·
≥ ρ∗ ϕ
C
k ∂lt ϕ ρ∗ ∂k−l
u
β|
u|q−1 u
d
x
t
l0
k l1
Ω
Ω
Ω
2
1 β|
u|q−1 ∂kt u
d
x
∂lt ϕ∂kt u
· ∂k−l
t
C ρ∗ ϕ
Ω
∂kt u
1 |
u|q−1 u
d
x
k
u|q−1 ∂k−l
∂lt 1 β|
d
x
t u
l1
ρ
k 2
d
x C|w|3 .
≥
∂t u
2 Ω
∗
2.21
Journal of Applied Mathematics
9
For the term on the right-hand side of 2.18 we have,
k
l
k−l
k
l i k−l
∂
x
w
·
∂
D∂
w
∂
w
·
∂
A
∂
w
t
,i d
t
t
t
t
t
t
Ω
k l i k−l ≤ C∂kt w 2 ∂lt D 4 ∂k−l
t wt 4 C∂t w 2 ∂t A 4 ∂t w,i L
L
L
L
L
L4
2.22
≤ C|w|3 ,
for l 1, 2 and l ≤ k ≤ 3. When l k 3, we get
k
l
k−l
k
l i k−l
∂t w · ∂t D∂t wt ∂t w · ∂t A ∂t w,i d
x
Ω
≤ C∂kt w 2 ∂lt D 2 wt L∞ C∂kt w 2 ∂lt Ai 2 w,i L∞ ≤ C|w|3 .
L
L
L
2.23
L
Combining 2.18 and 2.23, we get
d
dt
Ω
∂kt w
·
D∂kt wd
x
2
k
ρ
x ≤ C|w|3 ,
∂t ux, t d
∗
Ω
2.24
for k 1, 2, 3. Thus we prove Lemma 2.1.
For slip boundary condition, spatial derivatives can be controlled by temporal
derivatives and vorticity that is discussed below.
Lemma 2.2. Let |w| ≤ δ be sufficiently small and ϕ, u
be the solution of 1.11 and 1.2, then
there exists a constant C4 such that
2 3 2
2
l
∂lt w .
|w| ≤ C4
∂t ∇ × u
2.25
2
l0
2−l
l0
Proof. From the velocity equation 2.32 we have
1
u
t u
· ∇
u β|
u|q−1 u
.
∇ϕ − ∗
u
g ρ ϕ
2.26
Taking the L2 inner product of 2.26 with ∇, we get
1
q−1
u
·
∇
u
β|
u
u
· ∇ϕd
x
u
u
|
t
∗
Ω g ρ ϕ
q−1
≤ C ut 2 uL∞ ∇
uL∞ u2 β
u2
u2 ≤ C ut 2 C|w|r ,
u2 2
∇ϕ −
where r min{3, q 1} > 2.
2.27
10
Journal of Applied Mathematics
The continuity equation 2.31 implies
∇·u
−
Therefore, we have
1 · ∇ϕ .
ϕt u
ϕ ρ∗
2.28
2
2 ≤ C ϕt |w|3 .
∇ · u
2.29
Using Lemma 1.4 with s 1, we have
u21 ≤ C ∇ × u
2 u2
2 ∇ · u
2
≤ C ∇ × u
u2 ϕt |w|3 .
2 2.30
Next, we take time derivatives of 2.26 and 2.28. It is clear that every time derivative
!
2
up to order two of ∇ϕ and ∇ · u
is bounded by 3l0 ∂lt w . Furthermore, by using an
induction on the number of spatial derivatives, we get that the same is true for any derivative
up to order of two of ∇ϕ and ∇ · u
. Applying Lemma 1.4 with s 1, 2, 3, we complete the
proof of Lemma 2.2.
2
2
From Lemma 2.2, to prove Theorem 1.1 we only need to estimate ∂lt ∇ × u
2−l and
∂it w for l 0, 1, 2, i 0, 1, 2, 3. The following Lemma is contributed to the estimate of
∇×u
.
Lemma 2.3. Let |w| ≤ δ is sufficiently small, then for any solution w ϕ, u
of problem
1.111.2, one has
1 d
2 dt
2 2
l
∂t ∇ × u
l0
2 2
l
≤ C|wt|3 .
∂t ∇ × u
2.31
l0
Proof. Taking ∇× of the velocity equation in 2.3 and denoting the vorticity v
∇×u
, we
have
−∇ × u · ∇
u − ∇ × g ρ ∇ϕ − β∇ × |
v
t v
u|q−1 u
− β∇|
u|q−1 × u
.
v
· ∇
u−u
· ∇
v−v
∇ · u
− β|
u|q−1 v
2.32
Let ∂ denote any mixed time and spatial derivative of order at most 2, then by taking any
mixed derivative of the above equation, we have
∂
vt ∂
v −∂ ∇ × u · ∇
u β∇ × |
u|q−1 u
−∂−
v · ∇
uu
· ∇
vv
∇ · u
− ∂ β|
u|q−1 v
β∇|
u|q−1 × u
.
2.33
Journal of Applied Mathematics
11
Multiplying the above equation by ∂
v and integrating the result over Ω, similar the
calculations in 2.22 and 2.23, we get
1 d
vt2 ≤ C|w|3 ,
v2 ∂
∂
2 dt
2.34
by using the Sobolev inequality in Lemma 1.3. This completes the proof of Lemma 2.3.
The next Lemma gives the dissipation in density due to nonlinearity.
Lemma 2.4. There exists a positive constant C5 such that
d
−
dt
3
l1
l
∂l−1
ϕ∂
ϕ
d
x
t
t
Ω
3 l 2
ut 2 |w|r ,
u2 ∂t ϕ ≤ C5 2.35
l0
where r min{3, q 1}.
Proof. Due to the conservation of total mass, we know that
Using Poincaré’s inequality and 2.27, we have
"
Ω
ρ−ρ∗ d
x 0, that is,
2
ϕ ≤ C∇ϕ2 ≤ C ut |w|r .
u2 "
Ω
ϕd
x 0.
2.36
Taking the t derivative of 2.31 and applying ∇· to 2.32 , we get
ϕtt ∇ϕ · u
t ϕt ∇ · u
ϕ∇ · u
t 0,
2.37
t ∇ · u · ∇
u ∇ · u
∇ · g ρ ∇ϕ −β∇ · |
.
u|q−1 u
∇ · u
2.38
2.37 − 2.38 × ϕ yields
ϕtt ∇ϕ · u
t ϕt ∇ · u
0,
− ϕ∇ · u
· ∇
u g ρ ∇ϕ u
β|
u|q−1 u
2.39
ϕtt ∇ϕ · u
t ϕt ∇ · u
ϕ∇ · u
t 0.
2.40
that is,
Multiplying 2.40 by ϕ, and integrating the result over Ω, we have
d
dt
ϕϕt d
x−
∇ϕt · u
ϕd
x
∇ϕ · u
t ϕd
x
Ω
Ω
Ω
∇ · ϕ2 u
t d
x 0.
ϕt ∇ · u
t − ∇ ϕ2 · u
ϕd
x
ϕ2t d
x
Ω
Ω
Ω
2.41
12
Journal of Applied Mathematics
Using the Divergence Theorem, we obtain
d
dt
ϕϕt d
x−
ϕ2t d
x
∇ϕt · u
ϕdx ∇ϕ · u
t ϕd
x
Ω
Ω
Ω
Ω
ϕ2 u
ds − 2
ϕt ∇ · u
t · n
ϕ∇ϕ · u
t d
x 0.
ϕd
x
Ω
Ω
2.42
Ω
Then we have
d
−
dt
Ω
ϕϕt d
x
Ω
ϕ2t d
x ≤ C|w|3 .
2.43
Furthermore, applying the derivative ∂t to 2.40 we have
t ϕtt ∇ · u
tt ϕt ∇ · u
t 0. 2.44
ϕttt ∇ϕtt · u
ϕt ∇ · u
t ϕ∇ · u
∇ϕ · u
tt 2∇ϕt · u
Taking L2 inner product of 2.44 with ϕt , we get
−
d
dt
Ω
ϕtt ϕt d
x
Ω
ϕ2tt d
x−
Ω
∇ · ϕtt u
ϕ
utt ϕt d
x−
Ω
x−
∇ · ϕ2t u
t d
ϕ2t ∇
x 0,
ut d
Ω
2.45
that is,
−
d
dt
Ω
ϕtt ϕt d
x
Ω
ϕ2tt d
x ≤ C|w|3 ,
2.46
where we have used the boundary condition and the Sobolev inequality in Lemma 1.3. Next,
apply the derivative ∂t to 2.44 and times the result with ϕtt to get
d
−
dt
Ω
ϕttt ϕtt d
x
Ω
ϕ2ttt d
x ≤ C|w|3 ,
2.47
in which we have used
ϕtt ∇ · ϕttt u
ϕtt ∇ · ϕ
uttt ∇ · ϕttt ϕtt u
− ∇ϕtt ϕttt u
∇ · ϕϕtt u
ttt − ∇ϕtt ϕ
uttt ,
2.48
and the similar method as 2.46. Combining 2.43, 2.46, and 2.47, we finish the proof of
Lemma 2.4.
Journal of Applied Mathematics
13
Combining Lemmas 2.1 and 2.4, we can characterize the total dissipation of
2
l
l0 ∂t w . Let
!3
#
$
ρ∗ ρ∗ g ρ∗ /2
, ,
λ min
,
2C5 4
4
3 3 l
∂lt w · D∂lt w − λ
∂l−1
ϕ∂
ϕ
d
x.
Et t
t
Ω
We can easily see that
B such that
!3
l0
l0
2.49
l1
2
∂lt w and Et are equivalent, that is, there exist constants A and
3 3 2
2
A ∂lt w ≤ Et ≤ B ∂lt w .
l0
2.50
l0
Calculating 2.9 λ2.35, we have
d
dt
3 Ω
3 3 l 2
r
l
∂lt w · D∂lt w − λ
∂l−1
ϕ∂
ϕ
d
x
λ
w
∂
t
t ≤ C|w| ,
t
l0
l0
2.51
l0
where r min{3, r}, that is,
d
λ
Et Et ≤ C|w|r .
dt
B
2.52
Then Gronwall’s inequality and the equivalent relationship 2.50 deduce that
t
3 3 2
−λ/Bt
l 2
l
w
≤
C
w0
e
C
es−t |ws|r ds.
∂t ∂t
l0
2.53
0
l0
This inequality combines Lemmas 2.2 and 2.3 deduces the result of Theorem 1.1.
3. About Porous Medium Equations
In this section, we investigate the large time behavior of classical solutions to problem
1.31.4. As indicated in the introduction, we expect that 1.1 and 1.2 are captured by
1.3 and 1.4 time asymptotically if
Ω
ρ0 d
x
Ω
ρ0 d
x ρ∗ .
3.1
Noticing Theorem 1.1 and the triangle inequality, we only need to show that the large time
asymptotic state of 1.3 and 1.4 is also the constant state ρ∗ , 0.
14
Journal of Applied Mathematics
Consider
ρt ΔP ρ ,
with the initial data
Ω
ρ
x, 0 ρ0 x,
t ≥ 0,
∇P ρ · n
|∂Ω 0,
3.2
ρ0 d
x ρ∗ ,
3.3
ρ0 x ∈ L∞ Ω.
Here we assume ρ0 is uniform bounded, that is, there exists a constant ρ∗ < ρ < ∞ such that
0 ≤ ρ0 x ≤ ρ.
3.4
The global existence and the large time behavior of solutions to 3.2 and 3.3 have
been established in 18. The method we used in this section is similar with which in 15.
Yet, since the pressure is more general in this paper, we can not use the corresponding result
in 15 directly.
−∇P ρ.
Lemma 3.1. Let ρ be the global solution of problem 3.2 and 3.3, M
Then there exist
positive constants C > 0, η > 0, independent of time t, such that
ρ − ρ∗ M
≤ Ce−ηt ,
1
3.5
for big enough t.
Proof. From 18, we know that there exists a positive constant T > 0 such that the problem
3.2 and 3.3 has a classical solution ρ
x, t for t > T , and
ρ
x, t >
ρ∗
,
2
for t > T.
3.6
On the other hand, due to the comparison principle
0 ≤ ρ
x, t ≤ ρ,
for any x, t ∈ Ω × 0, L∞ .
3.7
Δ P ρ − P ρ∗ .
3.8
For t > T , We consider
ρ − ρ∗
t
Journal of Applied Mathematics
15
Taking L2 inner product of 3.8 with ρ − ρ∗ , we have
1 d
0
2 dt
Ω
ρ − ρ
∗ 2
1 d
ρ − ρ∗ 2 2 dt
d
x−
Ω
Δ P ρ − P ρ∗
ρ − ρ∗ d
x
2
∇ P ρ − P ρ∗ d
x,
P ρ
Ω
3.9
where we have used boundary condition 3.23 . Combining the increasing property of P ρ
and the inequality 3.7, we have
1 d
ρ − ρ∗ 2 1 ∇ P ρ − P ρ∗ 2 ≤ 0.
2 dt
P ρ
3.10
Multiplying 3.21 with P ρ,
we have
that is,
P ρ t − P ρ ΔP ρ 0,
3.11
P ρ − P ρ∗ t − P ρ Δ P ρ − P ρ∗ 0.
3.12
Define
ψ ρ − ρ∗ ,
φ P ρ − P ρ∗ .
3.13
Taking L2 inner product of 3.12 with Δφ, we get
∗
1 d
∇φ2 P ρ Δφ2 ≤ 0.
2 dt
2
3.14
1 d ψ 2 ∇φ2 C
∇φ2 Δφ2 ≤ 0,
2 dt
3.15
3.10 plus 3.14 deduce
min{P ρ∗ /2, 1/P ρ}. Since the conservation of total mass and the Poincaré’s
where C
inequality, we have
2
2
2
1
ψ ≤ C∇ψ 2 ≤ C
3.16
∗ ∇φ .
P ρ /2
Then inequality 3.15 turns into
1 d ψ 2 ∇φ2 C ψ 2 ∇φ2 Δφ2 ≤ 0.
2 dt
The Gronwall inequality and 3.16 deduce Lemma 3.1.
3.17
16
Journal of Applied Mathematics
Acknowledgments
This project is supported by the National Natural Science Foundation of China Grant no.
10901095 and the Promotive Research Fund for Excellent Young and Middle-Aged Scientists
of Shandong Province Grant no. BS2010SF025.
References
1 C. M. Dafermos, “A system of hyperbolic conservation laws with frictional damping,” Zeitschrift für
Angewandte Mathematik und Physik, vol. 46, pp. 294–307, 1995.
2 C. M. Dafermos and R. Pan, “Global BV solutions for the p-system with frictional damping,” SIAM
Journal on Mathematical Analysis, vol. 41, no. 3, pp. 1190–1205, 2009.
3 X. X. Ding, G. Q. Chen, and P. Z. Luo, “Convergence of the fractional step Lax-Friedrichs scheme and
Godunov scheme for the isentropic system of gas dynamics,” Communications in Mathematical Physics,
vol. 121, no. 1, pp. 63–84, 1989.
4 L. Hsiao and T. P. Liu, “Convergence to nonlinear diffusion waves for solutions of a system of
hyperbolic conservation laws with damping,” Communications in Mathematical Physics, vol. 143, no.
3, pp. 599–605, 1992.
5 F. M. Huang, P. Marcati, and R. H. Pan, “Convergence to the Barenblatt solution for the compressible
Euler equations with damping and vacuum,” Archive for Rational Mechanics and Analysis, vol. 176, no.
1, pp. 1–24, 2005.
6 F. M. Huang and R. H. Pan, “Asymptotic behavior of the solutions to the damped compressible Euler
equations with vacuum,” Journal of Differential Equations, vol. 220, no. 1, pp. 207–233, 2006.
7 F. M. Huang and R. H. Pan, “Convergence rate for compressible Euler equations with damping and
vacuum,” Archive for Rational Mechanics and Analysis, vol. 166, no. 4, pp. 359–376, 2003.
8 K. Nishihara and T. Yang, “Boundary effect on asymptotic behaviour of solutions to the p-system with
linear damping,” Journal of Differential Equations, vol. 156, no. 2, pp. 439–458, 1999.
9 K. Nishihara, W. Wang, and T. Yang, “Lp-convergence rate to nonlinear diffusion waves for p-system
with damping,” Journal of Differential Equations, vol. 161, no. 1, pp. 191–218, 2000.
10 R. Pan, “Darcy’s law as long-time limit of adiabatic porous media flow,” Journal of Differential
Equations, vol. 220, no. 1, pp. 121–146, 2006.
11 C. K. Lin, C. T. Lin, and M. Mei, “Asymptotic behavior of solution to nonlinear damped p-system
with boundary effect,” International Journal of Numerical Analysis and Modeling B, vol. 1, no. 1, pp.
70–92, 2010.
12 M. Mei, “Nonlinear diffusion waves for hyperbolic p-system with nonlinear damping,” Journal of
Differential Equations, vol. 247, no. 4, pp. 1275–1296, 2009.
13 C. T. Zhu and M. N. Jiang, “Lp -decay rates to nonlinear diffusion waves for p-system with nonlinear
damping,” Science in China A, vol. 49, no. 6, pp. 721–739, 2006.
14 T. C. Sideris, B. Thomases, and D. H. Wang, “Long time behavior of solutions to the 3D compressible
Euler equations with damping,” Communications in Partial Differential Equations, vol. 28, no. 3-4, pp.
795–816, 2003.
15 R. Pan and K. Zhao, “The 3D compressible Euler equations with damping in a bounded domain,”
Journal of Differential Equations, vol. 246, no. 2, pp. 581–596, 2009.
16 W. Wang and T. Yang, “The pointwise estimates of solutions for Euler equations with damping in
multi-dimensions,” Journal of Differential Equations, vol. 173, no. 2, pp. 410–450, 2001.
17 Y. Guo and W. Strauss, “Stability of semiconductor states with insulating and contact boundary
conditions,” Archive for Rational Mechanics and Analysis, vol. 179, no. 1, pp. 1–30, 2006.
18 J. L. Vázquez, The Porous Medium Equation, The Clarendon Press Oxford University Press, Oxford,
UK, 2007.
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