Global solution and large-time behavior of the 3D compressible Euler equations with damping

J. Differential Equations 254 (2013) 1686–1704

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Journal of Differential Equations www.elsevier.com/locate/jde

Global solution and large-time behavior of the 3D compressible Euler equations with damping Zhong Tan, Yong Wang ∗,1 School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 23 July 2012 Available online 22 November 2012 MSC: 35Q31 76N10 35Q35 35B40 Keywords: Euler equations with damping Global solution Optimal decay rates Interpolation

We construct the global unique solution to the compressible Euler equations with damping in R3 . We assume the H 3 norm of the initial data is small, but the higher order derivatives can ˙ −s norm (0  s < 3/2) or B˙ −s be arbitrarily large. When the H 2,∞ norm (0 < s  3/2) of the initial data is finite, by a regularity interpolation trick, we prove the optimal decay rates of the solution. As an immediate byproduct, the L p –L 2 (1  p  2) type of the decay rates follow without requiring that the L p norm of initial data is small. © 2012 Elsevier Inc. All rights reserved.

1. Introduction We consider the three-dimensional compressible Euler equations with damping

⎧ ⎪ ⎨ ∂t ρ + div(ρ u ) = 0, ∂t (ρ u ) + div(ρ u ⊗ u ) + ∇ p (ρ ) + aρ u = 0, ⎪ ⎩ (ρ , u )(x, t )|t =0 = (ρ0 , u 0 )(x), x ∈ R3 ,

(1.1)

which simulate that the compressible fluid flows through a porous medium. Here the unknown functions ρ , u represent the density, velocity of the fluid respectively, and the pressure p = p (ρ ) satisfies

*

Corresponding author. E-mail addresses: [email protected] (Z. Tan), [email protected] (Y. Wang). 1 Supported by the National Natural Science Foundation of China – NSAF (No. 10976026) and the National Natural Science Foundation of China (Grant No. 11271305). 0022-0396/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2012.10.026

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the γ -law: p (ρ ) = A ρ γ , with constants A > 0 and γ > 1 the adiabatic exponent. a > 0 is the damping constant and 1/a may be regarded as the relaxation time for some physical flows. For one-dimensional case, there are a lot of mathematical results about the existence and asymptotics of the solutions to the compressible Euler equations with damping. For the Cauchy problems, we can refer to [2,10] for existence of the global BV solutions, to [3,15] for L ∞ entropy weak solutions and to [7,8,23] for small smooth solutions. For the initial–boundary value problems, we can refer to [28,29] for existence of L ∞ entropy weak solutions and to [11,12,21,26] for existence of small smooth solutions. For the asymptotics of solutions, we refer to [14–17] for L ∞ entropy weak solutions and to [7,24,25,38] for small smooth solutions. Besides, there are some results on the non-isentropic compressible Euler equations with damping, see [9,13,22,27]. For multi-dimensional case: Wang and Yang [34] showed the global existence and pointwise estimates of the solutions by a detailed analysis of the Green function of the linearized system and some energy estimates; Sideris et al. [30] proved that if the initial data is small in an appropriate norm, then damping can prevent the development of singularities and the Cauchy problem has a unique global smooth solution which decays in the maximum norm to the background state at a rate of t −(3/2) ; when the initial data is near its equilibrium, Pan and Zhao [29] proved global existence and uniqueness of classical solutions to the initial–boundary value problem for the 3D damped compressible Euler equations on bounded domain with slip boundary condition and showed that the classical solutions converge to steady state exponentially fast in time; Liao et al. [19] showed the L p convergence rates of planar diffusion waves for multi-dimensional Euler equations with damping by using frequency decomposition and Green function based energy method; Fang and Xu [4] showed the existence and asymptotic behavior of C 1 solutions to the multi-dimensional compressible Euler equations with damping on the framework of Besov space by the spectral localization method; Tan and Wu [33] showed the long-time behavior and optimal decay rates of global strong solutions to the isentropic compressible Euler equations with damping when the regular initial data belong to the s space H  (R3 ) ∩ B˙ 1−,∞ (R3 ) with   4 and s ∈ [0, 1]. Moreover, the readers can also refer to [18,35,37] and references therein. The main purpose of this paper is to study the global existence and time-asymptotic behavior of small smooth solutions to the system (1.1) by a refined pure energy method. There are two main differences between the analysis of this paper and the known results for Navier–Stokes equations [6,36]. For Navier–Stokes(–Poisson) equations, there are strong dissipative terms (i.e. viscous of fluid) in the momentum equations, which is helpful for energy estimates. However, there is only a weak dissipative term (i.e. damping term) in the momentum equations of Euler equations, which causes troubles for energy estimates which are overcome by a detailed analysis. Another difference is that s we introduce the homogeneous Besov space B˙ 2−,∞ (0 < s  3/2) to consider the optimal time decay rates, as in [31]. At this moment, we can directly obtain the optimal L p –L 2 (1  p < 2) type of the optimal decay rates by the embedding lemma A.4. In particular, we do not require that the L p norm of initial data is sufficiently small. We reformulate the Cauchy problem of the compressible Euler system (1.1) as in [30]. The main point is to obtain a symmetric system. Introduce the sound speed

μ(ρ ) =



p  (ρ ),

¯ = μ(ρ¯ ) corresponding to the sound speed at a background density ρ¯ > 0. and set μ Define n=

2

γ −1





μ(ρ ) − μ¯ .

Then the Euler equations (1.1) are transformed into the following system:

⎧ ¯ div u = −u · ∇ n − ν n div u , ⎨ ∂t n + μ ¯ ∇ n = −u · ∇ u − ν n ∇ n , ∂ u + au + μ ⎩ t (n, u )|t =0 = (n0 , u 0 ),

(1.2)

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where

ν :=

γ −1 2

,



2

n0 =

γ −1



μ(ρ0 ) − μ¯ .

(1.3)

Notation. In this paper, we use H s (R3 ), s ∈ R, to denote the usual Sobolev spaces with norm · H s and L p (R3 ), 1  p  ∞, to denote the usual L p spaces with norm · L p . ∇  with an integer   0 stands for the usual any spatial derivatives of order . When  < 0 or  is not a positive integer, ∇  stands for Λ defined by Λ f := F −1 (|ξ | F f ), where F is the usual Fourier transform operator ˙ s (R3 ), s ∈ R, to denote the homogeneous Sobolev spaces on R3 and F −1 is its inverse. We use H with norm · H˙ s defined by f H˙ s := Λs f L 2 . We then recall the homogeneous Besov spaces. Let φ ∈ C 0∞ ( R 3ξ ) be such that φ(ξ ) = 1 when |ξ |  1 and φ(ξ ) = 0 when |ξ |  2. Let ϕ (ξ ) = φ(ξ ) − φ(2ξ ) ˙ and ϕ j (ξ ) = ϕ (2− j ξ ) for j ∈ Z. Then by the construction, k∈Z ϕ j (ξ ) = 1 if ξ = 0. We define  j f := F −1 (ϕ j ) ∗ f , then for s ∈ R and 1  p , r  ∞, we define the homogeneous Besov spaces B˙ sp,r (R3 ) with norm · B˙ s defined by p ,r

f B˙ sp,r :=



˙ j f rL p 2  rsj

1r .

j ∈Z

Particularly, if r = ∞, then

˙ j f L p . f B˙ sp,∞ := sup 2sj  j ∈Z

(1.4)

Throughout this paper we let C denote some positive universal constants. We will use a  b if a  Cb. We use C 0 to denote the depending on the initial data. For simplicity, we write

constants ( A , B ) X := A X + B X and f := R3 f dx. Our main results are stated in the following theorems: Theorem 1.1. Assume that (n0 , u 0 ) ∈ H N for an integer N  3. Then there exists a constant δ0 > 0 such that if

  (n0 , u 0 )

H3

 δ0 ,

(1.5)

then the problem (1.2) admits a unique global solution (n, u )(t ) satisfying that for all t  0,

  (n, u )(t )2 N + H

t

      ∇ n(τ )2 N −1 + u (τ )2 N dτ  C (n0 , u 0 )2 N . H H H

(1.6)

0

˙ −s for some s ∈ [0, 3/2) Theorem 1.2. Under the assumptions of Theorem 1.1, and assuming that (n0 , u 0 ) ∈ H s or (n0 , u 0 ) ∈ B˙ 2−,∞ for some s ∈ (0, 3/2], then for all t  0,

  (n, u )(t ) ˙ −s  C 0 H

(1.7)

  (n, u )(t ) ˙ −s  C 0 , B

(1.8)

or

2,∞

and

   ∇ (n, u )(t ) N −  C 0 (1 + t )− +2 s H

for  = 0, 1, . . . , N − 1.

(1.9)

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Note that the Hardy–Littlewood–Sobolev theorem (cf. Lemma A.3) implies that for p ∈ (1, 2], L p ⊂ ˙ −s with s = 3( 1 − 1 ) ∈ [0, 3/2) and Lemma A.4 implies that for p ∈ [1, 2), L p ⊂ B˙ −s with s = H 2,∞ p 2 3( 1p − 12 ) ∈ (0, 3/2]. Then by Theorem 1.2, we have the following corollary of the usual L p –L 2 type of the optimal decay results:

˙ −s or B˙ −s assumption by Corollary 1.3. Under the assumptions of Theorem 1.2 except that we replace the H 2,∞ p that (n0 , u 0 ) ∈ L for some p ∈ [1, 2], then the following decay results hold: +s p    ∇ (n, u )(t ) N −  C 0 (1 + t )− 2 H

for  = 0, 1, . . . , N − 1.

(1.10)

Here the number s p is defined by s p := 3( 1p − 12 ). The followings are several remarks for Theorem 1.1, Theorem 1.2 and Corollary 1.3. Remark 1.4. Notice that for the global existence of the solution in Theorem 1.1 we only assume that the H 3 norm of initial data is small, while the higher order Sobolev norm can be arbitrarily large.

˙ −s and B˙ −s are preserved along Remark 1.5. From (1.7) and (1.8), we prove that the norms of H 2,∞ with the evolution of time, which plays a key role in the proof of time decay results. However, in Corollary 1.3 it is difficult to show that the L p norm of the solution can be preserved. We claim that the decay results of (n, u ) in Theorem 1.2 are optimal in the sense that they are consistent with those in the linearized case. Note that the L 2 decay rate of the higher order spatial derivatives of the solution is obtained. Then the general optimal L q (2  q  ∞) decay rates of the solution follow by the Sobolev interpolation. The present paper is structured as follows. In Section 2, we will establish the refined energy estimates and derive the negative Sobolev and Besov estimates. We will prove Theorem 1.1 and Theorem 1.2 in Section 3. The analytic tools used in this paper will be collected in Appendix A. 2. Nonlinear energy estimates 2.1. Energy estimates In this subsection, we will derive the a priori nonlinear energy estimates for the system (1.2). Hence we assume a priori assumption that for a sufficiently small δ > 0,

  (n, u )(t )

H3

 δ.

(2.1)

We show the first type of energy estimates which contains the dissipation estimate for u. Lemma 2.1. Assume that 0  k  N − 1, then we have

d dt

        ∇ k (n, u )22 + C ∇ k u 22  δ ∇ k+1n22 + ∇ k u 22 . L L L L

(2.2)

Proof. For 0  k  N − 1, applying ∇ k to (1.2)1 , (1.2)2 and then multiplying the resulting identities by ∇ k n, ∇ k u respectively, summing up and integrating over R3 , we obtain

1 d 2 dt



   k  ∇ (n, u )2 + a∇ k u 22 = − L

 ∇ k (u · ∇ n + ν n div u )∇ k n + ∇ k (u · ∇ u + ν n∇ n) · ∇ k u

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Z. Tan, Y. Wang / J. Differential Equations 254 (2013) 1686–1704



 ∇ k (u · ∇ n)∇ k n −

=−

∇ k (u · ∇ u ) · ∇ k u

 ∇ k (n div u )∇ k n + ∇ k (n∇ n) · ∇ k u

−ν

:= I 1 + I 2 + ν I 3 .

(2.3)

We shall estimate the three terms on the right-hand side of (2.3). First we estimate I 1 . By Hölder’s and Sobolev’s inequalities, we obtain





I1 = −

0k



   ∇ k− u · ∇∇ n

C k ∇ k− u · ∇∇  n∇ k n 

L 6/5

 k  ∇ n 

L6

0k

     ∇ k− u · ∇∇  n 6/5 ∇ k+1n 2 . L L

(2.4)

0k

If 0    [ 2k ], by Hölder’s inequality and Lemma A.1, we have

 k−      ∇ u · ∇∇  n 6/5  ∇ k− u  2 ∇ +1n 3 L L L   1 −   1 −     u Lk2 ∇ k u  L 2 k ∇ α n L 2 k ∇ k+1n Lk2       δ  ∇ k +1 n  L 2 +  ∇ k u  L 2 ,

where

(2.5)

α is defined by +

3 2



  + (k + 1) × =α× 1− k

k

⇒

α=

3k − 2 2k − 2

 ∈

3 2

,3

since  

k 2

;

(2.6)

if [ 2k ] + 1    k, by Hölder’s inequality and Lemma A.1 again, we have

 k−  ∇ u · ∇∇ n

L 6/5

     ∇ k− u  L 3 ∇ +1n L 2 k−   +1   k−  +1   ∇ α u Lk2+1 ∇ k u Lk2+1 n Lk2+1 ∇ k+1n Lk2+1       δ ∇ k+1n L 2 + ∇ k u  L 2 ,

where

(2.7)

α is defined by

k−+

1 2

=α×

+1 k− +k× k+1 k+1

⇒

α=

3k − 2 + 1 2 + 2

 ∈

1 2

,3

since  

k+1 2

. (2.8)

In light of (2.5) and (2.7), we deduce from (2.4) that







 

2 2 I 1  δ ∇ k+1n L 2 + ∇ k u  L 2 .

Next, we estimate the term I 2 . By Lemma A.2 and Hölder’s inequality, we obtain

(2.9)

Z. Tan, Y. Wang / J. Differential Equations 254 (2013) 1686–1704

 I2 = −



 2  k   1 ∇ , u ∇ u + u · ∇∇ k u · ∇ k u  ∇ u L ∞ ∇ k u  L 2 −

 2 1  ∇ u L ∞ ∇ k u  L 2 +



2

1691



u · ∇ ∇k u · ∇k u

2

2  ∇ · u ∇ k u · ∇ k u  δ ∇ k u  L 2 .



(2.10)

Finally, we estimate the term I 3 . We split I 3 as:

 ∇ k (n div u )∇ k n + ∇ k (n∇ n) · ∇ k u

I3 = −

 =− 

 0k



=−  −



C k ∇ k−n∇  ∇ · u ∇ k n + ∇ k−n∇ +1 n · ∇ k u Ck ∇ 

k−





k

n∇ ∇ · u ∇ n − 

0k−1



C k ∇ k−n∇ +1 n · ∇ k u

0k−1

n ∇ · ∇ k u ∇ k n + n ∇ k +1 n · ∇ k u

:= I 31 + I 32 + I 33 .

(2.11)

First we estimate I 33 . By Hölder’s, Sobolev’s and Cauchy’s inequalities, we obtain

 I 33 = −

n ∇ · ∇ k u ∇ k n + n ∇ k +1 n · ∇ k u = −









n∇ · ∇ k u ∇ k n =

∇ n∇ k u ∇ k n

    2 2     ∇ n L 3 ∇ k u  L 2 ∇ k n L 6  δ ∇ k+1n L 2 + ∇ k u L 2 .

(2.12)

Next we estimate the term I 31 . By Hölder’s and Sobolev’s inequalities, we obtain





I 31 = −

0k−1







C k ∇ k−n∇  ∇ · u ∇ k n 

 k−   ∇ n ∇ ∇ · u 

L 6/5

 k  ∇ n 6 L

0k−1

 k−     ∇ n∇ ∇ · u  6/5 ∇ k+1n 2 . L L

(2.13)

0k−1

If 0    [ 2k ], by Hölder’s inequality and Lemma A.1, we have

 k−   ∇ n ∇ ∇ · u 

L 6/5

     ∇ k−n L 2 ∇ +1 u  L 3 +1   k−   k−   +1  n Lk2+1 ∇ k+1n Lk2+1 ∇ α u  Lk2+1 ∇ k u Lk2+1       δ ∇ k+1n 2 + ∇ k u  2 ,

L

where

(2.14)

L

α is defined by +

3 2

=α×

k− k+1

+k×

+1 k+1

⇒

α=

k + 2 + 3 2k − 2

∈

1 2

,3

if [ 2k ] + 1    k − 1, by Hölder’s inequality and Lemma A.1 again, we have

since  

k 2

;

(2.15)

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Z. Tan, Y. Wang / J. Differential Equations 254 (2013) 1686–1704

 k−   ∇ n ∇ ∇ · u 

L 6/5

     ∇ k−n L 3 ∇ +1 u  L 2 k−−1    +1   k−−1  +1  ∇ α n L 2k ∇ k+1n L 2 k u L 2 k ∇ k u L 2k       δ ∇ k+1n L 2 + ∇ k u  L 2 ,

where

(2.16)

α is defined by k−+

+1

k−−1 + (k + 1) × k  k 3 k+1 α=1+ ∈ ,3 since   . 2 + 2 2 2

1 2

⇒

=α×

k

(2.17)

In light of (2.14) and (2.16), we deduce from (2.13) that







 

2 2 I 31  δ ∇ k+1n L 2 + ∇ k u  L 2 .

(2.18)

Finally, we estimate the term I 32 . By Hölder’s inequality, we obtain





I 32 = −

C k ∇ k−n∇ +1 n · ∇ k u 

0k−1



 k− +1   k  ∇ n ∇ n  2 ∇ u  2 . L L

(2.19)

0k−1

If 0    [ 2k ], by Hölder’s inequality and Lemma A.1, we have

 k− +1  ∇ n ∇ n 

L2

     ∇ k−n L 6 ∇ +1n L 3    1−    1−      n Lk2+1 ∇ k+1n L 2 k+1 ∇ α n L 2 k+1 ∇ k+1n Lk2+1    δ ∇ k+1n 2 ,

(2.20)

L

where

α is defined by  + + =α× 1− 2 k+1 3



⇒

α=

3k + 3 2k − 2 + 2

 ∈

3 2

,3

since  

k 2

;

(2.21)

if [ 2k ] + 1    k − 1, by Hölder’s inequality and Lemma A.1 again, we have

 k− +1  ∇ n ∇ n 

L2

     ∇ k−n L 3 ∇ +1n L 6    1 −   1 −       ∇ α n Lk2−1 ∇ k+1n L 2 k−1 ∇ 2n L 2 k−1 ∇ k+1n Lk2−1    δ ∇ k+1n L 2 ,

where

(2.22)

α is defined by

  + (k + 1) × 1 − 2 k−1 k−1  −k + 1 3 k+1 ⇒ α = 2 + ,3 since   . ∈ 2 2 2

k−+

1

=α×

(2.23)

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In light of (2.20) and (2.22), we deduce from (2.19) that







 

2 2 I 32  δ ∇ k+1n L 2 + ∇ k u  L 2 .

(2.24)

Hence, by (2.12), (2.18) and (2.24), we deduce from (2.11) that







 

2 2 I 3  δ  ∇ k +1 n  L 2 +  ∇ k u  L 2 .

(2.25)

Summing up the estimates for I 1 − I 3 , i.e., (2.9), (2.10) and (2.25), we deduce (2.2) for 0  k  N − 1. 2 Next, we derive the second type of energy estimates which contains the dissipation estimate for u. Lemma 2.2. Assume that 0  k  N − 1, then we have

d dt

        ∇ k+1 (n, u )22 + C ∇ k+1 u 22  δ ∇ k+1n22 + ∇ k+1 u 22 . L L L L

(2.26)

Proof. Let 0  k  N − 1. Applying ∇ k+1 to (1.2)1 , (1.2)2 and multiplying the resulting identities by ∇ k+1 n, ∇ k+1 u respectively, summing up and then integrating over R3 by parts, we obtain

1 d



   k +1  ∇ (n, u )2 + a∇ k+1 u 22 L

2 dt



=−  =−

∇ k+1 (u · ∇ n + ν n div u )∇ k+1n + ∇ k+1 (u · ∇ u + ν n∇ n) · ∇ k+1 u ∇ k+1 (u · ∇ n)∇ k+1n + ∇ k+1 (u · ∇ u ) · ∇ k+1 u 

−ν

∇ k+1 (n∇ · u )∇ k+1n + ∇ k+1 (n∇ n) · ∇ k+1 u

:= M 1 + ν M 2 .

(2.27)

We shall estimate M 1 and M 2 . By Lemma A.2, Hölder’s and Cauchy’s inequalities, we obtain

 M1 = −

 =−  −

∇ k+1 (u · ∇ n)∇ k+1 n + ∇ k+1 (u · ∇ u ) · ∇ k+1 u 

    ∇ k +1 , u · ∇ n ∇ k +1 n + ∇ k +1 , u · ∇ u · ∇ k +1 u 



u · ∇∇ k+1 n∇ k+1 n + u · ∇∇ k+1 u · ∇ k+1 u

     2     ∇ u L ∞ ∇ k+1n L 2 + ∇ k+1 u L 2 ∇ n L ∞ ∇ k+1n L 2 + ∇ u L ∞ ∇ k+1 u  L 2    1 − u · ∇ ∇ k +1 n ∇ k +1 n + ∇ k +1 u · ∇ k +1 u 2

   ∇(n, u )

L∞

 k +1  ∇ (n, u )22 + 1 L

2 2     δ ∇ k+1n L 2 + ∇ k+1 u L 2 ,

2



∇ · u ∇ k +1 n ∇ k +1 n + ∇ · u ∇ k +1 u · ∇ k +1 u (2.28)

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Z. Tan, Y. Wang / J. Differential Equations 254 (2013) 1686–1704

and

 M2 = −

 =−

∇ k+1 (n∇ · u )∇ k+1n + ∇ k+1 (n∇ n) · ∇ k+1 u 

   ∇ k +1 , n ∇ · u ∇ k +1 n + ∇ k +1 , n ∇ n · ∇ k +1 u −



n ∇ · ∇ k +1 u ∇ k +1 n + n ∇ k +2 n · ∇ k +1 u

        ∇ n L ∞ ∇ k+1 u  L 2 + ∇ k+1n L 2 ∇ u L ∞ ∇ k+1n L 2 −    2  ∇(n, u ) L ∞ ∇ k+1 (n, u ) L 2 +







n ∇ · ∇ k +1 u ∇ k +1 n



∇ n ∇ k +1 u ∇ k +1 n

2 2     δ  ∇ k +1 n  L 2 +  ∇ k +1 u  L 2 .

(2.29)

Summing up the estimates for M 1 and M 2 , i.e., (2.28) and (2.29), we deduce (2.26) for 0  k  N − 1. 2 Now, we will use Eqs. (1.2) to recover the dissipation estimate for n. Lemma 2.3. Assume that 0  k  N − 1, then we have

d



dt

 2 2 2   ∇ k u · ∇∇ k n + C ∇ k+1n L 2  ∇ k u  L 2 + ∇ k+1 u L 2 .

(2.30)

Proof. Let 0  k  N − 1. Applying ∇ k to (1.2)2 and then multiplying it by ∇∇ k n, we obtain

 2 μ¯ ∇ k+1n L 2  −



        ∇ k ∂t u · ∇∇ k n + C ∇ k u  L 2 ∇ k+1n L 2 + ∇ k (u · ∇ u + ν n∇ n) L 2 ∇ k+1n L 2 . (2.31)

The delicate first term in the right-hand side of (2.31) involves ∇ k ∂t u, and the key idea is to integrate by parts in the t-variable and use the continuity equation. Thus integrating by parts for both the t- and x-variables, we obtain

 −

∇ k ∂t u · ∇∇ k n = − =−

d dt d dt



 ∇ k u · ∇∇ k n −



∇ k div u · ∇ k ∂t n

 2 ¯ ∇ k div u  L 2 + ∇ k u · ∇∇ k n + μ

 ∇ k div u · ∇ k (u · ∇ n + ν n div u ). (2.32)

Next, we will estimate the last two terms in (2.32) by



 ∇ k div u · ∇ k (u · ∇ n) =



C k ∇  u · ∇∇ k−n · ∇ k div u

0k

     ∇  u · ∇∇ k−n 2 ∇ k+1 u  2 .  L L 0k

If  = 0, then

(2.33)

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1695

        u · ∇∇ k n 2 ∇ k+1 u  2  u L ∞ ∇ k+1n 2 ∇ k+1 u  2 L L L L  k +1  2   k+1 2  δ ∇ n  2 + ∇ u  2 ; L

(2.34)

L

if 1    [ 2k ], by Hölder’s inequality and Lemma A.1, we have

   ∇ u · ∇∇ k−n

L2

     ∇ k+1−n L 6 ∇  u L 3 −1   k−+2   k−+2   −1  n Lk2+1 ∇ k+1n L 2k+1 ∇ α u L 2k+1 ∇ k+1 u  Lk2+1       δ ∇ k+1n 2 + ∇ k+1 u  2 ,

L

where

(2.35)

L

α is defined by +

1 2

k−+2

=α×

k+1

+−1

⇒

α=



3k + 3 2k − 2 + 4

∈

3 2

,3

since  

k 2

;

(2.36)

if [ 2k ] + 1    k, by Hölder’s inequality and Lemma A.1 again, we have

   ∇ u · ∇∇ k−n

L2

     ∇ k+1−n L 3 ∇  u L 6 k−    +1   k−  +1  ∇ α n Lk2+1 ∇ k+1n Lk2+1 u Lk2+1 ∇ k+1 u  Lk2+1       δ ∇ k+1n L 2 + ∇ k+1 u L 2 ,

where

(2.37)

α is defined by +1 k−+ =α× +k− 2 k+1 3

⇒

α=

3k + 3 2 + 2

 ∈

3 2

,3

since  

k+1 2

.

(2.38)

Hence, by (2.34), (2.35) and (2.37), we have



2 2    ∇ k div u · ∇ k (u · ∇ n)  δ ∇ k+1n L 2 + ∇ k+1 u L 2 .

(2.39)

 2 2   ∇ k div u · ∇ k (n div u )  δ ∇ k+1n L 2 + ∇ k+1 u  L 2 .

(2.40)

Similarly, we also have



Thus, in view of (2.32)–(2.40), together with Cauchy’s inequality, we obtain

 −

∇ k ut · ∇∇ k n  −

d dt



 2 2  ∇ k u · ∇∇ k n + C ∇ k+1 u  L 2 + δ ∇ k+1n L 2 .

(2.41)

On the other hand, by a method similar to the above, we have

 k  ∇ (u · ∇ u + ν n∇ n)

L2

      δ  ∇ k +1 n  L 2 +  ∇ k +1 u  L 2 .

(2.42)

Plugging the estimates (2.41)–(2.42) into (2.31), by Cauchy’s inequality, since δ is small, we then obtain (2.30). 2

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2.2. Negative Sobolev estimates In this subsection, we will derive the evolution of the negative Sobolev norms of the solution. In order to estimate the nonlinear terms, we need to restrict ourselves to that s ∈ (0, 3/2). We will establish the following lemma. Lemma 2.4. For s ∈ (0, 1/2], we have

d dt

     (n, u )2˙ −s + C u 2 −s  ∇(n, u )2 1 (n, u ) ˙ −s ; ˙ H H H H

(2.43)

and for s ∈ (1/2, 3/2), we have

   1/2     (n, u )2˙ −s + C u 2 −s  (n, u )s− ∇(n, u )5/12−s (n, u ) ˙ −s . ˙ H H L2 H H

d dt

(2.44)

Proof. Applying Λ−s to (1.2)1 , (1.2)2 and multiplying the resulting identities by Λ−s n, Λ−s u respectively, summing up and then integrating over R3 by parts, we obtain

1 d 2 dt

 (n, u )2˙ −s + a u 2 −s = ˙ H H



Λ−s (−ν n div u − u · ∇ n)Λ−s n − Λ−s (u · ∇ u + ν n∇ n) · Λ−s u

 n div u + u · ∇ n H˙ −s n H˙ −s + u · ∇ u + n∇ n H˙ −s u H˙ −s .

(2.45)

We now restrict the value of s in order to estimate the nonlinear terms on the right-hand side of (2.45). If s ∈ (0, 1/2], then 1/2 + s/3 < 1 and 3/s  6. Then using the estimate (A.11) in Lemma A.3 and the Sobolev interpolation of Lemma A.1, together with Hölder’s and Young’s inequalities, we obtain

n div u H˙ −s  n div u

1

L 1/2+s/3

1/2+s  

 ∇ n L 2

 n L 3/s ∇ u L 2

1/2−s ∇ 2n L 2 ∇ u L 2

 ∇ n 2H 1 + ∇ u 2L 2 .

(2.46)

u · ∇ n H˙ −s  ∇ u 2H 1 + ∇ n 2L 2 ;

(2.47)

Similarly, we can bound

u · ∇ u H˙ −s 

∇ u 2H 1

+ ∇ u 2L 2 ;

(2.48)

n∇ n H˙ −s  ∇ n 2H 1 + ∇ n 2L 2 .

(2.49)

Hence, plugging the estimates (2.46)–(2.49) into (2.45), we deduce (2.43). Now if s ∈ (1/2, 3/2), we shall estimate the right-hand side of (2.45) in a different way. Since s ∈ (1/2, 3/2), we have that 1/2 + s/3 < 1 and 2 < 3/s < 6. Then using the (different) Sobolev interpolation, we have s−1/2

n div u H˙ −s  n L 3/s ∇ u L 2  n L 2 

n 2H 1

3/2−s

∇ n L 2

+ ∇ u 2L 2 ;

u · ∇ n H˙ −s  u 2H 1 + ∇ n 2L 2 ;

∇ u L 2 (2.50) (2.51)

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1697

u · ∇ u H˙ −s  u 2H 1 + ∇ u 2L 2 ;

(2.52)

n∇ n H˙ −s  n 2H 1 + ∇ n 2L 2 .

(2.53)

Hence, plugging the estimates (2.50)–(2.53) into (2.45), we deduce (2.44).

2

2.3. Negative Besov estimates In this subsection, we will derive the evolution of the negative Besov norms of the solution. The argument is similar to the previous subsection. Lemma 2.5. For s ∈ (0, 1/2], we have

d dt

     (n, u )2˙ −s + C u 2 −s  ∇(n, u )2 1 (n, u ) ˙ −s ; B B H B˙ 2,∞

(2.54)

2,∞

2,∞

and for s ∈ (1/2, 3/2], we have

d dt

    1/2    (n, u )2˙ −s + C u 2 −s  (n, u )s− ∇(n, u )5/12−s (n, u ) ˙ −s . B B L2 H B˙ 2,∞

(2.55)

2,∞

2,∞

˙ j of (1.2)1 , (1.2)2 and multiplying the resulting identities by  ˙ j n,  ˙ j u respectively, Proof. We take  summing up and then integrating over R3 by parts, we obtain

  ˙ j u 22 = ˙ j (n, u )22 + a  L L

1 d 2 dt



˙ j (−ν n div u − u · ∇ n) ˙ jn −  ˙ j (u · ∇ u + ν n ∇ n ) ·  ˙ j u.  (2.56)

Further, multiplying the inequality above by 2−2sj and then taking the supremum over j ∈ Z, we have

1 d 2 dt

 (n, u )2˙ −s + a u 2 −s B B˙ 2,∞

 sup 2−2sj j ∈Z



2,∞

   ˙ jn +  ˙ j (u · ∇ u + ν n ∇ n ) ·  ˙ j u ˙ j (ν n div u + u · ∇ n)

 n div u + u · ∇ n B˙ −s n B˙ −s + u · ∇ u + n∇ n B˙ −s u B˙ −s . 2,∞

2,∞

2,∞

2,∞

(2.57)

s Then, as in the proof of Lemma 2.4, applying Lemma A.4 instead to estimate the B˙ 2−,∞ norm, we complete the proof of Lemma 2.5. Notice that we allow s = 3/2. 2

3. Proof of theorems 3.1. Proof of Theorem 1.1 In this subsection, we shall combine the energy estimates that we have derived in Section 2 to prove Theorem 1.1. We first close the energy estimates at each -th level in our weaker sense. Let N  3 and 0    m − 1 with 3  m  N. Summing up the estimates (2.2) of Lemma 2.1 from k =  to m − 1, and then

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adding the resulting estimates to the estimates (2.26) of Lemma 2.2 for k = m − 1, by changing the index and since δ is small, we obtain

d dt

      ∇ k (n, u )22 + C 1 ∇ k u 22  C 2 δ L L km

km



 k 2 ∇ n 2 . L

(3.1)

+1km

Summing up the estimates (2.30) of Lemma 2.3 from k =  to m − 1, we have

d dt





∇ k u · ∇∇ k n + C 3

km−1



   k 2  ∇ n 2  C 4 ∇ k u 22 . L L

+1km

(3.2)

km

Multiplying (3.2) by 2C 2 δ/C 3 , adding it to (3.1), since δ > 0 is small, we deduce that there exists a constant C 5 > 0 such that for 0    m − 1,

d dt

   k  ∇ (n, u )22 + 2C 2 δ L C3

km





 ∇ k u · ∇∇ k n

km−1

    k 2  ∇ n 2 + ∇ k u 22  0. L L



+ C5



+1km

(3.3)

km

Next, we define Em (t ) to be C 5−1 times the expression under the time derivative in (3.3). Observe that since δ is small Em (t ) is equivalent to ∇  (n, u )(t ) 2H m− , that is, there exists a constant C 6 > 0 such that for 0    m − 1,



2



2

C 6−1 ∇  (n, u )(t ) H m−  Em (t )  C 6 ∇  (n, u )(t ) H m− .

(3.4)

Then we may write (3.3) as that for 0    m − 1,

2  d m  +1 2 E + ∇ n H m−−1 + ∇  u  H m−  0. dt 

(3.5)

Taking  = 0 and m = 3 in (3.5), and then integrating directly in time, in light of (3.4), we obtain

    (n, u )(t )2 3  C 6 E 3 (t )  C 6 E 3 (0)  C 2 (n0 , u 0 )2 3 . 6 0 0 H H

(3.6)

By a standard continuity argument, this closes the a priori estimates (2.1) if we assume (n0 , u 0 ) H 3  δ0 is sufficiently small. This in turn allows us to take  = 0 and 3  m  N in (3.5), and then integrate it directly in time, to obtain

  (n, u )(t )2 m + H

t 0

This proves (1.6).

      ∇ n(τ )2 m−1 + u (τ )2 m dτ  C 2 (n0 , u 0 )2 m . 6 H H H

(3.7)

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3.2. Proof of Theorem 1.2 In this subsection, we will prove the optimal time decay rates of the unique global solution to system (1.2) obtained in Theorem 1.1. First we prove (1.9). Assume for the moment that we have proved (1.7) and (1.8). Using Lemma A.5, we have for  = 0, 1, . . . , N − 1,

   ∇ n 

L2

1   +s   +s 1+s  +1  +1+s  n H+  C 0 ∇ +1n L+2 1+s . ∇ n L2 ˙ −s

(3.8)

Using Lemma A.6, we similarly have for  = 0, 1, . . . , N − 1,

   ∇ n 

L2

1   +s   +s  n B˙+−1s+s ∇ +1n L+2 1+s  C 0 ∇ +1n L+2 1+s .

(3.9)

2,∞

This together with (1.6) implies in particular that for  = 0, 1, . . . , N − 1,

 +1 2      1 ∇ n N −−1 + ∇  u 2 N −  C 0 ∇  (n, u )2 N − 1+ +s . H H H

(3.10)

In view of (3.4) and (3.10), we then deduce from (3.5) with m = N the following time differential inequality

d dt

 1+ +1 s EN + C 0 EN  0 for  = 0, 1, . . . , N − 1.

(3.11)

Solving this inequality directly gives, together with (3.7),

EN (t )  C 0 (1 + t )−(+s) for  = 0, 1, . . . , N − 1.

(3.12)

Consequently, in view of (3.4), we obtain from (3.12) that

   ∇ (n, u )(t )2 N −  C 0 (1 + t )−(+s) for  = 0, 1, . . . , N − 1. H

(3.13)

Thus, by (3.13), (1.6) and the interpolation, we prove the optimal decay (1.9). Now, we turn back to prove (1.7) and (1.8). First, we prove (1.7) by Lemma 2.4. However, we are not able to prove it for all s ∈ [0, 3/2) at this moment. We must distinct the arguments by the value of s. First, for s ∈ [0, 1/2], integrating (2.43) in time, by (1.6), we obtain that for s ∈ (0, 1/2],

    (n, u )(t )2˙ −s  (n0 , u 0 )2˙ −s + C H H

t

    ∇(n, u )2 1 (n, u )(τ ) ˙ −s dτ H

H

0

     C 0 1 + sup (n, u )(τ ) H˙ −s . 0τ t

(3.14)

This implies (1.7) for s ∈ [0, 1/2], that is,

  (n, u )(t ) ˙ −s  C 0 for s ∈ [0, 1/2], H

(3.15)

and thus verifies (1.9) for s ∈ [0, 1/2]. Next, we prove (1.7) for s ∈ (1/2, 3/2). Notice that the arguments for the case s ∈ [0, 1/2] cannot ˙ −1/2 since H˙ −s ∩ L 2 ⊂ H˙ −s for be applied to this case. However, observing that we have n0 , u 0 ∈ H

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any s ∈ [0, s], we then deduce from what we have proved for (1.7) and (1.9) with s = 1/2 that the following decay result holds:

   ∇ (n, u )(t )2 N −  C 0 (1 + t )−(+1/2) for − 1    N − 1. H

(3.16)

2

Hence, by (3.16), we deduce from (2.44) that for s ∈ (1/2, 3/2),

    (n, u )(t )2˙ −s  (n0 , u 0 )2˙ −s + C H H

t

 1/2      ∇(n, u )5/12−s (n, u )(τ ) ˙ −s dτ (n, u )s− 2 L

H

H

0

t  C0 + C0

  (1 + τ )−(7/4−s/2) dτ sup (n, u )(τ ) H˙ −s 0τ t

0

     C 0 1 + sup (n, u )(τ ) H˙ −s .

(3.17)

0τ t

This implies (1.7) for s ∈ (1/2, 3/2), that is,

  (n, u )(t ) ˙ −s  C 0 for s ∈ (1/2, 3/2), H

(3.18)

and thus verifies (1.9) for s ∈ (1/2, 3/2). Now we will prove (1.8). As in the proof of (1.7), we also must distinct the arguments by the value of s. First, for s ∈ (0, 1/2], integrating (2.54) of Lemma 2.5 in time, by (1.6), we obtain that for s ∈ (0, 1/2],

    (n, u )(t )2˙ −s  (n0 , u 0 )2˙ −s + C B B 2,∞

t

2,∞

    ∇(n, u )2 1 (n, u )(τ ) ˙ −s dτ H B 2,∞

0

     C 0 1 + sup (n, u )(τ ) B˙ −s . 0τ t

(3.19)

2,∞

This gives (1.8) for s ∈ (0, 1/2] and thus verifies (1.9) for s ∈ (0, 1/2]. Next, let s ∈ (1/2, 3/2]. Ob−1/2 s s  ∩ L 2 ⊂ B˙ − serving that we have (n0 , u 0 ) ∈ B˙ 2,∞ since B˙ 2−,∞ 2,∞ for any s ∈ [0, s], where we have used Lemma A.7 and Lemma A.8, then as in the proof of (3.17) we deduce from (2.55) that for s ∈ (1/2, 3/2],

    (n, u )(t )2˙ −s  (n0 , u 0 )2˙ −s + C B B 2,∞

t

2,∞

 1/2      ∇(n, u )5/12−s (n, u )(τ ) ˙ −s dτ (n, u )s− L2 H B 2,∞

0

t  C0 + C0

  (1 + τ )−(7/4−s/2) dτ sup (n, u )(τ ) B˙ −s 0τ t

0

     C 0 1 + sup (n, u )(τ ) B˙ −s . 0τ t

2,∞

This gives (1.8) for s ∈ (1/2, 3/2] and thus verifies (1.9) for s ∈ (1/2, 3/2].

2,∞

(3.20)

Z. Tan, Y. Wang / J. Differential Equations 254 (2013) 1686–1704

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Appendix A. Analytic tools Lemma A.1. Let 2  p  +∞ and 0  m, α  ; when p = ∞ we require further that m    α + 2. Then we have that for any f ∈ C 0∞ (R3 ),

α + 1 and

 α      ∇ f  p  ∇ m f 12−θ ∇  f θ 2 , L L L

(A.1)

where 0  θ  1 and α satisfy



α+3

1 2

−

1



p

= m(1 − θ) + θ.

(A.2)

Proof. For 2  p < ∞, it follows from the classical Sobolev embedding theorem that, refer to [1, p. 29, Theorem 1.38] for instance,

 α  ∇ f 

Lp

   ∇ β f 

L2

with β = α + 3

1 2

−

1 p

.

(A.3)

By the Parseval theorem and Hölder’s inequality, we have

 β  ∇ f 

L2

1−θ  θ   ∇ m f  L 2 ∇  f  L 2 ,

(A.4)

where θ is defined by (A.2). Hence when 2  p < ∞, (A.1) follows by (A.3)–(A.4). Now for p = ∞, by [32, p. 10, Proposition 3.8], we have

 α      ∇ f  ∞  ∇ α +1 f 12/2 ∇ α +2 f 12/2 . L L L

(A.5)

Since m  α + 1 and   α + 2, we can employ (A.1) with p = 2 to have

 α +1  ∇ f

1− α+1−m   α+1−m   ∇ m f L 2 −m ∇  f L 2−m ,

(A.6)

 α +2  1− α+2−m   α+2−m  ∇ f  L 2  ∇ m f  L 2 −m ∇  f  L 2−m .

(A.7)

L2

and

Substituting (A.6)–(A.7) into (A.5), we have

 α   α+3/2−m   α+3/2−m  ∇ f  ∞  ∇ m f 12− −m ∇  f  2 −m . L L L This is exactly (A.1) with p = ∞ by denoting θ =

α +3/2−m −m

.

(A.8)

2

We recall the following commutator estimate: Lemma A.2. Let k  1 be an integer and define the commutator



 ∇ k , g h = ∇ k ( gh) − g ∇ k h.

(A.9)

Then we have

 k    ∇ , g h

L2

       C k ∇ g L ∞ ∇ k−1 hL 2 + ∇ k g L 2 h L ∞ .

Proof. It can be proved by using Lemma A.1, see Lemma 3.4 in [20] (p. 98) for instance.

(A.10)

2

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We have the following L p embedding lemmas: Lemma A.3. Let 0  s < 3/2, 1 < p  2 with 1/2 + s/3 = 1/ p, then

f H˙ −s  f L p .

(A.11)

Proof. It follows from the Hardy–Littlewood–Sobolev theorem, see [5].

2

s Lemma A.4. Suppose that s > 0 and 1  p < 2. We have the embedding L p ⊂ B˙ q−,∞ with 1/2 + s/3 = 1/ p. In particular we have the estimate

f B˙ −s  f L p .

(A.12)

2,∞

Proof. See Lemma 4.6 in [31].

2

We will give the special interpolation estimate: Lemma A.5. Let s  0 and   0, then we have

   ∇ f 

L2

1−θ   ∇ +1 f L 2 f θH˙ −s ,

where θ =

1

+1+s

(A.13)

.

Proof. By the Parseval theorem and Hölder’s inequality, we have

   ∇ f 

L2

 1−θ  θ 1−θ    = |ξ | fˆ  L 2  |ξ |+1 fˆ L 2 |ξ |−s fˆ  L 2 = ∇ +1 f L 2 f θH˙ −s .

2

(A.14)

Lemma A.6. Suppose k  0 and s > 0, then we have

 k  ∇ f 

L2

Proof. See Lemma 4.5 in [31].

1−θ   ∇ k+1 f L 2 f θB˙ −s ,

where θ =

2,∞

1

+1+s

.

(A.15)

2

The following embedding lemma are known: Lemma A.7. If 1  r1  r2  ∞, then

B˙ sp ,r1 ⊂ B˙ sp ,r2 .

(A.16)

We will give the special Besov interpolation estimate: Lemma A.8. Fix m >   k, and 1  p  q  r  ∞. We have

g B˙   g θB˙ k g 1B˙−θ m . 2,q

(A.17)

2, p

2,r

Here these parameters satisfy the following restrictions

 = kθ + m(1 − θ), Proof. See Lemma 4.8 in [31].

2

1 q

=

θ r

+

1−θ p

.

(A.18)

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References [1] H. Bahouri, J.Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, 2011. [2] C.M. Dafermos, R.H. Pan, Global BV solutions for the p-system with frictional damping, SIAM J. Math. Anal. 41 (2009) 1190–1205. [3] X.X. Ding, G.Q. Chen, P.Z. Luo, Convergence of the fraction step Lax–Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Comm. Math. Phys. 121 (1989) 63–84. [4] D.Y. Fang, J. Xu, Existence and asymptotic behavior of C 1 solutions to the multi-dimensional compressible Euler equations with damping, Nonlinear Anal. 70 (2009) 244–261. [5] L. Grafakos, Classical and Modern Fourier Analysis, Springer, 2004. [6] Y. Guo, Y.J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations 37 (2012) 2165–2208. [7] L. Hsiao, T.P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys. 143 (1992) 599–605. [8] L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, 1998. [9] L. Hsiao, T. Luo, Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media, J. Differential Equations 125 (1996) 329–365. [10] L. Hsiao, T. Luo, T. Yang, Global B V solutions of compressible Euler equations with spherical symmetry and damping, J. Differential Equations 146 (1998) 203–225. [11] L. Hsiao, R.H. Pan, Initial boundary value problem for the system of compressible adiabatic flow through porous media, J. Differential Equations 159 (1999) 280–305. [12] L. Hsiao, R.H. Pan, The damped p-system with boundary effects, Contemp. Math. 255 (2000) 109–123. [13] L. Hsiao, R.H. Pan, Global existence of solutions for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal. 27 (1996) 70–77. [14] F.M. Huang, P. Marcati, R.H. Pan, Convergence to Barenblatt solution for the compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal. 176 (2005) 1–24. [15] F.M. Huang, R.H. Pan, Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, J. Differential Equations 220 (2006) 207–233. [16] F.M. Huang, R.H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal. 166 (2003) 359–376. [17] F.M. Huang, R.H. Pan, Z. Wang, L 1 convergence to the Barenblatt solution for compressible Euler equations with damping, Arch. Ration. Mech. Anal. 200 (2011) 665–689. [18] Y.Q. Liu, W.K. Wang, Well-posedness of the IBVP for 2-D Euler equations with damping, J. Differential Equations 245 (2008) 2477–2503. [19] J. Liao, W.K. Wang, T. Yang, L p convergence rates of planar waves for multi-dimensional Euler equations with damping, J. Differential Equations 247 (2009) 303–329. [20] A.J. Majda, A.L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. [21] P. Marcati, M. Mei, Convergence to nonlinear diffusion waves for solutions of the initial boundary value problem to the hyperbolic conservation laws with damping, Quart. Appl. Math. 58 (2000) 763–783. [22] P. Marcati, R.H. Pan, On the diffusive profiles for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal. 33 (2001) 790–826. [23] T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D’Orsay 78 (1978) 46–53. [24] K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations 131 (1996) 171–188. [25] K. Nishihara, W.K. Wang, T. Yang, L p -convergence rate to nonlinear diffusion waves for p-system with damping, J. Differential Equations 161 (2000) 191–218. [26] K. Nishihara, T. Yang, Boundary effect on asymptotic behavior of solutions to the p-system with damping, J. Differential Equations 156 (1999) 439–458. [27] R.H. Pan, Darcy’s law as long time limit of adiabatic porous media flows, J. Differential Equations 220 (2006) 121–146. [28] R.H. Pan, K. Zhao, Initial boundary value problem for compressible Euler equations with damping, Indiana Univ. Math. J. 57 (2008) 2257–2282. [29] R.H. Pan, K. Zhao, The 3D compressible Euler equations with damping in a bounded domain, J. Differential Equations 246 (2009) 581–596. [30] T.C. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations 28 (2003) 795–816. [31] V. Sohinger, R.M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in Rnx , preprint, arXiv: 1206.0027, 2012. [32] M.E. Taylor, Partial Differential Equations III. Nonlinear Equations, 2nd ed., Springer, 2011. [33] Z. Tan, G.C. Wu, Large time behavior of solutions for compressible Euler equations with damping in R3 , J. Differential Equations 252 (2012) 1546–1561. [34] W.K. Wang, T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differential Equations 173 (2001) 410–450. [35] W.K. Wang, T. Yang, Existence and stability of planar diffusion waves for 2-D Euler equations with damping, J. Differential Equations 242 (2007) 40–71.

1704

Z. Tan, Y. Wang / J. Differential Equations 254 (2013) 1686–1704

[36] Y.J. Wang, Decay of the Navier–Stokes–Poisson equations, J. Differential Equations 253 (2012) 273–297. [37] X.F. Yang, W.K. Wang, The suppressible property of the solution for three-dimensional Euler equations with damping, Nonlinear Anal. Real World Appl. 8 (2007) 53–61. [38] H.J. Zhao, Convergence to strong nonlinear diffusion waves for solutions of p-system with damping, J. Differential Equations 174 (2001) 200–236.