Global classical solutions to the one-dimensional compressible fluid models of Korteweg type with large initial data

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Global classical solutions to the one-dimensional compressible fluid models of Korteweg type with large initial data Zhengzheng Chen a , Xiaojuan Chai a , Boqing Dong a , Huijiang Zhao b,c,∗ a School of Mathematical Sciences, Anhui University, Hefei 230601, China b School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China c Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China

Received 6 February 2015; revised 27 May 2015

Abstract This paper is concerned with the global existence of classical solutions with large initial data away from vacuum to the Cauchy problem of the one-dimensional isothermal compressible fluid models of Korteweg type with density-dependent viscosity coefficient and capillarity coefficient. The case when the viscosity coefficient μ(ρ) = ρ α and the capillarity coefficient κ(ρ) = ρ β for some parameters α, β ∈ R is considered. Under some conditions on α, β, we first show the global existence of large solutions around constant states if the far-fields of the initial data are the same, while if the far-fields of the initial data are different, we prove the global stability of rarefaction waves with large strength. Here global stability means the initial perturbation can be arbitrarily large. Our analysis is based on the elementary energy method and the technique developed by Y. Kanel’ [29]. © 2015 Elsevier Inc. All rights reserved. MSC: 35Q35; 35L65; 35B40 Keywords: Navier–Stokes–Korteweg system; Large initial data; Rarefaction waves; Global stability; Energy estimates

* Corresponding author at: Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China.

E-mail address: [email protected] (H. Zhao). http://dx.doi.org/10.1016/j.jde.2015.05.023 0022-0396/© 2015 Elsevier Inc. All rights reserved.

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1. Introduction The one-dimensional isothermal compressible fluid models of Korteweg type with densitydependent viscosity coefficient and capillarity coefficient read in the Eulerian coordinates: 

ρτ + (ρu)y = 0, (ρu)τ + (ρu2 + p(ρ))y = (μ(ρ)uy )y + Ky ,

τ > 0, y ∈ R.

(1.1)

Here τ and y represent the time variable and the spatial variable, respectively. K is the Korteweg tensor given by K = ρκ(ρ)ρyy +

 1  ρκ (ρ) − κ(ρ) ρy2 . 2

The unknown functions ρ > 0, u and p = p(ρ) denote the density, the velocity, and the pressure of the fluids respectively, and μ = μ(ρ) > 0 and κ = κ(ρ) > 0 are the viscosity coefficient and capillary coefficient, respectively. In what follows, we will assume that μ(ρ) = μρ ˜ α , κ(ρ) = κρ ˜ β with the parameters α, β ∈ R and take the constants μ˜ = κ˜ = 1 for simplicity. System (1.1) can be used to model the motions of compressible isothermal viscous fluids with internal capillarity, see [3–5] for details on the derivation of the Korteweg model. Notice that when κ = 0, system (1.1) is reduced to the compressible Navier–Stokes system. There have been many mathematical results on the compressible fluid models of Korteweg type. In two or more space dimensions, Hattori and Li proved the local existence and global existence of smooth solutions for the isothermal [6,7] and nonisothermal [8] compressible fluid models of Korteweg type in Sobolev space. Bresch, Desjardins, and Lin [1] showed the global existence of weak solutions for an isothermal fluid in a periodic box or strip domain and such a result was later improved by Haspot in [9]. Danchin and Desjardins [2] studied the existence and uniqueness of suitably smooth solutions in critical Besov space and the local existence of strong solutions in a bounded domain of Rn was proved by Kotschote [10]. Recently, Wang and Tan [11], and Tan and Zhang [12] obtained the optimal temporal decay estimates of global smooth solution without external force. Li [13] and Wang et al. [14] discussed the global existence and optimal L2 -decay rate of smooth solutions with potential external force. Moreover, Chen and Zhao [27] proved the existence and nonlinear stability of stationary solution to the non-isothermal compressible fluid models of Korteweg type effected by the given mass source, the external force of general form, and the energy source. Notice that the studies of [2,10,11,13,14] are concentrated on the isothermal fluid, and all of the above results are obtained under small initial perturbation except [1,10]. In one space dimension, Tsyganov [15] studied the global existence and asymptotic convergence of weak solution for an isothermal fluid with the viscosity coefficient μ(ρ) ≡ 1, the capillarity coefficient κ(ρ) = ρ −5 and large data on the interval [0, 1]. Charve and Haspot [16] proved the global existence of large strong solution for the isothermal system with μ(ρ) = ερ and κ(ρ) = ε2 ρ −1 in R. Moreover, they showed that the strong solution converges to a weak-entropy solution of the compressible Euler system as the parameter ε tends to zero. Haspot [9] obtained the global existence of weak solution of an isothermal compressible fluid model of Korteweg type with a general density-dependent viscosity coefficient and a specific type of capillarity coefficient under large initial perturbation in the energy space. Recently, Germain and LeFloch [17]

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studied the global existence of weak solution with large initial data for the isothermal compressible Korteweg system with general density-dependent viscosity and capillarity coefficients. They shown the global existence of weak solutions which may contain vacuum in the whole space R. Moreover, the global existence of non-vacuum weak solutions was also obtained on both the torus and the whole space in [17]. Finally, for the existence and nonlinear stability of some elementary waves (such as rarefaction waves, viscous shock profiles and contact discontinuity wave) to the isothermal or non-isothermal compressible fluid models of Korteweg type with small initial data, we refer to [22–26]. However, few results have been obtained for smooth and non-vacuum solutions of the compressible fluid models of Korteweg type with large initial data up to now. This paper is devoted to this problem and as a first step toward this goal, we are concerned with the global existence of classical solutions with large initial data away from vacuum to the initial value problem of system (1.1). Here unlike the small initial perturbation solutions, the viscosity coefficient μ(ρ) = ρ α and capillarity coefficient κ(ρ) = ρ β make a significant influence on the global existence of large solutions. We hope that such a study can shed some light on the construction of global classical, large solutions to the high-dimensional compressible fluid models of Korteweg type. We point out that there are a lot of results on the construction of non-vacuum, large solutions to the one-dimensional compressible Navier–Stokes equations in various contexts, see [28–37] and the references therein. The method in the present paper is essentially motivated by the work of Y. Kanel’ [29]. Now we begin to formulate our main results. Since it is convenient for us to study such a problem in the Lagrangian coordinates, we make the following coordinate transformation: y x=

ρ(τ, z)dz,

t = τ.

0

Then the system (1.1) in the Lagrangian coordinates becomes ⎧ ⎪ ⎨ vt − ux = 0, ⎪ ⎩ ut + p(v)x = where v =

1 ρ



u  t > 0, x ∈ R, vxx β + 5 vx2 x + − + , α+1 β+5 β+6 2 v v v x x

(1.2)

> 0 denotes the specific volume. We supplement system (1.2) with the initial data

(v(0, x), u(0, x)) = (v0 (x), u0 (x)),

lim (v0 (x), u0 (x)) = (v± , u± ),

x→±∞

(1.3)

where v± > 0 and u± are given constants. The main purpose of this paper is to show the global existence of classical, non-vacuum solutions to the Cauchy problem (1.2)–(1.3). To this end, we first consider the case when the far-fields of the initial data (v0 (x), u0 (x)) are equal, i.e. (v− , u− ) = (v+ , u+ ) := (v, ¯ u). ¯ Then our first theorem is concerned with the global existence of classical solutions to problem (1.2)–(1.3) around the constant state (v, ¯ u). ¯ Theorem 1.1 (Global existence around constant states). Let the pressure p(v) be a smooth function satisfying p (v) < 0 for v > 0, and the constants α, β satisfy one of the following conditions:

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α < −2β − 4,

(i)

3 β ≥− , 2

(iii)

5 3 α < −β − , −2 ≤ β < − , 2 2 −∞ < α < +∞, −3 ≤ β < −2,

(iv)

α > −2β − 5,

(ii)

β < −3.

(1.4)

Assume that the initial data v0 − v¯ ∈ H 4 (R), u0 − u¯ ∈ H 3 (R), and there exist two positive constants V¯ , V such that V ≤ v0 (x) ≤ V¯ for all x ∈ R. Then the Cauchy problem (1.2)–(1.3) admits a unique global classical solution (v, u)(t, x), which satisfies for any given constant T > 0 that C1−1 (T ) ≤ v(t, x) ≤ C1 (T ),

(v

2 − v)(t) ¯ H 4 (R)

2 + (u − u)(t) ¯ H 3 (R)

+

T

∀ (t, x) ∈ [0, T ] × R,

(1.5)

 ¯ x (s)2H 3 (R) ds (v − v) ¯ x (s)2H 4 (R) + (u − u)

0

≤ C2 (T ).

(1.6)

Here C1 (T ) is a positive constant depending only on T , α, β, V , V¯ , v0 − v ¯ H 1 (R) and ¯ L2 (R) , and C2 (T ) is a positive constant depending only on T , α, β, V , V¯ , v0 − v ¯ H 4 (R) u0 − u and u0 − u ¯ H 3 (R) . Remark 1.1. Some remarks concerning Theorem 1.1 are listed below: • The conditions (i)–(iv) are used to derive the positive lower and upper bounds of the specific volume v(t, x) by using Kanel’s method [29], see the proof of Lemmas 2.2 and 2.4 for details. • The global existence of weak solutions away from vacuum for the system (1.1) with μ(ρ) = ρ α and κ(ρ) = ρ β was also discussed by Germain and LeFloch in [17]. There the parameters α and β should satisfy the condition: 1 0≤α< , 2

2α − 4 ≤ β ≤ 2α − 1,

or α ≥ 0,

β < −2,

2α − 4 ≤ β ≤ 2α − 1,

which means that −4 ≤ β < 0, thus compared to the main results in [17], we obtain the global existence of classical solutions for the cases of β < −4 and β ≥ 0 in Theorem 1.1. Next, we consider the case when (v− , u− ) = (v+ , u+ ). In this situation, the left state (v− , u− ) can be connected to the right state (v+ , u+ ) by a rarefaction wave, and we are concerned with the global existence of smooth solutions to problem (1.2)–(1.3) around the rarefaction wave.

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Since rarefaction waves are expansive, the right hand side of (1.2) decays faster than each individual term on the left hand side. Therefore, the large time behavior of solutions to system (1.2) may be approximated by the following Riemann problem of the compressible Euler equations 

vt − ux = 0,

(1.7)

ut + p(v)x = 0 with the Riemann data  (v(t, x), u(t, x))|t=0 = (v0R , uR 0 )(x) =

(v− , u− ), x < 0, (v+ , u+ ), x > 0.

(1.8)

It is well known that there are two families of rarefaction waves which are solutions to the Riemann problem (1.7)–(1.8) (see [18]). Here, we only consider the 1-rarefaction wave. Similar argument can be applied to study the case when the solutions of the Riemann problem (1.7)–(1.8) consisting the superposition of 1-rarefaction wave and 2-rarefaction wave. That is, we assume that (v+ , u+ ) ∈ R1 (v− , u− ), where ⎧ ⎨

⎫  v  ⎬  R1 (v− , u− ) = (v, u)u = u− + −p  (z)dz, u ≥ u− ⎩ ⎭

(1.9)

v−

is the 1-rarefaction wave curve. The 1-rarefaction wave (V R , U R )(t, x) = (V R , U R )( xt ) is defined by ⎧ −∞ ≤ xt ≤ λ1 (v− ), (v− , u− ), ⎪ ⎪ ⎞ ⎛ ⎪ ⎪ x ⎪ λ−1 ⎪ 1 ( t ) ⎨⎜ x ⎟   x (V R , U R )( ) = ⎜λ−1 x , u− − λ1 (s) ds ⎟ ¯ 1 t ⎠ , λ1 (v− ) ≤ t ≤ λ1 (v), ⎝ ⎪ t ⎪ ⎪ ⎪ v− ⎪ ⎪ ⎩ (v, ¯ u), ¯ λ1 (v) ¯ ≤ xt ≤ +∞,

(1.10)

 with λ1 (v) = − −p  (v). Notice that rarefaction waves are only Lipschitz continuous. A smooth approximation to the above Riemann solution (V R , U R )(t, x) can be constructed as follows. As in [21], let w(t, x) be the unique global smooth solution to the following Cauchy problem ⎧ ⎨ wt + wwx = 0, ⎩ w(t, x)|t=0 = w0 (x) =

w + + w − w+ − w − + tanh( x), 2 2

(1.11)

where > 0 is a sufficiently small constant to be determined later. Then the smooth approximate wave (V , U )(t, x) of (V R , U R )(t, x) can be defined by

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⎧ w± = λ1 (v± ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ λ(V (t, x)) = w(t + 1, x), V(t,x) ⎪  ⎪ ⎪ ⎪ −p  (z)dz. ⎪ U (t, x) = u− + ⎪ ⎩

(1.12)

v−

The second aim of this paper is to show the time-asymptotic nonlinear stability of the 1-rarefaction wave (V R , U R )(t, x) for the Cauchy problem (1.2)–(1.3) with arbitrarily large initial data. Depending on whether the initial perturbation is small or large, such type nonlinear stability is usually called local stability or global stability. Then our second result on the global stability of the 1-rarefaction wave (V R , U R )(t, x) is stated as follows. Theorem 1.2 (Global stability of rarefaction waves). Assume that p(v) is a smooth function satisfying p  (v) < 0, p  (v) > 0 for v > 0, and the solution (V R , U R )(t, x) to the Riemann problem (1.7)–(1.8) is given by (1.10). Let (V , U )(t, x) be a smooth approximation of the Riemann solution (V R , U R )(t, x) constructed by (1.12), and β = 2α − 3,

−3 ≤ β < −2.

(1.13)

Suppose further that there exist two positive constants V , V¯ such that V ≤ v0 (x), V (t, x) ≤ V¯

(1.14)

for all (t, x) ∈ R+ × R, and v0 (x) − V (0, x) ∈ H 4 (R), u0 (x) − U (0, x) ∈ H 3 (R). Then the Cauchy problem (1.2)–(1.3) admits a unique global smooth solution (v, u)(t, x) satisfying C3−1 ≤ v(t, x) ≤ C3 ,

(v

− V )(t)2H 4 (R)

+ (u − U )(t)2H 3 (R)

+

t

∀ (t, x) ∈ [0, ∞) × R,

(1.15)

 (v − V )x (s)2H 4 (R) + (u − U )x (s)2H 3 (R) ds

0

 ≤ C4 (v0 (·) − V (0, ·)2H 4 (R) + u0 (·) − U (0, ·)2H 3 (R) + 1 ,

∀ t > 0,

(1.16)

and  

  lim sup  v(t, x) − V R (t, x), u(t, x) − U R (t, x)  = 0.

t→+∞ x∈R

(1.17)

Here C3 > 0 is a constant depending only on V , V¯ , v0 (·) − V (0, ·)H 1 (R) and u0 (·) − U (0, ·)L2 (R) , and C4 > 0 is a constant depending only on V , V¯ , v0 (·) − V (0, ·)H 4 (R) and u0 (·) − U (0, ·)H 3 (R) .

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Remark 1.2. Several remarks on Theorems 1.1–1.2 are given as follows. • In Theorem 1.1, if the parameters α and β satisfy (1.13), then by using the same argument as that in the proof of Theorem 1.2, we can also show that the Cauchy problem (1.2)–(1.3) admits a unique global classical solution (v, u)(t, x) which satisfies some similar estimates as (1.15) and (1.16). Consequently, we have the following asymptotic behavior of solutions, i.e., (v − v, ¯ u − u)(t) ¯ L∞ → 0 as t → +∞. • In Theorem 1.2, the strength of rarefaction wave δ := |v+ − v− | + |u+ − u− | is only required to be finite and has no need to be restricted to small. • In Theorem 1.2, we obtain the global stability of strong rarefaction waves for the onedimensional isothermal compressible fluid models of Korteweg type with density-dependent viscosity coefficient and capillarity coefficient, which partially answers an open problem proposed by the first author in [24]. However, the global stability of strong rarefaction waves for the one-dimensional isothermal compressible fluid models of Korteweg type with constant viscosity and capillarity coefficients is still open. • In Theorems 1.1–1.2, we obtain the global existence of classical solutions to the onedimensional isothermal compressible fluid models of Korteweg type for a variety of densitydependent viscosity coefficient and capillarity coefficient. However, our results cannot cover the case when both the viscosity coefficient and the capillarity coefficient are constants. Indeed, even for the one-dimensional case, the global existence of weak solution of the compressible fluid models of Korteweg type with constant viscosity and capillarity coefficients and large initial data in the whole space is still open up to now. This interesting problem will be pursued by the authors in the future. • In Theorems 1.1–1.2, we show the global existence of large solutions for the onedimensional isothermal compressible fluid models of Korteweg type. It is interesting to study the global existence of large solutions for the non-isothermal compressible fluid models of Korteweg type, which is left for the future. Now we outline the main ideas used in proving our main results. The main difficulty in the proof of Theorem 1.1 is to deduce the positive lower and upper bounds for the specific volume v(t, x). To achieve this, we mainly use the method of Y. Kanel’ [29]. Firstly, due to effect of the  vx2 Korteweg tensor, an estimate of R v β+5 dx appears in the basic energy estimate (see Lemma 2.1). √ Based on this and the estimate of  (v, v)(t) ¯ given in (2.7), the lower and upper bound of v(t, x) for the case (iii) of Theorem 1.1 follows easily by using Kanel’s method (see Lemma 2.2).  vx2 Secondly, we can derive the estimate of R v 2α+2 dx from equation (2.1)2 under some suitable assumptions on the parameters α and β (see Lemma 2.3). This, together with the estimate of √  (v, v)(t) ¯ and Kanel’s method gives the lower and upper bound of v(t, x) for the case (i), t  vx4 (ii) and (iv) of Theorem 1.1. Here we remark that a difficult term 0 R v α+β+8 dxdτ appears in the  vx2 estimate of R v 2α+2 dx. We deal with this term by using the Sobolev inequality and the estimate  vx2 dx repeatedly (see (2.22)–(2.24)). Notice that the global existence of weak solution of R v β+5 for the isothermal compressible fluid models of Korteweg type with large initial data and the viscosity coefficient μ(ρ) = ρ α and capillarity coefficient κ(ρ) = ρ β is also studied by Germain and LeFloch in [17]. However, the method in [17] cannot be used here to treat this difficult term since the analysis there relies crucially on a nonlinear Sobolev inequality, which leads to the global existence of weak solutions for another variety of parameters α and β. Having obtained

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the lower and upper bound of the specific volume v(t, x), the higher order energy estimates of solutions can be deduced by using the lower order energy estimates and Gronwall’s inequality. Then as usual, Theorem 1.1 follows by combining the local existence result and the a priori estimates. The proof of Theorem 1.2 is motivated by [19–21] for the global stability of rarefaction waves of the compressible Navier–Stokes equations. First, we make the a priori assumption (3.6) for the specific volume v(t, x). Then under this a priori assumption and (1.13), we can show the uniformin-time lower and upper bounds for v(t, x) and some uniform-in-time energy-type estimates of solutions to problem (3.3)–(3.4) by using some delicate  t estimates and Kanel’s method [29]. Here in order to derive the uniform-in-time estimate of 0 φx (τ )21 dτ , we perform an estimate of the Korteweg tensor in a style different from (2.20). And in this process, we need the parameters α and β satisfy the relation (1.13)1 , see (3.30)–(3.33) for details. Finally, the global existence of solutions can be obtained by combining the local existence result and energy estimates. The remainder of this paper is organized as follows. Section 2 is devoted to the proof of Theorem 1.1. In Section 3, we first recall some basic properties of the smooth approximate rarefaction wave, and then give the proof of Theorem 1.2. Notations. Throughout this paper, for simplicity, we will omit the variables t, x of functions if it does not cause any confusion. C denotes some generic constants which may vary in different estimates. If the dependence need to be explicitly pointed out, the notation C(·, · · · , ·) or Ci (·, · · · , ·)(i ∈ N) is used. H l (R) is usual l-th order Sobolev Space with its norm  f l =

l 

 12 ∂xi f 2

with

 ·    · L2 ,

i=0

and Lp (R) with 1 ≤ p ≤ +∞ denotes the standard Lebesgue space with the norm  ·Lp . Finally,  · L∞ stands for the norm  · L∞ ([0,T ]×R) . T ,x 2. Proof of Theorem 1.1 This section is devoted to proving Theorem 1.1. First, set φ(t, x) = v(t, x) − v, ¯

ψ(t, x) = u(t, x) − u. ¯

Then we can deduce from (1.2)–(1.3) that ⎧ ⎪ ⎨ φt − ψx = 0, ⎪ ¯ x= ⎩ ψt + [p(v) − p(v)]



u  t > 0, x ∈ R vxx β + 5 vx2 x + − + , α+1 β+5 β+6 2 v v v x x

(2.1)

with the initial data (φ, ψ)|t=0 = (φ0 , ψ0 )(x) = (v − v, ¯ u − u)(0, ¯ x). To prove Theorem 1.1, it suffices to show the following theorem.

(2.2)

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Theorem 2.1. Under the assumptions of Theorem 1.1, the Cauchy problem (2.1)–(2.2) admits a unique smooth solution (φ, ψ)(t) on [0, T ] × R for any arbitrarily large but fixed time T > 0. Moreover, it holds C1−1 (T ) ≤ v(t, x) ≤ C1 (T ), φ(t)24

+ ψ(t)23

∀ (t, x) ∈ [0, T ] × R,

(2.3)

T

 + φx (s)24 + ψx (s)23 ds ≤ C2 (T ),

(2.4)

0

where C1 (T ) > 0 is constant depending only on T , α, β, V , V¯ , φ0 1 and ψ0 , and C2 (T ) > 0 is constant depending only on T , α, β, V , V¯ , φ0 4 and ψ0 3 . For some positive constants M ≥ m > 0 and T > 0, we define the set of functions X(0, T ; m, M) for which the solutions to the problem (2.1)–(2.2) are sought as follows ⎧   φ(t, x) ∈ C(0, T ; H 4 (R)) ∩ C 1 (0, T ; H 2 (R)) ⎪ ⎪  ⎪ ⎪  ⎨  ψ(t, x) ∈ C(0, T ; H 3 (R)) ∩ C 1 (0, T ; H 1 (R)) X(0, T ; m, M) = (φ, ψ)(t, x)  ⎪ ⎪  (φx , ψx )(t, x) ∈ L2 (0, T ; H 4 (R) × H 3 (R)) ⎪ ⎪  ⎩  m≤v≤M

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

.

Under the assumptions listed in Theorem 1.1, we have the following: Proposition 2.1 (Local existence). Under the assumptions in Theorem 1.1, there exists a sufficiently small positive constant t1 depending only on α, β, V , V¯ , φ0 4 and ψ0 3 such that the V Cauchy problem (2.1)–(2.2) admits a unique smooth solution (φ, ψ)(t, x) ∈ X(0, t1 ; 2 , 2V¯ ) and sup {φ(t)4 + ψ(t)3 } ≤ b{φ0 4 + ψ0 3 },

[0,t1 ]

where b > 1 is a positive constant depending only on V , V¯ . The proof of Proposition 2.1 is standard, which is similar to that of Theorem 1.1 in [6] and thus omitted here for brevity. To prove Theorem 2.1, it remains to show the following a priori estimates by the standard continuation argument. Proposition 2.2 (A priori estimates). Under the assumptions of Theorem 1.1, suppose that (φ, ψ)(t, x) ∈ X(0, T1 ; V0 , V1 ) is a solution of the Cauchy problem (2.1)–(2.2) for some positive constants 0 < T1 ≤ T and V0 , V1 > 0. Then there exist two positive constants C1 (T ) and C2 (T ) which are independent of V0 and V1 such that the following two estimates hold: C1−1 (T ) ≤ v(t, x) ≤ C1 (T ), φ(t)24

+ ψ(t)23

∀ (t, x) ∈ [0, T1 ] × R,

t

 + φx (s)24 + ψx (s)23 ds ≤ C2 (T ), 0

(2.5) ∀ t ∈ [0, T1 ].

(2.6)

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Proposition 2.2 can be obtained by a series of lemmas below. We first give the following key lemma. Lemma 2.1 (Basic energy estimates). Under the assumptions of Proposition 2.2, it holds   t   vx2 u2x ψ2 dx + dxdτ dx + (v, v) ¯ + β+5 2 2v v α+1

R

R

  ¯ + (v0 , v)

= R

2 ψ02 v0x + β+5 2 2v0

0 R



(2.7)

dx

for all t ∈ [0, T1 ], where the function (v, v) ¯ is defined by (2.9)1 . Proof. Multiplying (2.1)1 by [p(v) ¯ − p(v¯ + φ)], (2.1)2 by ψ , then combining the two resulting equations, we have 

ψ2 (v, v) ¯ + 2

+ Hx + t



u2x v α+1

vxx β + 5 vx2 = ψ − β+5 + 2 v β+6 v

,

(2.8)

x

where v (v, v) ¯ = p(v)φ ¯ −

p(s) ds, (2.9)

v¯

H = [p(v) ¯ − p(v¯ + φ)]ψ −

ux v α+1

ψ.

Using equation (2.1)1 , we have

vxx β + 5 vx2 ψ − β+5 + 2 v β+6 v

x



β + 5 vx2 vxx = {· · ·}x − ψx − β+5 + 2 v β+6 v

β + 5 vx2 vxx = {· · ·}x − φt − β+5 + 2 v β+6 v  2 vx . = {· · ·}x − 2v β+5 t

(2.10)

Here and hereafter, {· · ·}x denotes the terms which will disappear after integrating with respect to x. Combining (2.8) and (2.10), and integrating the resultant equation over [0, t] × R yields (2.7) immediately. This completes the proof Lemma 3.1. 2 Based on Lemma 3.1, we now use Y. Kanel’s method to deduce the lower and upper bounds of v(t, x) for the case (iii) of Theorem 1.1.

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11

Lemma 2.2 (Lower and upper bounds of v(t, x) for the case (iii) of Theorem 1.1). Under the assumptions of Proposition 2.2, if we assume further that −3 ≤ β < −2, then there exists a positive constant C5 depending only on β, V , V¯ , φ0 , ψ0  and v0x  such that C5−1 ≤ v(t, x) ≤ C5

(2.11)

for all (t, x) ∈ [0, T1 ] × R. v Proof. Let v˜ = , then we have from (2.7) that v¯ ! ! ! ! ! ! ! v˜x ! ! vx β+3 ! ! vx ! ! !=! ! ! ! 2 ! β+5 ! ! β+5 v¯ ! ≤ C ! β+5 ! ≤ C. v˜ 2 v 2 v 2

(2.12)

Due to [21], (v, v) ¯ ≥ c0

(1 − vv¯ )2 := c0 (v) ˜ 1 + vv¯

(2.13)

for some constants c0 > 0. Define

ϒ(v) ˜ =

v˜ √ (s) 1

s

β+5 2

ds,

(2.14)

then we have  ϒ(v) ˜ −→

−∞,

as v˜ → 0, if β ≥ −3,

+∞,

as v˜ → +∞, if β < −2.

(2.15)

Moreover, it follows from (2.7), (2.12) and (2.13) that  x    x          v ˜ ∂ y    |ϒ(v(t, ˜ x))| =  (v) ˜ β+5 dy  ϒ(v(t, ˜ y)dy  =  ∂y     v˜ 2 −∞ −∞ ! ! ! ! v˜ ! ! x ! ! ! ¯ !! ≤ ! (v, v) ! β+5 ! ≤ C. v˜ 2

(2.16)

Thus if −3 ≤ β < −2, then (2.15) together with (2.16) implies (2.11). This completes the proof of Lemma 2.2. 2 Remark 2.1. From the proof of Lemma 2.2, we see that if β < −2, then v(t, x) has a uniform upper bound, i.e. v(t, x) ≤ C5 , and if β ≥ −3, then v(t, x) has a uniform lower bound, i.e. v(t, x) ≥ C5−1 , where C5 is given in Lemma 2.2.

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12

Lemma 2.3. Under the assumptions of Proposition 2.2, there exists a positive constant C(T ) > 0 depending only on T , α, β, V , V¯ , φ0 , ψ0  and v0x  such that for 0 ≤ t ≤ T1 ,  R

t 

vx2 v 2α+2

dx +

−p  (v)

0 R

vx2 v α+1

 t  " dxdτ + v

0 R

≤ C(V , V¯ )(φ0 , ψ0 , v0x )2 + C(T )

#2

vx

⎧ ! !α−2β−5 !1! ⎪ ⎪ ! ! ⎪ , ⎪ ⎪ ⎨ ! v !L∞ T ,x 1

−α+2β+5

vL∞ ⎪ ⎪ T1 ,x ⎪ ⎪ ⎪ ⎩ v2β+6 , L∞ T ,x

dxdτ

α+β+6 2

,

x

if α > 0 and β < −3, if α ≤ −1 and β ≥ −2,

(2.17)

if α > −1 and β ≥ −2.

1

Proof. Rewriting (2.1)2 as



v vxx β + 5 vx2 x  − ψ = p (v)v + − . x 2 v β+6 x v α+1 v β+5 t Multiplying (2.18) by 

vx v α+1

yields

vx vx2 − ψ α+1 2α+2 2v v =

u2x v α+1

(2.18)

− p  (v) t



vx2 v α+1

u¯ x ux vxx β + 5 vx2 − α+1 + β+5 − 2 v β+6 v v

x

vx v α+1

+ {· · ·}x .

(2.19)

By direct computation, we obtain

vx β + 5 vx2 vxx − β+5 β+6 α+1 2 v v x v

vxx β + 5 vx2 vx  = − β+5 − + {· · ·}x 2 v β+6 v v α+1 x =− =−

2 vxx

v α+β+6 " vx "

=−

v

α+β+6 2

vx α+β+6 2

v + {· · ·}x .

vx2 vxx vx4 − (α + 1)(β + 5) + {· · ·}x 2v α+β+7 2v α+β+8 #2   β + 5 vx3 x (α + 1)2 + (β + 5)2 vx4 − + + {· · ·}x 6 v α+β+7 4 v α+β+8 x #2  (α + 1)2 + (β + 5)2 (β + 5)(α + β + 7) vx4 + − 4 6 v α+β+8 x

+ (β + 2α + 7)

(2.20)

Substituting (2.20) into (2.19), and integrating the resultant equation in t and x over [0, t] × R, we have

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 R

t 

vx2

dx +

v 2α+2

−p  (v)

0 R

⎛



≤C⎝

R

 t  "

vx2 v α+1

v

0 R

α+β+6 2

⎞

2 v0x

dx + ψ(t) + ψ0  2α+2 2

2⎠

v0

#2

vx

dxdτ +

dxdτ x

t  + C(α, β) 0 R

13

vx4 v α+β+8

dxdτ,

(2.21)

where we have used the fact that  R

t  The term I1 := 0 R

ψvx 1 dx ≤ 4 v α+1

vx4 dxdτ α+β+8 v

 R

vx2 2α+2 v

dx + Cψ(t)2 .

can be estimated as follows.

If α > 0 and β < −3, then we deduce from Lemma 2.1 and the Sobolev inequality that t  I1 ≤ 0 R

! 2 ! ! vx ! ! ! dτ dx (τ ) ! ! ∞ β+5 α+3 v v L vx2

t ! ! ≤C ! ! 0

t ! ! ≤C ! ! 0

! ! ! 1 !−β−3 ! (τ )! ! v ! ∞ dτ L L∞

!2 ! ! (τ ) α+β+6 ! vx

v

2

! !  ! ! vx ! ! (τ )! · ! α+β+6 α+β+6 ! v 2 v 2 vx

! ! ! !1 ! !−β−3 ! (τ )! · ! dτ (τ )! ! ! v !L∞ x

t ! ! v x ! ≤ η ! α+β+6 ! v 2

!2 ! ! t  ! ! 1 !−2β−6 vx2 ! ! (τ )! dτ + Cη dx (τ )! dτ ! v α+β+6 ! v !L∞ x

t ! ! v x ! ≤ η ! α+β+6 ! v 2

!2 ! ! t  ! ! 1 !α+1−2β−6 vx2 ! ! (τ )! dτ + Cη dx ! (τ )! dτ β+5 ! v !L∞ v x

0

0

t ≤η 0

! ! vx ! ! α+β+6 ! v 2

0 R

0 R

!2 ! ! (τ )! dτ + Cη T ! x

! !α−2β−5 !1! ! ! . !v ! ∞ L T1 ,x

If α ≤ −1 and β ≥ −2, then similar to (2.22), we can obtain t  I1 ≤ C 0 R

! ! ! dx ! β+5 vx2

v

!2 ! ! v(τ )β+3 (τ ) α+β+6 L∞ dτ ! vx

v

2

L∞

(2.22)

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14

t ! ! ≤C ! ! 0

! !  ! ! vx ! ! (τ )! · ! α+β+6 α+β+6 ! v 2 v 2 vx

! ! ! β+3 (τ )! · v(τ )L∞ dτ ! x

t ! ! v x ! ≤ η ! α+β+6 ! v 2

!2 t  ! vx2 ! 2β+6 (τ )! dτ + Cη dx v(τ )L∞ dτ α+β+6 ! v x

t ! ! v x ! ≤ η ! α+β+6 ! v 2

!2 t  ! vx2 ! −α−1+2β+6 (τ )! dτ + Cη dx v(τ )L∞ dτ β+5 ! v x

t ! ! v x ! ≤ η ! α+β+6 ! v 2

!2 ! ! −α+2β+5 (τ )! dτ + Cη T vL∞ . T1 ,x ! x

0

0

0

0 R

0 R

(2.23)

If α > −1 and β ≥ −2, then we have from Lemma 2.1, the Sobolev inequality and Remark 2.1 that t ! ! I1 ≤ C ! ! 0

! !  ! ! vx !·! (τ ) α+β+6 α+β+6 ! ! ! v 2 v 2 vx

t ! ! v x ! ≤ η ! α+β+6 ! v 2 0

t ≤η 0

t ≤η 0

! ! ! β+3 (τ )! · v(τ )L∞ dτ ! x

!2 t  ! vx2 ! 2β+6 (τ )! dτ + Cη dx v(τ )L∞ dτ α+β+6 ! v x 0 R

! ! vx ! ! α+β+6 ! v 2

!2 ! ! t  ! ! 1 !α+1 vx2 ! 2β+6 ! (τ )! (τ )! dτ + Cη dx ! v ! ∞ v(τ )L∞ dτ β+5 ! v L x

! ! vx ! ! α+β+6 ! v 2

!2 ! ! 2β+6 (τ )! dτ + Cη T vL∞ . T1 ,x ! x

0 R

(2.24)

Here and hereafter, η > 0 denotes a suitably small constant and Cη is a positive constant depending only on η. Substituting (2.22)–(2.24) into (2.21) respectively, then (2.17) follows from Lemma 2.1 and the smallness of η. This completes the proof of Lemma 2.3. 2 Now we use Y. Kanel’s method again to deduce the following: Lemma 2.4 (Lower and upper bounds of v(t, x) for the case (i), (ii) and (iv) of Theorem 1.1). Under the assumptions of Proposition 2.2, if we assume further that the constants α, β satisfy the condition (i) or (ii), or (iv) of Theorem 1.1, then there exists a positive constant C6 (T ) depending only on T , α, β, V , V¯ , φ0 , ψ0  and v0x  such that C6−1 (T ) ≤ v(t, x) ≤ C6 (T ) for all (t, x) ∈ [0, T1 ] × R.

(2.25)

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15

Proof. Set v˜ √ (s) ϒ(v) ˜ = ds, s α+1

(2.26)

1

then it is easy to check that there exist positive constants A1, A2 such that   ϒ(v) ˜ ≥



A1 v˜ −α − A2 , A1 v˜

1 2 −α

− A2 ,

as v˜ → 0,

(2.27)

as v˜ → +∞.

For the case (iv): α > −2β − 5 and β < −3, we have α > 1 and v(t, x) ≤ C5 by Remark 2.1. Moreover, we deduce from (2.27)1 , (2.13), Lemmas 2.1 and 2.3 that  x        1 ∂     ≤ C + C ϒ( v) ˜ ≤ C + C ϒ( v(t, ˜ y)dy   vα ∂y   −∞ !! v ! ! !! x ! ! ¯ ! ! α+1 ! ≤ C + C ! (v, v) v ! ! α−2β−5 !1! 2 ! , ≤ C(V , V¯ ) ((φ0 , ψ0 , v0x ) + 1) + C(T ) ! !v ! ∞ L

(2.28)

T1 ,x

which implies that ! ! α−2β−5 ! !α !1! 2 !1! ! ! ! ! ¯ . ! v ! ∞ ≤ C(V , V ) ((φ0 , ψ0 , v0x ) + 1) + C(T ) ! v ! ∞ L L T1 ,x

(2.29)

T1 ,x

Consequently, if α > −2β − 5, then it follows from (2.29) and the Young inequality that there exists a positive constant C7 (T ) depending only on T , V , V¯ , φ0 , ψ0  and v0x  such that v(t, x) ≥ C7−1 (T ) for all (t, x) ∈ [0, T1 ] × R. For the case (i): α < −2β − 4 and β ≥ − 32 , it follows that α < −1 and v(t, x) ≥ C5−1 by Remark 2.1. Now we show the upper bound of v(t, x). We have from (2.27)2 , (2.13), Lemmas 2.1 and 2.3 that   1 v 2 −α ≤ C + C ϒ(v) ˜    x     ∂  ϒ(v(t, ˜ y)dy  ≤C +C ∂y   −∞

!! v ! ! !! x ! ! ¯ ! ! α+1 ! ≤ C + C ! (v, v) v −α+2β+5

≤ C(V , V¯ ) ((φ0 , ψ0 , v0x ) + 1) + C(T ) vL∞ 2

T1 ,x

,

(2.30)

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16

which implies that −α+2β+5

1

−α vL2 ∞ ≤ C(V , V¯ ) ((φ0 , ψ0 , v0x ) + 1) + C(T ) vL∞ 2 T1 ,x

.

(2.31)

T1 ,x

Thus if 12 − α > −α+2β+5 , i.e. α < −2β − 4, then (2.31) together with the Young inequality 2 implies that v(t, x) ≤ C8 (T ) for all (t, x) ∈ [0, T1 ] × R. Here C8 (T ) is a positive constant depending only on T , V , V¯ , φ0 , ψ0  and v0x . For the case (ii): α < −β − 52 and −2 ≤ β < − 32 , it holds that −1 < −β − 52 ≤ − 12 and v(t, x) ≥ C5−1 by Remark 2.1. To prove the upper bound for v(t, x), we divide this case into the following two subcases: Subcase (ii)1 : α ≤ −1 and −2 ≤ β < − 32 . In this subcase, we have α < −2β − 4. Then similar to the case (i), we see that (2.31) holds for this subcase, and consequently, v(t, x) ≤ C8 (T ) for all (t, x) ∈ [0, T1 ] × R. Subcase (ii)2 : −1 < α < −β − 52 and −2 ≤ β < − 32 . In this situation, similar to (2.31), we can deduce that 1

−α β+3 vL2 ∞ ≤ C(V , V¯ ) ((φ0 , ψ0 , v0x ) + 1) + C(T ) vL∞ . T1 ,x

(2.32)

T1 ,x

Thus if 12 − α > β + 3, i.e. α < −β − 52 , then (2.32) together with the Young inequality implies that v(t, x) ≤ C9 (T ) for all (t, x) ∈ [0, T1 ] × R, where C9 (T ) is a positive constant depending only on T , V , V¯ , φ0 , ψ0  and v0x . Then we finish the proof of Lemma 2.4 by letting C6 (T ) = max{C5 , C7 (T ), C8 (T ), C9 (T )}. 2 As a direct consequence of Lemmas 2.1, 2.2 and 2.4, we have: Corollary 2.1. Under the assumptions of Proposition 2.2, there exists a positive constant C10 (T ) depending on T , α, β, V , V¯ , φ0 , ψ0  and v0x  such that for 0 ≤ t ≤ T1 , t  (φ, ψ, vx )(t) +

u2x dxdτ ≤ C10 (T ).

2

(2.33)

0 R

From the estimates of the previous lemmas and Corollary 2.1, we can obtain Lemma 2.5. Under the assumptions of Proposition 2.2, there exists a positive constant C11 (T ) > 0 depending on T , α, β, V , V¯ , φ0 , ψ0  and v0x  such that for 0 ≤ t ≤ T1 , t vx (t) +

vx (τ )21 dτ ≤ C11 (T ).

2

0

(2.34)

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17

Proof. First, if the parameters α, β satisfy the condition (iii) of Theorem 1.1, then we have from (2.21) and Lemmas 2.1 and 2.2 that t vx (s)21 ds

vx (t) + 2

≤ C(φ0 , ψ0 , v0x ) + C 2

t 

 vx4 + |vxx vx |2 dxdτ, (2.35)

0 R

0

where the constant C > 0 depends only on α, β, V , V¯ , ϕ0 , ψ0  and v0x . And if α, β satisfy the condition (i) or (ii), or (iv) of Theorem 1.1, then we derive from Lemmas 2.3 and 2.4 that t vx (s)21 ds

vx (t) + 2

≤ C(T ) + C(T )

t 

 vx4 + |vxx vx |2 dxdτ.

(2.36)

0 R

0

Using the Cauchy inequality, the Sobolev inequality and Corollary 2.1, we have t 

t t   4 2 2 vx4 dxdτ vx + |vxx vx | dxdτ ≤ η vxx (τ ) dτ + Cη 0 R

0 R

0

t

t vxx (τ ) dτ + Cη

≤η

vx (τ )3 vxx (τ )dτ

2

0

0

t

t vxx (τ ) dτ + Cη { sup vx (t)}

≤ 2η

2

0≤t≤T

0

6

dτ 0

t vxx (τ )2 dτ + C(η, T ).

≤ 2η

(2.37)

0

Substituting (2.37) into (2.35) and (2.36), and using the smallness of η, we can obtain (2.34). This completes the proof of Lemma 2.5. 2 The next lemma gives the estimates on ux (t). Lemma 2.6. Under the assumptions of Proposition 2.2, there exists a positive constant C12 (T ) depending on T , α, β, V , V¯ , φ0 , ψ0 , u0x  and v0x 1 such that for 0 ≤ t ≤ T1 , t (ux , vxx )(t) +

uxx (τ )2 dτ ≤ C12 (T ).

2

0

(2.38)

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18

Proof. Multiplying (2.1)2 by −uxx and using equation (2.1)1 , we have 

2 vxx u2x + β+5 2 2v

+ t

2 u u2xx ux vx uxx vxx x  = p (v)v u + (α + 1) − (β + 5) x xx v α+1 v α+2 2v β+6

+ (β + 5)

vx vxx uxx vx3 uxx − (β + 5)(β + 6) + {· · ·}x . (2.39) 2v β+7 v β+6

Integrating (2.39) with respect to t and x over [0, t] × R, we get from Lemmas 2.2 and 2.4 that t (ux , vxx )(t) + 2

 uxx (τ )2 dτ ≤ C(T ) (v0xx , u0x )2 + I5 + I6 ,

(2.40)

0

where t     2  I5 = vxx ux  dxdτ, 0 R

I6 =

t  

    |vx uxx | + |vx vxx uxx | + |ux vx uxx | + vx3 uxx  dxdτ.

(2.41)

0 R

It follows from the Sobolev inequality, the Cauchy inequality, Corollary 2.1 and Lemma 2.5 that t I5 ≤

1

1

ux (τ ) 2 uxx (τ ) 2 vxx (τ )2 dτ 0

t ≤η

t uxx (τ ) dτ + Cη

0

t uxx (τ ) dτ + C

ux (τ ) dτ + Cη

0

0

0

t uxx (τ )2 dτ + Cη

0

vxx (τ )4 dτ + C(T ), 0

t uxx (τ )2 dτ + Cη

t 

 2 + vx6 dxdτ vx2 + u2x vx2 + vx2 vxx

0 R

0

t ≤η

vxx (τ )4 dτ

2

t

I6 ≤ η

t

2

≤η

8

0

t ≤η

2

ux (τ ) 3 vxx (τ ) 3 dτ

2

t

uxx (τ ) dτ + Cη vx (τ )2 + ux (τ )uxx (τ )vx (τ )2 2

0

0

 + vx (τ )vxx (τ )3 + vx (τ )4 vxx (τ )2 dτ

(2.42)

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t 

t ≤ 2η

uxx (τ )2 dτ + Cη

vx (τ )2 + ux (τ )2 { sup vx (τ )}4 0≤τ ≤t

0

0

+vx (τ )21 vxx (τ )2

+ { sup vx (τ )} vxx (τ )

t ≤ 2η

19

 4

0≤τ ≤t

2

dτ

⎛ t ⎞ 

 uxx (τ )2 dτ + C(η, T ) ⎝ vx (τ )21 + 1 vxx (τ )2 dτ + 1⎠ .

0

(2.43)

0

Putting (2.42)–(2.43) into (2.40), then we can get (2.38) by using the smallness of η and Gronwall’s inequality and Lemma 2.5. Thus the proof of Lemma 2.6 is completed. 2 Now, we estimate vxx (t). Lemma 2.7. Under the assumptions of Proposition 2.2, there exists a positive constant C13 (T ) > 0 depending on T , α, β, V , V¯ , φ0 , ψ0 , u0x  and v0x 1 such that for 0 ≤ t ≤ T1 , t vxx (t) +

vxx (τ )21 dτ ≤ C13 (T ).

2

(2.44)

0

Proof. Differentiating (2.1)2 twice with respect to x, then multiplying the resultant equation by vxx , and using equation (2.1)1 , we have v α+1 

2 vxx vxx − ux α+1 2α+2 2v v

− p  (v) t

2 2 vxx vxxx + = Q1 + Q2 + Q3 + Q4 + {· · ·}x , v α+1 v α+β+6

(2.45)

where Q1 = p  (v)

vx2 vxx , v α+1

Q2 = 2(α + 1) Q3 =

vx vxx uxx vx2 vxx ux − (α + 1)(α + 2) , v 2α+3 v 2α+4

u2xx vx vxx ux u2x vxx − (α + 1) + (α + 1) , v α+1 v α+2 v 2α+4

Q4 = (α + 2β + 11) +

2 v2 vx vxx vxxx vxx (β + 5)(β + 6) vx3 vxxx x − 2(α + 1)(β + 5) − v α+β+7 2 v α+β+8 v α+β+8

(α + 1)(β + 5)(β + 6) vx4 vxx . 2 v α+β+8

Integrating (2.45) with respect to t and x over [0, t] × R, by some similar estimates as those in the proof of Lemma 2.6, using Corollary 2.1, Lemmas 2.5–2.6 and the smallness of η, we can obtain (2.44). The details are omitted here for brevity. This completes the proof of Lemma 2.7. 2

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20

It follows from Corollary 2.1 and Lemmas 2.5–2.7 that there exists a positive constant C14 (T ) depending on T , α, β, V , V¯ , φ0 , ψ0 , u0x  and v0x 1 such that for 0 ≤ t ≤ T1 ,

φ(t)22

+ ψ(t)21

+

t

 φx (τ )22 + ψx (τ )21 dτ ≤ C14 (T ).

(2.46)

0

By the same argument as above, we can also get

φxxx (t)21

+ ψxx (t)21

+

t

 φxxx (τ )22 + ψxxx (τ )21 dτ ≤ C15 (T ),

(2.47)

0

where C15 (T ) is a positive constant depending on T , α, β, V , V¯ , φ0 4 and ψ0 3 . Proof of Proposition 2.2. Proposition 2.2 follows immediately from (2.46) and (2.47).

2

3. Proof of Theorem 1.2 This section is devoted to proving Theorem 1.2 and organized as follows. First, we study the properties of the approximate rarefaction wave (V , U )(t, x) defined by (1.12). Then we reformulate our original problem (1.2)–(1.3) into a perturbation one around the approximate rarefaction wave, and focus on deducing the uniform-in-time energy estimates of solutions to the reformulate system (3.3)–(3.4). Finally, the proof of Theorem 1.2 is given at the end of this section. As [19,21], we start with the Riemann problem for the typical Burgers equation: ⎧ ⎨ wt + wwx = 0, w− , x < 0, ⎩ w(0, x) = w+ , x > 0.

(3.1)

If w− < w+ , then the Riemann problem (3.1) admits a unique rarefaction wave solution:

w R (t, x) = w R

x  t

⎧ ⎪ − , x ≤ w− t, ⎨w x , w− t < x < w+ t, = ⎪ ⎩ t w+ , x ≥ w+ t.

(3.2)

We approximate w R (t, x) by the solution w(t, x) of (1.11). Then by the method of characteristic, w(t, x) has the following properties. Lemma 3.1. (See [18,21].) Let w− < w+ and w˜ = w+ − w− . Then the Cauchy problem (1.11) admits a unique global smooth solution w(t, x) satisfying (i) w− < w(t, x) < w+ , wx > 0 for (t, x) ∈ R+ × R. (ii) For any 1 ≤ p ≤ +∞, there exists a constant Cp > 0 depending only on p such that for t ≥ 0,

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1

21

−1+ 1

p ,w p }, wx (t)Lp ≤ Cp min{w ˜ ˜ pt ! ! j ! !∂ j − p1 j −1− p1 −1 ! ! ˜ , t }, j ≥ 2. ! ∂x j w(t)! p ≤ Cp min{w L

(iii)

    lim sup w(t, x) − w R (t, x) = 0.

t→+∞ x∈R

Notice that (V , U )(t, x) defined by (1.12) satisfies the Euler equations (1.7). Based on this, (1.12) and Lemma 3.1, we can deduce the following: Lemma 3.2. Let δ = |v+ − v− | + |u+ − u− |, the smooth approximations (V , U )(t, x) constructed in (1.12) have the following properties: (i) Vt = Ux > 0, |(Vt , Ut )| ≤ C|(Vx , Ux )|, ∀ (t, x) ∈ R+ × R. (ii) For any 1 ≤ p ≤ +∞, there exists a constant Cp > 0 depending only on p such that for t ≥ 0,   1 1− 1 −1+ p1 (Vx , Ux )(t)Lp ≤ Cp min δ p , δ p (1 + t) , ! j !   ! ∂ ! 1 1 ! ! ≤ Cp min δ j − p , j −1− p (1 + t)−1 , j ≥ 2. (V , U )(t) ! ∂x j ! p L (iii)

    lim sup (V , U )(t, x) − (V R , U R )(t, x) = 0.

t→+∞ x∈R

Now we reformulate our original problem (1.2)–(1.3). Define ¯ x) = v(t, x) − V (t, x), φ(t,

¯ x) = u(t, x) − U (t, x). ψ(t,

¯ ψ) ¯ by (φ, ψ) in the rest of this subsecFor notational simplicity, we denote the perturbation (φ, tion. Then the problem (1.2)–(1.3) can be rewritten as ⎧ ⎪ ⎨ φt − ψx = 0, ⎪ ⎩ ψt + [p(φ + V ) − p(V )]x =



ux Ux − v α+1 V α+1

+F

(3.3)

x

with the initial data (φ, ψ)|t=0 = (φ0 , ψ0 )(x) = (v − V , u − U )(0, x),

(3.4)

where

vxx β + 5 vx2 F = − β+5 + 2 v β+6 v



+ x

Ux V α+1

. x

(3.5)

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22

The local solvability of the Cauchy problem (3.3)–(3.4) is similar to Proposition 2.1. Suppose that the local solution (φ, ψ)(t, x) has been extended to the time t = T1 ≥ t1 and satisfies the following a priori assumption: m1 ≤ v(t, x) ≤ M1

(3.6)

for all (t, x) ∈ [0, T1 ] × R and some positive constants m1 and M1 . Based on (3.6), if we can show that there exists a positive constant C3 which may depends on V , V¯ and the initial data but independent of m1 and M1 such that C3−1 ≤ v(t, x) ≤ C3

(3.7)

for all (t, x) ∈ [0, T1 ] × R, then Theorem 1.2 can be obtained by combining the a priori estimates and the local existence result. In the rest of this section, we pay our attention to deducing (3.7) and some uniform-in-time a priori estimates on (φ, ψ)(t, x). More precisely, we prove the following: Proposition 3.1 (A priori estimates). Under the assumptions of Theorem 1.2, suppose that (φ, ψ)(t, x) ∈ X(0, T1 ; m1 , M1 ) is a solution of the Cauchy problem (3.3)–(3.4) for some positive constants T1 , m1 and M1 . Then there exist a positive constants C3 depending only on V , V¯ , φ0 1 , ψ0  and a positive constants C4 depending only on V , V¯ , φ0 4 , ψ0 3 such that the following two estimates hold: C3−1 ≤ v(t, x) ≤ C3 ,

φ(t)24

+ ψ(t)23

+

t

∀ (t, x) ∈ [0, T1 ] × R,

(3.8)



 φx (τ )24 + ψx (τ )23 dτ ≤ C4 1 + φ0 24 + ψ0 23 ,

0

∀ t ∈ [0, T1 ].

(3.9)

To prove Proposition 3.1, we start with the following: Lemma 3.3. Under the assumptions of Proposition 3.1, there exists a uniform positive constant C such that   t  t  ψx2 φx2 ψ2  Vt {p(v) − p(V ) − p (V )φ}dxdτ + dxdτ + β+5 dx + (v, V ) + 2 2v v α+1

R

0 R

⎧  ⎫  ⎨ ⎬ 2 ψ02 φ0x ≤C + β+5 dx + 1 , (v0 , V0 ) + ⎩ ⎭ 2 2v0 R

for all t ∈ [0, T1 ], where the function (v, V ) is defined in (3.12).

0 R

(3.10)

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23

Proof. Multiplying (3.3)1 by [p(V ) − p(V + φ)], (3.3)2 by ψ , then combining the two resulting equations, we have  % $ ψx2 ψ2 + Vt p(v) − p(V ) − p  (V )φ + α+1 (v, V ) + 2 t v

 2 vxx β + 5 vx 1 1 = − β+5 + ψ + Ux ψ x − α+1 β+6 α+1 2 v v v V x

 +ψ

Ux V α+1

+ {· · ·}x , x

(3.11) where v (v, V ) = p(V )φ −

p(s) ds.

(3.12)

V

Using equation (3.3)1 , we have

β + 5 vx2 vxx − β+5 + 2 v β+6 v



ψ = {· · ·}x − x

φx2 2v β+5

+ t

ψx Vxx 5(β + 5)(φx2 Ux + ψx Vx2 ) − . v β+5 2v β+6 (3.13)

Putting (3.13) into (3.11), and integrating the resultant equation with respect to t and x over [0, t] × R, we get from Ux > 0 and the assumption −3 ≤ β < −2 that   R

φx2 ψ2 + β+5 (v, V ) + dx + 2 2v t

 

+ 0 R

t 

Vt {p(v) − p(V ) − p (V )φ}dxdτ

0 R

ψx2 (β + 5)φx2 Ux + dxdτ 2v β+6 v α+1

   3 2  φ0x ψ02 = Ji , + β+5 dx + (v0 , V0 ) + 2 2v0 i=1

(3.14)

R

where 

t  J1 =

ψ 0 R

Ux V α+1 

t  J2 =

Ux ψ x 0 R

dxdτ, x

1 v α+1

−

1 V α+1

dxdτ,

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24

t   J3 = 0 R

(β + 5)ψx Vx2 ψx Vxx − dxdτ. v β+5 2v β+6

It follows from (1.14), the a priori assumption (3.6), the Sobolev inequality, the Young inequality, the Cauchy inequality and Lemma 3.2 that |J1 | ≤ C(V , V¯ )

t  (|Uxx | + |Ux Vx |)|ψ|dxdτ 0 R

≤ C(V , V¯ )

!1 t ! ! ψx !2 ! ! (τ ) ! α+1 ! v 2

! α+1 ! ! 2 ! !v !

L∞ T1 ,x

0

1 ≤ 8

≤

1 8

t



0 R

t



0 R

1 ψx2 dxdτ + C(m1 , M1 , V , V¯ )ε 4 v α+1

t

 13 ψ(τ )2 + 1 (1 + τ )− 12 dτ

0

ψx2 dxdτ v α+1

⎞ ⎛ t  1 13 + C1 (m1 , M1 , V , V¯ )ε 4 ⎝ ψ(τ )2 (1 + τ )− 12 dτ + 1⎠ ,

(3.15)

0

|J2 | ≤ C(m1 , M1 , V , V¯ ) 1 ≤ 8 1 ≤ 8

≤

 1  ψ(τ ) 2 Uxx (τ )L1 + Vx (τ )L∞ Ux (τ )L1 dτ

1 8

t

 |Ux ψx φ| dxdτ

0 R

t 

ψx2 dxdτ v α+1

0 R

t 

+ C(m1 , M1 , V , V¯ )

Ux2 φ 2 dxdτ 0 R

ψx2 dxdτ + C(m1 , M1 , V , V¯ ) v α+1

0 R

t 

t

⎛

Ux (τ )L∞ ⎝

ψx2 dxdτ + C2 (m1 , M1 , V , V¯ )ε v α+1

0 R

⎞ Vt φ 2 dx ⎠ dτ

R

0

t 



t  Vt φ 2 dxdτ,

(3.16)

0 R

and 1 |J3 | ≤ 8 1 ≤ 8

≤

1 8

t  0 R

t



0 R

t



0 R

ψx2 dxdτ + C(m1 ) v α+1

t (Vxx (τ )2 + Vx (τ )4L4 )dτ 0

ψx2 dxdτ + C(m1 )ε v α+1

t

(1 + τ )−2 dτ

0

ψx2 dxdτ + C3 (m1 )ε. v α+1

(3.17)

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25

Finally, it is easy to get from the convexity assumption on p(v) that t 

t 



Vt {p(v) − p(V ) − p (V )φ}dxdτ = 0 R

0 R

Vt  p (θ v + (1 − θ )V )φ 2 dxdτ 2

≥ C0 (m1 , M1 , V , V¯ )

t  0 R

Vt 2 φ dxdτ, 2

(3.18)

where θ ∈ (0, 1) is a positive constant. Substituting (3.15)–(3.18) into (3.14) and choosing ε to be sufficiently small such that ⎧ 1 ⎪ C (m , M , V , V¯ )ε 4 < 1, ⎪ ⎨ 1 1 1 1 C2 (m1 , M1 , V , V¯ )ε < C0 (m1 , M1 , V , V¯ ), ⎪ ⎪ 2 ⎩ C3 (m1 )ε < 1, then we obtain   (v, V ) + R

t

ψ2 dx + sup 2 0≤τ ≤T1





φx2 dx 2v β+5

R

t 



Vt {p(v) − p(V ) − p (V )φ}dxdτ +

+ 0 R

0 R

ψx2 dxdτ v α+1

 t   2 ψ02 φ0x 13 ≤ + β+5 dx + ψ(τ )2 (1 + τ )− 12 dτ + 1. (v0 , V0 ) + 2 2v0 R

(3.19)

0

Thus (3.10) follows from (3.19) and Gronwall’s inequality. This completes the proof of Lemma 3.3. 2 Similar to Lemma 2.2, we have Lemma 3.4. Under the assumptions of Proposition 3.1, there exists a positive constant C16 which depends only on V , V¯ , φ0 , ψ0  and φ0x  such that −1 C16 ≤ v(t, x) ≤ C16 ,

for all (t, x) ∈ [0, T1 ] × R. Proof. Let v˜ =

v , then by direct computation, V v˜x v˜

β+5 2

=

φx v

β+5 2

V

β+3 2

−

V

β+1 2

v

β+5 2

Vx (v − V ),

(3.20)

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26

thus it follows from (3.6) that ! ! ! ! ! v˜x ! ! ! ! ! ≤ C(V , V¯ ) ! φx (t)! + C(V , V¯ )C(m1 , M1 )Vx (t) (t) ! β+5 ! ! β+5 ! v˜ 2 v 2 ! ! ! φx ! 1 ! ¯ ≤ C(V , V¯ ) ! β+5 (t)! ! + C(V , V )C(m1 , M1 )ε 2 v 2 ! ! ! φx ! ! ¯ ≤ C(V , V ) ! β+5 (t)! !+1 v 2

(3.21)

provided that ε > 0 is sufficiently small. On the other hand,  x    ! ! !!    ! ! v ˜ v˜x y !!   ! |ϒ(v(t, ˜ x))| =  (v) ˜ β+5 dy  ≤ ! (v, V )(t)! ! β+5 (t)! !,   v˜ 2 v˜ 2

(3.22)

−∞

where the functions (·) and ϒ(·) are defined in (2.13) and (2.14) respectively. (3.20) thus follows from (2.15), (3.10), (3.21) and (3.22). This completes the proof of Lemma 3.4. 2 By Lemmas 3.3–3.4, we have Corollary 3.1 (Basic energy estimates). Under the assumptions of Proposition 3.1, there exists a positive constant C17 depending only on V , V¯ , φ0 , ψ0  and φ0x  such that for 0 ≤ t ≤ T1 , t  (φ, ψ, φx )(t) +

t Vt φ dxdτ +

2

2

0 R

 ψx (τ )2 dτ ≤ C17 1 + (φ0 , ψ0 , φ0x )2 .

0

(3.23) t φx (τ )21 dτ , we have

For the estimate on 0

Lemma 3.5. Under the assumptions of Proposition 3.1, there exists a positive constant C18 depending on V , V¯ , φ0 , ψ0  and φ0x  such that for 0 ≤ t ≤ T1 , t φx (t) + 2

 φx (τ )21 dτ ≤ C18 1 + (φ0 , ψ0 , φ0x )2 .

(3.24)

0

Proof. From (3.3)2 , we have 

φx −ψ v α+1



t

Vx = − α+1 v



t

vxx β + 5 vx2 + β+5 − 2 v β+6 v

+ [p(φ + V ) − p(V )]x . x

(3.25)

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Multiplying (3.25) by 

φx 2v α+1

=

ψx2 v α+1

+

2

27

φx yields v α+1

φx − ψ α+1 v

 − p  (v) t

φx2 v α+1

ψψx Vx ψφx Ux Uxx φx φx Vx (ψx + Ux ) + (α + 1) α+2 − 2α+2 + (α + 1) α+2 v v v v 2α+3

2 φx β + 5 vx φx Vx − + [p  (φ + V ) − p  (V )] α+1 + {· · ·}x . (3.26) β+6 α+1 2 v v x v

− (α + 1)

vxx v β+5

Integrating (3.26) with respect to t and x over [0, t] × R, we have 1 4

 R

t 

φx2

v

dx + 2α+2

−p  (v)

0 R

⎛

≤ C ⎝ψ0 2 + ψ(t)2 +

φx2 dxdτ v α+1 ⎞



2 φ0x

v02α+2

R

⎠+

t  0 R

 ψx2 dxdτ + C Ji , v α+1 6

(3.27)

i=4

where t   φx Vx J4 = [p  (φ + V ) − p  (V )] α+1 v

dxdτ,

0 R

t  J5 = 0 R

J6 =

β + 5 vx2 vxx − β+5 2 v β+6 v

x

φx dxdτ, v α+1

          t    ψψx Vx   ψφx Ux   Uxx φx   φx ψx Vx   Ux Vx φx   + + + +   v α+2   v α+2   v 2α+2   v 2α+3   v 2α+3  dxdτ, 0 R

and we have used the fact that  R

ψφx 1 dx ≤ 4 v α+1

 R

φx2 v 2α+2

 dx + C

ψ 2 dx. R

By the Cauchy inequality, the mean value theorem, Lemmas 3.2 and 3.4 and Corollary 3.1, we have |J4 | ≤ C(V , V¯ )

t  |φφx Vx |dxdτ 0 R

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28

t 

1 ≤ 8

0 R

t 

1 ≤ 8

0 R

t 

1 ≤ 8

0 R

−p  (v)φx2 dxdτ + C(V , V¯ ) v α+1

t Vx (τ )2L∞ sup {φ(τ )2 }dτ 0≤τ ≤t

0

−p  (v)φx2 dxdτ + C(V , V¯ ) v α+1

t

(1 + τ )−2 dτ

0

−p  (v)φx2 dxdτ + C(V , V¯ ), v α+1

(3.28)

and t  0 R

1 + 8

≤

1 8

t

t



0 R

0 R

t

−p  (v)φx2 dxdτ + C v α+1

+ C(V , V¯ )

t



0 R



(Vx , Ux )(τ )2L∞ 0

1 ≤ 8

t



0 R

 |ψ 2 Vx2 | + |ψ 2 Ux2 | + |Uxx |2 + |Vx Ux |2 dxdτ

−p  (v)φx2 dxdτ + C(V , V¯ ) v α+1

0 R



t 

ψx2 dxdτ + C(V , V¯ ) v α+1

|J6 | ≤ C

t 

−p  (v)φx2 dxdτ + C v α+1

0 R

t 

ψx2 dxdτ v α+1

Vx (τ )2L∞ 0 R

ψx2 dxdτ v α+1  −2

sup {ψ(τ ) } + ε (τ + 1) 2

2

0≤τ ≤t

−3

+ ε (τ + 1) 3

ψx2 dxdτ + C(V , V¯ ). v α+1

dτ

(3.29)

For J5 , we deduce from integration by parts that t   J5 =

v β+2

0 R

t  =− 0 R

t



1

1 v β+2



=− 0 R

−vx v3

x

β − 1 vx2 2 v β+6

v   φ x x α−2 · v v α+1 x v α+1 &

1 v β−α+4 t

+



+ (α − 2) 0 R

φx v α+1

'2 x

Vx (φx + Vx ) v β+6

φx dxdτ v α+1

x

β −1 dxdτ + 2 x &

t  0 R



φx v α+1

t



dxdτ − x

v β+6

' t 

β −1 dxdτ + − (α − 2) 2

0 R

0 R

1 v β+2



vx2



φx v α+1 

vx2 v β+6

Vx v3

 x

dxdτ x

φx v α+1

φx v α+1

dxdτ x

dxdτ x

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t  =− 0 R

&

1 v β−α+4

φx v α+1

'2

 Vx (φx + Vx ) φx v β+6 v α+1 )*

t  dxdτ + (α − 2)

x

0 R

(

29

dxdτ x

+

J51

t  − 0 R



1 v β+2



Vx v3

x

φx v α+1

)*

(

(3.30)

dxdτ , x

+

J52

where in the last step of (3.30), we have used the assumption that β = 2α − 3. The Cauchy inequality, (3.6) and Lemma 3.2 imply that 1 J51 ≤ 4 1 ≤ 4

t  0 R

t  0 R

&

1 v β−α+4

&

1 v β−α+4

t  0 R

1 ≤ 4

0 R

&

1 v β−α+4

φx v α+1

'2

t  dxdτ + C

x

0 R

1 v α+β+8

'2

Vx2 (φx2 + Vx4 )dxdτ

t dxdτ + C(m1 , M1 )

x

Vx (τ )6L6 dτ 0

−p  (v)φx2 Vx (τ )2L∞ dxdτ v α+1

+ C(m1 , M1 ) t 

φx v α+1

φx v α+1

'2

⎛ dxdτ + C(m1 , M1 ) ⎝ε 2

x

t  0 R

⎞ −p  (v)φx2 dxdτ + ε ⎠ , v α+1 (3.31)

and 1 J52 ≤ 4

≤

1 4

t  0 R

t  0 R

&

1 v β−α+4

&

1 v β−α+4

t  + C(m1 , M1 ) 0 R

1 ≤ 4

t  0 R

1 v β−α+4

&

φx v α+1 φx v α+1

'2

t 

 2 dxdτ + C(m1 , M1 ) + Vx2 (Vx + φx )2 dxdτ Vxx

x

0 R

'2 dxdτ + C(m1 , M1 ) x

t

 Vxx (τ )2 + Vx (τ )4L4 dxdτ

0

−p  (v)φx2 Vx (τ )2L∞ dxdτ v α+1 φx v α+1

'2 x

⎛ dxdτ + C(m1 , M1 ) ⎝ε

t  2 0 R

⎞ −p  (v)φx2 dxdτ + ε ⎠ . v α+1 (3.32)

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30

Combining (3.30)–(3.32) yields 1 J5 ≤ − 2

t  0 R

&

1 v β−α+4

φx v α+1

'2

⎛ dxdτ + C(m1 , M1 ) ⎝ε 2

x

t  0 R

⎞ −p  (v)φx2 dxdτ + ε ⎠ . v α+1 (3.33)

Then it follows from (3.27)–(3.29), (3.33) and Lemma 3.3 that  R

t 

φx2

v

dx + 2α+2 0 R

⎛

−p  (v)φx2 dxdτ + v α+1

t  0 R

1 v β−α+4

&

φx v α+1

'2 dxdτ x

⎞    2 2 φ φ 0x 0x ≤C⎝ + β+5 (v0 , V0 ) + ψ02 + 2α+2 dx + 1⎠ v0 v0

(3.34)

R

provided that ε is sufficiently small such that C(m1 , M1 )ε 2 < 12 . (3.34) together with Lemma 3.4 implies that t φx (t) + 2

 φx (τ )2 dτ ≤ C(V , V¯ ) 1 + (φ0 , ψ0 , φ0x )2 .

(3.35)

0

On the other hand, we can also deduce from (3.34) and Lemma 3.4 that t φx (t) +

φx (τ )21 dτ

2

0

⎛

⎞ t 

 ≤ C(V , V¯ ) ⎝(φ0 , ψ0 , φ0x )2 + |φxx φx vx | + |φx2 vx2 | dxdτ ⎠ . 0 R

The last term on the right hand side of (3.36) can be estimated as follows C(V , V¯ )

t 

 |φxx φx vx | + |φx2 vx2 | dxdτ

0 R

1 ≤ 4

t

t  φxx (τ ) dτ + C 0 R

0

1 ≤ 4

|φx2 vx2 |dxdτ

2

t φxx (τ ) dτ + C 2

0

t 

0 R

 |φx4 | + |φx2 Vx2 | dxdτ

(3.36)

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1 ≤ 4

t φxx (τ ) dτ + C 2

0

1 ≤ 2

t

t φxx (τ ) dτ + Cε 2

1 2

t φx (τ ) dτ + C sup {φx (τ ) }

2

0

≤

 φxx (τ )φx (τ )3 + Vx (τ )2L∞ φx (τ )2 dτ

0

t

t

31

2

φx (τ )2 dτ

2 2

0≤τ ≤t

0

0

φxx (τ )2 dτ + C(V , V¯ )N1 (0)3 ,

(3.37)

0

where N1 (0) := 1 + (φ0 , ψ0 , φ0x )2 , and we have used (3.35) in the last step of (3.37). Combining (3.36) and (3.37), then we can obtain (3.24) by letting C18 = C(V , V¯ )N1 (0)2 . This competes the proof of Lemma 3.5. 2 Next, we derive the higher order energy type estimates on (φ, ψ). Lemma 3.6. Under the assumptions of Proposition 3.1, there exists a positive constant C19 depending on V , V¯ , φ0 1 and ψ0  such that for 0 ≤ t ≤ T1 , t (ψx , φxx )(t) +

ψxx (τ )2 dτ ≤ C19 (1 + φ0 22 + ψ0 21 ).

2

(3.38)

0

Proof. Multiplying (3.3)2 by −ψxx and using equation (3.3)1 , we have 

2 φxx ψx2 + β+5 2 2v

+ t

2 2 U ψxx (β + 5)φxx x + α+1 β+6 v 2v

= [p  (φ + V ) − p  (V )]Vx ψxx + p  (φ + V )φx ψxx −

Uxx ψxx (α + 1)ψx ψxx (φx + Vx ) + α+1 v v α+2

2 ψ (α + 1)Ux ψxx (φx + Vx ) (5 + β)φxx Vxxx (β + 5)(φx + Vx )φxx x + − + β+5 − v β+ 6 v α+2 2v β+6 v

2(β + 5)(φx + Vx )Vxx (β + 5)(β + 6)(φx + Vx )3 − + ψxx . 2v β+7 v β+6

(3.39)

Integrating (3.39) with respect to t and x over [0, t] × R, we get from (3.20) that 

t (ψx , φxx )(t) +

ψxx (τ ) dτ ≤ C (φ0xx , ψ0x ) +

2

2

0

where

2

8  i=6

 Ii ,

(3.40)

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32

t 

  % $  [p (φ + V ) − p  (V )]Vx ψxx  + p  (φ + V )φx ψxx  dxdτ,

J6 = 0 R

J7 =

t   ,

 |Uxx | + |Ux (ψx + Vx )| + |ψx (φx + Vx )| |ψxx | + |φxx |2 |ψx | dxdτ,

0 R

J8 =

 t  .

 / |Vxxx | + |(φx + Vx )φxx | + |(φx + Vx )Vxx | + |(φx + Vx )|3 |ψxx | dxdτ.

0 R

Similar to the estimates of Ji , i = 1, · · · , 5, we have t |J6 | ≤ η

 t  2 2 2 ψxx (τ ) dτ + Cη φx (τ ) + Vx (τ )L∞ sup {φ(τ ) } dτ 2

0

t ≤η

0≤τ ≤t

0

⎛

ψxx (τ ) dτ + Cη N2 (0) ⎝

0

t |J7 | ≤ η

t

2

⎞ φx (τ )2 dτ + 1⎠ ,

(3.41)

0

t

ψxx (τ ) dτ + Cη (Uxx , Vx Ux )(τ )2 + (Ux , Vx )(τ )2L∞ ψx (τ )2 2

0

0

 1 2

1 2

+ψx (τ )ψxx (τ ) sup {φx (τ ) } + ψx (τ ) ψxx (τ ) φxx (τ ) 2

0≤τ ≤t

t ≤ 2η

2

dτ

⎞ ⎛ t 

 ψxx (τ )2 dτ + Cη N2 (0)2 ⎝ (φx , ψx )(τ )2 + φxx (τ )4 dτ + 1⎠ ,

0

0

(3.42) t 

t |J8 | ≤ η

ψxx (τ ) dτ + Cη

(Vxxx , Vx Vxx )(τ )2 + sup {φx (τ )}φxx (τ )3

2

0

0≤τ ≤t

0



+(Vx , Vxx )(τ )2L∞ (φx , φxx )(τ )2 t ≤η 0

+ Vx (τ )6L6

+ sup {φx (τ ) }φxx (τ ) 4

0≤τ ≤t

2

dτ

⎞ ⎛ t 

 ψxx (τ )2 dτ + Cη N2 (0)2 ⎝ (φx , ψx , φxx )(τ )2 + φxx (τ )4 dτ + 1⎠ , 0

(3.43)   where the constant N2 (0) := C17 1 + (φ0 , ψ0 , φ0x )2 .

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33

Substituting (3.41)–(3.43) into (3.40), then (3.38) follows by using Gronwall’s inequality, Corollary 3.1, Lemma 3.5 and the smallness of η. This completes the proof of Lemma 3.6. 2 For the estimate of φxx (t), we have Lemma 3.7. Under the assumptions of Proposition 3.1, there exists a positive constant C20 depending only on V , V¯ , φ0 2 and ψ0 1 such that for 0 ≤ t ≤ T1 , t φxx (t) +

φxx (τ )21 dτ ≤ C20 (1 + φ0 22 + ψ0 21 ).

2

(3.44)

0

Proof. Differentiating (3.3)2 with respect to x, then multiplying the resultant equation by and using equation (3.3)1 , we have 

2 φxx φxx − ψx α+1 2α+2 2v v

− p  (v) t

2 2 φxx φxxx + = K1 + K2 + K3 + K4 + {· · ·}x , v α+1 v α+β+6

φxx , v α+1

(3.45)

where 2 ψxx (α + 1)ψx ψxx (φx + Vx ) (α + 1)ψx φxx (ψx + Ux ) − + , α+1 v v α+2 v α+2

 φ p  (v)φxx Vxx xx , K2 = p  (v)(φx + Vx )2 − p  (V )Vx2 − p  (V )Vxx α+1 + v v α+1 (α + 1)Vxx (ψx + Ux ) 2(α + 1)(ψxx + Uxx )(φx + Vx ) Uxxx + K3 = φxx − 2α+2 + v v 2α+3 v 2α+3

(α + 1)(α + 2)(φx + Vx )2 (ψx + Ux ) − , v 2α+4

2(β + 5)(φx + Vx )(φxx + Vxx ) (β + 5)(β + 6)(φx + Vx )3 Vxxx − K4 = φxxx − α+β+6 + v α+β+7 v 2v α+β+8 φxxx + Vxxx 2(β + 5)(φx + Vx )(φxx + Vxx ) − + (α + 1)φxx (φx + Vx ) v α+β+7 v α+β+8

(β + 5)(β + 6)(φx + Vx )3 + . 2v α+β+9

K1 = −

Integrating (3.45) with respect to t and x over [0, t] × R, by the similar estimates as in the proof previous lemmas, we can get (3.44). The details are omitted here. Thus the proof of Lemma 3.7 is completed. 2 Combining Corollary 3.1 and Lemmas 3.5–3.7, we have Lemma 3.8. Under the assumptions of Proposition 3.1, there exists a positive constant C21 depending only on V , V¯ , φ0 2 and ψ0 1 such that for 0 ≤ t ≤ T1 ,

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34

φ(t)22

+ ψ(t)21

t   +  Vt φ(τ )2 + φx (τ )22 + ψx (τ )21 dτ 0



≤ C21 1 + φ0 22

 + ψ0 21 .

(3.46)

By the same argument as above, we also obtain Lemma 3.9. Under the assumptions of Proposition 3.1, there exists a positive constant C22 depending only on V , V¯ , φ0 4 and φ0 3 such that for 0 ≤ t ≤ T1 ,

φxxx (t)21

+ ψxx (t)21

+

t

 φxxx (τ )22 + ψxxx (τ )21 dτ

0



 ≤ C22 1 + φ0 24 + ψ0 23 . Proof of Proposition 3.1. Proposition 3.1 follows immediately from Lemmas 3.8–3.9.

(3.47) 2

Proof of Theorem 1.2. With Proposition 3.1 and the local existence result in hand, we can extend the local-in-time solution to be a global one (i.e., T1 = +∞) by the standard continuity argument. Thus, (1.15) and (1.16) follows from (3.8) and (3.9), respectively. Moreover, the estimate (1.16) and the system (3.3) imply that +∞

   d 2  (φx , ψx )(t) +  (φx , ψx )(t)  dt < ∞, dt 2

(3.48)

0

which together with (1.16) and the Sobolev inequality leads to the following asymptotic behavior of solutions: lim sup {|(v(t, x) − V (t, x), u(t, x) − U (t, x))|} = 0.

t→+∞ x∈R

(3.49)

From (3.49) and (iii) of Lemma 3.2, we have (1.17) at once. This completes the proof of Theorem 1.2. 2 Acknowledgments The authors thank the anonymous referee for his/her many helpful comments and suggestions on the draft version of this manuscript. ZZC was supported by the Tian Yuan Foundation of China (Grant No. 11426031) and the Doctoral Scientific Research Funds of Anhui University (Grant No. J10113190005). HJZ was support by two grants from the National Natural Science Foundation of China under contracts 10925103 and 11271160 respectively. This work was also supported by a grant from the National Natural Science Foundation of China under contract 11261160485 and the Fundamental Research Funds for the Central Universities.

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