Asymptotic stability of a composite wave of two viscous shock waves for the one-dimensional radiative Euler equations

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Asymptotic stability of a composite wave of two viscous shock waves for the one-dimensional radiative Euler equations Lili Fan a , Lizhi Ruan b,∗ , Wei Xiang c a School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, China b The Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079,

China c Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Received 12 April 2017; received in revised form 28 November 2017; accepted 25 March 2018

Abstract This paper is devoted to the study of the wellposedness of the radiative Euler equations. By employing the anti-derivative method, we show the unique global-in-time existence and the asymptotic stability of the solutions of the radiative Euler equations for the composite wave of two viscous shock waves with small strength. This method developed here is also helpful to other related problems with similar analytical difficulties. © 2018 Elsevier Masson SAS. All rights reserved. MSC: 35B35; 35M20; 35L67; 35Q35 Keywords: Radiative Euler equations; Viscous shock waves; Diffusion wave; Stability

1. Introduction The radiative Euler equations are a fundamental system to describe the motion of the compressible gas with radiation heat transfer phenomenon. Mathematically, the radiative Euler equations are a hyperbolic–elliptic coupled system with the following form: ⎧ ρt + div(ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (ρu)t + div(ρu ⊗ u) + ∇p = 0,       (1.1) |u|2 |u|2 ⎪ ⎪ + div ρu e + + pu + divq = 0, ρ e + ⎪ 2 2 ⎪ t ⎪ ⎩ −∇divq + aq + b∇θ 4 = 0, * Corresponding author.

E-mail addresses: [email protected] (L. Fan), [email protected] (L. Ruan), [email protected] (W. Xiang). https://doi.org/10.1016/j.anihpc.2018.03.008 0294-1449/© 2018 Elsevier Masson SAS. All rights reserved.

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where ρ, u, p, e, θ , and q are respectively the mass density, velocity, pressure, internal energy, absolute temperature, and the radiative heat flux. Positive constants a and b depend only on the concerned gas itself. As the classic compressible Euler equations, the first three equations in (1.1) stand for the conservation of mass, momentum and energy respectively. The fourth equation in (1.1) is related to the radiation heat transfer phenomenon, and one can refer [32,45,53] for more details. System (1.1) can also be derived by the non-relativistic limit (speed of light tending to +∞) from a hyperbolic–kinetic system, where the radiation is described by the photon density. The photon density satisfies a transport equation with the interaction kernel given by the Stefan–Boltzmann law. Details in this direction can be found in [1,9,17,30]. Let v = ρ1 be the specific volume, then one can rewrite the one-dimensional radiative Euler equations in the Lagrangian coordinates that ⎧ vt − ux = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ut + px = 0,   (1.2) 2 ⎪ ⎪ e + u2 + (pu)x + qx = 0, ⎪ ⎪ ⎪

 ⎩ qx  t − v x + avq + b θ 4 x = 0. In this paper, we will consider the polytropic gas, i.e., the compressible flow satisfies the following thermal relations that Rθ R p= , e= θ, (1.3) v γ −1 where γ > 1 is the adiabatic exponent and R > 0 is the specific gas constant. The solutions satisfy the following initial data and the far field behaviours that (v, u, θ ) (x, 0) = (v0 , u0 , θ0 ) (x), (1.4) (v, u, θ, q) (±∞, t) = (v± , u± , θ± , 0) , where the far field states are given constants and satisfy that v± > 0, θ± > 0, and u± ∈ R. Then, in this paper, we will study the global existence and the asymptotic stability of the Cauchy problem (1.2) and (1.4) with the far field states that (v± , u± , θ± , 0). As far as we know, there are very few results on the wellposedness of the Cauchy problem of the radiative Euler equations (1.2) due to the complexity and nonlinearity of the governing equations. Almost all the results are related to the analysis of the global in time stability of the viscous Riemann solutions. More precisely, if the radiation effect is neglected, the Riemann solution consists of elementary waves such as shock waves, rarefaction waves and contact discontinuities, which are scaling invariant solutions of the Riemann problem (Euler system): ⎧ vt − ux = 0, ⎪ ⎪ ⎪ ⎨ ut + px = 0, (1.5)   ⎪ ⎪ 2 ⎪ ⎩ e+ u + (pu) = 0, 2

x

t

with the Riemann-type initial data (v, u, θ ) (x, 0) =



R v0R , uR 0 , θ0



(x) :=

(v− , u− , θ− ) , x < 0, (v+ , u+ , θ+ ) , x > 0.

(1.6)

The global in time existence of solutions around a constant state was shown in [19]. For the analysis of the rarefaction wave, if the initial data is a small perturbation of a given rarefaction wave, it was proved in [26] that the solutions converge to the rarefaction wave as t → +∞. Then in [13], the authors showed that when the absorption 1 coefficient α tends to +∞, the solutions converge to the rarefaction wave with the convergence rate α − 3 |ln α|2 , where the absorption coefficient α is defined by the relationship a = 3α 2 and b = 4ασ for positive constants a, b and the Stefan–Boltzmann constants σ . The asymptotic stability of a single viscous contact wave was proved in [46,47]. The stability of the composite wave of rarefaction waves and a contact wave was investigated in [39,51]. However, for

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the analysis of the viscous shock wave, only the shock profile has been studied. The existence of the shock profile was studied in [27]. The asymptotic stability of the shock profile was studied in [28]. In [34], the authors discussed the stability for the case of general hyperbolic–elliptic systems including also the nonlinear stability for the radiative gas dynamics. Recently the formation of the Zeldovich spikes, which is the internal maximum in the profile of temperature, is given in [2,31]. Therefore, there is a natural question: How is the asymptotic stability of the composite wave consisting of two viscous shock waves for the radiative Euler equations (1.2)? We will give the positive answer on this problem in this paper. Our analysis is based on the anti-derivative method and energy estimates. As far as we know, it is the first time to use the anti-derivative method to study the asymptotic stability of the elementary waves for the radiative Euler equations. Since system (1.2) is less dissipative than other systems which were studied by the anti-derivative method, [5,12,14–16] for example, we need more subtle estimates to recover the regularity. We also construct a diffusion wave to eliminate the extra mass, such that (v, u, E) are conserved. We remark that we are also motivated by the related investigations on the simplified model (Hamer model), which gives a good approximation to the fundamental system (1.2) in a certain physical situation (see [10]). The stability of shock waves for the simplified model has been extensively studied in [19–21,24,23,33–38,43]. For the other related results, one can refer to [3,4,6–8,22,40–42,48–50,52]. The rest of the paper is organised as follows: In Section 2, we construct the viscous shock waves and the diffusion wave, and state the main results. In Section 3, basic properties of these viscous waves and the anti-derivative method are introduced. In Section 4, we show the local existence. Finally, in Section 5, the a priori estimates are established by the energy method. 2. Viscous waves and main theorem 2.1. Viscous shock waves In this subsection, we will construct the viscous shock waves of (1.2) based on [25,27]. For the shortness, we only sketch the arguments and omit the details. In this paper, we consider the situation when the Riemann solution of problem (1.5) and (1.6) consists of two shock waves, i.e., there exists an intermediate state (vm , um , θm ) such that (v− , u− , θ− ) connects with (vm , um , θm ) by the 1-shock wave with the shock speed s1 < 0, and (vm , um , θm ) connects with (v+ , u+ , θ+ ) by the 3-shock wave with the shock speed s3 > 0. It is well-known that for any given (v− , u− , θ− ) with v− > 0, such Riemann solution exists provided that (v+ , u+ , θ+ ) is located on a curved surface in a small neighbourhood of (v− , u− , θ− ). Let − be the neighbourhood of (v− , u− , θ− ), i.e.,



− := (v, u, θ) |(v − v− , u − u− , θ − θ− )| ≤ δ¯ , (2.1) where δ¯ is a positive constant depending only on (v− , u− , θ− ). In order to describe the strength of the shock waves, let δ1 = |vm − v− | + |um − u− | + |θm − θ− |, δ3 = |vm − v+ | + |um − u+ | + |θm − θ+ |,

(2.2)

and let δ = min{δ1 , δ3 }.

(2.3)

If δ¯ is small, then for (v+ , u+ , θ+ ) ∈ − , it holds that (cf. [44]), δ ≤ C |(v+ − v− , u+ − u− , θ+ − θ− )| ,

(2.4)

where C is a positive constant depending only on (v− , u− , θ− ). Moreover, we assume that there exists a constant C > 0, such that δ1 + δ3 ≤ Cδ.

(2.5)

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Next let us consider the viscous i-shock wave of (1.2) with the form Zi = (Vi , Ui , i , Qi )(x − si t), i = 1, 3. If i = 1, Z1 corresponds to the 1-shock wave with the shock speed s1 < 0 connecting the far field states z− = (v− , u− , θ− , 0) and zm = (vm , um , θm , 0). Since the decay rate of qxx is expected in a higher order, the viscous 1-shock wave of (1.2) is the solution with the form (v, u, θ, q) = (V1 , U1 , 1 , Q1 ) (ξ ), where ξ = x − s1 t , satisfying the following equation that ⎧ −s1 V1 − U1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −s1 U1 + P1 = 0, ⎪ ⎪ ⎪ ⎪     ⎪ ⎪ U12 ⎨ R 4b 3 −s1 γ −1 1 + 2 + (P1 U1 ) = aV1 1 1 , (2.6)

 ⎪ b b 3 ⎪ 4 ⎪ Q1 = − aV 1 = − aV 1 1 , ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ , U , , Q (−∞) = (V ) (v− , u− , θ− , 0) , 1 1 1 1 ⎪ ⎪ ⎪ ⎩ (V1 , U1 , 1 , Q1 ) (+∞) = (vm , um , θm , 0) , where P1 = p (V1 , 1 ) , p± = p (v± , θ± ) , e± = e (θ± ), pm = p (vm , θm ), and em = e (θm ). In addition, the shock speed s1 and the far field states of Z1 given in (2.6) must satisfy the Rankine–Hugoniot condition and the following entropy condition   λ1 (v− , θ− ) = − γp− /v− > s1 > − γpm /vm = λ1 (vm , θm ). (2.7) It is easy to see that the Rankine–Hugoniot condition and entropy condition (2.7) imply that     γp− d− v− + vm d− s12 = 1− , θm = θ − 1 − , v− > v m , vm 1 + d− v− 1 + d−

(2.8)

vm −v− ¯ where d− = γ −1 2 vm < 0. Here we may assume that δ is suitably small to assure that |d− | < 1. By [25,27], we can construct a solution Z1 of (2.6), which is unique up to the shift of ξ . We omit the details, since the arguments can be found in [25,27]. Similarly, if i = 3, Z3 corresponds to the 3-shock wave of system (1.5) with the shock speed s3 > 0 connecting the far field states zm = (vm , um , θm , 0) and z+ = (v+ , u+ , θ+ , 0). The solution satisfies the following equation that ⎧ −s3 V3 − U3 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −s3 U3 + P3 = 0, ⎪ ⎪ ⎪ ⎪   ⎪   ⎪ U32 ⎨ R 4b 3 −s3 γ −1 3 + 2 + (P3 U3 ) = aV , 3 3 3 (2.9)

4  ⎪ b b 3 ⎪ ⎪ Q = − = − , 3 ⎪ 3 aV3 aV3 1 3 ⎪ ⎪ ⎪ ⎪ ⎪ (V3 , U3 , 3 , Q3 ) (−∞) = (vm , um , θm , 0) , ⎪ ⎪ ⎪ ⎩ (V3 , U3 , 3 , Q3 ) (+∞) = (v+ , u+ , θ+ , 0) ,

under the entropy condition   λ3 (vm , θm ) = γpm /vm > s3 > γp+ /v+ = λ3 (v+ , θ+ ). We omit the details, since the arguments can be found in [25,27] too. 2.2. Diffusion wave 

In this subsection, by using the method in [11], we will construct a diffusion wave Z d (x, t) = v d , ud , θ d , q d (x, t), which connects the same constant state zm = (vm , um , θm , 0) at the positive and negative infinity.

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First, we temporarily (the exact definition will be given in (2.16) and (2.17)) expect that v d and θ d are of the following form vd =

R ˜ , pm

˜ θd =

and the radiative heat flux q d is b  ˜ 4 q d := − d x av ˜ with the far field states that (±∞, ˜ for a function t) = θm . Then the first equation of (1.2) is   R ˜ − udx = 0. pm t The second and third equations together yield that   ˜x ˜ 3 R ˜ 4b d . t + pm ux = γ −1 a vd

(2.10)

(2.11)

x

˜ Substituting (2.10) into (2.11), we have a nonlinear diffusion equation for   ˜ t = γ − 1 4bpm ˜x . ˜ 2 x Rγ aR

(2.12)

In order to avoid the nonlinearity of the equation above, we further approximate the equation (2.12) by the following linear heat equation ˜ t =k ˜ xx ,

˜ where (±∞, t) = θm ,

k=

γ − 1 4bpm θm2 . Rγ aR

Let constant β2 be  ˜ β2 = ( (x, t) − θm )dx.

(2.13)

(2.14)

Thus, by employing the heat kernel formulation, the solution of (2.13) is ˜ = θm + √

x2 β2 e− 4k(1+t) . 4πk(1 + t)

Now define θ d to be  2 ˜ − γ − 1 ud − um , θd = 2R and v d and ud to be R ˜ aR ˜ , ud = um + x . vd = pm pm

(2.15)

(2.16)

(2.17)

2.3. Composite wave of two viscous shock waves and a diffusion wave Based on the construction of the viscous shock waves and the diffusion wave, we can construct a composite wave, which is the asymptotic states of the solutions of the initial value problem (1.2) and (1.4) as t → ∞. Set γ −1 2 and m(x, t) = (v, u, E) (x, t). (2.18) u 2R Following the arguments in [29], the asymptotic state is expected to be a composite wave M(x, t) = 

V , U, E β1 ,β3 (x, t), which is given by E=θ +

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V = V1 (x − s1 t + β1 ) + V3 (x − s3 t + β3 ) − vm + (v d (x, t) − vm ), U = U1 (x − s1 t + β1 ) + U3 (x − s3 t + β3 ) − um + (ud (x, t) − um ),

(2.19)

E β1 ,β3 = E1 (x − s1 t + β1 ) + E3 (x − s3 t + β3 ) − Em + (E d (x, t) − Em ), = E β1 ,β3 −

γ −1 2 U , 2R

Q = Q1 (x − s1 t + β1 ) + Q3 (x − s3 t + β3 ) + q d (x, t), where Ei = i +

γ −1 2 2R Ui (i

= 1, 3, m) and E d = θ d +

γ −1 d 2 2R (u ) .

It follows from (2.16) that

    ˜ − θm + γ − 1 ud − um um , E d − Em = R and  (E d − Em )dx = β2 . Similarly as done in [5], the constant vector (β1 , β2 , β3 ) is determined by  (m(x, 0) − M(x, 0)) dx = 0.

(2.20)

R

We remark that (2.20) allows us to apply the anti-derivative method. 2.4. Main theorem Before stating the main theorem, let us introduce several notations which we will use throughout this paper. Lp (R)(1 ≤ p ≤ ∞) represents the Lebesgue space on R with norm · Lp . For k ∈ Z+ , H k (R) represents the Sobolev space with the norm · k . It is easy to see that · 0 = · L2 . To simplify the notation, we set · := · 0 = · L2 . Finally, we denote by C k (I ; H p ) the k-times continuously differentiable functions in the interval I with the range in H p (R); and denote by L2 (I ; H p ) the space of L2 functions in the interval I with the range in H p (R). We are ready to introduce the main result of this paper. For any fixed (v− , u− , θ− , 0), we assume that (v+ , u+ , θ+ , 0) ∈ − and the Riemann solution of (1.5) and (1.6) with the initial data (v± , u± , θ± , 0) consists of two shock waves. Let v = V1 (x) + V3 (x) − vm , E = E1 (x) + E3 (x) − Em ,

u = U1 (x) + U3 (x) − um , θ =E−

γ −1 2 u , 2R

(2.21)

q = Q1 (x) + Q3 (x).

Suppose the initial data (v0 , u0 , θ0 , q0 ) satisfies that v0 − v, ¯ u0 − u, ¯ θ0 − θ¯ ∈ H 2 ∩ L1 ,

q0 − q¯ ∈ H 3 ∩ L1 .

Then (V , U, , Q) in (2.19) is well defined and satisfies (2.20). Furthermore, let

(2.22)

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x 0 (x) =

[v0 (y) − V (y, 0)] dy, −∞ x

0 (x) =

[u0 (y) − U (y, 0)] dy, −∞ x

W 0 (x) = −∞ x

0 (x) =



u2 R θ0 + 0 γ −1 2



  R U2 (y) − + (y, 0) dy, γ −1 2 

(2.23)

[(v0 q0 ) (y) − (V Q) (y, 0)] dy, −∞

and assume that (0 , 0 , W 0 , 0 ) ∈ H 3 .

(2.24)

Let I (v0 , u0 , θ0 , q0 ) = (v0 − V (·, 0), u0 − U (·, 0), θ0 − (·, 0)) H 2 ∩L1 + q0 − Q(·, 0)) H 3 ∩L1 + (0 , 0 , W 0 , 0 ) L2 . Then, our main result is the following theorem. Theorem 2.1. Assume that (2.2), (2.22) and (2.24) holds, and 1 < γ < 3. Then there exist positive constants δ0 and 0 such that if |(v+ − v− , u+ − u− , θ+ − θ− )| ≤ δ0 , I (v0 , u0 , θ0 , q0 ) ≤ 0 , then the Cauchy problem (1.2) and (1.4) admits a unique global in time solution (v, u, θ, q) satisfying that     (v − V , u − U, θ − ) ∈ C 0 [0, ∞); H 2 , q − Q ∈ C 0 [0, ∞); H 3     (v − V , u − U, θ − ) ∈ L2 [0, ∞); H 2 , q − Q ∈ L2 [0, ∞); H 3 , and the asymptotic behaviour that lim sup |(v − V , u − U, θ − , q − Q) (x, t)| = 0.

t→∞ x∈R

(2.25)

3. Preliminaries and mathematical reformulation In this section, we will introduce several properties of the viscous shock waves and the diffusion wave, and then introduce the anti-derivative method to reformulate the problem. 3.1. Preliminaries In this subsection, we will recall several properties of viscous shock waves and the diffusion wave. First, let us introduce two notations which will be frequently used throughout this paper. A  B means that there is a constant C > 0 which does not depend on δ such that A ≤ CB. Next, A ≈ B means that it holds that |A − B|  δ 2 e−cδ(|x|+t) +

|β2 | (1 + t)

cx 2

3 2

e− 1+t + δe−c(|x|+t) .

For the properties of viscous shock waves, as the ones listed in Proposition 2.1 of [5], we have

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Proposition 3.1. For any fixed (v− , u− , θ− , 0), suppose (v+ , u+ , θ+ , 0) ∈ − and suppose the Riemann solution of (1.5) consists of two shock waves whose strengths satisfy (2.4) and (2.5). Then problems (2.6) and (2.9) admit smooth solutions (V1 , U1 , 1 , Q1 ) (x − s1 t) and (V3 , U3 , 3 , Q3 ) (x − s3 t) respectively which are unique up to the spatial shift and satisfy the following: (1) Uix (x − si t) = −si Vix (x − si t) < 0, | ix (x − si t)|  |Vix (x − si t)| ,

(3.1)

x ∈ R, t ≥ 0,

i = 1, 3;

(2) There exist positive constants c and C such that for i = 1, 3, |(V1 − vm , U1 − um , 1 − θm ) (x − s1 t)|  δ1 e−cδ1 |x−s1 t| , |(V3 − vm , U3 − um , 3 − θm ) (x − s3 t)|  δ3 e−cδ3 |x−s3 t| , |(Vix , Uix , ix , Vixx , Uixx , ixx , Qi , Qix , Qixx ) (x − si t)|  δi2 e−cδi |x−si t| ,

2 γpm

 δi .

s −

i vm

(3.2)

The first three inequalities are respectively valid in the regions {x > s1 t, t ≥ 0}, {x < s3 t, t ≥ 0} and {x ∈ R, t ≥ 0}. Next,  wave obtained by (2.16) and (2.17), by the straightforward calculations, we know that vector

for the diffusion Z d = v d , ud , θ d , q d satisfies that ⎧ v d − udx = 0, ⎪ ⎪ ⎪ t ⎪

 ⎪ ⎪ ⎪ udt + pxd = R1d x , ⎪ ⎪ ⎪ 

 ⎪ 

 ⎨ R d (ud )2 θ + + p d ud x + qxd = R2d x , γ −1 2 (3.3) t ⎪ ⎪  d   ⎪ 

4 ⎪ q ⎪ − vxd + av d q d + b θ d = R3d x , ⎪ ⎪ ⎪ x x ⎪ ⎪ ⎪ ⎩ d d d d v , u , θ , q (±∞, t) = (vm , um , θm , 0) . Note that p d − pm = −

2 γ −1  d − u , u m 2v d

so the errors R1d , R2d and R3d are given by 2 Ra 2 ˜ γ −1  d xx − − u , u m pm 2Rv d    2 2 ˜ − θd 4bpm ˜ 2 ˜ x − γ − 1 ud − um ud + Ra um ˜ xx R2d = d d aR θ 2v pm R1d =

and R3d = b



θd

4



˜4 + −

 2 4bpm aR 2

˜ 2 ˜x

By (2.15), for x ∈ R, t ≥ 0, we have that

x2 |β2 | − 4a(1+t)

d

e ,

Z − zm  1 (1 + t) 2 and that

˜

(3.4)

 x

.

(3.5)

(3.6)

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∂ i  x2 |β2 | − 4a(1+t)

e , R1d , R2d , R3d 

∂x

(1 + t) 32

0 ≤ i ≤ 3.

9

(3.7)

Finally, let us introduce properties of the asymptotic state Z(x, t) = (V , U, , Q) (x, t) defied by (2.19). Let

 R 3 x = V Q − V1 Q1 − V3 Q3 − v d q d       = Q1 (V3 − vm ) + v d − vm + Q3 (V1 − vm ) + v d − vm + q d [(V1 − vm ) + (V3 − vm )] . Then one has  

 − QVx + aV Q + b 4 x x  



 = a R 3 x + aV1 Q1 + aV3 Q3 + av d q d + b 4 x − QVx   x   d 4



4 4 4 4 4 = a R 3 x + b − 1 + 3 − θm − θ − θm x   d qx Q1x Q3x Qx − V − V1 − V3 − v d . x

Therefore, it follows from (2.6), (2.9) and (3.3), that Z(x, t) approximately satisfies (1.2) with errors in the following way ⎧ Vt − Ux = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ U + Px = (R1 )x , ⎪ ⎨t  (3.8) R U2 + ⎪ γ −1 2 t + (P U )x + Qx = (R2 )x , ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎩ − Qx + aV Q + b 4  = (R3 ) , x V x x

where (V , U, , Q) (±∞, t) = (v± , u± , θ± , 0), P = R V , and the errors R1 , R2 and R3 are given by    R1 = P − P1 + P3 − pm + p d − pm + R1d ,    R2 = P U − P1 U1 + P3 U3 − pm um + p d ud − pm um + R2d ,        4 Q3x qxd Q1x Qx 4 4 4 4 d 4 R3 = aR 3 + b − 1 + 3 − θm − θ − θm − − − d . − V V1 V3 v For the errors, we can show the estimates that  i

 ∂ Rj ≈ 0, j = 1, 2, 3, 0 ≤ i ≤ 3. ∂x

(3.9)

(3.10)

Without loss of the generality, we only show the estimate of R1 when i = 0, since the proof of the other estimates is similar. Note that 

= 1 + 3 − θ m + θ d − θ m   



− γ R−1 (U1 − um ) (U3 − um ) + (U1 − um ) ud − um + (U3 − um ) ud − um . Then we have the estimate that 

2





d

d

|R1 |  u − um + |Z1 − zm | |Z3 − zm | + (|Z1 − zm | + |Z3 − zm |) z − zm + R1d .

(3.11)

The wave interaction terms at the right hand side of (3.11) can be estimated by applying Proposition 3.1 and (3.6), since we can see that   |Z1 − zm | |Z3 − zm |  δ1 δ3 e−cδ1 (|x|+t)+cδ1 |β1 | + e−cδ3 (|x|+t)+cδ3 |β3 | ,

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and



|Zi − zm | zd − zm  δi e−cδi (|x|+t)+cδi |βi |

|β2 |

cx 2

(1 + t)

1 2

e− 1+t + δi |β2 |e−c(|x|+t) , (i = 1, 3).

Note that δ1 and δ3 are small in the same order by (2.5), so |Z1 − zm | |Z3 − zm |  δ 2 e−cδ(|x|+t) ,



(Z1 − zm )zd , (Z3 − zm )zd  δ 2 e−cδ(|x|+t) +

|β2 | (1 + t)

cx 2

3 2

e− 1+t + δe−c(|x|+t) ,

|Z1x (Z3 − zm ), Z3x (Z1 − zm )|  δ 3 e−cδ(|x|+t) ,

(3.12)



Zix zd , (Z1 − zm )zxd , (Z3 − zm )zxd  δ 2 e−cδ(|x|+t) +

|β2 | (1 + t)

cx 2

3 2

e− 1+t + δe−c(|x|+t) .

It means that |R1 |  δ 2 e−cδ(|x|+t) +

|β2 | (1 + t)

cx 2

3 2

e− 1+t + δe−c(|x|+t) .

(3.13)

By the definition of ≈, estimate (3.13) is equivalent to that R1 ≈ 0. 3.2. Anti-derivative method In this subsection, we will introduce the anti-derivative method by reformulating the Cauchy problem (1.2) and (1.4) into an integrated system of (1.2). Set x (x, t) =

(v − V )(y, t) dy, −∞ x

(u − U )(y, t) dy,

(x, t) = −∞ x

W (x, t) = −∞



u2 R θ+ γ −1 2



 −

R U2 + γ −1 2

 (y, t) dy,

(3.14)

γ −1 (W − U )(x, t), R x (x, t) = (vq − V Q)(y, t) dy.

W (x, t) =

−∞

Obviously,

 Ux  − 12 2x ,

v − V = x ,

θ − = Wx +

γ −1 R

u − U = x ,

q −Q=

(x − Qx ) ,

1 V +x

(3.15)

where we expect (, , W, ) ∈ C([0, ∞); H 3 ). Then, substituting (3.15) to (1.2), subtracting (3.8), and integrating the deduced system with respect to x, we have the integrated system for (, , W, ) that

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⎧ t − x = 0, ⎪ ⎪ ⎪   ⎪

 ⎪ ⎪ + Wx + γ R−1 Ux  − 12 2x ⎪ ⎪ ⎪ t + R − = −R1 , ⎪ ⎪ V + x V ⎪ ⎪ ⎪ ⎪ 

 ⎪ ⎪ ⎪ + Wx + γ R−1 Ux  − 12 2x R ⎪ ⎪ ⎪ Wt + Ut  + R x ⎪ ⎪ V + x ⎪ ⎨ γ −1 x − Qx ⎪ = −R2 + U R1 − , ⎪ ⎪ ⎪ V + x ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ 1 Qx x x − Qx ⎪ ⎪ a + − ⎪ ⎪ V + x V + x x V (V + x ) ⎪ ⎪ ⎪ ⎪  ⎪   4 ⎪ ⎪ γ − 1 1 ⎪ ⎪ ⎪ = −b + Wx + − 4 − R 3 , Ux  − 2x ⎩ R 2

(3.16)

with the initial data that (, , W, )(0) = (0 , 0 , W0 , 0 ) ∈ H 3 .

(3.17)

Then we only need to study the Cauchy problem (3.16) and (3.17) instead of problem (1.2) and (1.4) to finish the proof of Theorem 2.1. 4. Local existence In this section, we will consider the local existence of the Cauchy problem (3.16) and (3.17). For any interval I ⊆ R+ , define the solution set X(I ) as  X(I ) = (, , W, w) ∈ C(I ; H 3 )| (x , x , Wx , x ) ∈ L2 (I ; H 2 ), w ∈ L2 (I ; H 3 ) , where

  γ −1 1 2 x − Qx ξ = θ − = Wx + . Ux  − x , w = q − Q = R 2 V + x We further choose a positive constant δ¯0 for the given (v− , u− , θ− , 0) such that if δ  δ¯0 , then sup |(V − v− , U − u− , − θ− ) (x, t)| ≤

x∈R,t≥0

1 min (v− , θ− ) , 4

where δ is given by (2.3). By the definition of Q in (2.19), it holds that

 sup |Q(x, t)|  sup Q 1 + Q 3 +  x  δ 2 e−cδ(|x|+t) + x∈R,t≥0

x∈R,t≥0

(4.1)

|β2 | 1 (1+t) 2

cx 2

e− 1+t  δ + |β2 |.

We also choose a positive constant ¯0 (≤ 12 min(v− , θ− )) such that if (, , W, )(t) 2 ≤ ¯0 , then

   

1

γ −1 1 2 x − Qx

sup , , Wx + Ux  − x , (x, t)

≤ min(v− , θ− ). R 2 V + x 2 x∈R

(4.2)

(4.3)

Then we have the following proposition about the local existence of the Cauchy problem (3.16) and (3.17). Proposition 4.1. (Local existence) For any fixed (v− , u− , θ− , 0), there exist positive constants ¯1 (≤ ¯0 ) and C¯ 1 (C¯ 1 ¯1 ≤ ¯0 ) such that the following statements hold. Under the assumption that δ  δ¯0 and that M ∈ (0, ¯1 ), there exists a positive constant t0 = t0 (M), such that if ((·, τ ), (·, τ ), W (·, τ ), w(·, τ )) H 3 ≤ M, then the Cauchy problem (3.16) with the initial condition (3.17) via replacing t = 0 by t = τ , admits a unique solution (, , W, w) ∈ X([τ, τ + t0 ]) satisfying sup

t∈[τ,τ +t0 ]

(, , W, w)(·, t) 3 ≤ C¯ 1 M.

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Proof. Since the main ideas and techniques are similar as the ones in [5], we sketch the process and only list the difference here for the shortness. Without loss of the generality, we only consider the case that τ = 0. First note that C¯ 1 M ≤ C¯ 1 ¯1 ≤ ¯0 is suitable small, then the Sobolev’s inequality with (4.1)–(4.3) together implies the local solutions constructed in Proposition 4.1 satisfy that 1 7 1 7 v− ≤ (V + x )(x, t) ≤ v− , θ− ≤ ( + ξ )(x, t) ≤ θ− . 4 4 4 4 Differentiate (3.16) with respect to x, and introduce the new variables (φ, ψ, ξ, w) that γ −1 1 (Ux  − 2x ), R 2 x − Qx . ψ = x , w = q − Q = V + x System (3.16) can be rewritten as follows ⎧ t = ψ, ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ t = − Rξ − P φ − R1 , ⎨ φ = x ,

(4.4)

ξ = θ − = Wx +

V +φ

V +φ

R( +ξ ) ⎪ R ⎪ ⎪ γ −1 Wt + U x = V +φ ψ − w + U R1 − R2 , ⎪ ⎪     ⎪   ⎪ Qφ Qx φ x 1 ⎩ a + 1 4 4 = + V (V V +φ V +φ V +φ V +φ +φ) − b ( + ξ ) − − R3 , x

(4.5)

(4.6)

x

with the initial data (, , W, )(x, 0) = (0 , 0 , W0 , 0 )(x). Moreover, (3.16) again can be rewritten as the following quasi-linear system of (φ, ψ, ξ, w) ⎧ φ − ψx = 0, ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ R( + ξ ) ⎪ ⎪ ⎪ ⎨ ψt − (V + φ)2 φx = F1 (φ, ξ, φx , ξx ), R( + ξ ) R ⎪ ⎪ ξt + ψx = F2 (φ, ξ, ψx , wx ), ⎪ ⎪ γ −1 V +φ ⎪ ⎪ ⎪ ⎪ 1 wx  ⎪ ⎪ = F3 (φ, ξ, φx , ξx ), ⎩ aw − V +φ V +φ x

(4.7)

(4.8)

where Px φ Rξx Rξ − P φ Vx − R1x , − + V +φ V +φ (V + φ)2 Rξ − P φ F2 (φ, ξ, ψ, w) := −wx − Ux + U R1x − R2x , V +φ 4b( + ξ )3 4b( + ξ )3 (2 + ξ ) x ξ ξx + 2 F3 (φ, ξ, φx , ξx ) := − (V + φ) (V + φ)     1 a( + ξ )5 Q V + φ V Qx φ − − − . V + φ (V + φ)V x V +φ ( + ξ )5 5 F1 (φ, ξ, φx , ξx ) :=

(4.9)

The initial data is (φ, ψ, ξ, w)(x, 0) = (φ0 , ψ0 , ξ0 , w0 )(x) 

 = 0 , 0 , W0x + γ R−1 Ux 0 − 12 20x ∈ H 2 . 0x − Q0x ∈ H 3. w(x, 0) = V + 0x

(4.10)

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Therefore, we only need to look for the existence of solutions of (4.8)–(4.10) for t ∈ [0, t0 ] with the estimate that   2 2 M0 = sup (φ, ψ, ξ )(t) 2 + w(t) 3 ≤  0 . t∈[0,t0 ]

Note that the left hand side of (4.8) for (φ, ψ, ξ ) is strictly hyperbolic, and the right hand side of (4.8) can be regarded as the lower order terms. Therefore, applying the method in [18,46], for any given function w (or ) with M0 ≤ ¯0 , we can show the local existence and uniqueness of solution (φ, ψ, ξ ) ∈ C([0, t0 ], H 2 ) for a suitably small t0 = t0 (M0 ) > 0, provided that (, , W, w)(t) 23 ≤ M0 ≤  1 small. The remaining equation, the last equation of (4.8) is a second order differential equation, so plugging the obtained solution (φ, ψ, ξ ) into (4.8)4 , we can easily have a solution w ∈ C([0, t0 ], H 3 ). Therefore, by applying the fixed point theorem and a straightforward argument, we can obtain a unique solution (φ, ψ, ξ ) ∈ C([0, t0 ], H 2 ) and w ∈ C([0, t0 ], H 3 ) for a suitably small time t0 = t0 (M0 ) > 0. Next, we can put (φ, ψ, ξ, w) into (4.6) and obtain the unique existence of solution (, , W, w) ∈ C([0, t0 ], H 3 ) of (4.6) and (4.7). So we complete the proof of this proposition. 2 5. The a priori estimates and the global existence Based on the local existence, Proposition 4.1, we should next establish an a priori estimate for the solution of (3.16) and (3.17) to show the global in time existence. Let   N(T ) = sup (, , W, )(t) 2 + (φ, ψ, ξ )(t) 22 + w(t) 23 . t∈[0,T ]

In this section, we will show the a priori estimate as follows. Proposition 5.1. (a priori estimates) Under the same assumptions in Theorem 2.1, there exist positive constants δ0 (≤ δ¯0 ) and 0 (≤ ¯1 ), such that if (, , W, ) ∈ X([0, T ]) is the solution of (3.16) and (3.17) for some T > 0, |β2 | + δ ≤ δ0 , and N (T ) ≤ 0 , then for t ∈ [0, T ], it holds that, t  N (t) + +

t 

1

(|U1x | + |U3x |) 2 (, W )(τ )

2

 + (, x , Wx )(τ ) 2 dτ

0

   1 (φ, ψ, ξ )(τ ) 22 + w(τ ) 23 dτ  N (0) + δ 2 + |β2 | .

(5.1)

0

First, we remark that based on Proposition 5.1, one can complete the proof of Theorem 2.1. Proof of Theorem 2.1. Choosing δ0 , and N (T ) ≤ 0 suitably small, we can construct the global solution for (, , W, ) ∈ X([0, +∞]) by combining Propositions 4.1 and 5.1, and can show the estimate (5.1) holds for all t ∈ [0, +∞). Furthermore, it holds ∞

d (x , x , x , Wx )(t) dt + dt

∞ (x , x , x , Wx )(t) 2 dt < +∞,

2

0

(5.2)

0

which together with the Sobolev inequality yields the asymptotic behaviour of the solution that lim sup |(, , W, , x , x , Wx , x )(x, t)| = 0.

(5.3)

t→∞ x∈R

Therefore, we obtain the global in time solution as in Theorem 2.1 with the asymptotic behaviour (2.25). Therefore, the remaining task is to show the a prior estimate, Proposition 5.1.

2

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Let (, , W, ) ∈ X([0, T ]) be a solution of (3.16) for some T > 0, δ ≤ δ0 (≤ δ¯0 ), and N (T ) ≤ 0 (≤ ¯1 ), where 0 and δ0 are suitably small and will be chosen later. By the Sobolev’s inequality with (4.1)–(4.3) together, V , V + x , , and + ξ are uniformly positive on [0, T ], and satisfy that 3v− v− 3θ− θ− , inf(V + x ) ≥ , inf ≥ , inf( + ξ ) ≥ . x,t x,t x,t 4 4 4 4 In order to derive the a prior estimate (5.1), we rewrite system (3.16) as ⎧ t − x = 0, ⎪ ⎪ ⎪ ⎪ ⎪ γ −1 P R ⎪ ⎪ ⎪ ⎨ t − V x + V Wx + V Ux  = N1 − R1 , Qx x R γ −1 Wt + P x + Ut  + V − V = N2 − R2 + U R1 , ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ x −Qx ⎪ + 4b 3 (Wx + γ R−1 Ux ) = N3 − R3 , ⎩ a − v inf V ≥ x,t

(5.4)

x

where

  γ −1 2 P R x − p − P + x − (θ − ) , 2V V V x (x − Qx ), N2 = (P − p)x − vV γ −1 2 Qx x N3 = − − 2b 3 x + o(ξ 2 ), vV R   Rθ γ −1 1 2 p= , v = V + x , ξ = θ − = Wx + Ux  − x . v R 2

N1 =

The initial data is (, , W, )(x, 0) = (0 , 0 , W0 , 0 )(x).

(5.5)

Notice that if (, , W ) 3 is small, we have that 

N1 = O(1) 2x + 2x + Wx2 + |Ux |2 ,

 N2 = O(1) 2x + 2x + Wx2 + |Ux |2 + |x ||x | , 

N3 = O(1) |Qx ||x | + 2x + Wx2 + |Ux |2 . Then the proof of Proposition 5.1 is divided into the proof of the following six lemmas. Lemma 5.1. Under the conditions listed in Proposition 5.1, if δ0 and 0 are suitably small, it holds that (, , W )(t) + 2

t  

  (|U1x | + |U3x |) 2 + W 2 + 2 + 2x dxdτ

0 R

t   (0 , 0 , W0 ) + δ + |β2 | + (0 + δ0 ) (x , x , Wx )(τ ) 2 dτ. 

2

1 2

(5.6)

0

Proof. Multiplying equation (5.4)1 by  and (5.4)2 by VP  respectively, and adding together, we have that   2     V γ −1 R V V 2  + − + +  Ux 2 + (−)x + Wx  =  (N1 − R1 ) . 2 2P 2P P P P t t Multiplying equation (5.4)3 by together, we obtain

R W P2

and (5.4)4 by

R  4bV P 2 3

(5.7)

respectively, and adding all the resulted equations

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R2 W2 2(γ − 1)P 2

15



R 2 Pt R R R W 2 + W  x + 2 Ut W  + QW x 3 P (γ − 1)P P V P2 t      aV x − Qx x − Qx V R

V 2 W x +  +  −  + 2 5 5 5 v v VP 4bR 4bR 4bR x x +

γ −1 V R Ux  = (N3 − R3 ) + 2 W (N2 − R2 + U R1 ). VP2 P 4bR 5 Combining (5.7) and (5.8) together, we get  2  R2 R 2 Pt aV V 2  2 2 W + A + + W2 + 2 +  + 2 3 5 2 2P 2(γ − 1)P (γ − 1)P 4bR t   V x − Qx V R R 2 +  + −  − + W + W  + J1 v P VP2 4bRv 5 x 4bR 5 x +

= where

V R V (N3 − R3 ), (N1 − R1 ) + 2 W (N2 − R2 + U R1 ) + P P 4bR 5

(5.8)

(5.9)



 V γ −1 A=− + Ux , 2P t P   RPx R R R γ −1 J1 = W  + 2 Ut W  + QW x + W + Ux  P2 P VP2 VP2 x VP2   Qx V x − Qx V x  − + v v 4bR 5 4bR 5 x   RR1x R R γ −1 = W + QW x + W + Ux  2 2 2 P VP VP x VP2   V Qx x − Qx V − x . + v v 4bR 5 4bR 5 x

We estimate the terms A and J1 one by one. By (3.7), (3.9) and (3.10), it holds that Pt ≈ P1t + P3t ,

Px ≈ P1x + P3x ,

Ux ≈ U1x + U3x .

Therefore

        V1 γ −1 γ −1 V3 A≈ − + U1x + − + U3x =: A1 + A3 . 2P1 t P1 2P3 t P3

By (3.8) and (3.10), we know that Pit = −si Pix ≈ si Uit = si2 (−Uix ) > 0. Then it follows from Proposition 3.1 and (3.8) that Vit Vi Pit γ −1 + + Uix 2Pi Pi 2Pi2   −Uix  2 ≈ V − γ P s + (3 − γ )P i i i i 2Pi2   ≥ c|Uix | (3 − γ )pm − Cδi .

Ai = −

Since γ ∈ (1, 3), choosing δ0 suitably small, we have that there exists a positive constant c, such that ˜ 2, A2 ≥ c(|U1x | + |U3x |)2 − R where and also in the followings R˜ stands for some error function which satisfies that R˜ ≈ 0.

(5.10)

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Next, for J1 , note that Q = Q1x + Q3x + Qcx 4b 3 4b 3 4b(θ c )3 c 1x − 3x − θ aV1 aV3 av c x ≈ 1x + 3x , =−

(5.11)

where we use the fact that (θxd )2 ≈ 0 and R1,x ≈ 0. Then by the fact that |Q|  δ0 , we have J1 ≥ −|QW x | − |Ux | − |x Qx | − O(1)(Vx , x )|x + W + Qx |       ≥ −δ0 2 + 2x + 2x − Cδ0 (|U1x | + |U3x |) 2 + W 2 − R˜ 2 + W 2 .

(5.12)

Next let us consider N1 , W N2 and N3 , which are the terms at the right hand side of (5.8) and (5.9). From the condition N(T ) ≤ 0 and the Sovolev’s inequality, it easily follows that   ˜ 2, |N1 |  0 2x + 2x + Wx2 + (|U1x | + |U3x |) 2 + R ˜ 2. |W N2 |  0 (2x + 2x + Wx2 + ξ 2 + (|U1x | + |U3x |)2 ) + R

(5.13)

|N3 |  0 (2x + ξx2 + 2x ). ˜ ˜ and RW ˜ at the right hand side of (5.7) and (2 + W 2 )R˜ at the right Finally, for all the error terms like RW x , R hand side of (5.8)–(5.12), recalling the fact that R1 ≈ 0 and R2 ≈ 0 by (3.10) and the definition of “≈”, we know that all the integrations of such terms on R × (0, t) are estimated by t

1

(x , Wx )(τ ) 2 dτ + δ 2 + |β2 |.

δ0

(5.14)

0

˜ ˜ we have that For example, for the terms RW x and R, t 

˜ |R|(|W x | + ||) dxdτ  δ0

0 R

t

t  Wx dτ + 2

˜ |R||| dxdτ,

(5.15)

0 R

0

and t 

˜ |R||| dxdτ 

0 R t

t  0

1

0 R

δ 2 e−cδ(|x|+τ ) || dxdτ

0 R



+

t 

|β2 | (1 + τ )

3

3 2

e

cx 2 − 1+τ

δ 2 e−cδτ  dτ +

 δ 2 + |β2 |.

t

t 

|| dxdτ + C 0 R

|β2 | 5

0

δe−c(|x|+τ ) || dxdτ

(1 + τ ) 4

 dτ + δ (5.16)

The other error terms can be similarly estimated. We omit the details for the shortness. Combining all the estimates (5.8)–(5.14), integrating (5.7) on R × (0, t) and choosing 0 and δ0 suitably small, we have the desired estimate (5.6). This completes the proof of Lemma 5.1. 2 We remark that in order to show (5.10), we need the assumption that γ ∈ (1, 3). In order to derive the estimates of higher order derivatives, we recall that from the perturbation equation (3.16), the equations of (φ, ψ, ξ, w)(x, t) are

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⎧ ⎪ ⎪ φt − ψx = 0, ⎪ ⎪     ⎪ ⎪ P R ⎪ ⎪ ⎪ ψ − + = −R1x , φ ξ ⎪ ⎨ t v v x x   R Rξ Pφ ⎪ ⎪ ξt + pψx + − Ux + wx = U R1x − R2x , ⎪ ⎪ γ −1 v v ⎪ ⎪       ⎪ ⎪ ξ ξ x a vq V Q Qx φ wx ⎪ ⎪ ⎩ − + − − − = 0, θ x θ vV x 4b θ 5 4bθ 5 v 5

17

(5.17)

with the conditions that (φ, ψ, ξ, w)(x, 0) = (φ0 , ψ0 , ξ0 , w0 )(x), (φ, ψ, ξ, w)(±∞, t) = (0, 0, 0, 0). Then we have the following lemma. Lemma 5.2. Under the conditions listed in the Proposition 5.1, if δ0 and 0 are suitably small, it holds that t (φ, ψ, ξ )(t) 21

+

(w, wx , wxx )(τ ) 2 dτ 0



(φ0 , ψ0 , ξ0 ) 21

t

1 2

+ δ + |β2 | + (0 + δ0 )

(φ, ψ, ξ )(τ ) 21 dτ.

(5.18)

0

Proof. Multiplying (5.17)1 by results together, we get

P v

φ, (5.17)2 by ψ , (5.17)3 by

ξ θ,

and (5.17)4 by w respectively, and combining the

 a v 2 Rξ 2 P 2 ψ2 + w φ + + wx2 + 5 2v 2 2(γ − 1)θ t 4bθ v 4bθ 5    R P ξ w wx Qx φ + + J2 ξ ψ − φψ + w − − v v θ vV 4bθ 5 v x ξ = −R1x ψ + (U R1x − R2x ) , θ 

where 

       P vt Pt ξ Rξ wx Pφ Qx φ 2 − + + w φ − U − x 2v 2 2v θ v v v vV 4bθ 5 x   V Rθt wx Qx φ x ξ w a Qw v − ξ2 − + + − θ b 2(γ − 1)θ 2 4bθ 5 vV 4θ 5 4 5   ≥ −C (0 + δ0 ) φ 2 + ξ 2 + w 2 + wx2 .

J2 =

Note that the terms at the right hand side of (5.19) can be dealt with in the same way as (5.16), such that t   0 R

 1 ξ −R1x ψ + (U R1x − R2x ) dxdτ  δ 2 + |β2 |. θ

Integrating (5.19) on R × (0, t), and choosing 0 and δ0 suitably small, we have the desired estimate that

(5.19)

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18

t (φ, ψ, ξ )(t) +

(w, wx )(τ ) 2 dτ

2

0

t

1 2

 (φ0 , ψ0 , ξ0 ) + δ + |β2 | + (0 + δ0 ) 2

(φ, ψ, ξ )(τ ) 2 dτ. 0

Similarly, multiplying (5.17)1x by the results together, we get that

P v

φx , (5.17)2x by ψx , (5.17)3x by

ξx θ , and (5.17)4x

by wx respectively, combining



 a v 2 R2 P 2 ψx2 2 + w + J3 φx + + ξx2 + wxx 5 2v 2 2(γ − 1)θ 4bθ v 4bθ 5 x t        ξ wx wx R Qx φ P + wx − + − ξ − φ ψx θ x vV x v v 4bθ 5 v x x ξx = −R1xx ψx + (U R1x − R2x )x , θ

(5.20)

where         ξ wxx vx wx Qx φ Qx φ wx J3 = 2 θx wxx + wx − + − v vV x vV x θ v2 4bθ 5 x 4bθ 5        x ξ ξx V a v R a Q v + − − 5 wx + ξ ψxx + px ψx + wwx − 5 5 θ 4b v x θ θ 4bθ x   2 + ξ 2 + ξx2 + ψx2 . ≥ −C (0 + δ0 ) φ 2 + φx2 + wx2 + wxx Integrating (5.20) on R × (0, t) and choosing 0 and δ0 suitably small, we have the estimate that t (φx , ψx , ξx )(t) +

(wx , wxx )(τ ) 2 dτ

2

0



(φ0 , ψ0 , ξ0 ) 21

t

1 2

+ δ + |β2 | + (0 + δ0 )

(φ, ψ, ξ )(τ ) 21 dτ.

(5.21)

0

Combining all the results together, we have the desired estimate (5.18). This completes the proof of Lemma 5.2.

2

For the estimate of second order derivatives, it holds that Lemma 5.3. Under the conditions listed in the Proposition 5.1, if δ0 and 0 are suitably small, it holds that t (φxx , ψxx , ξxx )(t) +

wxxx (τ ) 2 dτ

2

0



(φ0 , ψ0 , ξ0 ) 22

1 2

t

+ δ + |β2 | + (0 + δ0 )

(φ, ψ, ξ )(τ ) 22 dτ.

(5.22)

0

Proof. Multiplying (5.17)1xx by ( Pv φx )x , (5.17)2xx by ψxx , (5.17)3xx by get

ξxx θ ,

and (5.17)4xx by wxx respectively, we

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 2 ψ2 Rξxx P 2 + φxx + xx + w 2 + J4 + J5 2v 2 2(γ − 1)θ t 4bθ 5 v xxx        ξ wxx wx P Qx φ R + wxx − − ψxx − φ− ξ θ xx v vV xx v v xx 4bθ 5 x ξxx = −R1xxx ψxx + (U R1x − R2x )xx , θ

19



where

(5.23)



     P a vq V Q ξ x ξxx 2 J4 := − φ + (2px ψxx + pxx ψx ) − 5 − wxx + 2v t xx θ 4b θ 5 θ xx            ξxx Rξ Qx φ Pφ wxxx 1 1 + + wx − − Ux + 2wxx v v v x v xx vV xx 4bθ 5 xx θ 2 w 2 − Cwxx ≥− 16bθ 5 v xxx   −C (0 + δ0 ) |(φ, ψ, ξ, w)|2 + |(φx , ψx , ξx , wx )|2 + |(φxx , ψxx , ξxx , wxx )|2 ,

and



     P P Rξx vx 1 φψxxx + 2 φx ψxxx + ψ − Rξ ψxxx xxx 2 v xx v x v xx v     Pxx 2v 2 P Rξx vx Px vx = φx ψxxx + ψxxx − 2 2 + P 3x φψxxx + 2 v v x v v v2 2v 2 Rξ − P φ Rξ − P φ −Rξ 3x ψxxx + Vxx ψxxx + φxx ψxxx 2 v v v2 6 ! = J5i ,

J5 :=

(5.24)

i=1

where 

     Pxx 2v 2 Rξx vx 2vx2 Rξ − P φ Px vx P φx + − Rξ + V − 2 2 + P 3x φ + 2 ψ xx xx v v v v x v2 v3 v2 x i=1      Pxx 2vx2 Rξx vx 2vx2 Rξ − P φ Px vx P − φx + − Rξ 3 + Vxx ψxx , −2 2 +P 3 φ+2 v v v v x v2 v v2 x Rξ − P φ 6 φxx φtxx J5 = v2     Rξ − P φ 2 Rξ − P φ 2 = φxx − φxx . 2v 2 2v 2 t t 5 !

J5i =

Here we have used the equation that φt = ψx . Note that system (1.2) is less dissipative, so we need more careful estimates on the terms contain ψxxx in J5 . Integrating J4 , J5 on R × (0, t) and choosing 0 and δ0 suitably small, we have the estimate "t " 0

R (J4 + J5 )dxdτ ≥ − 16bθ 5 v

t

t wxxx (τ ) dτ − C

wxx (τ ) 2 dτ

2

0

0

t

− C(0 + δ0 ) φxx (t) − C(0 + δ0 )

(φ, ψ, ξ, w)(τ ) 22 dτ.

2

(5.25)

0

Therefore, integrating (5.23) on R × (0, t) and choosing 0 and δ0 , the terms on the right hand side of (5.23) can also be dealt with in the same way as (5.16) to get (5.22). This completes the proof of Lemma 5.3. 2

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20

Next, we should deal with the term

"t 0

(φ, ψ)(τ ) 22 dτ .

Lemma 5.4. Under the conditions listed in the Proposition 5.1, if δ0 and 0 are suitably small, it holds that t (φ, ψ)(τ ) 22 dτ

t

1 2

 N (0) + δ + |β2 | +

0

ξ(τ ) 22 dτ.

(5.26)

0 2

R Proof. Multiplying the equation (5.4)2 by − P2 x , (5.4)3 by x , and (5.4)4 by V 4b 3 Wx respectively, and adding the resulted equations, we have     P P R R W x − x  + t  − W t (γ − 1) 2 2 (γ − 1) t x P 2 2 RP R2 2 P 2 +  − x Wx + W +  x + J6 2V x 2V V x 2   P R R2 = Wx (N3 − R3 ) , x + Wx (N1 − R1 ) + x (N2 − R2 ) + (5.27) 2 (γ − 1) V 4b 3

where Px Pt γ −1 x x Qx x x + x − P Ux x + Ut x + − 2 2 2V V vV   2 γ − 1 R Wx Ux  + a − wx + 4b 3 R V 4b 3   ≥ −η 2x + 2x + Wx2 − c (|U1x | + |U3x |) 2 + 2 + wx2 .

J6 = −

By the positivity of P and V , it holds that P 2 2 RP R2 2 x − x Wx + W ≥ c(2x + Wx2 ). 2V 2V V x Then integrating (5.27) and choosing δ0 , η suitably small, we have t  

 2x + 2x + Wx2 dxdτ  (, W, x , x )(t) 2 + (0 , W0 , 0x , 0x ) 2

0 R

+

t  

  |U1,x | + |U3,x | 2 + 2 + wx2 dxdτ

0 R

t  +

[(|x | + |Wx |) |N1 − R1 | + |x | |N2 − R2 |

0 R  ˜ 2 dxdτ. + |Wx | |N3 − R3 | + |R|

(5.28)

Following the proof of Lemma 5.1 and also using (5.16) to control the last term of (5.28), (5.28) can be further estimated such that: "t " 2  1 2 2 2 2 2 0 R x + x + Wx dxdτ  (0 , 0 , W0 ) + (φ0 , ψ0 , ξ0 ) 1 + δ + |β2 | t (φ, ψ, ξ )(τ ) 21 dτ.

+ (0 + δ0 )

(5.29)

0

− P2 φx ,

Similarly, multiplying (5.17)2 by (5.17)3 by ψx , (5.17)2,x by − P2 φxx , and (5.17)3,x by ψxx respectively, adding them all and integrating the resulted formula, we can also get

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t  

+ ψx2

φx2

2 + φxx

2 + ψxx



21

t

1 2

dxdτ  N (0) + δ + |β2 | + (0 + δ0 )

0 R

(φ, ψ, ξ )(τ ) 22 dτ. 0

Putting the results together and taking δ0 and 0 suitably small, we have the desired estimate (5.26) immediately. This completes the proof of Lemma 5.4. 2 Now combining all the three lemmas above, we have the estimate that (, , W )(t)

2

+ (φ, ψ, ξ )(t) 22

+

 (φ, ψ)(τ ) 22 + w(τ ) 23 dτ

0

t 

1 2

(|U1x | + |U3x |) (, W )(τ )

+

t 

0

2

 + (, x , Wx )(τ ) 2 dτ

t

1 2

 N(0) + δ + +|β2 | + (0 + δ0 )

ξ(τ ) 22 dτ.

(5.30)

0

Then we need to deal with the term

"t 0

ξ(τ ) 22 dτ .

Lemma 5.5. Under the conditions listed in the Proposition 5.1, if δ0 and 0 are suitably small, it holds that t

1

ξ(τ ) 22 dτ  N (0) + δ 2 + |β2 |.

(5.31)

0

Proof. Since      γ −1 1 2 2 2 ξ = Wx + ≤ C Wx2 + Ux2 2 + 4x , Ux  − x R 2 and by (5.30), we have that t ξ(τ ) dτ  2

0

t  

  |U1,x | + |U3,x | 2 + Wx2 dxdτ + 0

0 R 1

t

 N (0) + δ 2 + +|β2 | + (0 + δ0 )

t x (τ ) 2 dτ 0

ξ(τ ) 22 dτ.

(5.32)

0

On the other hand, by (5.17)4 , we get     ξ ξx a v a Q v V ξ 2 x Qx φ wx ξx = − w − − + + − . θ θ vV x 4bθ θ 5 5 4bθ 4 v bθ 4 Multiplying equation (5.33) above by ξx , we get     ξ ξx2 ξ 2 ξx x a v a Q v V Qx φ wx 2 ξx = ξx − wξx − − + + − ξx θ θ vV x 4bθ θ 5 5 4bθ 4 v bθ 4     1 2  ξx2 + (0 + δ0 ) ξ 2 + φ 2 + φx2 + w 2 + wx2 + wxx . 4 Integrating (5.34), we have that

(5.33)

(5.34)

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22

t

t ξx (τ ) dτ 

0

t w(τ ) 22 dτ

2

+ (0 + δ0 )

0

(ξ, φ, φx )(τ ) 2 dτ 0

1 2

t

 N (0) + δ + |β2 | + (0 + δ0 )

ξ(τ ) 22 dτ.

(5.35)

0

Similarly, taking derivative ∂x on (5.33) and multiplying the resulted equation by ξxx , we have t

t

1 2

ξxx (τ ) dτ  N (0) + δ + |β2 | + (0 + δ0 ) 2

0

ξ(τ ) 22 dτ.

(5.36)

0

Putting (5.32), (5.35) and (5.36) together and taking δ0 and 0 suitably small, we have the desired estimate (5.31) immediately. This completes the proof of Lemma 5.5. 2 Finally, we will give the estimate of w(t) 23 . Lemma 5.6. If δ0 and 0 are suitably small, it holds that 1

(t) 21 + w(t) 23  N (0) + δ 2 + |β2 |. Proof. Multiplying the equation (5.4)4 by , it yields that   x − Qx 2 Qx x 2 a −  + x = − 4b 3 ξ  + (N3 − R3 ). v v v x Integrating the equation above by x, we get    2   x Qx x a2 dx + dx = − 4b 3 ξ  + (N3 − R3 ) dx v v R R R  2    1 x + ξx 2 + C(0 + δ) (φ, ψ, ξ ) 21 . + 0 ≤ , √ 4 v

(5.37)

(5.38)

(5.39)

By (5.30) and (5.31), it holds that 1

(t) 21  N(0) + δ 2 + |β2 |. On the other hand, (5.17)4 can be rewritten as   w    Qx φ x avw − =− − aQφ − 4bθ 3 ξx − 4b θ 3 − 3 x w − R3x . v x vV x Multiplying the equation above by w and integrating it, we have     2      wx Qx φ avw 2 dx + − aQφ − 4bθ 3 ξx − 4b θ 3 − 3 x w − R3x wdx dx = − v vV x R R R     1 + 0 (w, wx ) 2 + Q 21 + (φ, ξ ) 22 + (0 + δ) (φ, ψ, ξ ) 22 . ≤ 4

(5.40)

(5.41)

(5.42)

By (5.30) and (5.31), it holds that 1

w 21  N(0) + δ 2 + |β2 |. Differentiating (5.41) with respect to x, one has     w    Qx φ x 3 3 3 (avw)x − = − − aQφ − 4bθ ξx − 4b θ − x w − R3x . v xx vV x x

(5.43)

(5.44)

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23

Multiplying equation (5.44) by −wxxx and integrating it, we have   2 wxxx 2 avwxx dx + dx v R R        1 1 wxx wxxx + wx wxxx dx ≤ 2avx wx wxx + avxx wwxx + 2 v x v xx R       Qx φ + aQφ + 4bθ 3 ξx + 4b θ 3 − 3 x w + R3x wxxx dx + vV x x R    1 + 0 (wxx , wxxx ) 2 + Q 21 + (w, φ, ξ ) 22 + (0 + δ) (φ, ψ, ξ ) 22 .  8

(5.45)

Therefore it follows that 1

wxx 21  N (0) + δ 2 + |β2 |. Combining (5.40), (5.43) and (5.46) together, we have (5.37). This completes the proof of Lemma 5.6.

(5.46) 2

Now combining Lemma 5.1–Lemma 5.6 together, we have (5.1) and then finish the proof of the Proposition 5.1. Conflict of interest statement No conflict of interest. Acknowledgements The authors would like to thank the referee for the valuable suggestions and comments. L.Z. Ruan would like to express heartfelt thanks to Professor Zhouping Xin for his valuable discussions on this topics during he visited at IMS, The Chinese University of Hong Kong. The research of L.Z. Ruan was supported in part by the Natural Science Foundation of China #11771169, #11331005, Program for Changjiang Scholars and Innovative Research Team in University #IRT17R46, and the Special Fund for Basic Scientific Research of Central Colleges #CCNU16A02011. The research of Wei Xiang was supported in part by the Research Grants Council of the HKSAR, China (Project No. CityU 21305215, Project No. CityU 11332916, and Project CityU 11304817). The research of Lili Fan was supported in part by the Fundamental Research Grants from the Science Foundation of Hubei Province under the contract 2018CFB693. References [1] C. Buet, B. Despres, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transf. 85 (2004) 385–418. [2] J. Coulombel, T. Goudon, P. Lafitte, C. Lin, Analysis of large amplitude shock profiles for non-equilibrium radiative hydrodynamics: formation of Zeldovich spikes, Shock Waves 22 (2012) 181–197. [3] R. Duan, K. Fellner, C. Zhu, Energy method for multi-dimensional balance laws with non-local dissipation, J. Math. Pures Appl. 93 (2010) 572–598. [4] R. Duan, L. Ruan, C. Zhu, Optimal decay rates to conservation laws with diffusion-type terms of regularity-gain and regularity-loss, Math. Models Methods Appl. Sci. 22 (2012) 1250012, 39 pp.. [5] L. Fan, A. Matsumura, Asymptotic stability of a composite wave of two viscous shock waves for a one-dimensional system of non-viscous and heat-conductive ideal gas, J. Differ. Equ. 258 (2015) 1129–1157. [6] M. Francesco, Initial value problem and relaxation limits of the Hamer model for radiating gases in several space variables, Nonlinear Differ. Equ. Appl. 13 (2007) 531–562. [7] W. Gao, C. Zhu, Asymptotic decay toward the planar rarefaction waves for a model system of the radiating gas in two dimensions, Math. Models Methods Appl. Sci. 18 (2008) 511–541. [8] W. Gao, L. Ruan, C. Zhu, Decay rates to the planar rarefaction waves for a model system of the radiating gas in n-dimensions, J. Differ. Equ. 244 (2008) 2614–2640. [9] P. Godillon-Lafitte, T. Goudon, A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, SIAM J. Multiscale Model. Simul. 4 (2005) 1245–1279.

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