Global stability of rarefaction waves for a viscous radiative and reactive gas with temperature-dependent viscosity

Nonlinear Analysis: Real World Applications 53 (2020) 103056

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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa

Global stability of rarefaction waves for a viscous radiative and reactive gas with temperature-dependent viscosity Yongkai Liao Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

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Article history: Received 23 May 2019 Accepted 17 October 2019 Available online xxxx Keywords: Viscous radiative and reactive gas Rarefaction waves Temperature-dependent viscosity

abstract We study the nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional viscous radiative and reactive gas when the viscosity and heat conductivity coefficients depend on both density and absolute temperature. Our main idea is to use the smallness of the strength of the rarefaction waves to control the possible growth of its solutions induced by the nonlinearity of the system and the interactions of rarefaction waves from different families. The proof is based on some detailed analysis on uniform positive lower and upper bounds of the specific volume and the absolute temperature. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction and main result This paper is concerned with the precise description of the large time behavior of global smooth largeamplitude solutions to the Cauchy problem of a model for the one-dimensional viscous radiative and reactive gas. The model consists of the following equations in the Lagrangian coordinates corresponding to the conservation laws of the mass, the momentum and the energy coupling with a reaction–diffusion equation (see [1–4]) vt − ux = 0, ( ) µ (v, θ) ux ut + p (v, θ)x = , v x ( ) ( ) ( ) u2 µ (v, θ) uux κ (v, θ) θx e+ + (up(v, θ))x = + + λϕz, 2 t v v x x ( ) dzx − ϕz. zt = v2 x

(1.1)

Here x ∈ R is the Lagrangian space variable, t ∈ R+ the time variable and the primary dependent variables are the specific volume v = v (t, x), the velocity u = u (t, x), the absolute temperature θ = θ (t, x), and the E-mail address: [email protected]. https://doi.org/10.1016/j.nonrwa.2019.103056 1468-1218/© 2019 Elsevier Ltd. All rights reserved.

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Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

mass fraction of the reactant z = z (t, x). The positive constants d and λ are the species diffusion coefficient and the difference in the heat between the reactant and the product, respectively. The reaction rate function ϕ = ϕ (θ) is defined, from the Arrhenius law [4,5], by ( ) A ϕ (θ) = Kθβ exp − , (1.2) θ where positive constants K and A are the coefficients of the rates of the reactant and the activation energy, respectively, and β is a non-negative number. We treat the radiation as a continuous field and consider both the wave and photonic effect. Assume that the high-temperature radiation is at thermal equilibrium with the fluid. Then the pressure p and the specific internal energy e consist of a linear term in θ corresponding to the perfect polytropic contribution and a fourth-order radiative part due to the Stefan–Boltzmann radiative law [4,6]: Rθ aθ4 + , e(v, θ) = Cv θ + avθ4 , (1.3) v 3 where the positive constants R, Cv , and a are the perfect gas constant, the specific heat and the radiation constant, respectively. In particular, we have that Cv = 23 R for the radiative gas (see [6]). The radiation constant a is used to measure the amount of heat that is emitted by a blackbody, which absorbs all of the radiant energy that hits it, and will emit all the radiant energy. It is defined as (see [6]) p (v, θ) =

4σ 8π 5 k 4 = , c 15c3 h3 where σ is the Stefan–Boltzmann constant, c is the speed of light, k is Boltzmann constant, and h is Planck’s constant. Numerically, a = 7.5657 × 10−16 Jm−3 K−4 . In general, the value of radiation constant a is much smaller than the value of perfect gas constant R or the specific heat Cv . Based on the above fact, we can assume a is a very small number throughout this paper. Since the energy producing process inside the medium is taken into account in the system (1.1), that is, the gas consists of a reacting mixture and the combustion process is current at the high temperature stage, and the experimental results for gases at high temperatures in [7] show that both the viscosity coefficient µ and the heat conductivity κ may depend on the specific volume v and/or the absolute temperature θ. In this paper, as in [2,4,5,8–11], we focus on the case when the heat conductivity κ is supposed to take the following form κ (v, θ) = κ1 + κ2 vθb (1.4) a=

with κ1 , κ2 and b being some positive constants. As for the viscosity coefficient µ, motivated by the works [12,13], we assume that { v −ℓ1 , v → 0+ , 2 v |h′ (v)| ≤ Ch3 (v), (1.5) µ = µ (v, θ) = h (v) θα , h (v) ∼ ℓ2 v , v → ∞, where h(v) is a smooth function of v for v > 0 and α, ℓ1 , ℓ2 and C are positive constants. Here and in the rest of this paper, f (x) ∼ g(x) as x → x0 means that there exists a positive constant C ≥ 1 such that C −1 g(x) ≤ f (x) ≤ Cg(x) in a neighborhood of x0 . Moreover, the initial data is given by (v (0, x) , u (0, x) , θ (0, x) , z (0, x)) = (v0 (x) , u0 (x) , θ0 (x) , z0 (x))

(1.6)

for x ∈ R, which is assumed to satisfy the following far-field condition: lim (v0 (x) , u0 (x) , θ0 (x) , z0 (x)) = (v± , u± , θ± , 0).

|x|→∞

where v± > 0, u± and θ± > 0 are constants.

(1.7)

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The global solvability and large-time behaviors of the global solutions to the initial value problem and/or the initial–boundary value problems of the radiative and reactive gas equations (1.1)–(1.4), and (1.6) has been paid a lot of attention by many authors. To go to the theme of this paper, we will only review some results on the system (1.1)–(1.4), and (1.6) as follows: • When the viscosity coefficient µ is a positive constant, the well-posedness of global classical solutions was established by [1,8,14] for the Dirichlet–Neumann boundary conditions, and [4,5,9,11,15] for the free and pure Neumann boundary conditions. The global solvability and large-time behavior of solutions to the Cauchy problem (1.1)–(1.4), (1.6), and (1.7) has been obtained in [3,16] when (v− , u− , θ− , z− ) = (v+ , u+ , θ+ , z+ ). Recently, Gong et al. [17] have studied the nonlinear stability of rarefaction waves to the system (1.1)–(1.4), (1.6), and (1.7) provided the radiation constant a is small. For spherically or cylindrically symmetric solutions of the multidimensional compressible radiative and reactive gases, the global existence, uniqueness and exponential stability of spherically (cylindrically) symmetric solutions in a bounded concentric annular domain were obtained in [10,18,19], while the corresponding result on the global solvability and the precise description of the large time behavior of the global solutions constructed in an exterior domain was obtained recently in [20]; • For the case when the viscosity coefficient µ is a smooth, possible degenerate function of the specific volume v for v > 0, the two types of initial–boundary value problems of (1.1)–(1.4), and (1.6) in the bounded interval (0, 1) mentioned above are studied in [2,21], while the Cauchy problem is treated by [22]. It is worth pointing out that all the estimates obtained in [2,21,22] depend on the time variable and thus the problem on the large time behavior of global solutions constructed in [2,21,22] remains unsolved; • For the case when the viscosity coefficient µ depends on both density and temperature, He et al. [12] studied the large-time behavior of the system (1.1)1 –(1.1)3 , (1.2)–(1.7) when (v− , u− , θ− , z− ) = (v+ , u+ , θ+ , z+ ) under the assumptions b ≥ 7,

(

8 3 − 6− ℓ1 2ℓ2 + 1

)

ℓ1 > 1, ℓ2 > 1, 8 21 3 12 36 + > 0. b +18 − + 2+ − 2 2ℓ1 4ℓ1 (2ℓ2 + 1) 2ℓ2 + 1 ℓ1 (2ℓ2 + 1)

(1.8) (1.9)

From the results obtained above, a natural and interesting question is : For the Cauchy problem (1.1)–(1.4), (1.6), and (1.7) with density and temperature-dependent viscosity coefficients satisfying (1.5), can we obtain the global nonlinear stability of some elementary waves such as rarefaction waves, viscous shock waves, viscous contact waves, and some of their superpositions? Here “global nonlinear stability” means “the corresponding nonlinear stability result with large initial perturbation”. And our main purpose here is to deduce the global nonlinear stability of rarefaction waves with large initial perturbation. Since our interest is to show the nonlinear stability of the expansion waves for (1.1), it is convenient to work with the equations for the entropy s and the absolute temperature θ. Let s be the specific entropy, the constitutive relations (1.3) together with the Gibbs equation de = θds − pdv tells us that 4 s(v, θ) = Cv ln θ + avθ3 + R ln v. 3 We get from (1.1) (1.3), and (1.10) that ( ) κ(v, θ)θx κ(v, θ)θx2 µ(v, θ)u2x λϕz st = + + + 2 vθ vθ vθ θ x and θt + Here pθ :=

∂p(v,θ) ∂θ

=

R v

θpθ ux 1 = eθ eθ

+ 43 aθ3 and eθ :=

(

κ(v, θ)θx v

∂e(v,θ) ∂θ

) + x

µ(v, θ)u2x λϕz + . veθ eθ

= Cv + 4avθ3 .

(1.10)

(1.11)

(1.12)

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In fact, for smooth solutions, Eqs. (1.1)1 –(1.1)2 –(1.1)3 –(1.1)4 are equivalent to Eqs. (1.1)1 –(1.1)2 –(1.11)– (1.1)4 or (1.1)1 –(1.1)2 –(1.12)–(1.1)4 . In what follows, we will consider (1.1)1 –(1.1)2 –(1.11)–(1.1)4 with the initial data (v(t, x), u(t, x), s(t, x), z(t, x))|t=0 = (v0 (x), u0 (x), s0 (x), z0 (x)) → (v± , u± , s± , 0)

as x → ±∞.

(1.13)

3 Here v± > 0, u± , and s± := Cv ln θ± + 43 av± θ± + R ln v± are constants and s0 (x) := Cv ln θ0 (x) + 4 3 av (x)θ (x) + R ln v (x). Since we will focus on the expansion waves to (1.1), we assume that s+ = s− = s¯ 0 0 0 3 in the rest of this paper. For expansion waves, the right hand side of (1.1) decays faster than each individual term on the left hand side. Therefore, the compressible radiative and reactive gas equations (1.1) may be approximated, time-asymptotically, by the Riemann problem of the following equations: ⎧ vt − ux = 0, ⎪ ⎪ ⎨ ut + (˜ p(v, s))x = 0, (1.14) st = λϕz ⎪ ⎪ θ , ⎩ zt = −ϕz,

with Riemann data (v(0, x), u(0, x), s(0, x), z(0, x)) =

R R (v0R (x), uR 0 (x), s0 (x), z0 (x))

{ =

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

(1.15)

We consider the case when the Riemann problem (1.14)–(1.15) admits a unique global weak (rarefaction ( ) wave) solution V R ( xt ), U R ( xt ), S R ( xt ), 0 which consists of a rarefaction wave of the first family, denoted ( ) ( ) by V1R ( xt ), U1R ( xt ), s¯, 0 , and another of the third family, denoted by V3R ( xt ), U3R ( xt ), s¯, 0 . That is, there exists a unique constant state (vm , um ) ∈ R2 (vm > 0) such that (vm , um ) ∈ R1 (v− , u− ) and (v+ , u+ ) ∈ R3 (vm , um ). Here { } ∫ v √ R1 (v− , u− , s¯, 0) = (v, u, s, z)|u = u− + −˜ pξ (ξ, s¯)dξ, u ≥ u− , s = s¯, z = 0 , v− v

{ R3 (vm , um , s¯, 0) =

∫ (v, u, s, z)|u = um −

} √ −˜ pξ (ξ, s¯)dξ, u ≥ um , s = s¯, z = 0 .

vm

) In other words, the unique weak solution V R ( xt ), U R ( xt ), S R ( xt ), 0 to the Riemann problem (1.14)–(1.15) is given by ( (x) (x) (x) ) ( (x) (x) (x) (x) ) VR , UR , SR , 0 = V1R + V3R − vm , U1R + U3R − um , s¯, 0 , (1.16) t t t t t t t (

where (ViR ( xt ), UiR ( xt ), S R ( xt ), 0)(i = 1, 3) are determined by the following equations: ⎧ S R ( xt ) ⎪ ⎪ ⎪ ∫ V1R ( x ) √ ⎪ t ⎪ ⎪ ⎪ U1R ( xt ) − −˜ pξ (ξ, s¯)dξ ⎪ ⎪ ⎪ 1 ( R x ) ⎪ ⎪ ⎪ λ1x V1 ( t ), s¯ ⎨ λ1 (v, s) ⎪ ∫ V3R ( x ) √ ⎪ ⎪ t ⎪ ⎪ ⎪ U3R ( xt ) + −˜ pξ (ξ, s¯)dξ ⎪ ⎪ ⎪ 1 ( R x ) ⎪ ⎪ λ3x V3 ( t ), s¯ ⎪ ⎪ ⎩ λ3 (v, s)

= s¯, v−

∫ = u− −

√

−˜ pξ (ξ, s¯)dξ,

1

> 0,√ = − −˜ pv (v, s), ∫ vm √ = um + −˜ pξ (ξ, s¯)dξ, 1

> 0, √ = −˜ pv (v, s).

(1.17)

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To study the nonlinear stability of the expansion waves, as in [23], we first construct a smooth approximation to the above Riemann solution (1.16). For this purpose, let ωi (t, x) (i = 1, 3) be the unique global smooth solution to the following Cauchy problem: ⎧ ⎨ ωit + ωi ωix = 0, ∫ ϵx (1.18) w −w w +w (1 + y 2 )−q dy, ⎩ ωi (t, x)|t=0 = ωi0 (x) = i+ 2 i− + i+ 2 i− Kq 0

where q >

3 2,

∫ +∞ Kq = ( 0 (1 + y 2 )−q dy)−1 , ϵ > 0 is a positive constant which will be specified later, and √ ⎧ pv (v− , s¯), ω = λ (v , s ¯ ) = − 1− 1 − ⎪ √−˜ ⎪ ⎨ ω1+ = λ1 (vm , s¯) = √ − −˜ pv (vm , s¯), ⎪ ω = λ (v , s ¯ ) = −˜ p (vm , s¯), 3− 3 m v ⎪ √ ⎩ pv (v+ , s¯). ω3+ = λ3 (v+ , s¯) = −˜

Then, by setting ϵ = δ = |v− − v+ | + |u− − u+ | (the strength of the rarefaction waves), the smooth approximation of the rarefaction wave profile (V (t, x), U (t, x), S(t, x), 0) is constructed as follows: (V (t, x), U (t, x), S(t, x), 0) = (V1 (t + 1, x) + V3 (t + 1, x) − vm , U1 (t + 1, x) + U3 (t + 1, x) − um , s¯, 0) , where (Vi (t, x), Ui (t, x))(i = 1, 3) are defined by the following equations: ⎧ λi (Vi (t, x), s¯) = ωi√ (t, x), i = 1, 3, ⎪ ⎪ ⎪ ⎪ λ (v, s) = − pv (v, s), ⎪ 1 ⎪ √ −˜ ⎪ ⎪ ⎪ −˜ p λ (v, s) = v (v, s), 3 ⎨ ∫ V1 (t,x) √ −˜ pξ (ξ, s¯)dξ, U (t, x) = u + ⎪ 1 − ⎪ ⎪ v− ⎪ ⎪ ∫ V3 (t,x) ⎪ ⎪ √ ⎪ ⎪ U3 (t, x) = um − −˜ pξ (ξ, s¯)dξ, ⎩

(1.19)

(1.20)

vm

and Θ(t, x) is defined by ˜ (t, x), s¯). Θ(t, x) = θ(V We now turn to state our main result in this paper. To this end, for each given positive constant 0 < w ≤ 1, if we denote H(ω) := sup |(h(σ), h′ (σ), h′′ (σ), h′′′ (σ))| , (1.21) ω≤σ≤ω −1

then our main result can be summarized as follows: Theorem 1.1.

Suppose that

• The viscosity coefficient µ satisfies (1.5); • The parameters b, ℓ1 and ℓ2 are assumed to satisfy: b ≥ 7,

ℓ1 > 1,

0≤β
ℓ2 > 1,

3 ; 2ℓ2 + 1

(1.22)

• There exist positive constants Π0 and V0 ≤ 1 such that (v0 (x) − V (0, x), u0 (x) − U (0, x), θ0 (x) − Θ(0, x), z0 (x)) ∈ H 3 (R) , ∥(v0 (x) − V (0, x), u0 (x) − U (0, x), θ0 (x) − Θ(0, x), z0 (x))∥H 3 (R) ≤ Π0 , V0 ≤ v0 (x) , V (t, x), Θ(t, x) ≤ V0−1 , 1

z0 (x) ∈ L (R) ,

0 ≤ z0 (x) ≤ 1,

θ0 (x) ≥ V0 ,

∀x ∈ R.

(1.23)

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Then there exists ϵ0 > 0, which depends only on Π0 , V0 and H(C0 ) with positive constant C0 depending only on Π0 , V0 and H(V0 ), such that if |α| + δ + a ≤ ϵ0 (1.24) the Cauchy problem (1.1)–(1.5) with prescribed initial data (1.6) satisfying the far field condition (1.7) admits a unique solution (v(t, x), u(t, x), θ(t, x), z(t, x)) satisfying (v(t, x) − V (t, x), u(t, x) − U (t, x), θ(t, x) − Θ(t, x), z(t, x)) ∈ C([0, ∞), H 3 (R)), ∂ (v(t, x) − V (t, x)) ∈ L2 ([0, ∞); H 2 (R)), ∂x ∂ (u(t, x) − U (t, x), θ(t, x) − Θ(t, x), z(t, x)) ∈ L2 ([0, ∞); H 3 (R)), ∂x and {v (t, x) , θ (t, x)} > 0,

inf (t,x)∈[0,∞)×R

sup

{v (t, x) , θ (t, x)} < +∞,

0 ≤ z(t, x) ≤ 1.

(t,x)∈[0,∞)×R

In addition, we have the following large time behavior )⏐} {⏐( ⏐ ⏐ lim sup ⏐ v(t, x) − V R (t, x), u(t, x) − U R (t, x), s(t, x) − s, z(t, x) ⏐ = 0. t→+∞ x∈R

Remark 1.1. Several remarks concerning our main result are listed below: • Obviously, our main result has removed the condition (1.9) on the parameters b, ℓ1 and ℓ2 as required in [12]. Thus our result has improved the result obtained in [12]; • The result obtained in Theorem 1.1 can not be thought as a perturbation result of the case when the viscosity coefficient µ depends only on density. That is, if similar result holds for some density dependent µ = h(v), then similar result holds also for µ = h(v)θα for sufficiently small |α|. The main reason is that all the estimates to guarantee the global solvability results of the compressible radiative and reactive gas equations (1.1) with density dependent viscosity obtained in [2,21] depend on the time variable t and thus can not be used to yield the desired large behaviors. • As we can see in the proof of Theorem 1.1, the smallness assumption of the radiation constant a is only used to ensure that p˜(v, s) (we choose (v, θ) or (v, s) as independent variables and write (p, e, s) = ˜ s))) is convex with respect to (v, s) in Section 2, (p(v, θ), e(v, θ), s(v, θ)), or (p, e, θ) = (˜ p(v, s), e˜(v, s), θ(v, and we have not used the above assumption elsewhere. In fact, if we make the most use of the smallness of a, the result of Theorem 1.1 can be improved to the case: ℓ1 > 1, ℓ2 > 1, b > 25 + ℓ11 + 2ℓ23+1 , 0 ≤ β < b + 3 − 2ℓ23+1 . In fact, we have the following corollary: Corollary 1.1. If we make the most use of the smallness of the radiation constant a, Theorem 1.1 still holds when ℓ1 > 1, ℓ2 > 1, b > 52 + ℓ11 + 2ℓ23+1 , 0 ≤ β < b + 3 − 2ℓ23+1 . Note that when ℓ1 and ℓ2 are large enough, the result in Corollary 1.1 covers the most physically interesting radiation case b = 3([5,8,24,25],). The proof of Corollary 1.1 will be given in Section 6. Now we outline the main difficulties of the problem and our strategy to yield the above result. As pointed out in [17,26], the main difficulty to obtain the global stability result with large initial perturbation is to control the possible growth of the solutions induced by the nonlinearity of the equations, the interactions of rarefaction waves from different families and the interactions between the solutions and the rarefaction waves. The key point is to deduce the uniform-in-time positive lower and upper bounds on the specific volume v(t, x) and the absolute temperature θ(t, x). Motivated by the works of [3,27], Gong et al. [17] used a cut-off function φ(x) (see (2.14) in [3]; see also (3.5) in [17]) to derive an explicit expression of v(t, x) to deduce

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

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uniform-in-time positive lower and upper bounds on the specific volume. Then the authors introduce the following auxiliary functions ∫ t∫ ( ) 2 ˜ X(t) : = 1 + θb (s, x) |χt | (s, x), 0 R∫ ( ) 2 ˜ Y (t) : = sup 1 + θ2b (s, x) |χx | (s, x)dx, s∈(0,t) R ∫ 2 ˜ : = sup Z(t) |ψxx | (s, x)dx, s∈(0,t)

R

(we set (φ(t, x), ψ(t, x), χ(t, x), ξ(t, x)) = (v(t, x) − V (t, x), u(t, x) − U (t, x), θ(t, x) − Θ(t, x), s(t, x) − s¯)) to deduce uniform-in-time positive upper bound on the absolute temperature. It is worth to mention that the trick used in [17] to deduce the desired upper bound estimate on θ(t, x) relies highly on the uniform bounds on v (t, x) obtained before. Since the viscosity coefficient µ is not a positive constant but depends on the specific volume v, the above argument can not be used any longer. In fact, for such case we can not deduce a similar explicit repression for v(t, x) and consequently we can not deduce the desired positive lower and upper bounds on v(t, x) first. Besides, due to the fact that the viscosity coefficient µ depends on the absolute temperature, the identity similar to (1.18) in [12] which plays an essential role in deducing the desired uniform estimate on the density becomes ( ) ( ) µ (v, θ) φx µv Vx Ux µUxx µux Vx µθ (φx θt − θx ux ) = ψt + (p(v, θ) − P (V, Θ))x + g(V, Θ)x − + − + . 2 v v v v v t (1.25) The last term in (1.25) is a highly nonlinear term, the temperature dependence of the viscosity µ has a strong influence on the solution and leads to difficulty in mathematical analysis for global solvability with large data. Moreover, it is worth to point out that the smallness assumption imposing on the radiation constant a also brings about additional mathematical difficulties in our analysis. In [12], the authors used ∫ 4 the boundedness of R v (θ − 1) (s, x)dx to deduce the following inequality (see (3.2) in [12]) 1

∥θ∥L∞ (R) ≤ C + CY (T ) 2b+6−ς1 −2ς2 .

(1.26)

to control the upper bound of θ(t, x) (The definitions of Y (T ), ς1 , and ς2 are the same as that in (3.3) ∫ and (4.1) in our paper). Since R vχ4 (s, x)dx is not bounded any longer due to the smallness assumption ( ) ∫ on the radiation constant a, we can only use the boundedness of R Φ Θθ dx (see (2.3) and (2.18)) to deduce (4.2). This makes the exponential coefficient of Y (T ) in (4.2) larger than that in (1.26). Besides, the exponential(coefficient of θ(t, x) in X(t) (see (4.1)) ) will be smaller than that in the counterpart term ∫t∫ θ b+3 (τ,x) 1 ˜ X(t) := + θ2 (τ, x) in [12] due to the same reason. The above ς ς 0

R

1+∥θ∥ 2∞ L

([0,T ]×R)

1+∥θ∥ 1∞ L

t

([0,T ]×R)

changes make it harder to deduce the upper bound of θ(t, x): the term I21 (see (4.7)) can not be controlled in a similar way by using the method developed in [12] (Notice that the counterpart term of I21 is the term H7 in [12]). Our strategy to overcome the above difficulties can be summarized as in the following: • Motivated by [12,13], we apply Kanel’s method [28] to deduce the lower and the upper bounds of v(t, x) in terms of ∥θ∥L∞ ([0,T ]×R) simultaneously in Lemma 3.2. We use the smallness of the strength of the rarefaction waves to estimate the nonlinear terms arising from the nonlinearities of equations, the interactions of rarefaction waves from different families and the interaction between the solutions and the rarefaction waves; Moreover, we also use the smallness of |α| to control the last term in (1.25); • Due to (4.8) and (4.9), we introduce the auxiliary functions X(t), Y (t), and Z(t) (see (4.1)) to deduce the upper bound of θ(t, x) in Lemmas 4.1–4.4. Thus the lower and the upper bounds of v(t, x) follows

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

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from Lemma 3.2. Instead of integration by parts, we will estimate the term I21 directly. To∫ control the   t∫ ∫t∫ 4  µ(v,θ)φx 2 2 nonlinear term 0 R ψx involved in I21 and I23 , we first derive bounds on  v (t) and ψxx in 0

R

terms of ∥θ∥L∞ ([0,T ]×R) (see Corollary 3.3 and (3.8)). In view of (3.8), Sobolev and Gagliardo–Nirenberg ∫t∫ inequality , the term 0 R ψx4 can be estimated as follows: ∫ t∫ ∫ t ∫ t 2 2 3 ψx4 ≤ C ∥ψx ∥L∞ (R) ∥ψx ∥L2 (R) dτ ≤ C ∥ψx ∥L2 (R) ∥ψxx ∥L2 (R) dτ 0 R (0 )∫ t ∫ ∫0 t 2 3 v µψxx 2 ∥ψx ∥L2 (R) ∥ψxx ∥L2 (R) dτ ≤ C 1 + ∥θ∥L2 ∞ ([0,T ]×R) ≤C · v µ 0 0 R ( ) 9 2 ≤ C 1 + ∥θ∥L∞ ([0,T ]×R) . (1.27) Owning to (1.27), we can avoid using the auxiliary function W (t) (see (3.1) in [12]) to bound the terms I21 and I23 . This is also the main reason why we can remove the condition (1.9) as required in [12]; • We adopt the method in [3,12] to deduce the positive local-in-time lower bound of θ(t, x) and energy estimates of higher-order derivatives in Sections 4 and 5. Then by using the continuation argument designed in [13], we can then prove Theorem 1.1. Remark 1.2. The nonlinear stability of combination of viscous contact wave with rarefaction wave for the compressible Navier–Stokes equations (1.1)1 -(1.1)3 , (1.3), (1.6), and (1.7) (λ = 0, a = 0) has been considered in [29] when the transport coefficients µ and κ both takes the form (see also (1.22)-(1.24) in [29]) µ = κ = h(v)θα . Here the smooth function h(v) satisfies (li > 0, i = 1, . . . , 4) C −1 (v l1 + v −l2 ) ≤ h(v) ≤ C(v l3 + v −l4 ), and

9(1 − l2 )+ 1 37(1 − l1 )+ + + max 4(1 + 2l1 ) 2l2 4

h′ (v)2 v ≤ Ch(v)3 , {

(l3 − 1)+ (l4 − 1)+ , 1 + 2l1 2l2

∀v ∈ (0, ∞), } < 2.

We recall that Huang and Liao [29] adopted the technique developed by Li and Liang [30] to deduce the uniform upper bound of θ(t, x). Note that the above method relies on the following Sobolev inequality (see Lemma 3.6 in [29]) max θ2 ≤ 2 max χ2 + 2 max Θ 2 ≤ C + C max ∥χ(t)∥L2 (R) ∥χx (t)∥L2 (R) x,t

x,t

≤C+

x,t

t

∗ C∥θ∥ℓL∞ ([0,T ]×R) .

∗

(0 < ℓ < 2)

However, such a method loses its power due to the special form of κ(v, θ) considered in our case (cf. (1.4)): we can not control ∥χx (t)∥L2 (R) properly due to the nonlinear terms caused by κ(v, θ). Before concluding this section, we notice that there has been extensive literature on the stability analysis of viscous wave pattern to the one-dimensional compressible Navier–Stokes equations. We refer to [31–34] for the viscous shock wave, [23,26,35–39] for the rarefaction wave, [40–44] for the viscous contact wave, and [29,45–47] for the superpositions of the above three wave patterns. The rest of the paper is arranged as follows: we first give some properties of the smooth approximation of the rarefaction wave solutions and some basic energy estimates in Section 2. The pointwise bounds on the specific volume will be derived in Section 3. In Section 4, we will derive pointwise bounds on the absolute temperature. Moreover, some high order energy type estimates and the proof of our main result will be given in Section 5. Finally, we will prove Corollary 1.1 in Section 6.

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

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Notations. Throughout this paper, C ≥ 1 or Ci ≥ 1(i = 1, 2, . . .) is used to denote a generic positive constant which is independent of t, δ, a, and x but may depends on v± , u± , θ± , Π0 , V0 and H(V0 ), where Π0 , V0 and H are given by (1.21) and (1.23), respectively. Note that these constants may vary from line to line. C(·, ·) stands for some generic constant depending only on the quantities listed in the parenthesis. ϵ < 1 represents some small positive constant. For function spaces, Lq (R) (or Lq ) (1 ≤ q ≤ ∞) denotes the usual Lebesgue space on R with norm ∥ · ∥Lq (R) , while H q (R) denotes the usual Sobolev space in the L2 sense with norm ∥·∥H q (R) . We denote by C(I; H q (R)) the space of continuous functions on the internal I with values in H q (R) and L2 (I; H q (R)) stands for the space of L2 -functions on I with values in H q (R). For simplicity, we use ∥ · ∥∞ to denote the norm in L∞ ([0, T ] × R) with T > 0 being some given positive constant, ∥ · ∥ and ∥ · ∥q are used to denote the norm ∥ · ∥L2 (R) and the norm ∥·∥H q (R) , respectively. Finally, A ≲ B (or B ≳ A) means that A ≤ CB holds uniformly for some generic positive constant C. 2. Preliminaries In this section, we derive some basic energy estimates which will be used later on. First, we list some properties of the global smooth functions (V (t, x), U (t, x), S(t, x), Θ(t, x), 0) constructed in (1.18) and (1.19). According to (1.1), (1.17), and (1.19), we know that (V (t, x), U (t, x), S(t, x), 0) solves the following problem: ⎧ Vt − Ux = 0, ⎪ ⎪ ⎪ ⎪ U + p(V, Θ)x = g(V, Θ)x , t ⎪ ) ⎨ ( U2 e(V, Θ) + 2 + (U p(V, Θ))x = q(V, Θ), t ⎪ ⎪ Θ (V,Θ) ⎪ Θt + Θp ⎪ ⎪ eΘ (V,Θ) Ux = r(V, Θ), ⎩ St = 0, where

⎧ g(V, Θ) = p(V, Θ) − p(V1 , Θ1 ) − p(V3 , Θ3 ) − p(v(m , θm ), ⎪ ⎪ ) ⎪ 2 2 2 ⎪ ⎪ ⎨ q(V, Θ) = (e(V, Θ) − e(V1 , Θ1 ) − e(V3 , Θ3 )) + U − U1 − U3 t 2 2 2 t ⎪ ⎪ + (U p(V, Θ) − U p(V , Θ ) − U p(V , Θ )) , ⎪ 1 1 1 3 3 3 x ⎪ ⎪ ⎩ r(V, Θ) = ΘpΘ (V,Θ) U − Θ1 pΘ (V1 ,Θ1 ) U − Θ3 pΘ (V3 ,Θ3 ) U , x 1x 3x e (V,Θ) e (V1 ,Θ1 ) e (V3 ,Θ3 ) Θ

Θ

Θ

˜ m , s¯). and θm = θ(v Since ω0 (x) is a strictly increasing function, we have the following lemma. Lemma 2.1. For each i ∈ {1, 3}, the Cauchy problem (1.18) admits a unique global smooth solution ωi (t, x) which satisfies the following properties: (i). ωi− < ωi (t, x) < ωi+ ,

ωix (t, x) > 0 for each (t, x) ∈ R+ × R;

(ii). For any p with 1 ≤ p ≤ ∞, there exists a constant Cp.q , depending only on p, q, such that { p−1 p } ⎧ p ∥ωix (t)∥LP ≤ Cp,q min { ϵ ω ˜i , ω ˜ i t−p+1 , ⎪ ⎪ } ⎪ p−1 ⎪ 1 − 2q −p− p−1 ⎪ p 2p−1 p (p−1)(1− 2q ) ⎪ 2q ω ˜i , ϵ ω ˜i t , ⎪ ⎪ ∥ωixx (t)∥LP ≤ Cp,q min ϵ ⎨ { } 1

−

2p−1

2p−1

∥ωixxx (t)∥LP ≤ Cp,q min ϵ3p−1 ω ˜ ip , ϵ(2p−1)(1− 2q ) ω ˜ i 2q t−p− 2q , { } 3p−1 3p−1 1 − p ∥ωixxxx (t)∥LP ≤ Cp,q min ϵ4p−1 ω ˜ ip , ϵ(3p−1)(1− 2q ) ω ˜ i 2q t−p− 2q ; p

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

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(iii). If 0 < ωi− (< ωi+ ) and q is suitably large, then ⎧ )− 3q ( )− q ( ⎪ 2 ⎨ |ωi (t, x) − ωi− | ≤ C ω , x ≤ 0, ˜ i 1 + (ϵx)2 3 1 + (ϵωi− t) q ( ) − 1 ( )− 2 ⎪ 2 ⎩ |ωix (t, x)| ≤ Cϵ˜ ωi 1 + (ϵx)2 2 1 + (ϵωi+ t) , x ≤ 0; (iv). If (ωi− ) < ωi+ ≤ 0 and q is suitably large, then ⎧ )− 3q ( )− 3q ( ⎪ 2 2 ⎨ 1 + (ϵωi− t) , x ≤ 0, |ωi (t, x) − ωi+ | ≤ C ω ˜ i 1 + (ϵx) q 1 ( ( ) ) − − ⎪ 2 2 2 2 ⎩ |ωix (t, x)| ≤ Cϵ˜ ωi 1 + (ϵx) 1 + (ϵωi+ t) , x ≤ 0; ⏐ ( )⏐ (v). limt→+∞ supx∈R ⏐ωi (t, x) − ωiR xt ⏐ = 0; ( ) Here ω ˜ i = ωi+ − ωi− > 0 and ωiR xt is the unique rarefaction wave solution of the corresponding Riemann problem of (1.14)1 , i.e., ⎧ ωi− , ⎨ ωi− , ξ ≤ ξ, ωi− ≤ ξ ≤ ωi+ , ωiR (ξ) = ⎩ ωi+ , ξ ≥ ωi+ . Based on the results obtained in Lemma 2.1 and from (1.19) and (1.20), we can deduce that Lemma 2.2. Letting ϵ = δ, q = 2, the smooth approximations (V (t, x), U (t, x), Θ(t, x), 0) constructed in (1.19) and (1.20) have the following properties: (i). Vt (t, x) = Ux (t, x) > 0 for each (t, x) ∈ R+ × R; (ii). For any p with 1 ≤ p ≤ ∞ there exists a constant Cp , depending only on p, such that ⎧ { } p −p+1 ∥(Vx , Ux , Θx ) (t)∥Lp ≤ Cp min δ 2p−1 , δ(t + 1) , ⎪ ⎪ { } ⎪ p−1 5p−1 ⎪ p ⎪ , ∥(Vxx , Uxx , Θxx ) (t)∥Lp ≤ Cp min δ 3p−1 , δ 2 (t + 1)− 4 ⎨ { } 2p−1 6p−1 p 4p−1 2 (1 + t)− 4 ≤ C min δ , δ ∥(V , U , Θ ) (t)∥ , ⎪ p p xxx xxx xxx L ⎪ ⎪ { } ⎪ 3p−1 7p−1 ⎪ p − 5p−1 ⎩ , δ 2 (1 + t) 4 ∥(Vxxxx , Uxxxx , Θxxxx ) (t)∥Lp ≤ Cp min δ . It is obvious that ∥Vx (t)∥2L2 is not integrable with respect to t. However we can get for any r > 0 and p > 1 that ⎧ ∫ ∞ ⎪ 2+r ⎪ ∥(Vx , Ux , Θx ) (t)∥L2+r dt ≤ C(r)δ, ⎪ ⎪ ⎪ ⎪ ∫0 ∞ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ∥(Vxx , Uxx , Θxx ) (t)∥Lp dt ≤ C(p)δ 4 (1− p ) , ⎨ ∫0 ∞ 1 1 ⎪ ⎪ ⎪ ∥(Vxxx , Uxxx , Θxxx ) (t)∥Lp dt ≤ C(p)δ 4 (2− p ) , ⎪ ⎪ ⎪ ⎪ ∫0 ∞ ⎪ ⎪ 1 1 ⎪ ⎪ ∥(Vxxxx , Uxxxx , Θxxxx ) (t)∥Lp dt ≤ C(p)δ 4 (3− p ) ; ⎩ 0

(iii). For each p ≥ 1, 2

4

∥(g(V, Θ)x , r(V, Θ), q(V, Θ)) (t)∥Lp ≤ C(p)δ 3 (1 + t)− 3 . Especially, ∫ 0

∞

1

∥(g(V, Θ)x , r(V, Θ), q(V, Θ)) (t)∥Lp dt ≤ C(p)δ 3 ;

⏐ ( ( ) ( ) ( ))⏐ (iv). limt→+∞ supx∈R ⏐(V (t, x), U (t, x), Θ(t, x)) − V R xt , U R xt , Θ R xt ⏐ = 0;

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

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(v). |(Vt (t, x), Ut (t, x), Θt (t, x))| ≤ O(1) |(Vx (t, x), Ux (t, x), Θx (t, x))|. We can easily verify that (φ(t, x), ψ(t, x), χ(t, x), ξ(t, x), z(t, x)) solves ⎧ φt − ψx = 0, ⎪ ⎪ ⎪ x ⎪ ψt + [p(v, θ) − p(V, Θ)]x = ( µ(v,θ)u ⎪ v ( )x − g(V, Θ)x , ⎪ ) ⎪ ⎪ µ(v,θ)u2 κ(v,θ)θx 1 θ (v,θ) x ⎪ χt + θp ψ = + ( ) + λϕz ⎨ x x eθ (v,θ) eθ (v,θ) v v) ( θpθ (v,θ) ΘpΘ (V,Θ) Ux − r(V, Θ), − − ⎪ eθ (v,θ) ⎪ ⎪ ( eΘ (V,Θ) ) ⎪ 2 ⎪ µ(v,θ)ux κ(v,θ)θ 2 x ⎪ ξt = + κ(v,θ)θ + vθ2 x + λϕz ⎪ ⎪ vθ vθ θ , ⎪ x ( dz ) ⎩ x − ϕz, zt = v2 x

(2.1)

with initial data (φ(t, x), ψ(t, x), χ(t, x), ξ(t, x), z(t, x)) |t=0 = (φ0 (x), ψ0 (x), χ0 (x), ξ0 (x), z0 (x)) ( ) ¯ z0 (x) . = v0 (x) − V (0, x), u0 (x) − U (0, x), θ0 (x) − Θ(0, x), s0 (x) − S, On the other hand, it is easy to see that ( ) (v) 1 ) θ avχ2 ( 2 η(v, u, θ; V, U, Θ) = Cv ΘΦ + RΘΦ + ψ2 + 3θ + 2θΘ + Θ 2 , Θ V 2 3

(2.2)

Φ(y) = y − ln y − 1 (2.3)

is a convex entropy to Eq. (1.1) around the smooth rarefaction wave profile (V (t, x), U (t, x), Θ(t, x), 0) which satisfies the following identity ( ) µ(v, θ)Θψx2 κ(v, θ)Θχ2x ηt (v, u, θ, V, U, Θ) + ((p(v, θ) − p(V, Θ)) ψ)x + + vθ vθ2 λϕzΘ + (˜ p(v, s) − p˜(V, s¯) − p˜v (V, s¯)φ − p˜s (V, s¯)ξ) Ux + θ ( { ) µ(v, θ)ψψx κ(v, θ)χχx 2µ(v, θ)Ux χψx µ(v, θ)Ux ψφx κ(v, θ)Θx χφx = + + − − 2 v vθ vθ v v2 θ x } ( ) ( ) 2 κ(v, θ)Θx χχx µ(v, θ)ψUxx κ(v, θ)χΘxx µ(v, θ)Ux χ µ(v, θ)Ux Vx ψ κ(v, θ)Vx Θx χ + + + + − − vθ2 v vθ vθ v2 v2 θ ( ) κx (v, θ)χΘx + g(V, Θ)x U − q(V, Θ) − g(V, Θ)x ψ − r(V, Θ)ξ + λϕz + vθ ( ) µv (v, θ)φx ψUx µv (v, θ)ψVx Ux µθ (v, θ)ψΘx Ux µθ (v, θ)ψχx Ux + + + + . (2.4) v v v v We first define the following set of functions for which solutions to Cauchy problem (2.1) and (2.2) will be sought. { X(s, t; m1 , m2 , N ) := (φ(τ, x), ψ(τ, x), χ(τ, x), z(τ, x)) : (φ(τ, x), ψ(τ, x), χ(τ, x), z(τ, x)) ∈ C([s, t]; H 3 (R)), φx (τ, x) ∈ L2 (s, t; H 2 (R)), (ψx (τ, x), χx (τ, x), zx (τ, x)) ∈ L2 (s, t; H 3 (R)), z(τ, x) ∈ C([s, t]; L1 (R)), 0 ≤ z(τ, x) ≤ 1, } E(s, t) ≤ N 2 , V (τ, x) + φ(τ, x) ≥ m1 , Θ(τ, x) + χ(τ, x) ≥ m2 ∀ (τ, x) ∈ [s, t] × R , for some positive constants N , mi , s, and t (i = 1, 2, t ≥ s), where { } ∫ t[ ] E(s, t) := sup ∥(φ, ψ, χ, z)(τ )∥23 + ∥z(τ )∥2L1 (R) + ∥φx (τ )∥22 + ∥(ψx , χx , zx )(τ )∥23 dτ. s≤τ ≤t

s

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

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The existence and uniqueness of solution (φ(t, x), ψ(t, x), χ(t, x), ξ(t, x), z(t, x)) to the Cauchy problem (2.1)–(2.2) in the set of functions X(0, t1 ; m1 , m2 , N ) for some sufficiently small t1 > 0 and certain positive constants m1 , m2 and N is guaranteed by the well-established local existence result for hyperbolic– parabolic system, cf. [48]. Suppose that the local solution (φ(t, x), ψ(t, x), χ(t, x), ξ(t, x), z(t, x)) to the Cauchy problem (2.1)–(2.2) has been extended to the time step t = T for some positive constant T > 0 and (φ(t, x), ψ(t, x), χ(t, x), ξ(t, x), z(t, x)) ∈ X(0, T ; m1 , m2 , N ) for some positive constants T , mi ≤ 1 (i = 1, 2) and N ≥ 1. Then in order to prove Theorem 1.1, we only need to derive certain a priori estimates on the solution (φ(t, x), ψ(t, x), χ(t, x), ξ(t, x), z(t, x)) in terms of the initial data (φ0 (x), ψ0 (x), χ0 (x), ξ0 (x), z0 (x)) but independent of the constants mi ≤ 1 (i = 1, 2) and N ≥ 1. Applying Sobolev’s inequality yields m1 ≤ v(t, x) ≤ 4N,

m2 ≤ θ(t, x) ≤ 4N,

∀ (t, x) ∈ [0, T ] × R.

(2.5)

We then try to deduce some a priori estimates independent of the constant mi ≤ 1 (i = 1, 2) and N ≥ 1 provided that the constants α, δ and a are assumed to be sufficiently small such that −|α|

m2

≤ 2,

N |α| ≤ 2,

Ξ (m1 , m2 , N ) (|α| + δ + a) ≤ ϵ1 ,

where

(2.6)

]800+800b

[ −1 Ξ (m1 , m2 , N ) := m−1 1 + m2 + N +

sup

h(σ) + 1

,

m1 ≤σ≤4N

The following lemma ensures the convexity of p˜(v, s) with respect to (v, s). Lemma 2.3. Suppose that (φ(t, x), ψ(t, x), χ(t, x), z(t, x)) ∈ X(0, T ; m1 , m2 , N ) be a solution to the Cauchy problem (2.1) and (2.2) satisfying the a priori assumption (2.5) and (2.6), then p˜(v, s) is convex with respect to v and s. Moreover, we have p˜(v, s) − p˜(V, s¯) − p˜v (V, s¯)φ − p˜s (V, s¯)ξ ≥ 0. Proof . The proof is similar to that of Lemma 2.4 in [17]. For the sake of completeness, we sketch the proof here. In fact, we only need to prove ( 2 )2 ∂ 2 p˜(v, s) ∂ 2 p˜(v, s) ∂ 2 p˜(v, s) ∂ 2 p˜(v, s) ∂ p˜(v, s) ≥ 0, ≥ 0, − ≥ 0. (2.7) ∂v 2 ∂s2 ∂s2 ∂v 2 ∂v∂s For this purpose, we can deduce from Gibbs equation, (1.3) and (1.10) that ˜ s) pθ ∂ θ(v, =− , ∂v sθ

˜ s) ∂ θ(v, 1 = . ∂s sθ

(2.8)

Differentiating (2.8) with respect to v and s, respectively, we can infer that ˜ s) ∂ 2 θ(v, sθθ (θ˜v )2 + 2pθθ θ˜v + pvθ =− , 2 ∂v sθ

˜ s) ∂ 2 θ(v, sθθ (θ˜s )2 =− , 2 ∂s sθ

˜ s) ∂ 2 θ(v, sθθ θ˜v θ˜s + pθθ θ˜s =− . ∂v∂s sθ

(2.9)

On the other hand, we can deduce from the chain rule, (2.8), and (2.9) that ) ∂ 2 p˜(v, s) ∂ ( pv + pθ θ˜v = sθθ (θ˜v )3 + 3pθθ (θ˜v )2 + 3pvθ θ˜v + pvv = 2 ∂v ∂v −sθθ p3θ + 3pθθ p2θ sθ − 3pvθ pθ s2θ + pvv s3θ = , s3θ ∂ 2 p˜(v, s) ∂ ( ˜) = pθ θs = sθθ (θ˜s )2 θ˜v + pθθ (θ˜s )2 ∂s2 ∂s (θ˜s )2 (pθθ sθ − sθθ pθ ) = , sθ

(2.10)

(2.11)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

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∂ 2 p˜(v, s) ∂ ( ˜) = pθ θs = pθ θ˜vs + pθθ θ˜v θ˜s + pvθ θ˜s ∂v∂s ∂v = sθθ (θ˜v )2 θ˜s + 2pθθ θ˜v θ˜s + pvθ θ˜s .

(2.12)

Combining (2.7)–(2.12), after simple calculation, we obtain ( ) [ 40aCv R2 + 28aCv2 R − 8aR3 θ 1 Cv R3 + 3Cv2 R2 + 2Cv3 R ∂ 2 p˜(v, s) = 3 + ∂v 2 sθ v3 θ2 v2 ( ) ( ) ] 496a2 Cv R + 192a2 R2 θ4 640a3 Cv + 7488a3 R θ7 1792a4 vθ10 + + + , (2.13) 3v 27 27 ( [ ) ] (θ˜s )2 Cv R 16aCv ∂ 2 p˜(v, s) 16a2 vθ4 = + − 8aR θ + , (2.14) ∂s2 sθ vθ2 3 3 and ) ( ( 2 )2 [ 32aCv2 R − 52aCv R2 − 24aR3 θ ∂ p˜(v, s) (θ˜s )2 Cv R3 + Cv2 R2 ∂ 2 p˜(v, s) ∂ 2 p˜(v, s) − = + ∂s2 ∂v 2 ∂v∂s s2θ θ2 v4 3v 3 ( ) ] 448a2 Cv R − 1200a2 R2 θ4 320a3 Rθ7 256a4 θ10 + − − . (2.15) 9v 2 9v 9 Combine the fact that sθ =

Cv θ

+ 4avθ2 > 0, (2.5), (2.6), (2.13), (2.14), and (2.15) to derive (2.7). □

We present the basic energy estimates in the next lemma, which will play a fundamental role in our analysis. Lemma 2.4 (Basic Energy Estimates). Under the assumptions stated in Theorem 1.1, we have for all 0 ≤ t ≤ T that ∫ ∫ t∫ z(t, x)dx + ϕ(τ, x)z(τ, x) ≲ 1, (2.16) R ) ∫ ∫ t ∫0 (R d 2 z 2 (t, x)dx + z + ϕz 2 (τ, x) ≲ 1, (2.17) 2 x v R 0 R ) ∫ t∫ ∫ ∫ t∫ ( κ(v, θ)Θχ2x µ(v, θ)Θψx2 λΘϕz + η(t, x)dx + + 2 vθ vθ θ R 0 R 0 R ∫ t∫ + (˜ p(v, s) − p˜(V, s¯) − p˜v (V, s¯)φ − p˜s (V, s¯)ξ) Ux ≲ 1, (2.18) 0

R

Proof . Identities (2.16) and (2.17) follows directly from (2.1)5 and integration by parts. Integrating (2.4) over (0, t) × R, we have ) ∫ t∫ ∫ ∫ t∫ ( 2 2 µ(v, θ)Θ ψx κ(v, θ)Θ χx + vθ vθ2

η(t, x)dx + 0

R

R

+ 0

R

λΘ ϕz θ

∫ t∫ (˜ p(v, s) − p˜(V, s¯) − p˜v (V, s¯)φ − p˜s (V, s¯)ξ) Ux

+ 0

R

∫ t∫ (

∫ η0 dx +

=

0

R

∫ t∫ ( +

0

R

R

2µ(v, θ)Ux χψx µ(v, θ)Ux ψφx κ(v, θ)Θx χφx κ(v, θ)Θx χχx − − + vθ v2 v2 θ vθ2





I1

µ(v, θ)Uxx ψ κ(v, θ)Θxx χ + v vθ

 I2

) ∫ t∫ ( +

 0

R

)

µ(v, θ)Ux2 χ µ(v, θ)Ux ψVx κ(v, θ)Θx χVx − − vθ v2 v2 θ

 I3

) 

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

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∫ t∫ (−q(V, Θ ) − g(V, Θ )x ψ + g(V, Θ )x U − r(V, Θ )ξ)

+

0

R





I4

∫ t∫ (

)

λϕz +

+

0

R

µv (v, θ)φx ψUx µv (v, θ)ψVx Ux µθ (v, θ)ψ Θx Ux µθ (v, θ)ψχx Ux κx (v, θ)χΘx + + + + (2.19) vθ v v v v





I5

In view of Lemma 2.2, the a priori assumption (2.5), (2.6), (2.16), and Cauchy–Schwarz’s inequality, Ij (j = 1, 2, 3, 4, 5) can be estimated as follows: ) ∫ t∫ ( 1 µ(v, θ)Θψx2 µ(v, θ)θφ2x κ(v, θ)Θχ2x + I1 ≲ δ 4 + v3 vθ vθ2 0 R  ∫ t    )∫ t [ ( 2 ] ( )  κ (v, θ)   κ(v, θ)  1  h(v)  2 2 2 2 2      + δ− 4  ∥U ∥ ∥ψ∥ + ∥χ∥ dτ + + ∥Θ ∥ ∥χ∥ dτ x L∞ x L∞  vθ   vθ3 h(v)   vθ2  0 ∞ 0 ∞ ∞ ) ∫ t∫ ( 1 µ(v, θ)θφ2x µ(v, θ)Θψx2 κ(v, θ)Θχ2x ≲ δ4 + + v3 vθ vθ2 0 R ∫ t 3 −3 2 4 (1 + τ ) 2 ∥(ψ, χ) (τ )∥ dτ, (2.20) + δ Ξ (m1 , m2 , N ) 0

   )  ∫ t ∫ t(  κ(v, θ)χ 2  h(v) 2 2     I2 ≲ ∥Uxx ∥ + ∥Θxx ∥ + ∥Θxx ∥  dτ +  ∥Uxx ∥ ∥ψ∥ dτ vθ  v ∞ 0 0 1

∫

1

t

≲ δ 4 + δ 4 Ξ (m1 , m2 , N )

9 −8

(1 + τ )

2

∥(ψ, χ) (τ )∥ dτ,

(2.21)

0

∫

t

∥(Vx , Ux , Θx )∥

I3 ≲

∫ t∫ (

5 2 5

dτ +

L2

0

⏐ ⏐ ⏐ µ(v, θ)ψ ⏐2 ⏐ ⏐ + |Ux | 32 |Vx | ⏐ v2 ⏐

0

R

3 2

⏐ ⏐ ⏐ µ(v, θ)χ ⏐2 3 ⏐ ⏐ 2 ⏐ vθ ⏐ + |Θx |

⏐ ⏐ ) ⏐ κ(v, θ)χ ⏐2 ⏐ ⏐ ⏐ v2 θ ⏐

(      )∫ t  µ(v, θ) 2  µ(v, θ) 2  κ(v, θ) 2 ( )      ≲ δ+  ∥Vx ∥2L∞ ∥ψ∥2 + ∥Ux ∥2L∞ ∥χ∥2 + ∥Θx ∥2L∞ ∥χ∥2 dτ  v 2  +  vθ  +  v 2 θ  ∞

∞

∫

1

≲ δ + δ 2 Ξ (m1 , m2 , N )

t

∞

0

5

(1 + τ )− 4 ∥(ψ, χ) (τ )∥2 dτ,

(2.22)

0

I4 ≲

∫ t( ) 2 2 ∥q (V, Θ)∥L1 + ∥g (V, Θ)x ∥L1 + ∥g (V, Θ)x ∥ + ∥r (V, Θ)∥ + ∥g (V, Θ)x ∥ ∥ψ∥ + ∥r (V, Θ)∥ ∥ξ∥ dτ 0

1

2

∫

≲ δ3 + δ3

t

(1 + τ )

4 −3

2

∥(ψ, ξ) (τ )∥ dτ,

0

and ∫ t∫ (

⏐ ⏐ θb |φx Θx χ| θb |Vx Θx χ| + + θb−2 |Θx χx χ| + θb−2 ⏐Θx2 χ⏐ vθ vθ 0 R ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐) ⏐ µv (v, θ)Ux φx ψ ⏐ ⏐ µv (v, θ)Ux Vx ψ ⏐ ⏐ αµ(v, θ)Ux Θx ψ ⏐ ⏐ αµ(v, θ)Ux χx ψ ⏐ ⏐+⏐ ⏐+⏐ ⏐+⏐ ⏐ + ⏐⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ v v vθ vθ ) ∫ t ∫ t∫ ( 5 1 µ(v, θ)θφ2x κ(v, θ)Θ χ2x ≲ δ4 + + ∥(Vx , Ux , Θx )∥ 2 5 dτ 3 2 L2 v vθ 0 R 0

I5 ≲

(2.23)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

  (  2b−2    v |µ (v, θ)|2   vθ  vθ2b−3     v       h(v)  +  κ(v, θ)  +    θh(v) ∞ ∞

[ + 1+δ

− 14

 2  )]  h (v)   +α   vκ(v, θ)  2

∞

∞

·

∫ t(

2

)

2

15

2

∥Ux ∥L∞ + ∥Θx ∥L∞ ∥(ψ, χ) (τ )∥ dτ

0

(      )∫ t ( )  µv (v, θ) 2  µ(v, θ) 2  b−2 2  θb−1 2 3 3 2 2    +  +α   ∥Vx ∥L2 ∞ + ∥Θx ∥L2 ∞ ∥(ψ, χ) (τ )∥ dτ +       v  + θ ∞ v vθ ∞

1

∞

∫ t∫ (

≲ 1 + δ4

0

0

∞

) ∫ t 1 κ(v, θ)Θ χ2x µ(v, θ)θφ2x −5 2 2 Ξ (m , m , N ) + + δ (1 + τ ) 4 ∥(ψ, χ) (τ )∥ dτ. 1 2 3 2 v vθ 0

R

(2.24)

Substituting the above estimates into (2.19), we can get that ∫ t∫ (

∫ η(t, x)dx +

0

R

R

κ(v, θ)Θχ2x µ(v, θ)Θψx2 + vθ vθ2

∫ t∫

) +

0

∫ t∫

R 1

ϕz θ ∫ t∫

(˜ p(v, s) − p˜(V, s¯) − p˜v (V, s¯)φ − p˜s (V, s¯)ξ) Ux ≲ 1 + δ 4

+ 0

0

R

∫t∫ Now we turn to bound the term 0 R x multiplying (2.1)2 by µ(v,θ)φ , one has v 1 2 ( =

+

{(

µ(v, θ)φx v

)2 }

)

(

µ(v, θ)ψφx v

+ t

− t

µ(v,θ)θφ2 x v3

R

µ(v, θ)θφ2x . v3

(2.25)

appeared on the right hand side of (2.25). To this end,

µ(v, θ)Rθφ2x v3

µ(v, θ)ψψx v

) + x

µ(v, θ)ψx2 µ(v, θ)φx + g(V, Θ)x v v

µ(v, θ)ψφx Ux µv (v, θ)ψψx Vx µv (v, θ)ψφx Ux µ(v, θ)ψψx Vx + − − v2 v v v2

µ(v, θ)RΘφx Vx µ(v, θ)RΘx φx φ µ(v, θ)Rφx χx µ(v, θ)Rθφx Vx − + − 2 2 2 vV v V v v3 ( 2 ) 4aµ(v, θ) θ + θΘ + Θ 2 φx χΘx 4aµ(v, θ)θ3 φx χx + + 3v 3v +

−

µv (v, θ)µ(v, θ)ψx φx Vx µv (v, θ)µ(v, θ)φx Vx Ux µ2 (v, θ)φx Uxx − − 2 2 v v v2

+

µ2 (v, θ)ψx φx Vx µ2 (v, θ)φx Vx Ux + 3 v v3

−

µθ (v, θ)θt φx ψ µθ (v, θ)ψψx Θx µθ (v, θ)ψψx χx µθ (v, θ)µ(v, θ)vx φx θt + + + v v v v2

−

µθ (v, θ)µ(v, θ)θx ux φx µθ (v, θ)µ(v, θ)Vx φx θt − . v2 v2

Integrating the above identity with respect to t and x over [0, t] × R, we obtain ∫ ( R

µ(v, θ)φx v

)2

∫ t∫ ( (t, x)dx + 0

R

µ(v, θ)φ2x v3

) (τ, x) ≲ 1 +

11 ∑ j=6

|Ij |,

(2.26)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

16

here ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

) ) ∫ t∫ ( µ(v, θ)ψx2 µ(v, θ)ψφx (τ, x)dx + (τ, x), v 0 R ) ∫Rt ∫ ( v µ(v, θ)ψφx Ux µv (v, θ)ψφx Ux µv (v, θ)ψψx Vx µ(v, θ)ψψx Vx − (τ, x), + − v2 v v v2 ) ∫0 t ∫R ( µ(v, θ)Rφx χx µ(v, θ)RΘx φx φ µ(v, θ)Rθφx Vx µ(v, θ)RΘφx Vx − + (τ, x), − v2 v2 V ( v3 ) vV 2 ) ∫0 t ∫R ( 4aµ(v, θ) θ2 + θΘ + Θ 2 φx χΘx 4aµ(v, θ)θ3 φx χx (τ, x), + 3v 3v 0 R ∫ t∫ ( µ(v, θ)g(V, Θ)x φx µv (v, θ)µ(v, θ)φx Vx Ux µv (v, θ)µ(v, θ)ψx φx Vx − − 2 v v v2 0 R )

∫ ( I6 = I7 = I8 = I9 =

⎪ ⎪ ⎪ ⎪ I10 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ I11 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2

2

2

x Uxx x φx Vx x Vx Ux − µ (v,θ)φ + µ (v,θ)ψ + µ (v,θ)φ (τ, x) v2 v3 v3 ∫ t∫ ( µθ (v, θ)θt φx ψ µθ (v, θ)ψψx χx µθ (v, θ)µ(v, θ)vx φx θt µθ (v, θ)ψψx Θx − + + v v v v2 0 R ) x ux φx − µθ (v,θ)µ(v,θ)θ − v2

µθ (v,θ)µ(v,θ)Vx φx θt v2

(τ, x).

Now we deal with I6 , I7 , I8 , I9 , I10 , and I11 term by term. For this purpose, by applying the Cauchy–Schwarz inequality, we can get from Lemma 2.2, the a priori assumption (2.5) and (2.6) that )2 ( ) ∫ t∫ ∫ ( µ(v, θ)φx µ(v, θ)Θψx2 2 dx + C(ϵ) ∥ψ(t)∥ + ∥θ∥∞ , (2.27) I6 ≤ ϵ v vθ R 0 R ) ∫ t∫ ( µ(v, θ)θφ2x µ(v, θ)Θψx2 I7 ≤ ϵ + v3 vθ 0 R ) ∫ t∫ ( 2 2 2 2 |µv (v, θ)| ψ 2 Ux2 v |µv (v, θ)| θψ 2 Vx2 µ(v, θ)ψ 2 Vx2 θ µ(v, θ)ψ Ux + C(ϵ) + + + , vθ µ(v, θ)θ µ(v, θ)v v3 0 R ) ∫ t∫ ( ∫ t µ(v, θ)θφ2x µ(v, θ)Θψx2 −3 2 ≤ϵ + + C(ϵ)δΞ (m , m , N ) (1 + τ ) 2 ∥ψ(τ )∥ dτ, (2.28) 1 2 3 v vθ 0 R 0 ∫ t∫ ( ∫ t∫ µ(v, θ)φ2 Θx2 µ(v, θ)χ2x µ(v, θ)θφ2x + C(ϵ) + I8 ≤ ϵ v3 vV 2 θ vθ 0 R 0 R ) 2 2 2 2 2 2 µ(v, θ)φ Θ (v + V ) Vx µ(v, θ)χ Vx + + v3 V 4 θ v3 θ [∫ t ( ) ∫ t∫ 3 3 µ(v, θ)θφ2x 2 2 2 ≤ϵ + C(ϵ)Ξ (m , m , N ) ∥V ∥ + ∥Θ ∥ ∥(φ, χ) (τ )∥ dτ ∞ ∞ 1 2 x L x L 3 v 0 R 0 ] ∫ t∫ κ(v, θ)Θχ2x + vθ2 0 R [ ∫ t ∫ t∫ µ(v, θ)θφ2x −3 2 ≤ϵ + C(ϵ)Ξ (m , m , N ) δ (1 + τ ) 2 ∥(φ, χ) (τ )∥ dτ 1 2 3 v 0 R 0 ] ∫ t∫ κ(v, θ)Θχ2x + , (2.29) vθ2 0 R ) ( ) ∫ t∫ ∫ t∫ ( a2 µ(v, θ)vχ2 1 + θ4 Θx2 µ(v, θ)θφ2x 2 5 2 I9 ≤ ϵ + C(ϵ) a µ(v, θ)vθ χx + v3 θ 0 R 0 R [ ] ∫ t∫ ∫ t 3 µ(v, θ)θφ2x −2 2 ≤ϵ + C(ϵ)Ξ (m , m , N ) 1 + δ (1 + τ ) ∥χ(τ )∥ dτ , (2.30) 1 2 v3 0 R 0

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

∫ t∫

∫ t∫ (

2

17

2

µ(v, θ) |g(V, Θ)x | v |µv (v, θ)| µ(v, θ)ψx2 Vx2 + θ vθ 0 R 0 R ) 2 2 2 3 2 3 2 2 3 µ (v, θ)Uxx µ (v, θ)ψx Vx µ (v, θ)Vx2 Ux2 |µv (v, θ)| µ(v, θ)Vx Ux + + + + vθ vθ v3 θ v3 θ ∫ t∫ 2 µ(v, θ)θφx ≤ϵ v3 0 R [∫ t ( ] ∫ t∫ ) µ(v, θ)Θψx2 2 2 2 2 4 ∥Vx ∥L∞ ∥Ux ∥ + ∥Uxx ∥ + ∥g(V, Θ)x ∥ dτ + δ + C(ϵ)Ξ (m1 , m2 , N ) vθ 0 0 R [ ] ∫ t∫ ∫ t∫ 2 2 1 µ(v, θ)θφx µ(v, θ)Θψx ≤ϵ + C(ϵ)Ξ (m1 , m2 , N ) δ 2 + δ 4 , (2.31) 3 v vθ 0 R 0 R

I10 ≤ ϵ

µ(v, θ)θφ2x + C(ϵ) v3

and ) ∫ t∫ ( 2 α µ(v, θ)ψ 2 Θx2 µ(v, θ)Θψx2 α2 µ(v, θ)ψ 2 χ2x µ(v, θ)θφ2x + + C(ϵ) + ≤ϵ v3 vθ vθ vθ 0 R 0 R 2 3 α µ (v, θ)Ux2 Θx2 + vθ3 ) α2 µ3 (v, θ)χ2x Ux2 α2 µ3 (v, θ)ψx2 χ2x α2 vµ(v, θ)ψ 2 θt2 α2 µ3 (v, θ)φ2x θt2 α2 µ3 (v, θ)ψx2 Θx2 + + + + + vθ3 vθ3 vθ3 θ3 vθ3 ) [∫ ∫ t∫ ( t( ) 2 2 µ(v, θ)θφx µ(v, θ)Θψx 2 2 2 2 2 ≤ϵ + + C(ϵ)α Ξ (m , m , N ) ∥Θ ∥ ∥ψ∥ + ∥U ∥ ∥Θ ∥ dτ ∞ ∞ 1 2 x x x L L v3 vθ 0 R 0 ) ] ∫ ∫ ∫ t∫ ( t ) ( 2 ) κ(v, θ)Θχ2x ( µ(v, θ)Θψx2 2 2 2 + 1 + U + Θ + α Ξ (m , m , N ) ψ + φ2x θt2 . (2.32) + 1 2 x x 2 vθ vθ 0 R  0 R  ∫ t∫ (

I11

H

To estimate the last term on the right hand side of (2.32), we deduce from (1.1)3 that ] [( ) κ(v, θ)θx κ(v, θ)θxx µ(v, θ)u2x κv (v, θ)θx κθ (v, θ)θx2 1 − + + − θp (v, θ)u + λϕz , v + θt = θ x x eθ (v, θ) v v2 v v v (2.33) where κv (v, θ) = κ2 θb ,

κθ (v, θ) = κ2 bvθb−1 R 4 3 eθ (v, θ) = cv + 4avθ3 , pθ (v, θ) = + aθ . v 3 Similar to the proof of Lemma 2.2 in [13], we can obtain )] [ ∫ t∫ ( µ(v, θ)θφ2x µ(v, θ)Θψx2 κ(v, θ)Θχ2x κ(v, θ)χ2xx 2 H ≲ α Ξ (m1 , m2 , N ) 1 + + + + . v3 vθ vθ2 v 0 R

(2.34)

(2.35)

On the other hand, multiplying (2.1)3 by χxx yields ( 2) χx κ(v, θ)χ2xx ∂t + − (χt χx )x 2 veθ (v, θ) ( ) θpθ (v, θ)ψx χxx θpθ (v, θ) ΘPΘ (V, Θ) µ(v, θ)u2x χxx κ(v, θ)Θxx χxx λϕzχxx = + − Ux χxx − − − eθ (v, θ) eθ (v, θ) EΘ (V, Θ) veθ (v, θ) veθ (v, θ) eθ (v, θ) ( ) κ(v, θ) κv (v, θ) κθ (v, θ) 2 + − (Θx + χx ) (Vx + φx ) χxx − (χx + Θx ) χxx v 2 eθ (v, θ) veθ (v, θ) veθ (v, θ) + r (V, Θ) χxx . (2.36) We integrate the above identity over [0, t] × R and apply Cauchy’s inequality to infer [ )] ∫ t∫ ∫ t∫ ( κ(v, θ)χ2xx µ(v, θ)Θψx2 κ(v, θ)Θχ2x ∥χx (t)∥2 + ≲ 1 + Ξ (m1 , m2 , N ) 1 + + . vθ vθ2 0 R veθ (v, θ) 0 R

(2.37)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

18

Plugging (2.35), (2.37) into (2.32) and using Lemma 2.2, we obtain ∫ t∫ (

I11

) µ(v, θ)Θψx2 µ(v, θ)θφ2x + ≤ϵ v3 vθ 0 R )] [ ∫ t∫ ( µ(v, θ)θφ2x µ(v, θ)Θψx2 κ(v, θ)Θχ2x 2 + . + C(ϵ)α Ξ (m1 , m2 , N ) 1 + + v3 vθ vθ2 0 R

(2.38)

Substituting (2.27)–(2.38) into (2.26) and choosing ϵ > 0 small enough, we arrive at ∫ ( R

µ(v, θ)φx v

)2

∫ t∫ ( (t, x)dx + 0

R

µ(v, θ)θφ2x v3

) (τ, x)

∫ t∫

µ(v, θ)Θψx2 vθ R 0 [ )] ∫ t∫ ( κ(v, θ)Θχ2x µ(v, θ)Θψx2 + + Ξ (m1 , m2 , N ) 1 + . vθ vθ2 0 R 2

≲ ∥ψ(t)∥ + ∥θ∥∞

Then (2.18) follows from (2.6), (2.25), and (2.39) directly. This completes the proof of our lemma.

(2.39) □

Since the pointwise bounds of z(t, x) can be established by a standard maximum principle, we omit the proof for brevity (see, for instance, [49]). Lemma 2.5. Assume that the conditions listed in Theorem 1.1 hold. Then 0 ≤ z(t, x) ≤ 1

∀ (t, x) ∈ [0, T ] × R.

(2.40)

3. Pointwise bounds for the specific volume This Section is devoted to deducing the lower and upper bounds on the specific volume v(t, x)in terms   µ(v,θ)φx 2 (t) , which of ∥θ∥∞ . For this purpose, we first present the following lemma concerning the term  v will be frequently used later on. Lemma 3.1. Assume that the conditions listed in Theorem 1.1 hold. Then    ℓ1 +1 ∫ t∫  µ(v, θ)φx 2 1 µ(v, θ)θφ2x ℓ +2   +   (t) ≲ 1 + + ∥v∥∞2 + ∥θ∥∞ .   v 3 v v 0 R ∞

(3.1)

Proof . Based on (1.5), Lemmas 2.2, and 2.4, we can re-estimate the terms I6 -I11 as follows: ∫ ( I6 ≤ ϵ R

∫ t∫

µ(v, θ)φx v

)2

∫ t∫ dx + C(ϵ) (1 + ∥θ∥∞ ) ,

I7 ≤ ϵ 0

R

µ(v, θ)θφ2x + C(ϵ), v3

[ ∫ t µ(v, θ)θφ2x −3 2 + C(ϵ) Ξ (m , m , N )δ (1 + τ ) 2 ∥(φ, χ) (τ )∥ dτ 1 2 3 v 0 R 0 ] ∫ t∫ κ(v, θ)Θχ2x µ(v, θ)θ + · vθ2 κ(v, θ) 0 R ( )   )  ℓ1 +1 ( ∫ t∫ ∫ t∫ 2  µ(v, θ)θ  1 µ(v, θ)θφx µ(v, θ)θφ2x ℓ2     ≤ϵ + C(ϵ) 1 +  + C(ϵ) 1 +   + ∥v∥∞ , ≤ϵ v3 κ(v, θ) ∞ v3 v ∞ 0 R 0 R

I8 ≤ ϵ

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

∫ t∫ (

) ( ) a2 µ(v, θ)vχ2 1 + θ4 Θx2 κ(v, θ)Θχ2x a2 µ(v, θ)v 2 θ7 · I9 ≤ ϵ + vθ2 κ(v, θ) θ 0 R 0 R  ( ) ∫ t∫ ∫ t 2 7 2  h(v)v θ  µ(v, θ)θφx 2 2   ≤ϵ + C(ϵ)  + Ξ (m1 , m2 , N ) ∥Θx ∥L∞ ∥χ(τ )∥ dτ v3 κ(v, θ) ∞ 0 R 0 ( )  ℓ1 −1 ∫ t∫ 1 µ(v, θ)θφ2x ℓ +2   ≤ϵ + C(ϵ) + ∥v∥∞2 1 + , v 3 v 0 R ∞ ( ) ∫ t∫ ∫ t∫ ∫ t∫ 1 µ(v, θ)θφ2x µ(v, θ)Θψx2 µ(v, θ)θφ2x 4 2 + C(ϵ)Ξ (m , m , N ) δ + C(ϵ), ≤ϵ + δ ≤ ϵ 1 2 3 v vθ v3 0 R 0 R 0 R ∫ t∫

I10

19

µ(v, θ)θφ2x + C(ϵ) v3

and ∫ t∫ (

) µ(v, θ)θφ2x µ(v, θ)Θψx2 + v3 vθ 0 R )] [ ∫ t∫ ( κ(v, θ)Θχ2x µ(v, θ)θφ2x + + C(ϵ)α2 Ξ (m1 , m2 , N ) 1 + v3 vθ2 0 R ∫ t∫ µ(v, θ)θφ2x ≤ϵ + C(ϵ). v3 0 R

I11 ≤ ϵ

It is worth to point out that we have used the assumption b ≥ 1 to bound the term I8 and the assumption b ≥ 7 to control the term I9 . Then collecting all the above estimates and choosing ϵ > 0 small enough, we can deduce (3.1). □ With the above lemma in hand, we can derive an estimate on the bound of the specific volume v(t, x) in terms of ∥θ∥∞ by employing Kanel′ technique (cf. [28]). Lemma 3.2. Assume that the conditions listed in Theorem 1.1 hold. Then   1   ≲ 1 + ∥θ∥ς∞1 , ∥v∥∞ ≲ 1 + ∥θ∥ς∞2 , v

(3.2)

∞

where ς1 :=

1 2ℓ1 ,

ς2 :=

1 2ℓ2 +1 .

(3.3)

Proof . To begin with, we set ∫ Ψ (v) := 1

v

√

Φ(z) h(z)dz. z

One can easily conclude from (1.5) and (1.23) that ( v )−ℓ1 ( v )ℓ2 v (t, x)) , h (v(t, x)) ≳ v −ℓ1 + v ℓ2 ≳ + := v˜−ℓ1 + v˜ ℓ2 ≳ h (˜ V V and  ℓ1 1 1 2 +ℓ2  ∥v∥∞ +  sup |Ψ (˜ v (t, x))|. v ≲ 1 + (t,x)∈[0,T ]×R ∞ vx Therefore we can get from Lemma 2.2, (2.6), (3.1), (3.4), and the fact that ˜ = φvx − ˜ v ⏐ ) ⏐∫ x ⏐ ∫ ⏐⏐( √ ⏐ ⏐ ⏐ Φ(˜ v ) ⏐ ⏐ |Ψ (˜ v )| = ⏐⏐ Ψ (˜ v (t, y))y dy ⏐⏐ ≤ h (˜ v ) v˜x (t, x)⏐ dx ⏐ ⏐ ⏐ v ˜ −∞ R ⏐ ⏐ ( ) √   ( ) ∫ ⏐ ⏐  −α  Φ(˜ v) v˜x ⏐ ⏐  ≲ h(v)˜ vx (t, x)⏐ dx ≲  θ µ(v, θ) (t) ⏐  ⏐ v˜ v˜ R⏐     ℓ1 +1 ℓ2 +2 1  µ(v, θ)φx  1 2 2 2    ≲ 1+ (t) ≲ 1 +  + ∥v∥ + ∥θ∥∞ . ∞   v v ∞

φVx vV

(3.4)

(3.5) that

(3.6)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

20

From (1.22), (3.5), and (3.6), we obtain (3.2) immediately. □ A direct corollary follows from Lemmas 3.1 and 3.2. Corollary 3.3. Assume that the conditions listed in Theorem 1.1 hold. Then for any 0 ≤ t ≤ T , we have   ∫ t∫  µ(v, θ)φx 2 µ(v, θ)θφ2x   + ≲ 1 + ∥θ∥∞ . (3.7) (t)   v v3 0 R ∫ t∫ ∫ t∫ 2 Before concluding this section, we will derive the estimate on the terms ψxx and ψx4 , which plays a key role in removing the condition (1.9) as required in [12].

0

R

0

R

Lemma 3.4. Under the assumptions listed in Theorem 1.1, we have for any 0 ≤ t ≤ T that ∫ t 2 2 ∥ψx (t)∥ + ∥ψxx (τ )∥ dτ ≲ 1 + ∥θ∥3∞ ,

(3.8)

0

and

∫ t∫ 0

9

2 ψx4 ≲ 1 + ∥θ∥∞ .

(3.9)

R

Proof . Multiplying (2.1)2 by −ψxx , we can obtain ( 2) 2 ψx µ(v, θ)ψxx ∂t + − (ψt ψx )x 2 v µ(v, θ)ψxx Uxx = (p(v, θ) − P (V, Θ))x ψxx + g (V, Θ)x ψxx − v µ(v, θ) (ψx φx ψxx + ψx Vx ψxx + Ux φx ψxx + Vx Ux ψxx ) µv (v, θ)φx ψx ψxx µv (v, θ)Vx ψx ψxx + − − v2 v v µv (v, θ)φx Ux ψxx µv (v, θ)Vx Ux ψxx αµ(v, θ)χx ψx ψxx αµ(v, θ)Θx ψx ψxx − − − − v v vθ vθ αµ(v, θ)χx Ux ψxx αµ(v, θ)Θx Ux ψxx − − . vθ vθ Integrating the above identity over (0, t) × R, we have ∫ ∫ t∫ 14 2 ∑ µ(v, θ)ψxx 2 |Ij |, (3.10) ψx dx + ≲1+ v R 0 R j=12 here ⎧ ⎪ ⎪ I12 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ I13 = ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ I14 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∫ t∫ (p(v, θ) − P (V, Θ))x ψxx , ∫0 t ∫R [ µ(v, θ)ψxx Uxx g (V, Θ)x ψxx − v 0 R ] +Ux φx ψxx +Vx Ux ψxx ) + µ(v,θ)(ψx φx ψxx +ψx Vx ψvxx , 2 ∫ t∫ ( µv (v, θ)φx ψx ψxx µv (v, θ)Vx ψx ψxx µv (v, θ)φx Ux ψxx µv (v, θ)Vx Ux ψxx − + + + v v v v ) 0 R x ψx ψxx + αµ(v,θ)χ + vθ

αµ(v,θ)Θx ψx ψxx vθ

+

αµ(v,θ)χx Ux ψxx vθ

+

αµ(v,θ)Θx Ux ψxx vθ

.

We have from (1.3) that (p(v, θ) − P (V, Θ))x =

Rχx RΘx φ RΘVx φ (v + V ) Rχφx RχVx RΘφx − + − − − 2 2 2 2 v vV v V v v v2 ( 2 ) 4 3 4 + aθ χx + aΘx χ θ + θΘ + Θ 2 . 3 3

(3.11)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

21

Then we exploit (1.22), Lemmas 2.2, 2.4, (2.6), (3.7), and (3.11) to find that ∫ t∫ 2 µ(v, θ)ψxx I12 ≤ ϵ v 0 R ) ∫ t∫ ( 2 Θx2 φ2 θ2 φ2x χ2 Vx2 (1 + v 2 )φ2 Vx2 χx 2 6 2 2 4 2 2 + + + + + a θ χ + a (1 + θ )χ Θ + C (ϵ) x x v2 v2 v4 v4 v4 0 R [ ( {∫ ) ] ∫ ∫ t∫ t 2 µ(v, θ)ψxx κ(v, θ)Θχ2x θ2 θ vθ8 µ(v, θ)θφ2x ≤ϵ · + C (ϵ) + + v vθ2 vκ(v, θ) κ(v, θ) v3 vµ(v, θ) 0 R 0 R ∫ t( ) } 2 2 ∥Vx ∥L∞ + ∥Θx ∥L∞ dτ + Ξ (m1 , m2 , N ) 0 ) (  2 ∫ t∫ 2 1 µ(v, θ)ψxx 2  ≤ϵ (3.12) + C (ϵ) 1 +   v  + ∥v∥∞ + ∥θ∥∞ , v 0 R ∞ ∫ t∫ 2 µ(v, θ)ψxx I13 ≤ ϵ v 0 R ) ∫ t∫ ( 2 2 µ(v, θ)Uxx µ(v, θ)ψx2 φ2x ψx2 Vx2 µ(v, θ)Ux2 φ2x µ(v, θ)Vx2 Ux2 v |g(V, Θ)x | + + + + + + C (ϵ) µ(v, θ) v v3 v3 v3 v3 0 R [ ∫ t∫ ∫ ∫ t t 2 3 1 µ(v, θ)ψxx −4 −9 ≤ϵ + C (ϵ) δ 2 (1 + τ ) 3 dτ + Ξ (m1 , m2 , N )δ 2 (1 + τ ) 4 dτ v 0 R 0 0     ∫ t∫ ∫ t ∫ t 2  µ(v, θ)φx 2  µ(v, θ)φx 2 µ(v, θ)Θψ θ 2 x 4    dτ  · 2+ + ∥Ux ∥L∞  ∥ψx ∥ ∥ψxx ∥   dτ + δ  v vθ v v 0 R 0 0 ] ∫ t 2 2 + Ξ (m1 , m2 , N ) ∥Vx ∥L∞ ∥Ux ∥ dτ 0 (  4 ) ∫ t∫ ∫ t 2  µ(v, θ)ψxx µ(v, θ)φ x 2  dτ ≤ 2ϵ + C (ϵ) 1 + ∥ψx ∥    v v 0 R 0 ∫ t∫ ( ) 2 µ(v, θ)ψxx 3 + C (ϵ) 1 + ∥θ∥∞ , (3.13) ≤ 2ϵ v 0 R and ∫ t∫ I14 ≤ ϵ 0

R

2 µ(v, θ)ψxx + C (ϵ) v 2

2

0

2

2

2

|µv (v, θ)| φ2x ψx2 |µv (v, θ)| Vx2 ψx2 |µv (v, θ)| φ2x Ux2 + + vµ(v, θ) vµ(v, θ) vµ(v, θ)

R µ(v, θ)χ2x ψx2 vθ2

) α |µv (v, θ)| α2 µ(v, θ)ψx2 Θx2 α2 µ(v, θ)χ2x Ux2 α2 µ(v, θ)Ux2 Θx2 + + + + vµ(v, θ) vθ2 vθ2 vθ2 [∫ ]   ∫ t∫ t 2  µ(v, θ)φx 2 ( 4 ) µ(v, θ)ψxx 2   ≤ϵ + C (ϵ) ∥ψx ∥ ∥ψxx ∥   dτ + δ + α Ξ (m1 , m2 , N ) v v 0 R 0 [ ] ∫ t∫ ( ) ∫ t ∫ µ(v, θ)Θψ 2 2 µ(v, θ)ψxx vθ 2 x ≤ 2ϵ + C (ϵ) 1 + 1 + ∥θ∥∞ · v vθ µ(v, θ) 0 R 0 R ∫ t∫ ( ) 2 µ(v, θ)ψxx 3 ≤ 2ϵ + C (ϵ) 1 + ∥θ∥∞ . (3.14) v 0 R +

Vx2 Ux2

∫ t∫ (

Inserting (3.12)–(3.14) into (3.10) and choosing ϵ > 0 small enough, we can obtain (3.8). Moreover, we employ (3.8), Gagliardo–Nirenberg, Sobolev and H¨older inequality to see that ∫ t∫ ∫ t ∫ t 3 2 2 ψx4 ≲ ∥ψx ∥L∞ ∥ψx ∥ dτ ≲ ∥ψx ∥ ∥ψxx ∥ dτ 0 R 0 ∫0 t 9 2 2 . ≲ ∥ψx ∥ ∥ψxx ∥ dτ ≲ 1 + ∥θ∥∞ 0

This completes the proof of our lemma.

□

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

22

4. Pointwise bounds for the absolute temperature In this section, we will obtain a uniform-in-time upper bound and a local-in-time lower bound for the absolute temperature θ. Motivated by [12,17], we set ) ∫ t∫ ( 1 b + θ χ2t (τ, x), X(t) : = 1 + ∥θ∥ς∞2 0 R ) ∫ ( 1 2b + θ (t, x) χ2x (t, x), Y (t) := 2 1 + ∥θ∥2ς R ∞ } { Y (t) := sup Y (τ ) , 0≤τ ≤t ∫ 2 Z(t) := sup ψxx (τ, x)dx, (4.1) 0≤τ ≤t

R

and then try to deduce certain estimates among them by employing the special structure of system (2.1)–(2.2). Lemma 4.1. Assume that the conditions listed in Theorem 1.1 hold, then we can get that 1

∥θ∥∞ ≲ 1 + Y (T ) 2b+3−2ς2 , 1 2

sup ∥ψx (τ )∥2 ≲ 1 + Z(t) ,

(4.2) 3 8

∥ψx ∥∞ ≲ 1 + Z(T ) .

(4.3)

τ ∈(0,t)

Proof . Without loss of generality, we can assume that x ∈ [−k − 1, k + 1] (k ∈ Z). Noticing from (2.3) and (2.18) that Φ(θ)(t, ·) ∈ L1 (R), we can employ Jensen’s inequality to find ν(t) ∈ [−k − 1, k + 1] such that θ(t, ν(t)) ∼ 1. Then ∫ x 2b+3 2b+2 (θ(t, x) − Θ(t, x)) = (θ(t, ν(t)) − Θ(t, ν(t)))2b+3 + (2b + 3) (θ − Θ) χx (t, y)dy ξ(t) 2b+3

( ( )) ] 12 [∫ ( ) ] 21 θ 1 ς 2b 2 1+Φ dx (1 + ∥θ∥∞2 ) + θ χ (t, x)dx x 2 Θ 1 + ∥θ∥2ς −k−1 R ∞

[∫

2 ≲ 1 + ∥ (θ − Θ) (t)∥L∞ 2b+3

k+1

2b+3

1

2 2 ≲ 1 + ∥ (θ − Θ) (t)∥L∞ Y 2 (t) + ∥ (θ − Θ) (t)∥L∞

+ς2

1

Y 2 (t),

from which we can deduce (4.2) by using Cauchy inequality and the fact that χ(t, x) ∈ C([0, t1 ]; H 3 (R)). Estimates (4.3) follow directly by applying Gagliardo–Nirenberg and Sobolev inequalities. □ With the above preparations in hand, our next result is to show that X(T ) and Y (T ) can be controlled by Z(T ). Lemma 4.2. Under the assumptions listed in Theorem 1.1, we have X(T ) + Y (T ) ≲ 1 + Z(T )λ1 ,

(4.4)

where

3(2b + 3 − 2ς2 ) . 8(2b + 1 − 2ς1 − 2ς2 ) It is easy to verify that 0 < λ1 < 1 when the assumption (1.22) holds. λ1 =

Proof . As in [3,17,50], we set ∫ K(v, θ) = 0

θ

κ(v, ξ) κ1 θ κ2 θb+1 dξ = + . v v b+1

(4.5)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

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Then it is easy to verify that Kt (v, θ) = Kv (v, θ)ψx + Kθ (v, θ)χt + Kv (v, θ)Ux + Kθ (v, θ)Θt , Kx (v, θ) = Kv (v, θ)φx + Kθ (v, θ)χx + Kv (v, θ)Vx + Kθ (v, θ)Θx , Kxt (v, θ) = (Kθ (v, θ)χx )t + [Kvv (v, θ)(ψx + Ux ) + Kvθ (v, θ) (χt + Θt )] φx + Kv (v, θ)ψxx + [Kvv (v, θ)(ψx + Ux ) + Kvθ (v, θ) (χt + Θt )] Vx + Kv (v, θ)Uxx + [Kvθ (v, θ)(ψx + Ux ) + Kθθ (v, θ)(χt + Θt )] Θx + Kθ (v, θ)Θxt , θ , v2

|Kv (v, θ)| ≲

|Kvv (v, θ)| ≲

θ , v3

|Kθ (v, θ)| ≲

1 + θb , v

|Kvθ (v, θ)| ≲

1 , v2

|Kθθ (v, θ)| ≲ θb−1 .(4.6)

Hereafter we abbreviate the terms K(v, θ), p(v, θ), e(v, θ), p(V, Θ), e(V, Θ), µ(v, θ), κ(v, θ) by K, p, e, P, E, µ, κ for brevity. Then multiplying (2.1)3 by Kt and integrating the result identity over (0, t)×R, we arrive at ∫ t∫ ∫ t∫ ∫ t∫ eθ Kθ χ2t + (Kθ χx ) (Kθ χx )t − Kθ Kvθ Ux χ2x 0

0

R

∫ t∫ = 0

(

0

R

R

Kθ Kθθ χt Θx2 + K2θ χt Θxx + Kθ Kvθ χt Θx Vx

)

R

∫ t∫ {

) } ( eθ ΘPΘ Ux Kθ χt Kθ Kθθ χ2x Θt + Kθ Kvθ χt φx Θx − Kθ Kvθ χx χt Vx − θpθ − EΘ 0 R ∫ t∫ ∫ t∫ Kθ Kvθ χ2x ψx − Kθ Kvθ χt φx χx + +

0

0

R

R

∫ t∫ (λϕz − eθ r(v, θ)) Kθ χt −

+ 0

:=

∫ t∫

21 ∑

θpθ Kθ ψx χt + 0

R

∫ t∫

R

0

R

µu2x Kθ χt v

Ik .

(4.7)

k=15

First of all, we can conclude from (2.34) that ( ) ∫ t∫ ∫ t∫ ( ) 1 2 3 b e θ Kθ χ t ≳ 1 + avθ + θ χ2t ≳ X(t), v 0 R 0 R

(4.8)

and ∫ t∫ 0

∫ ∫ 1 1 2 2 (Kθ χx ) (Kθ χx )t = (Kθ χx ) (t, x)dx − (Kθ χx ) (0, x)dx 2 R 2 R R ) ) ∫ ( ∫ ( 1 1 2b 2 2b ≳ +θ χx dx − 1 ≳ χ2x dx − 1 2ς2 + θ 2 v 1 + ∥θ∥∞ R R = Y (t) − 1.

(4.9)

We now turn to control Ik (k = 15, . . . , 21) term by term. To begin with, Lemma 2.2, (2.5), (2.6) tell us that ] (1 ) )2 ( ) ∫ t ∫ [( 2 2b 2 b 2 2 2 + θ |Θ | 1 1 θ V Θ xx 2 ς2 4 x x I15 ≤ ϵX(T ) + C(ϵ) (1 + ∥θ∥∞ ) + θb θb−2 |Θx | + v + + 2 v 1 + θb v3 v 1 + θb 0 R ∫ t( ) 4 2 2 2 ≤ ϵX(T ) + C(ϵ)Ξ (m1 , m2 , N ) ∥Θx ∥L4 + ∥Θxx ∥ + ∥Vx ∥L∞ ∥Θx ∥ dτ 0

1

≤ ϵX(T ) + C(ϵ)δ 2 Ξ (m1 , m2 , N ) ≤ ϵX(T ) + C(ϵ).

(4.10)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

24

In view of (2.18), (2.34), and Taylor’s formula, for 0 < ω < 1, we can obtain that ⏐ ⏐ ⏐ θpθ ( ) (( ) ) ΘPΘ ⏐⏐ θ|φ| ⏐ ≲ − + a 1 + θ3 |χ| + |χ| + a 1 + θ2 |χ| + θ3 |φ| , ⏐ eθ ⏐ EΘ v

(4.11)

and ∫

(

2

φ +χ

R

2

)

( ) ] ∫ [ ( ) θ v 2 2 (ωV + (1 − ω)v) + Φ (ωΘ + (1 − ω)θ) dx ≲ 1 + N 2 . dx = 2 Φ V Θ R

(4.12)

Then it follows from Lemma 2.2, (2.5), (2.6), (2.18), (3.7), (4.11), and (4.12) that ∫ t∫

( b+1 ) θ vθ2b+1 κΘχ2x + 2 1 + vθb 1 + vθb 0 R vθ ) )( 2 2 ∫ t ∫ [( φx Θx + χ2x Vx2 1 θ2b ς + C(ϵ) (1 + ∥θ∥∞2 ) + v6 v4 1 + θb 0 R ( )( ) ] ) 2 1 + v 2 θ6 θ2 φ2 ( 1 6 6 2 2b + + 1+θ χ +θ φ +θ Ux2 1 + θb v2 v2 ) ( ∫ t∫ [ Θx2 µθφ2x 1 ς2 2b 2 + θ ≲ ϵX(T ) + δ Ξ (m1 , m2 , N ) + C(ϵ) (1 + ∥θ∥∞ ) v3 v2 θ + θb+1 0 R ) ( ) ] ( ) ( ) 2 θ2 φ2 ( θ2b vθ2 Vx2 1 + v2 θ6 1 κΘχ2x 1 2b 6 6 2 + 4 + +θ + 1 + θ χ + θ φ Ux2 + vθ2 v6 v (1 + θb ) (1 + vθb ) 1 + θb v2 v2 ( ) ≤ ϵX(T ) + C(ϵ) 1 + δ 4 Ξ (m1 , m2 , N ) ≤ ϵX(T ) + C(ϵ). (4.13)

I16 ≤ ϵX(T ) + δ 2

On the other hand, we utilize (2.18), (3.2), and (4.3) to get that  2 )  3 ( ) ( ∫ t∫     κΘχ2x θ2 |ψx | 1 1 b   + θ ∥ψx ∥∞ I17 ≤ · + θ ≲ 1 +   2 v (1 + vθb ) v v ∞  v ∞ 0 R vθ )( ) ( ) 2+2ς2 ( )( 3 3 2+2ς2 ≲ 1 + ∥θ∥∞ 1 + Z(T ) 8 ≲ 1 + Y (T ) 2b+3−2ς2 1 + Z(T ) 8 ( ) 3(2b+3−2ς2 ) ≤ ϵY (T ) + C(ϵ) 1 + Z(T ) 8(2b+1−2ς1 −2ς2 ) .

(4.14)

For the term I18 , we can infer from (1.5), (2.18), (3.7), (4.6), H¨older and Sobolev inequality that I18

( ) ] ∫ t ∫ [⏐ ⏐ 1 θ2b v4 µ2 φ2x ⏐ κχx ⏐2 1 ≤ ϵX(T ) + C(ϵ) (1 + + 4 ⏐ ⏐ v 1 + θb v6 v (1 + v 2 θ2b ) µ2 v 2 0 R   ∫ t ( )     1  1 1  κχx 2  µφx 2 ς 2b  ≤ ϵX(T ) + C(ϵ) (1 + ∥θ∥∞2 )  + θ     dτ  1 + θb v2 (1 + v 2 θ2b ) µ2 ∞ 0 v L∞ v ς ∥θ∥∞2 )

(

1+ς ∥θ∥∞ 2

) (∫ t ∫ θ2 κ ⏐( κχ ) ⏐2 ) 21 (∫ t ∫ κΘχ2 ) 21 ⏐ ⏐ x x ⏐ ⏐ 2 v v vθ x 0 R 0 R

(

1+ς ∥θ∥∞ 2

( ( ) ⏐2 ) 21 ) ∫ t ∫ θ2 κ ⏐⏐( κθ ) κΘx ⏐⏐ x ⏐ − ⏐ v ⏐ v 0 R v x x   

≤ ϵX(T ) + C(ϵ) 1 +

≤ ϵX(T ) + C(ϵ) 1 +

1

J2

(4.15)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

25

On the other hand, on account of (1.1)3 , (1.22), Lemma 2.2, (2.6), (3.2), Lemma 3.4, and (4.11), one has ] )[ ( )2 ∫ t∫ ( 2 θ µ2 Ux4 θpθ ΘPΘ µ2 ψx4 2 2 b+2 2 2 2 2 2 2 2 2 2 +ϕ z J ≲ +θ eθ χt + θ pθ ψx + eθ − Ux + eθ r (V, Θ) + 2 + v eθ EΘ v v2 0 R ) ⏐ ) ⏐( ∫ t∫ ( 2 ⏐ κΘx ⏐2 θ ⏐ + + θb+2 ⏐⏐ ⏐ v v 0 R x    Ja ( 2 ) ( ) ∫ t∫ ( ) 1 θ 1 ς b+2 2 2 6 b ≲ J a + (1 + ∥θ∥∞2 ) + θ 1 + a v θ + θ χ2t b v 1 + ∥θ∥ς∞2 0 R 1+θ ) ( ( 2 )( ) ∫ t∫ [ ( ) µΘψx2 θ4 vθ θ 1 2 6 b+2 + + a θ + θb+4 + + θ 1 + a2 v 2 θ6 r2 (V, Θ) 2 vθ v v h(v) v 0 R ) ( ) ] ( 2 ( ) ( ) (( ) ) 2 2 θ|φ| θ + θb+2 1 + a2 v 2 θ6 + a 1 + θ3 |χ| + |χ| + a 1 + θ2 |χ| + θ3 |φ| Ux + v v ( 2  2( 2 ) ∫ t ( )  )  θ   µ θ 2 2 4 b+2  b+2  ∥ψ ∥ ∥ψ ∥ + ∥U ∥ dτ + + θ + θ θβ  + x L∞ x x L4  v    v2 v ∞ ( ) ∞ 0 ( ) 4 8+3ς b+11 b+ 13 +2(ℓ1 +2)ς1 ≲ J a + 1 + ∥θ∥∞ 2 X(T ) + 1 + ∥θ∥∞ + δ 4 + δ 3 + δ Ξ (m1 , m2 , N ) + ∥θ∥∞ 2 b+β+2

+ ∥θ∥∞

) ( b+11 b+β+2 8+3ς , ≲ J a + 1 + 1 + ∥θ∥∞ 2 X(T ) + ∥θ∥∞ + ∥θ∥∞

(4.16)

and ∫ t∫ (

) ( 2b 2 2 θ2 θ φx Θx θ2b Vx2 Θx2 b+2 J ≲ +θ + + θ2b−2 Θx4 + θ2b−2 χ2x Θx2 v v2 v2 0 R ) 2 κ2 Θxx κ2 φ2x Θx2 κ2 Vx2 Θx2 + + + v2 v4 v4  2b ( 2 [ ( 2 )] ) ∫ t ∫ t∫ 2b+1 2   + vθ3b+1 κ2 θ θ µθφx θ 2 b+2 3 θ b+2  4 + + θ + δ + θ ∥Vx ∥L∞ dτ ≲δ   3 2 v h(v) h(v)vθ v v v 0 R ∞ 0 [ ∫ t ∫ t( ) ] 2 4 2 + Ξ (m1 , m2 , N ) δ 4 + δ 3 ∥Vx ∥L∞ dτ + ∥Θx ∥L4 + ∥Θxx ∥ dτ 0 0 ( ) 1 ≲ Ξ (m1 , m2 , N ) δ 4 + δ 3 + δ + δ 2 ≲ 1. (4.17) a

The combination of (4.16) and (4.17) gives ( ) 8+3ς b+11 b+β+2 J ≲ 1 + 1 + ∥θ∥∞ 2 X(T ) + ∥θ∥∞ + ∥θ∥∞ .

(4.18)

Inserting (4.18) into (4.15) and making use of the assumption b ≥ 7, 0 ≤ β < 3b + 2 − 6ℓ2 , we can deduce that I18 ≤ ϵ (X(T ) + Y (T )) + C (ϵ) . (4.19) As for the term I19 , it follows from Lemma 2.2, (2.17), (4.6), and the fact that 0 ≤ β < b + 3 − 3ℓ2 that ) ( )] ∫ t∫ [ 2 2 ( 2 ) 1 ϕ z 1 |r (V, Θ)| ( ς2 2b 2 2 6 2b I19 ≤ ϵX(T ) + C(ϵ) (1 + ∥θ∥∞ ) +θ + 1+a v θ +θ b v2 1 + θb v2 0 R 1+θ (  [ ) ] ∫ t 1 ϕ  ς2  2 2b   ≤ ϵX(T ) + C(ϵ) (1 + ∥θ∥∞ )  2 + θ + Ξ (m1 , m2 , N ) ∥r (V, Θ)∥ dτ v 1 + θb ∞ 0 ( ) ( ) b+β+ς2 b+β+ς2 ≤ ϵX(T ) + C (ϵ) 1 + ∥θ∥∞ ≤ ϵX(T ) + C (ϵ) 1 + Y (T ) 2b+3−2ς2 ≤ ϵ (X(T ) + Y (T )) + C (ϵ) .

(4.20)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

26

Furthermore, one can conclude from (1.5), (2.18), (2.34), (4.6), and the fact that b > 6 + 3ς2 that )( ) ∫ t∫ ( 1 1 + aθ3 + θb |ψx χt | I20 ≲ θ v v 0 R )( ) ( ∫ t∫ µΘψx2 θ2 1 vθ ς 2 8 2b + a θ + θ ≤ ϵX(T ) + C(ϵ) (1 + ∥θ∥∞2 ) 2 2 vθ v v h(v)(1 + θb ) 0 ( )R b+β+9 ≤ ϵX(T ) + C(ϵ) 1 + ∥θ∥∞ ≤ ϵ (X(T ) + Y (T )) + C (ϵ) . (4.21) It suffices to control the term I21 . For this purpose, we utilize (1.5), Lemmas 2.2, 3.4, and (4.6) to deduce that ( ) ∫ t∫ ( 4 ) h2 (v) 1 ς 2b I21 ≤ ϵX(T ) + C(ϵ) (1 + ∥θ∥∞2 ) + θ ψx + Ux4 b 2 2 (1 + θ )v v 0 R2 ( ) ∫ t ∫   ( 4 ) h (v) 1 ς 2b  ≤ ϵX(T ) + C(ϵ) (1 + ∥θ∥∞2 )  ψx + Ux4 + θ  (1 + θb )v 2 v 2  ∞ 0 )R ( b+ 9 +(2ℓ1 +2)ς1 +ς2 2 ≤ ϵX(T ) + C(ϵ) 1 + δΞ (m1 , m2 , N ) + ∥θ∥∞ ≤ ϵ (X(T ) + Y (T )) + C (ϵ) . (4.22) Plugging (4.8)–(4.22) into (4.7) and choosing ϵ > 0 small enough yields (4.4). □ Our next lemma is to show that Z(T ) can be bounded by X(T ) and Y (T ). Lemma 4.3. Under the assumptions listed in Theorem 1.1, we have Z(T ) ≲ 1 + Y (T ) + Z(T )λ2 + Z(T )λ3 , where λ2 =

3(2b + 3 − 2ς2 ) , 4 (2b + 2 − 2ς2 − (ℓ1 + 1) ς1 )

λ2 =

(4.23)

3(2b + 3 − 2ς2 )2 . 8(2b + 1 − 2ς1 − 2ς2 )(2b + 3 − 4ς2 )

It is easy to verify that 0 < λ2 , λ3 < 1 when the assumption (1.22) holds. Proof . Differentiating (2.1)2 with respect to t, multiplying it by ψt , and integrating the above identity over (0, t) × R, we arrive at ∫ t∫ ∫ 2 µψtx ψt2 dx + 2 v ) ∫ t∫ ( 2 ∫R 2 ∫0 t ∫R µψx ψtx ψ0t µv ψx2 ψtx = dx + (p − P )t ψtx + − v2 v R 2  0 R   0 R   I22

∫ t∫ ( + 0

R

I24

∫ t∫ ( − 0

I23

) µψtx Ux2 + 2µψtx ψx Ux µv Ux2 ψtx + µψtx Utx + 2µv ψx Ux ψtx − + g (V, Θ)t ψtx v2 v  

R

) αµUx θt ψtx αµψx θt ψtx + . vθ vθ  

(4.24)

I25

Now we turn to bound the term Ik (k = 22, 23, 24, 25). First of all, one can conclude from (1.3) that ) ( ) ( 2 ) ( ( ) 1 + χ2 ψx2 + χ2 Ux2 + Ux2 φ2 (1 + v 2 ) 1 φ 2 2 2 6 2 2 2 4 |(p − P )t | ≲ + a θ χt +|Θt | +a χ 1+θ + . (4.25) v2 v2 v4

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27

Then (1.5), Lemma 2.2, (2.6), (2.18), (4.12), and (4.25) shows that ∫ t∫ 2 µψtx I22 ≤ ϵ v 0 R ] ( ) 2 ∫ t∫ [ 2 2 2 2 2 ( ) ( ( ) ) 1 + χ ψ + χ U + U φ x x x 1 + θ6 χ2t + φ2 + 1 + θ4 χ2 Θt2 + + C (ϵ) + Ux2 φ2 v2 0 R [ ∫ t∫ ∫ t ( ( ) ) 2 µψtx ς2 2 2 4 2 ≤ϵ + C (ϵ) (1 + ∥θ∥∞ ) X(T ) + ∥Θt ∥L∞ ∥φ∥ + 1 + ∥θ∥L∞ ∥χ∥ dτ v 0 R 0  2 ∫ t ] ∫ t ∫ t∫ ( ) 3 2 1 µΘψx θ + θ 2 2 2 2 2  ∥U ∥ ∥χ∥ + ∥φ∥ dτ + ∥U ∥ ∥χ∥ dτ · + + ∞ ∞ x x L L vθ vh(v)  v ∞ 0 0 0 R ∫ t∫ [ ] 2 µψtx ς 3 ≤ϵ + C (ϵ) 1 + δΞ (m1 , m2 , N ) + (1 + ∥θ∥∞2 ) X(T ) + ∥θ∥∞ v 0 R ∫ t∫ 2 µψtx ς + C (ϵ) [1 + (1 + ∥θ∥∞2 ) X(T ) + Y (T )] . (4.26) ≤ϵ v 0 R On the other hand, according to (1.5), (1.22), Lemma 3.4, and (4.2), we have ) ∫ t∫ ∫ t∫ ( 4 2 2 µψtx µψx |h′ (v)| ψx4 I23 ≤ ϵ + C (ϵ) + v v3 vh(v) 0 R 0 R (      )∫ t∫ ∫ t∫ 2  h(v)   1 2  h(v) 2 µψtx      ≤ϵ + C (ϵ)  ψx4  v  v +  v  v 0 R 0 R ∞ ∞ ∞ ∫ t∫ ≤ϵ 0

R

∫ t∫ ≤ϵ 0

R

( )( ) 2 9 µψtx (ℓ +3)ς1 2(ℓ +1)ς1 2 + C (ϵ) 1 + ∥θ∥∞1 + ∥θ∥∞ 1 1 + ∥θ∥∞ v 2 µψtx + C (ϵ) (1 + Y (T )) . v

(4.27)

Moreover, one can infer from Lemma 2.2, (2.6), (2.18), and (2.33) that ∫ t∫ 2 µψtx I24 + I25 ≤ ϵ v 0 R ∫ t∫ ( 2 2 2 |µv | (Ux4 + ψx2 Ux2 ) µUxt v |g(V, Θ)t | + C (ϵ) + + vµ v µ 0 R ( )) α2 µθt2 ψx2 + Ux2 µUx4 + µψx2 Ux2 + + v3 vθ2 ∫ t∫ ≤ϵ 0

R

2 ( ) µψtx + C (ϵ) δ + δ 4 + α2 Ξ (m1 , m2 , N ) ≤ ϵ v

∫ t∫ 0

R

2 µψtx + C (ϵ) . v

Inserting (4.26)–(4.28) into (4.24) and choosing ϵ > 0 small enough, we arrive at ∫ t∫ 2 µψtx 2 ς ∥ψt (t)∥ + ≲ 1 + (1 + ∥θ∥∞2 ) X(T ) + Y (T ). v 0 R Meanwhile, we take advantage of (2.1)2 to get [ ] v µx ux µux vx µUxx ψxx = ψt + (p − P )x + g(V, Θ)x − + − . µ v v2 v

(4.28)

(4.29)

(4.30)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

28

Then it follows from (1.5), Lemma 2.2, (2.6), (3.7), (3.11), (4.12), and (4.29) that ∫ ∫ 2[ ⏐ µ u ⏐2 ⏐ µu v ⏐2 µ2 U 2 ] v ⏐ x x⏐ ⏐ x x⏐ 2 2 xx 2 2 ψt + |(p − P )x | + |(g(V, Θ))x | + ⏐ dx ψxx dx ≲ ⏐ +⏐ 2 ⏐ + 2 v v v2 R R µ ∫ 2 ) |h′ (v)| ( 2 2 ς ≲ 1 + (1 + ∥θ∥∞2 ) X(T ) + Y (T ) + φx ψx + ψx2 Vx2 + φ2x Ux2 + Vx2 Ux2 2 R h (v) ∫ [( ) 2 ( 2 ) 2 ( ) ( ) ] 2 6 2 + 1 + a θ χx + Vx + Θx φ + 1 + θ2 φ2x + χ2 Vx2 + a2 χ2 1 + θ4 Θx2 dx R ∫ ∫ ψx2 χ2x + Θx2 ψx2 + Ux2 χ2x + Ux2 Θx2 ψx2 φ2x + ψx2 Vx2 + φ2x Ux2 + Vx2 Ux2 2 +α dx + dx 2 θ v2 R R ( ) 3(2b+3−2ς2 ) ς 2ς ≲ 1 + (1 + ∥θ∥∞2 ) X(T ) + 1 + ∥θ∥∞2 Y (T ) + Z(T ) 4(2b+2−2ς2 ) ( ) 3(2b+3−2ς2 ) 2ς (4.31) ≲ 1 + 1 + ∥θ∥∞2 (X(T ) + Y (T )) + Z(T ) 4(2b+2−2ς2 ) . Then the combination of (4.1), (4.4), and (4.31) gives (4.23). □ Combining Lemmas 4.2–4.3, we can deduce that X(T ) + Y (T ) + Z(T ) ≲ 1. Then the desired upper bound on θ(t, x) follows from (4.2) immediately. As a consequence, we can conclude from Lemmas 2.1–4.3 that Lemma 4.4. Under the assumptions listed in Theorem 1.1, there exist positive constants C1 and C2 , which depend only on Π0 and V0 , such that θ (t, x) ≤ C1 ,

∀ (t, x) ∈ [0, T ] × R.

C2−1 ≤ v (t, x) ≤ C2 ,

(4.32)

∀ (t, x) ∈ [0, T ] × R.

(4.33)

Moreover, we have ∫ t (√ )2   ∥(φ, ψ, χ, z, φx , ψt , ψx , χx , ψxx ) (t)∥ +  θφx , ψx , χx , zx , χt , ψxt , ψxx  (τ ) dτ ≲ 1, 0 ∫ t∫ ψx4 ≲ 1, ∥ψx ∥∞ ≲ 1. 2

0

(4.34)

R

∫ t∫ Before concluding this section, let us deduce nice bounds on 0

later on. In fact, we have the following lemma.

2

χ2xx and ∥zx (t)∥ , which will be used

R

Lemma 4.5. Under the assumptions listed in Theorem 1.1, we have for 0 ≤ t ≤ T that ∫ t 2 2 ∥χx (t)∥ + ∥χxx (τ )∥ dτ ≲ 1,

(4.35)

0

and 2

∫

∥zx (t)∥ +

t

2

∥zxx (τ )∥ dτ ≲ 1.

(4.36)

0

Proof . Integrating (2.36) over (0, t) × R and making use of (1.3), Lemma 2.2, (2.6), Lemma 2.4, (2.40), (4.11), Lemma 4.4, and Sobolev inequality, we arrive at ∫ t∫ κχ2xx ∥χx (t)∥2 + 0 R veθ ( ∫ t∫ ∫ t∫ [ 2 2 κχxx µΘψx2 v 2 θ3 p2θ veθ (φ2 + χ2 )Ux2 + veθ |r(V, Θ)| ≤ϵ + C(ϵ) 1 + · + vθ µκeθ κ 0 R veθ 0 R

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

2 µ2 (ψx4 + Ux4 ) + κ2θ (χ4x + Θx4 ) κΘxx vϕ2 z 2 + + + + φ2x Θx2 + Vx2 Θx2 + φ2x χ2x + χ2x Vx2 veθ κ veθ κeθ ( ) ∫ t∫ ∫ t κχ2xx ≤ϵ + C(ϵ) 1 + δΞ (m1 , m2 , N ) + ∥χx ∥ ∥χxx ∥ dτ 0 R veθ 0 ∫ t∫ κχ2xx ≤ 2ϵ + C(ϵ). 0 R veθ

29

])

(4.37)

Choosing ϵ > 0 small enough, we can deduce (4.35). On the other hand, we multiply (2.1)5 by zxx to conclude ( 2) 2dvx zx zxx zx dz 2 − (zt zx )x = + ϕzzxx . ∂t + xx 2 v2 v3 Integrating the above identity over (0, t) × R and utilizing Lemma 2.2, (2.17), Lemma 4.4, and Sobolev inequality, we can get that [ ] ∫ ∫ t∫ ∫ t∫ ∫ t∫ 2 2 ( 2 2 ) dzxx dzxx 2 2 2 zx dx + ≤ϵ + C (ϵ) 1 + vx zx + ϕ z 2 2 R 0 R v 0 R v 0 R ∫ t∫ ∫ ∫ t 2 (( 2 ) ) dzxx ≤ϵ + C (ϵ) Vx + φ2x zx2 + ϕz 2 2 v (0 R ) ∫ t ∫0 t ∫R 2 dzxx 4 + C (ϵ) 1 + δ + ≤ϵ ∥zx (τ )∥ ∥zxx (τ )∥ dτ 2 0 0 R v ∫ t∫ 2 dzxx + C (ϵ) . ≤ 2ϵ 2 0 R v Taking ϵ > 0 small enough, we can finish the proof of our lemma.

□

As a result of Lemmas 2.1–4.5, we can obtain the following corollary immediately. Corollary 4.6. Under the assumptions listed in Theorem 1.1, there exists a positive constant C3 , such that ] ∫ t [√ 2   2 2 ∥(φ, ψ, χ, z) (t)∥1 + (4.38)  θφx (τ ) + ∥(ψx , χx , zx ) (τ )∥1 dτ ≤ C32 . 0

The next lemma will give a local-in-time lower bound on the absolute temperature θ (t, x). For this purpose, we can deduce by repeating the method used in [3] that Lemma 4.7. Under the assumptions stated in Theorem 1.1, for each 0 ≤ s ≤ t ≤ T and x ∈ R, there exists a positive constant C4 , such that θ (t, x) ≥

C4 inf x∈R {θ(s, x)} . 1 + (t − s) inf x∈R {θ(s, x)}

(4.39)

5. Estimates of high-order derivatives In this section, to simplify the presentation, we introduce A ≲h B if A ≤ Ch B holds uniformly for some constant Ch , depending only on Π0 , V0 and H(C2 ) with C2 given in Lemma 4.4. The letter C(m2 ) will be employed to denote some positive constant which depends only on m2 , Π0 , V0 and H(C2 ). We note from (1.21) and (4.33) that sup

|(h(v(t, x)), h′ (v(t, x)), h′′ (v(t, x)), h′′′ (v(t, x)))| ≤ H(C2 ).

(5.1)

(t,x)∈[0,T ]×R

The next lemma is concerned with the second-order derivatives of (φ (t, x) , χ (t, x)) with respect to the space variable x.

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

30

Lemma 5.1. Under the assumptions listed in Theorem 1.1, for any 0 ≤ t ≤ T , we have ∫ t ∫ t 2 2 2 2 ∥(ψxx , χxx ) (t)∥ + ∥(ψxxx , χxxx ) (τ )∥ dτ ≲h C(m2 ) + ∥φxx (τ )∥ dτ + sup ∥φxx (t)∥ . 0

(5.2)

0≤t≤T

0

Proof . First of all, differentiating (2.1)2 with respect to x, and multiplying the resulting identity by ψxxx , one has ] ( ) [ 2 1 2 µψxxx µψxxx ( µux ) ψxxx + g(V, Θ)xx ψxxx . ψxx − (ψxt ψxx )x + = (p − P )xx ψxxx + − 2 v v v xx t Integrating the above identity over [0, t] × R and by using Lemma 2.2, (4.33), and Cauchy’s inequality, we arrive at ) ⏐ ∫ t∫ ∫ t∫ ( ( µu ) ⏐⏐2 ⏐ µψ xxx x 2 2 ⏐ . (5.3) ∥ψxx (t)∥2 + − ψxxx ≲1+ |(p − P )xx | + ⏐⏐ v v xx ⏐ 0 R 0 R Since (p − P )xx =

Rθxx RΘxx 2Rvx θx 2RVx Θx Rθvxx RΘVxx 2Rθvx2 2RΘVx2 − − + − + + − v V v2 V2 v2 V2 v3 V3 4 4 3 3 2 2 2 2 + 4aθ θx − 4aΘ Θx + aθ θxx − aΘ Θxx , 3 3

(5.4)

we have ( )( ) 2 2 2 |(p − P )xx | ≲ φ2xx + χ2xx + φ4x + χ4x + φ2x χ2x + φ2x + χ2x Vx2 + Θx2 + Vx4 + Θx4 + Vxx + Θxx + Vx2 Θx2 . (5.5) On the other hand, in light of Lemmas 4.4 and 4.7, one has ∫ t∫ φ2x ≤ C(m2 ). 0

(5.6)

R

Thus we have from Lemmas 2.2, 4.6, (5.5), (5.6), and Sobolev inequality that ∫ t∫ 2 |(p − P )xx | ∫0 t ∫R ( 2 ( )( ) ) 2 2 ≲ φxx + χ2xx + φ4x + χ4x + φ2x χ2x + φ2x + χ2x Vx2 + Θx2 + Vx4 + Θx4 + Vxx + Θxx + Vx2 Θx2 0 ∫R t ∫ t∫ ( ) ≲ 1+ ∥φx ∥3 ∥φxx ∥ + ∥χx ∥3 ∥χxx ∥ + ∥χx ∥∥χxx ∥∥φx ∥2 dτ ≲ C(m2 ) + φ2x . (5.7) 0

0

R

( ) ⏐⏐2 ∫ t ∫ ⏐⏐ To estimate the term 0 R ⏐ µψvxxx − µuv x xx ⏐ , we first make some estimate of θα . According to [12], for general smooth functions f (v), we have |(f (v)θα )x | ≲ |(f, f ′ )(v)||(vx , θx )|, [ ] 2 |(f (v)θα )xx | ≲ |(f, f ′ , f ′′ )(v)| |(vx , θxx )| + |(vx , θx )| , [ 2 3] |(f (v)θα )xxx | ≲ |(f, f ′ , f ′′ , f ′′′ )(v)| |(vxxx , θxxx )| + |(vxx , θxx )||(vx , θx )| + |(vx , θx )| . Taking f (v) =

h(v) v ,

we can combine the identity ( µu ) (µ) (µ) µ x = ux + 2 uxx + uxxx v xx v xx v x v

and the assumption (1.5) to conclude ⏐( µu ) ⏐ µ ⏐ ⏐ x 3 − ψxxx ⏐ ≲h |(vx , ux , θx )| + |vxx ∥ux | + |(uxx , θxx )∥(vx , ux , θx )| + |Uxxx |. ⏐ v xx v

(5.8)

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

From the above estimate and Lemma 2.2, we can derive ∫ t ∫ ⏐( ∫ t∫ ( ⏐2 ) ) µ ⏐ µux ⏐ 6 2 2 2 − ψxxx ⏐ ≲h 1 + |(vx , ux , θx )| + u2x vxx + |(uxx , θxx )| |(vx , ux , θx )| . ⏐ v xx v 0 R 0 R

31

(5.9)

We employ Lemmas 2.2, 4.4, Corollary 4.6, (5.6), and Sobolev inequality to get ∫ t∫ ( ∫ t∫ ) 6 6 6 6 6 6 6 |φx | + |ψx | + |χx | + |Vx | + |Ux | + |Θx | |(vx , ux , θx )| ≲ 0 0 R ∫R t ( ) ∥φx ∥4L∞ (R) ∥φx ∥2 + ∥ψx ∥4L∞ (R) ∥ψx ∥2 + ∥χx ∥4L∞ (R) ∥χx ∥2 dτ ≲ 1+ ∫0 t ( ) ∥φx ∥4 ∥φxx ∥2 + ∥ψx ∥4 ∥ψxx ∥2 + ∥χx ∥4 ∥χxx ∥2 dτ ≲ 1+ ∫0 t ∫ φ2xx , (5.10) ≲ 1+ 0

∫ t∫ 0

R

∫ t∫

2 u2x vxx =

) 2 2 ψx2 φ2xx + ψx2 Vxx + Ux2 φ2xx + Ux2 Vxx 0 R ∫ t∫ ∫ t∫ ≲ δ (1 + Ξ (m1 , m2 , N )) + φ2xx ≲ 1 + φ2xx ,

R

(

0

and ∫ t∫

2

0

R

(5.11)

R

2

|(uxx , θxx )| |(vx , ux , θx )| [ ] ∫ t∫ ( 2 )( 2 ) 2 2 2 2 2 2 2 2 ψxx + χxx + Uxx + Θxx φx + ψx + χx + Vx + Ux + Θx = 0 R ∫ t ∫ t∫ ∫ t∫ ( ) ( 2 ) ( 2 ) 2 2 2 2 4 2 2 ≲ ∥φx ∥L∞ + ∥χx ∥L∞ + ∥χxx ∥L∞ + ∥Uxx ∥L∞ dτ + δ ψxx + χxx + Uxx + ψxx + χ2xx 0 0 R ∫ t∫ ∫ t∫ ∫0 t ∫R ∫ t∫ ( ) ≲ϵ χ2xxx + C(ϵ) + φ2xx + φ2xx ≲ C(ϵ, m2 ) + ϵ χ2xxx + φ2xx . (5.12) 0

R

0

R

0

0

R

0

R

R

Plugging (5.10)–(5.12) into (5.9), we find that ∫ t∫ ∫ t∫ ∫ t ∫ ⏐( ⏐2 ) µ ⏐ ⏐ µux − ψxxx ⏐ ≲h C(ϵ, m2 ) + ϵ χ2xxx + φ2xx . ⏐ v xx v 0 R 0 R 0 R The combination of (5.3), (5.7), and (5.13) shows that ∫ t∫ ∫ t∫ ∫ t∫ 2 2 2 ∥ψxx (t)∥ + ψxxx ≲h C(ϵ, m2 ) + ϵ χxxx + φ2xx . 0

0

R

0

R

(5.13)

(5.14)

R

Next, we differentiate (2.1)2 with respect to x and multiply the result by χxxx to deduce 1 2 κχ2xxx (χxx )t − (χtx χxx )x + 2 ve ( ) ( 2)θ [ ( ( ) ) ] θpθ ψx µux κχxxx 1 κθx = χxxx − χxxx + − χxxx eθ veθ x veθ eθ v x x x ( ) [( ) ] λϕz θpθ ΘPΘ − χxxx + − Ux χxxx + r (V, Θ)x χxxx . eθ x eθ EΘ x Integrating the above identity over [0, t] × R, we employ Lemmas 2.2, 4.4, Corollary 4.6, and Cauchy’s inequality to get that ⏐( ⏐ ) ⏐ ) ⏐ ( ( ) ) ⏐2 ∫ t∫ ∫ t ∫ [⏐( ⏐ θpθ ψx ⏐2 ⏐ µu2x ⏐2 ⏐ κχxxx ⏐ 1 κθx 2 ⏐ ⏐ ⏐ +⏐ ⏐ +⏐ − ∥χxx (t)∥ + χ2xxx ≲ 1 + ⏐ ⏐ ⏐ ⏐ ⏐ eθ veθ x veθ eθ v x x⏐ 0 R 0 R x ⏐( ) ⏐2 ⏐[( ) ] ⏐2 ] ⏐ ϕz ⏐ ⏐ θpθ ⏐ ΘPΘ ⏐ ⏐ ⏐ +⏐ − Ux ⏐⏐ . +⏐ (5.15) ⏐ eθ eθ EΘ x

x

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32

Then we can compute from (2.34) that ⏐( ) ⏐ ⏐ θpθ ψx ⏐2 ( ) 2 ⏐ ⏐ ≲ φ2x + χ2x + Vx2 + Θx2 ψx2 + ψxx , ⏐ ⏐ eθ x ⏐( 2 ) ⏐2 ⏐ µux ⏐ ( )( ) ( )( 2 ) 2 ⏐ ⏐ ≲h φ2x + χ2x + Vx2 + Θx2 ψx4 + Ux4 + ψx2 + Ux2 ψxx + Uxx , ⏐ veθ ⏐ x ⏐ ( ( ) ) ⏐2 ⏐ κχxxx ⏐ ( )( ) ( )( ) 1 κθx ⏐ ⏐ ≲ χ2x + Θx2 φ4x + χ4x + φ2xx + χ2 + Θx2 Vx4 + Θx4 − ⏐ veθ ⏐ eθ v x x ( ) 2 ( )( 2 ) 2 2 2 + χ2x + Θxx Vxx + χ2xx + Θxx φx + χ2x + Vx2 + Θx2 + Θxxx , ⏐( ) ⏐2 2 2 2 2 2 2 2 2 2 2 2 2 ⏐ ϕz ⏐ ⏐ ⏐ ≲ ϕ z χx + ϕ z Θx + ϕ z χx + ϕ z Θx + ϕ2 zx2 ⏐ eθ ⏐ θ2 θ4 x ( ) + ϕ2 z 2 φ2x + χ2x + Vx2 + Θx2 , ⏐[( ) ] ⏐2 ⏐ θpθ ⏐ ( ) ( ( ) ) ΘPΘ ⏐ − Ux ⏐⏐ ≲ φ2 φ2x + χ2x + Vx2 + Θx2 + χ2 φ2x + χ2x + Vx2 + Θx2 Ux2 ⏐ eθ EΘ x ( ) 2 + φ2x + χ2x + φ2 + χ2 Uxx . Then by Lemmas 2.2, 2.4, 2.5, (4.34), Corollary 4.6, (5.6), and Sobolev inequality, we can obtain ) ⏐ ∫ t∫ ∫ t∫ ∫ t∫ ∫ t ∫ ⏐( ⏐ θpθ ψx ⏐2 ( 2 ) 2 2 4 2 ⏐ ≲ ⏐ φ + χ + ψ + δ ψ ≲ 1 + φ2x ≤ C(m2 ), x xx x ⏐ ⏐ e θ 0 R 0 R 0 R 0 R x

(5.16)

∫ t ∫ ⏐( 2 ) ⏐2 ∫ t( ∫ t∫ ∫ t∫ ) ⏐ µux ⏐ ( 2 ) 4 4 2 2 4 ⏐ ⏐ ≲h 1 + ∥ψx ∥L∞ (R) + ∥Ux ∥L∞ (R) + ∥Uxx ∥ dτ + ψxx + χx + δ ψx4 ⏐ veθ ⏐ 0 R 0 0 R 0 R x ∫ t ≲h 1 + ∥ψx ∥2 ∥ψxx ∥2 ≲h 1, (5.17) 0

) ) ⏐2 ( ( ∫ t∫ ⏐ ⏐ ⏐ κχxxx 1 κθx ⏐ ⏐ − ⏐ veθ eθ v x x⏐ 0 R ∫ t( ∫ t∫ ) ( 2 ) 4 4 2 2 2 4 ≲ 1+ ∥φx ∥L∞ (R) + ∥χx ∥L∞ (R) + ∥χxx ∥L∞ (R) + ∥χx ∥L∞ (R) ∥φxx ∥ dτ + δ φxx + χ2xx 0 0 R ∫ ∫ ∫ t ∫ t ∫ t ( 6 ) t 8 2 2 3 4 4 2 + δ +δ χx + ∥Θxx ∥L∞ (R) dτ + δ ∥φx ∥L∞ (R) dτ + δ ∥χx ∥L∞ (R) ∥χx ∥2 dτ R 0 0 0 0 ∫ ∫ t { } t ( ) 2 2 2 2 ≲ 1+ ∥φx ∥ ∥φxx ∥ + ∥χx ∥ ∥χxx ∥ + ∥χxx ∥∥χxxx ∥ dτ + sup ∥φxx (t)∥2 ∥χx ∥∥χxx ∥dτ 0≤t≤T

0

∫ t∫ ≲ C(ϵ) + ϵ 0

R

χ2xxx +

∫ t∫ 0

R

φ2xx + sup

{

∥φxx (t)∥

} 2

,

0

(5.18)

0≤t≤T

∫ t∫ ∫ t∫ ∫ t∫ ∫ t ∫ ⏐( ) ⏐2 ⏐ ϕz ⏐ ( 2 ) 2 4 2 ⏐ ≲ C(m2 ) ⏐ χx + δ Ξ (m1 , m2 , N ) ϕz + φx + χ2x + zx2 ≤ C(m2 ), (5.19) ⏐ ⏐ eθ 0 R 0 R 0 R 0 R x and ) ] ⏐2 [∫ t )] ∫ t ∫ ⏐[( ∫ t∫ ( ⏐ θpθ ⏐ ΘPΘ κΘχ2x vθ2 4 2 2 2 ⏐ ⏐ U ∥χ ∥ dτ + φ + χ + · − ≲ δ x x L∞ (R) x x ⏐ ⏐ eθ EΘ vθ2 κ 0 R 0 0 R x ∫ t [( ) ] + ∥Vx ∥2L∞ (R) + ∥Θx ∥2L∞ (R) ∥Ux ∥2L∞ (R) + ∥Uxx ∥2L∞ (R) dτ 0 ∫ t∫ ( 2 ) 4 ≲ δ (1 + Ξ (m1 , m2 , N )) + δ 4 χx + χ2xx ≲ 1. (5.20) 0

R

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

Putting (5.15)–(5.20) together, we arrive at ∫ t∫ ∫ t∫ ∫ t∫ } { ∥χxx (t)∥2 + χ2xxx (τ, x) ≲h C(ϵ, m2 ) + ϵ χ2xxx + φ2xx + sup ∥φxx (t)∥2 . 0

0

R

R

0

R

33

(5.21)

0≤t≤T

Combining (5.14) and (5.21) and taking ϵ > 0 small enough, we can get (5.2). □ We next obtain a m2 -dependent bound for the second-order derivatives with respect to x of the solution (φ (t, x) , ψ (t, x) , χ (t, x) , z (t, x)). Lemma 5.2. Under the assumptions listed in Theorem 1.1, for any 0 ≤ t ≤ T , we have ∫ t 2 2 ∥(φxx , ψxxx , χxxx , zxxx ) (τ )∥ dτ ≤ C(m2 ). ∥(φxx , ψxx , χxx , zxx ) (t)∥ +

(5.22)

0

( ) Proof . Differentiate (2.1)2 with respect to x and multiply the result by µφv x x to find [ ( ] [ ( µφ ) ] [ ( µφ ) ] 1 µφx )2 x x − ψx + ψx 2 v x t v x t v t x ( ( ) )( ( µφ ) ( µφ ) ( µφ ) [ µ ] µVx µφx ) x x x θ − ψxx + (vx θt − θx ux ) + g (V, Θ)xx − . = (p − P )xx v x v t v x v v tx v x x We integrate the above identity over [0, t] × R, and Cauchy’s inequality to obtain ( µφ ) 2 ∫ t ∫ µRθφ2   x xx (t) +  3 v x v 0 R ∫ t∫ [ ( µφ ) ( µφ ) ( µφ ) [ µ ] µRθφ2xx x x x θ + − ψ + (v θ − θ u ) ≤C+ (p − P )xx xx x t x x v x v3 v t v x v x ( 0 R ( ) )( ] ) µVx µφx + g (V, Θ)xx − . (5.23) v tx v x Now we turn to bound the terms on the right hand side of (5.23). Firstly, owing to (1.3), we have RΘxx φ 2Rφx χx 2RVx χx 2Rφx Θx 2RφVx Θx 2RφVx Θx RχVxx Rχxx − − − − + + − v vV v2 v2 v2 vV 2 v2 V v2 RΘφVxx RΘφVxx 2Rθφ2x 2RθVx2 4Rθφx Vx 2RΘVx2 + + + + + − + 4aθ2 χ2x + 4aθ2 Θx2 vV 2 v2 V v3 v3 v3 v3 ) 4 Rθχxx 4 ( + 8aθ2 χx Θx − 4aΘ 2 Θx2 + aχ θ2 + θΘ + Θ 2 (χxx + Θxx ) + aΘ 3 χxx − 3 3 v2 Rθχxx := M − . v2

(p − P )xx =

We then conclude from Lemmas 2.2, 4.4, Corollary 4.6, (5.6), and Sobolev inequality that ] ∫ t∫ [ ( µφ ) µRθφ2xx x (p − P )xx + v x v3 0 R ( ) ] ∫ t∫ [ µφxx M Rθφxx ( µ ) = + M− φ x v v2 v x 0 R [ ∫ t∫ ∫ t∫ 2 ) ( )( 2 ) ( 2 )( ) µRθφxx µv ( 2 ≤ϵ + C(ϵ) 1 + χ2 χ2xx + φ2 + χ2 Vxx + Θxx + φx + χ2x Vx2 + Θx2 3 v 0 R 0 R θ ] ∫ t∫ [ ( ) 2 4 4 4 4 2 2 2 + χ Vx + Θx + φx + χx + φ Vx Θx + C(ϵ) |φx | (|φ| + |χ|) (|Vxx | + |Θxx |) + |φx χxx | 0 R ( ) ( 2 2 + |φφx Vx Θx | + |φx χ| |Vx | + |Θx | + |φx | (|φx | + |χx |) (|Vx | + |Θx |) + |φx | φ2x + χ2x

34

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

] + |χχxx | + |φxx |) · (|φx | + |χx | + |Vx | + |Θx |) ( ) ∫ t ∫ t∫ µRθφ2xx + C(ϵ, m2 ) 1 + δΞ (m1 , m2 , N ) + (∥φx ∥∥φxx ∥ + ∥χx ∥∥χxx ∥) dτ ≤ϵ v3 0 0 R ∫ t∫ µRθφ2xx ≤ 2ϵ + C(ϵ, m2 ). v3 0 R On the other hand, we can compute from (1.5) that ( µφ ) αµφx θt µφx Ux µψxx µh′ (v)φx ψx µh′ (v)φx Ux µφx ψx x = − + + + − . v t vθ vh(v) vh(v) v2 v2 v

(5.24)

(5.25)

Then it follows from Lemma 2.2, (2.6), (2.33), Lemma 4.4, and Cauchy inequality that ∫ t∫ ∫ t∫ 2 2 ∫ t∫ ⏐ ( µφ ) ⏐ ( 2 ) φx θt ⏐ ⏐ x 2 φx + ψxx + φ2x Ux2 + α2 ≲ C(m2 ) + α2 Ξ (m1 , m2 , N ) ≤ C(m2 ). ⏐ ≲h ⏐ψxx 2 v t 0 R 0 R θ 0 R (5.26) Simultaneously, in view of (µ) ( µφ ) µ x = φxx + φx , v x v v x we can obtain from (1.5), Lemma 2.2, (4.33), and (5.6) that ∫ t∫ ⏐ ( µφ ) ⏐ ⏐ ⏐ x ⏐g (V, Θ)xx ⏐ v x ∫0 t ∫R [ ] ≲h |g (V, Θ)xx | |φxx | + φ2x + |φx | (|χx | + |Vx | + |Θx |) 0 ) ] ∫ t ∫ [( ∫ t R∫ ( 2 ) 1 µRθφ2xx 2 2 2 2 2 + C(ϵ) |g (V, Θ) | + φ 1 + φ + χ + V + Θ ≤ ϵ x x x x x xx v3 θ 0 R 0 R [ ] ∫ t∫ ∫ t ( ) µRθφ2xx ≤ ϵ + C(ϵ) 1 + ∥φx ∥3 ∥φxx ∥ + ∥φx ∥∥φxx ∥ dτ 3 v 0 R 0 ∫ t∫ µRθφ2xx + C(ϵ, m2 ). (5.27) ≤ 2ϵ v3 0 R Moreover, it is easy to see that [ ] ] [µ µθ µθθ θx + µθv vx µθ vx θ (vx θt − θx ux ) = (vxx θt + vx θxt − θxx ux − θx uxx ) + − 2 (vx θt − θx ux ) , v v v v x which implies ⏐[ µ ] ⏐ ⏐ θ ⏐ (vx θt − θx ux ) ⏐ ⏐ v x ≲h |α| [|θt | (|φxx | + |Vxx |) + |θxt | (|φx | + |Vx |) + (|χxx | + |Θxx |) (|ψx | + |Ux |) + (|χx | + |Θx |) (|ψxx | + |Uxx |)] + |α| (|φx | + |χx | + |Vx | + |Θx |) [|θt | (|φx | + |Vx |) + (|χx | + |Θx |) (|ψx | + |Ux |)] .

(5.28)

Furthermore, it follows from the identity ( 2) ( ) ( ) ( ) ( ) ( ) µux θpθ ux κv vx θx + κθ θx2 κθxx κθx vx λϕz θtx = − + + − + , veθ x eθ veθ veθ x v 2 eθ x eθ x x x and (1.2), (4.32), and (4.33) that |θtx | ≲ (|χxxx | + |Θxxx |) (1 + |φx | + |χx | + |Vx | + |Θx |) + (|ψxx | + |Uxx |) (1 + |ψx | + |Ux |) [ ] z z (|χx | + |Θx |) + (|χx | + |Θx |) (|φxx | + |Vxx |) + ϕ + + |zx | + z (|φx | + |χx | + |Vx | + |Θx |) θ θ2 ( 2 ) + (1 + |φx | + |ψx | + |χx | + |Vx | + |Ux | + |Θx |) φx + ψx2 + χ2x + Vx2 + Ux2 + Θx2 (5.29)

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35

By employing Lemmas 2.2, 2.4, 2.5, (2.6), Corollary 4.6, (5.6), (5.28), (5.29), and Sobolev inequality, one has ∫ t ∫ ⏐( ) [µ ] ⏐ ⏐ µφx ⏐ θ (vx θt − θx ux ) ⏐ ⏐ v x v x 0 R )] ∫ t∫ [ ∫ t∫ ⏐[ µ ] ⏐2 ( 1 µRθφ2xx ⏐ θ ⏐ 4 2 2 2 2 + C(ϵ) φ 1 + ≲h ϵ + φ χ + φ V + (v θ − θ u ) ⏐ ⏐ x t x x x x x x x v3 v θ x 0 R [0 R ∫ t( ∫ t∫ ) ] 2 µRθφxx 3 2 2 ∥φ ∥ ∥φ ∥ + ∥χ ∥∥χ ∥ + ∥V ∥ + C(ϵ) α Ξ (m , m , N ) + ≲h ϵ ∞ x xx x xx x 1 2 L (R) dτ v3 0 0 R ∫ t∫ µRθφ2xx ≤ 2ϵ + C(ϵ, m2 ). (5.30) v3 0 R Similarly, we can deduce from (1.5), (2.6), and (4.33) that ( ) 2 ⏐( ) ⏐ |αθ | |V | + |φ V | + |V | t x x x xx ⏐ µVx ⏐ ⏐ ⏐ ≲h |αθt Vx | (|χx | + |Θx |) + |αθtx Vx | + ⏐ v ⏐ θ2 θ θ tx ( ) 2 + |Vx | |ψxx | + φx + |φxx | ⏐ ⏐] |α| [ + (|χx Vx | + |Vx Θx |) (|ψx | + |Ux |) + (|χx | + |Θx |) (|Uxx | + |φx Vx |) + ⏐χx Vx2 ⏐ (θ ) + |φx | |Vx | + |Vx2 | (|ψx | + |Ux |) + |Uxx | (|φx | + |ψx | + |Vx |) + |Uxxx | + |φx Vxx | .

(5.31)

Then it follows from Lemma 2.2, (2.6), (2.33), Corollary 4.6, (5.29), and (5.31) that ⏐ ) ( ∫ t ∫ ⏐( ⏐ µVx µφx ) ⏐⏐ ⏐ ≲h 1. ⏐ v v x⏐ 0 R tx

(5.32)

Inserting (5.24), (5.26), (5.27), (5.30), (5.32) into (5.23), then choosing ϵ > 0 small enough and utilizing (4.33), we can get ∫ t∫ 2 ∥φxx (t)∥ + φ2xx (s, x) ≲h C(m2 ). (5.33) 0

R

The combination of (5.33) and Lemma 5.1 yields that ∫ t 2 2 ∥(φxx , ψxx , χxx ) (t)∥ + ∥(φxx , ψxxx , χxxx ) (τ )∥ dτ ≲h C(m2 ).

(5.34)

0

It suffices to deduce a nice bound on ∥zxx (t)∥2 . To this end, we differentiate (2.1)5 with respect to x and multiply the result by zxxx to get ( ) 4dvx zxx zxxx 1 2 dz 2 6dvx2 zx zxxx 2dvxx zx zxxx zxx + xxx − (zxt zxx )x = − + 2 3 4 2 v v v v3 t ( ) β A + ϕθx zzxxx + 2 + ϕzx zxxx . θ θ Integrating the above identity over (0, t) × R and using Lemma 2.2, (2.6), Lemmas 2.4, 4.4, Corollary 4.6, (5.33), and Sobolev inequality, we arrive at ∫ t∫ 2 ∥zxx (t)∥2 + zxxx (s, x) 0 R ∫ t∫ [ ( ) ] ≲ 1+ (|φx | + |Vx |) |zxx zxxx | + φ2x + Vx2 + |φxx | + |Vxx | |zx zxxx | + |ϕzx zxxx | 0

R

Y. Liao / Nonlinear Analysis: Real World Applications 53 (2020) 103056

36

∫ t∫

) |χx | + |Θx | |χx | + |Θx | + + |ϕzzxxx | θ θ2 0 R [ ∫ t∫ ∫ t( ) 2 2 2 ≲ϵ zxxx (s, x) + C(ϵ, m2 ) 1 + ∥φx ∥ ∥φxx ∥ + ∥zxx ∥ ∥zxxx ∥ + ∥zx ∥ ∥zxx ∥ dτ 0 R 0 ( ) )] ∫ t∫ ( ∫ t∫ κΘχ2x 1 ϕ2 z 2 2 2 + 1+ 2 zxxx (s, x) + C(ϵ, m2 ). + zx ≲ 2ϵ vθ2 θ κ 0 R 0 R (

(5.35)

Choosing ϵ > 0 small enough, we can get ∥zxx (t)∥2 +

∫ t∫ 0

2 zxxx (s, x) ≤ C(m2 ).

(5.36)

R

We combine (5.34) and (5.36) to deduce Lemma 5.2. □ Similarly, we can deduce the third-order derivatives whose proof is omitted for brevity. Lemma 5.3. Under the assumptions listed in Theorem 1.1, we have ∫ t 2 ∥(φxxx , ψxxx , χxxx , zxxx )(t)∥ + ∥(φxxx , ψxxxx , χxxxx , zxxxx )(τ )∥2 dτ ≤ C(m2 )

(5.37)

0

holds for all t ∈ [0, T ]. In light of Lemmas 2.1–5.3 , we can get the following corollary. Corollary 5.4. Under the assumptions listed in Theorem 1.1, there exists a positive constant C(m2 ) > 0 which depends only on m2 , Π0 , V0 and H(C2 ) with C2 being given in Lemma 4.4, such that for all t ∈ [0, T ], ∫ t[ ] 2 2 2 (5.38) ∥(φ, ψ, χ, z) (t)∥3 + ∥φx (τ )∥2 + ∥(ψx , χx , zx ) (τ )∥3 dτ ≤ C(m2 ). 0

With Corollary 5.4 in hand, we can deduce Theorem 1.1 by using the continuation argument deduced in [13] and we omit the details for brevity. 6. The proof of Corollary 1.1 Now we are in a position to prove Corollary 1.1. Since most of the terms are estimated in the same way as that in the proof of Theorem 1.1, we only need to re-estimate the terms related to the radiation constant a.  2 To begin with, we first re-estimate  µφv x (t) . In light of (2.6), (2.18), and (2.30), I9 can be re-estimated as ] [∫ t ∫ ∫ t∫ ∫ t 3 µRθφ2x κΘχ2x a2 µθ7 v 2 −2 I9 ≤ ϵ + C(ϵ) · + δaΞ (m1 , m2 , N ) (1 + τ ) dτ 2 v3 κ 0 R 0 R vθ 0 ∫ t∫ µRθφ2x ≲ϵ + C(ϵ). (6.1) v3 0 R Note that I6 , I7 , I8 , I10 , and I11 are bounded in the same way as that in Section 2, and we only need the assumption b ≥ 1 to control the term I8 . Plugging (2.27)–(2.29), (6.1), (2.31), (2.38) into (2.26), we arrive at  ℓ1 +1 ∫ t∫  µφ  1 µθφ2x  x 2 ℓ  (t) + ≲1+ + ∥v∥∞2 + ∥θ∥∞ .  v 3 v 0 R v ∞

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Then by repeating the argument used in Section 3, we can still deduce (3.2), (3.7), and (3.8) under the assumptions ℓ1 > 1, ℓ2 > 1, and b ≥ 1. On the other hand, on account of (1.5), (2.6), (2.18), (3.2), and (4.2), one has   ( 2 ) ) ∫ t∫ ( 2  1 ( ) θ θ ς b+2 2 2 6  + θb+2 e2θ χ2t ≲ (1 + ∥θ∥∞2 )  + θ 1 + a v θ  1 + θb v  X(T ) v 0 R ∞  ( ) ( ) 1 2 ς2 ς2 +2   ≲ (1 + ∥θ∥∞ )   + ∥θ∥∞ X(T ) ≲ 1 + ∥θ∥∞ X(T ), (6.2) v ∞ and

∫ t∫ ( 0

R

( 4  ) ) )(  θ vθ  θ2 1 2 6 b+2 2 2 2 b+4   ≲ 1 + ∥θ∥b+5 . +a θ +θ θ p θ ψx ≲  +θ ∞ v v v2 h(v) ∞

Then we can obtain from (4.16), (4.17), (6.2), and (6.3) that ) ( b+ 13 +2(ℓ1 +2)ς1 2+ς b+β+2 J ≲ 1 + 1 + ∥θ∥∞ 2 X(T ) + ∥θ∥∞ 2 + ∥θ∥∞ . Then (4.19) follows from (4.15) and (6.4) under the assumptions b > 67 + 3b + 2 − 2ℓ26+1 . Moreover, we can conclude from (1.5), (2.6), (3.2), and the assumption b >

2 3ℓ1

+

3 2ℓ2 +1

(6.3)

(6.4) 2 2ℓ2 +1 ,

and 0 ≤ β <

that

  )( ) ( 2  1 1 vθ  θ ς 2 8 2b  + a θ + θ I20 ≤ ϵX(T ) + C(ϵ) (1 + ∥θ∥∞2 )   1 + θb v2 v2 h(v) ∞ ( ) b+β+3 ≤ ϵX(T ) + C(ϵ) 1 + ∥θ∥∞ ≤ ϵ (X(T ) + Y (T )) + C (ϵ) .

(6.5)

The other terms are bounded in the same way as that in Section 4, and we need the assumption 0 ≤ β < b + 3 − 2ℓ23+1 and b > 52 + ℓ11 + 2ℓ23+1 to control the terms I19 and I21 , respectively. Thus we can complete the proof of Corollary 1.1 by repeating the argument as that in the proof of Theorem 1.1 and we omit the details for brevity. Acknowledgment The research of Yongkai Liao is supported by National Postdoctoral Program for Innovative Talents of China No. BX20180054. References [1] B. Ducomet, A model of thermal dissipation for a one-dimensional viscous reactive and radiative gas, Math. Methods Appl. Sci. 22 (15) (1999) 1323–1349. [2] Y.-K. Liao, H.-J. Zhao, Global solutions to one-dimensional equations for a self-gravitating viscous radiative and reactive gas with density-dependent viscosity, Commun. Math. Sci. 15 (5) (2017) 1423–1456. [3] Y.-K. Liao, H.-J. Zhao, Global existence and large-time behavior of solutions to the Cauchy problem of one-dimensional viscous radiative and reactive gas, J. Differential Equations 265 (5) (2018) 2076–2120. [4] M. Umehara, A. Tani, Global solution to one-dimensional equations for a self-gravitating viscous radiative and reactive gas, J. Differential Equations 234 (2) (2007) 439–463. [5] J. Jiang, S.-M. Zheng, Global solvability and asymptotic behavior of a free boundary problem for the one-dimensional viscous radiative and reactive gas, J. Math. Phys. 53 (2012) 1–33. [6] D. Mihalas, B.W. Mihalas, Foundations of Radiation Hydrodynamics, Oxford Univ. Press, New York, 1984. [7] Y.B. Zel’dovich, Y.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, vol. II, Academic Press, New York, 1967. [8] J. Jiang, S.-M. Zheng, Global well-posedness and exponential stability of solutions for the viscous radiative and reactive gas, Z. Angew. Math. Phys. 65 (2014) 645–686. [9] Y.-M. Qin, G.-L. Hu, T.-G. Wang, L. Huang, Z.-Y. Ma, Remarks on global smooth solutions to a 1D self-gravitating viscous radiative and reactive gas, J. Math. Anal. Appl. 408 (1) (2013) 19–26.

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