Global solutions for a one-dimensional compressible micropolar fluid model with zero heat conductivity

Accepted Manuscript Global solutions for the 1-D compressible micropolar fluid model with zero heat conductivity

Ran Duan

PII: DOI: Reference:

S0022-247X(18)30206-3 https://doi.org/10.1016/j.jmaa.2018.03.009 YJMAA 22085

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Journal of Mathematical Analysis and Applications

Received date:

25 November 2017

Please cite this article in press as: R. Duan, Global solutions for the 1-D compressible micropolar fluid model with zero heat conductivity, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2018.03.009

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Global solutions for the 1-D compressible micropolar fluid model with zero heat conductivity Ran Duan∗a a

School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China.

Abstract We study global solutions for the 1-D compressible micropolar fluid model with zero heat conductivity, which is a hyperbolic-parabolic system. The pressure, velocity and angular velocity are dissipative because of viscosity, whereas the entropy is non-dissipative due to the absence of heat conductivity. Compared with the classical Navier-Stokes equations, there is an extra angular velocity ω in micropolar fluid model which brings both benefit and trouble. The benefit lies in the fact that the term −vω is a damping term which provides extra regularity of ω,while the trouble is brought by the term vω 2 which increase the nonlinearity of the system. The global solutions are obtained by combining the local existence and a priori estimates if H 2 − norm of the initial perturbation around a constant state is small enough. The asymptotic behavior is also obtained in this paper. Keywords: micropolar fluids; global strong solution; zero-heat conductivity. 2000 MSC: 35Q35;, 76D03 1. Introduction 1.1. Formulation in Lagrangian coordinates The model of micropolar fluids which respond to micro-rotational motions and spin inertia was first introduced by Eringen [1] in 1966. This model is more proper than Navier-Stokes equations to describe the motions of a large variety of complex fluids consisting of diploe elements such as the suspensions, animal blood, liquid crystal, etc. For more physical background, please refer to [2],[3]. In Euler coordinates, it was formulated in [4] as follows: ⎧ ρ˙ + ρdivu = 0, x ∈ R, t > 0; ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρu˙ = divT + ρf, ⎪ ⎪ ρj ω˙ = divM + Tαx + ρm, ⎪ ⎪ ⎪ ⎩ ρe˙ = T · ∇u + M · ∇ω − 2Tαx · ω + divq + ρr,

(1.1)

where α˙ denotes material derivative of a field α: α˙ = ∗

E-mail:[email protected]

∂α + (∇α)u, ∂t

(1.2)

with notation: ρ − mass density v − velocity D(v) − stretching, D(v) = sym∇v p − pressure T − stress tensor T Tax − an axial vector with the cartesian components(Tax )i = eijk 2kj , where eijk is the alternating tensor ω − microrotation velocity ωskw − a skew tensor with the Cartesian components(ωskw )ij = eijk ωk j − microinertia density (a positive scalar field) M − couple stress tensor θ − absolute temperature e − internal energy density q − heat flux density vector f − body force density m − body couple density r − body heat density.

(1.3)

The linear constitutive equations for stress tensor, couple stress tensor and heat flux density vector are, respectively, of the forms: T = −pI + λ(divu)I + 2μD(u) + ξ(∇u + ωskw ), M = α(div)ω)I + β(∇ω)T + γ∇(ω), q = κ∇θ,

(1.4)

where λ, μ, ξ, α, β, γ and κ are scalar material coefficients, depending generally on mass density and temperature and satisfying the condition [1, 5]: 3λ + 2μ + ξ ≥ 0, 2μ + ξ ≥ 0, ξ ≥ 0, 3α + β + γ ≥ 0, |β| ≤ 0, κ ≥ 0.

(1.5)

Much attention has been paid to this model by many mathematicians among which Mujakovi´c initialed the study on this model for compressible flow. First, for one-dimensional compressible flow, she made a series of efforts in studying the local-in-time existence and uniqueness, the global existence and regularity of solutions to an initial-boundary value problem with both homogenous [4, 6, 7] and non-homogenous [8, 9, 10, 11] boundary conditions respectively. Besides, she also analyzed large time behavior of the solutions and the stabilization of solutions to the cauchy problem [12, 13, 14]. Second, for three-dimensional model, Mujakovi´c and her collaborator Draˇzi´c studied the local existence, global existence, uniqueness and large time behavior of spherical symmetry solutions [15, 16, 17, 18, 19]. There are other authors who contribute to the study of this model such as Chen [20] who proved the global existence of strong solutions to the Cauchy problem with initial vacuum for one-dimensional model. Huang and Nie [21] obtained exponential stability with homogeneous boundary conditions. There are other papers [22, 23, 24] which studied the local stability of rarefaction wave, contact wave as well as viscous shock wave. For the three-dimensional model, Chen[25], Chen, Huang and Zhang[26] proved a blow up criterion of strong solutions to the Cauchy problem. Chen, Xu and Zhang [27] established the global weak solutions with discontiuous initial data and vacuum. Recently, Liu and Zhang [28] have obtained the optimal time decay of the three-dimensional compressible flow. Huang and Kong [29] proved the regularity and exponential stability of the solutions in the domain bounded by two concentric spheres. For incompressible flow, we refer to [30, 31, 32, 33] and references cited therein.

We note that all the above analysis take advantage of the fact that the entropy is dissipative brought by non-zero heat conductivity. A natural question is that how should one close the energy estimates and then obtain global existence of solutions or even get the asymptotic behavior with zero heat conductivity? Let us study this problem in one-dimension first, and consider the polytropic gas   Rθ γ−1 Rθ −γ p(v, θ) = = Av exp s , e = Cv θ = , (1.6) v R γ−1 where the specific gas constants A, R and the specific heat at constant volume Cv are positive constants and γ > 1 is the adiabatic constant. In this case, by taking κ = 0 and ρ = v1 in the equations reformulated in Lagrangian coordinates in [4], we have the form: ⎧ vt − ux = 0, x ∈ R, t > 0; ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ ut + px = μuv x x , (1.7)

μuux  ωx2 1 2 ⎪ 2 ⎪ u ) + (pu) = + + vω , (e + t x ⎪ 2 v v x ⎪ ⎪

  ⎪ ⎩ ωt = B ωvx x − vω , where μ denotes the viscosity coefficient and B = j −1 (λ + 2μ + ξ)−1 (α + β + γ), both of which are assumed to be positive constants. This system is supplemented with initial data (v, u, θ, ω) |t=0 = (v0 , u0 , θ0 , ω0 )

(1.8)

and the boundary data at far fields lim (v0 , u0 , θ0 , ω0 ) = (v∞ , 0, θ∞ , 0)

x→±∞

Due to lack of heat conductivity, the estimates of the term 0

t

(1.9) θx (τ )2H 1 dτ could hardly be

obtained by applying usual L2 − energy estimates which just plays an essential role in closing the energy t estimates. Inspiringly, in [34], S.kawashima and M.Okada observed that the esti px (τ )2H 1 + ux 2H 1 dτ is enough to close the energy estimates with no aid of the mates 0 t estimates of the term θx (τ )2H 1 dτ , by considering the equations as a system of unknowns of 0

pressure, velocity and entropy instead of one of unknowns of density, velocity and temperature. And then they obtained smooth global solutions for the one-dimensional equations in Magenetohydrodynamics without heat conductivity. Motivated by which, the large-time behavior for one-dimensional compressible Navier-Stokes equations was obtained in [35], and the global existence as well as convergence rates for that equations was obtained in R3 in [36]. Like those, we first convert system (1.7) into system of unknowns p, u, θ, ω with zero heat conductivity. It is well known in thermodynamics that by using any given two of the five thermodynamical variables, v, p, e, θ, and s, the remaining three variables are their functions due to the second law of thermodynamics asserting that θds = de + pdv.

From which, if we choose (v, θ) as independent variables and write (p, e, s) = (p(v, θ), e(v, θ), s(v, θ)), then we can deduce that ⎧ sv (v, θ) = pθ (v, θ), ⎪ ⎪ ⎨ (1.10) , sθ (v, θ) = eθ (v,θ) θ ⎪ ⎪ ⎩ ev (v, θ) = θpθ (v, θ) − p(v, θ). Therefore, the energy equation (1.7)3 can be converted into the equations for the entropy s or the absolute temperature θ by the second law of thermodynamics. That is   θx θ2 u2 ω2 v st = + x2 + x2 + x + ω 2 , (1.11) vθ x vθ vθ vθ θ   θx θpθ (v, θ) κ μ u2x ωx2 θt + ux = + + vω 2 . + (1.12) eθ (v, θ eθ (v, θ) θ x eθ (v, θ) v v Therefore, we obtain a system of unknowns p, u, s, ω for ideal polytropic gas with zero heatconductivity by taking B = 1 for simplicity: ⎧ 2 ⎪ pt + γ Rθ u = γ−1 ωx2 + (γ − 1)ω 2 + (γ − 1)μ uvx2 , x ∈ R, t > 0; ⎪ v2 x v2 ⎪ ⎪  u ⎪ ⎪ ⎪ ⎨ ut + px = μ vx x , (1.13) ωx2 u2x vω 2 ⎪ s = + + μ , t ⎪ vθ θ vθ ⎪ ⎪ ⎪

 ωx  ⎪ ⎪ ⎩ ωt = v x − vω ,

and

with the initial condition (p, u, s, ω) |t=0 = (p0 , u0 , s0 , ω0 ) → (p∞ , 0, s∞ , 0) as x → ±∞.

(1.14)

Here p∞ = p(v∞ , θ∞ ) and s∞ = s(v∞ , θ∞ ). The main purpose of this paper is to obtain global existence of strong solution to the Cauchy problem (1.7) and (1.8) as well as study its asymptotic behavior when the initial data is a small perturbation of a constant state. Theorem 1.1. (Global strong solutions for micropolar fluid with zero heat conductivity ) For polytropic gas, let the initial data (v0 , u0 , θ0 , ω)(x) be such that (v0 − v∞ , u0 , θ0 − 0 θ∞ , ω0 )(x)H 2 ≤ √2C where 0 is sufficiently small such that 

v∞ θ∞ 1 √ , √ ,

0 ≤ min , (1.15) 4 2 4 2 4C and C ≥ 1 is a constant depending only on v∞ , θ∞ defined in Proposition 2.2, then the Cauchy problem (1.7) and (1.8) admits a unique global strong solution (v, u, θ, ω)(x, t) in time, satisfying v − v∞ , θ − θ∞ , u, ω ∈ L∞ ([0, T ]; H 2 (R)) And for each (x, t) ∈ R × [0, T ],   sup (v − v∞ , u, θ − θ∞ , ω)(t)22 ; t ∈ [0, T ] +



T 0

(1.16)

px (τ )21 + ux (τ )22 + ω(τ )23 dτ ≤ C (1.17)

and lim sup {|(p, u, ω)(x, t) − (p∞ , 0, 0)|} = 0.

t→∞ x∈R

(1.18)

Remark 1.1. We can only obtain the asymptotic behavior of p, u and ω but that of v, θ and s due to the the lack of dissipation on v, θ and s. If the microstructure of the fluid is not taken into account, that is to say the effect of the angular velocity fields of the particle’s rotation is omitted, i.e., ω = 0, then the equations (1.7) reduces to the classical Navier-Stokes equations. Compared with the classical Navier-Stokes equations, the angular velocity ω in this model brings both benefit and trouble. The benefit lies in the fact that the term −vω in equation (1.7)4 is a damping term which provides extra regularity of ω, while the trouble is brought by the term vω 2 in equation (1.7)3 which increase the nonlinearity of the system. This paper is arranged as follows: The proof of Theorem 1.1 is given in Section 2, and the energy estimates as well as the proof of Proposition 2.2 is given in Sections 3. Notations: Throughout the rest of this paper, C or O(1) will be used to denote a generic positive constant independent of t and x and Ci (·, ·) (i ∈ Z+ ) stands for some generic constants depending only on the quantities listed in the parenthesis. Note that all these constants may vary from line to line. H l (R)(l ≥ 0) denotes the usual Sobolev space with norm  · l and  · 0 =  ·  will be used   p1 to denote the usual L2 −norm. For 1 ≤ p ≤ +∞, f (x) ∈ Lp (R, Rn ), f Lp = |f (x)|p dx . R

It is easy to see that f L2 =  · . Finally,  · L∞ and  · L∞ are used to denote  · L∞ (R) and t,x  · L∞ ([0,t]×R) respectively. 2. Proof of Theorem 1.1 Setting (ϕ(t, x), ψ(t, x), u(t, x), φ(t, x), ξ(t, x), ω(t, x)) = (v(t, x) − v∞ , p(t, x) − p∞ , u(t, x) − 0, θ(t, x) − θ∞ , s(t, x) − s∞ , ω(t, x) − 0).

(2.1)

We deduce that (ϕ(t, x), u(t, x), φ(t, x), ω(t, x)) solve ⎧ ϕt − ux = 0, ⎪ ⎪ ⎪ ⎪  ux ⎪ ⎪ ⎪ ⎨ ut + ψx = μ v x , 2

⎪ φt + γ−1 θux = ωvx + vω 2 + ⎪ v ⎪ ⎪

 ω  ⎪ ⎪ ⎪ ⎩ ωt = vx x − vω ,

(2.2)

μ(γ−1) u2x , R v

with initial data (ϕ(t, x), u(t, x), φ(t, x), ω(t, x))|t=0 = (ϕ0 (x), u(0, x), φ0 (x), ω0 (x))

(2.3)

= (v0 (x) − v∞ , u0 (x), θ0 (x) − θ∞ , ω0 (x)). And (ϕ(t, x), φ(t, x)) yields 

u = ψt + γ Rθ v2 x ξt =

ωx2 vθ

+

vω 2 θ

γ−1 2 ωx v2 2

+ μ uvθx ,

2

+ (γ − 1)ω 2 + (γ − 1)μ uvx2 ,

(2.4)

with initial data (ψ(t, x), ξ(t, x))|t=0 = (ψ0 (x), ξ0 (x)) = (p0 (x) − p∞ , s0 (x) − s∞ ).

(2.5)

For convenience of presentation, in what follows we will choose (ϕ, u, φ, ω) as independent variables and for some fixed T > 0, we define the solution space of (2.2) and (2.3) by ⎫ ⎧   (ϕ, u, φ, ω) (t, x) ∈ C 0 (0, T ; H 2 (R)) ⎪ ⎪ ⎪ ⎪  ⎬ ⎨  2 2 . X(0, T ) := (ϕ, u, φ, ω) (t, x)  ux (t, x) ∈ L (0, T ; H (R)) , ⎪ ⎪ ⎪ ⎪  ⎭ ⎩  ω(t, x) ∈ L2 (0, T ; H 3 (R)) Then the iteration scheme: ⎧ (ϕ0 , u0 , φ0 , ω 0 ) (t, x) = (ϕ0 , u0 , φ0 , u0 )(x), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = uk+1 ϕk+1 ⎪ t x ⎪ ⎪ ⎨  k+1  uk+1 − μuvxk = −ψx (ϕk , φk ), t ⎪ x ⎪ ⎪ k k uk ⎪ (γ−1)θ∞ (ωxk )2 μ(γ−1)(ukx )2 x k+1 x ⎪ ⎪ − (γ−1)u φ = + + , φk+1 k k k t ⎪ v v v Rv k ⎪ ⎪  ⎪ ⎩ ω k+1 − 1 ω k+1 + v k ω k+1 = 0, t

vk

x

(2.6)

x

together with a standard contracting-mapping argument yield the following local existence result: Proposition 2.1. (Local existence result) Under the assumptions stated in Theorem 1.1, the Cauchy problem (1.13),(1.14) admits a unique solution (ϕ(t, x), u(t, x), φ(t, x), ω(t, x)) ∈ X(0, t1 ) for some sufficiently small t1 > 0 and (ϕ(t, x), u(t, x), φ(t, x), ω(t, x)) satisfies  v ∞ ≤ ϕ(t, x) + v∞ ≤ 3v2∞ 2 (2.7) θ∞ ≤ φ(t, x) + θ∞ ≤ 3θ2∞ , 2 and  (ϕ, u, φ, ω) (t, x)22



T

+ 0

ψx (τ )21 + ux (τ )22 + ω(τ )23 dτ ≤ 4 (ϕ0 , u0 , φ0 , ω0 ) (x)22 . (2.8)

Here t1 depends only on  (ϕ0 , u0 , φ0 , ω0 ) (x)2 . To extend the local solution obtained in proposition 2.1 globally, we need only to get H 2 −norm a-priori estimates on the solution which can be stated in the following: Proposition 2.2. Under the assumptions stated in Theorem 1.1, suppose that (ϕ(t, x), u(t, x), φ(t, x), ω(t, x)) obtained in Proposition 2.1 has been extended to the time t = T ≥ t1 , i.e., (ψ(t, x), u(t, x), ξ(t, x), ω(t, x)) ∈ X(0, T ) and satisfies the following a priori assumption T   2 2 ψx (τ )21 + ux (τ )22 + ω(τ )23 dτ < 4 20 , (2.9) E(T ) = sup (ϕ, u, φ, ω)(t)2 ; t ∈ [0, T ] + 0

for some positive constant 0 > 0, then we can obtain the energy estimate in the form  E 2 (T ) ≤ C E 2 (0) + E 3 (T ) .

(2.10)

Here C ≥ 1 is a positive constant independent of T, 0 , which depends only on v∞ and θ∞ .

The proof is a combination of a sequence of energy estimates which will be given at the end of Section 3. Now we are ready to extend the local solutions to global ones, that is the proof of Theorem 1.1, in the following procedure. Proof of Theorem 1.1 Firstly, we take the time t = 0 as initial data. Employing local existence result Proposition 2.1, there exists a positive constant t1 = t1 ((ϕ0 , u0 , φ0 , ω0 )2 ) = t1 (E(0)) > 0 , which depends only on E(0), such that there exists a unique solution (ϕ(t, x), u(t, x), φ(t, x), ω(t, x)) ∈ X(0, t1 ) to (2.2),(2.3) satisfying E(T ) ≤ 2E(0), 0 ≤ t ≤ t1 , (2.11) By the assumption on E(0) stated in Theorem 1.1, we have  2

0 ≤ 2 0 , 0 ≤ t ≤ t1 , E(T ) ≤ 2E(0) = C

(2.12)

and then we have from Proposition 2.2 that for 0 ≤ t ≤ t1 , (1 − CE(T ))E 2 (T ) ≤ CE 2 (0).

(2.13)

By (2.12) and the assumptions on 0 (1.15), we observing that for 0 ≤ t ≤ t1 , 1 CE(T ) ≤ 2C 0 ≤ . 2

(2.14)

1 2 E (T ) ≤ (1 − CE(T ))E 2 (T ) ≤ CE 2 (0), 2

(2.15)

Therefore, we get from (2.13)

that is E(T ) ≤ for 0 ≤ t ≤ t1 . Especially, we have

√ 2CE(0) ≤ 0 ,

E(t1 ) ≤ 0 .

(2.16) (2.17)

Next, we take time t = t1 as the initial time, applying Proposition 2.1 again, there exists a t2 = t2 (E(t1 )) = t2 ( 0 ) > 0, which depends only on E(t1 ) such that there exists a unique solution (ϕ(t, x), u(t, x), φ(t, x), ω(t, x)) ∈ X(t1 , t1 + t2 ) to (2.2),(2.3) satisfying E(T ) ≤ 2E(t1 ) ≤ 2 0

(2.18)

which is exactly the estimates (2.12) and the condition (2.9) in Proposition2.2. And then applying Proposition2.2 again, we again have (2.13),(2.14),(2.15) and (2.16) for t1 ≤ t ≤ t1 +t2 . Especially, we have E(t1 + t2 ) ≤ 0 . (2.19) From (2.19), if we take t = t1 + t2 as the new initial time, applying local existence result Proposition 2.1, there exists a t3 ( 0 ) = t2 ( 0 ) > 0 such that there exists a unique solution (ϕ(t, x), u(t, x), φ(t, x), ω(t, x)) ∈ X(t1 , t1 + 2t2 ) to (2.2),(2.3) satisfying E(T ) ≤ 2E(t1 + t2 ) ≤ 2 0 , t1 + t2 ≤ t ≤ t1 + 2t2 .

(2.20)

Repeating the above procedure we can thus extend the solution (ϕ(t, x), u(t, x), φ(t, x), ω(t, x)) step by step to a global one. Now to complete the proof of Theorem 1.1, we only need to prove that (1.18) holds. Before that, we need the following lemma whose proof can be found in [37, 38].

Lemma 2.1. Suppose that g(t) ≥ 0, g(t) ∈ L1 (0, +∞) and g  (t) ∈ L1 (0, +∞), then g(t) → 0 as t → +∞. Now we turn to prove (1.18). For simplicity, here we only show the asymptotic behavior of uL∞ as t → +∞. Similar analysis can be used to show the asymptotic behavior of (ψ, ω)L∞ x x as t → +∞. To this end, let g(t)  ux 2 ≥ 0. From (1.17), we can easily check that g(t) ∈ L1 (0, +∞) and g  (t) = 2 < ux , utx >= −2 < ut , uxx > = 2 < (μ uvx )x − ψx , ux x >= 2 <

μuxx v

− μ ϕxv2ux − ψx , uxx >

(2.21)

∈ L1 (0, +∞), where < ·, · > is the L2 −inner product. Hence, ux (t) → 0 as t → +∞. By Sobolev’s inequality, we have √ 1 1 uL∞ ≤ 2u 2 ux  2 → 0, as t → +∞. (2.22) x Thus (1.18) is proved and we finish the proof of Theorem1.1. 3. The Priori estimate Under the a priori assumption: E(T ) ≤ 2 0 , we know that |(ψ(t, x), ξ(t, x))|L∞ ≤

(3.1)

√ √ 2(ψ, ξ)1 ≤ 2 2 0 .

(3.2)

v∞ θ∞ , }, 2 2

(3.3)

Therefore, for 0 satisfying (1.15) |(ψ(t, x), ξ(t, x))|L∞ ≤ 2 0 ≤ min{ we may deduce v∞ 3v∞ θ∞ 3θ∞ ≤ v(t, x) ≤ , ≤ θ(t, x) ≤ , 2 2 2 2

∀(t, x) ∈ [0, T ] × R.

(3.4)

We first give a lemma on the estimates of (ψ, ξ)2 since we need to control the bound of |(ψ, ξ)|L∞ t,x and |(ψx , ξx )|L∞ , that is t,x Lemma 3.1. (ψ, ξ)(t)2 = O(1)(ϕ, φ)(t)2 . Proof. Observing that ψ = p − p∞ = p(v, θ) − p(v∞ , θ∞ ) = p(v, θ) − p(v∞ , θ) + p(v∞ , θ) − p(v∞ , θ∞ ) = pv (v + α1 (v − v∞ ), θ)(v − v∞ ) + pθ (θ∞ , θ + α2 (θ − θ∞ ))(θ − θ∞ ),

(3.5)

where αi ∈ [0, 1], i = 1, 2, ψx = px = pv (v, θ)ϕx + pθ (v, θ)θx ,

(3.6)

and ψxx = pxx = pvv (v, θ)ϕ2x + pv (v, θ)ϕxx + pθθ (v, θ)θx2 + pθ (v, θ)θxx + 2pvθ (v, θ)ϕx φx ,

(3.7)

we see ψ2 ≤ C(v∞ , θ∞ )(ϕ, θ)(t)2

(3.8)

due to the bound of (v, θ) (3.4). And ξ2 ≤ C(v∞ , θ∞ )(ϕ, θ)(t)2

(3.9)

can be proved in the same way. Thus we see (ψ, ξ)(t)2 ≤ C(v∞ , θ∞ )(ϕ, θ)(t)2 ,

(3.10)

from which we may deduce bound of p and s by constants depend on p∞ and s∞ respectively. And the converse inequality (ϕ, φ)(t)2 ≤ C(p∞ , s∞ )(ψ, ξ)(t)2

(3.11)

could be deduced in the same way considering v = v(p, s), θ = θ(p, s). And this finishes the proof of Lemma 3.1. In the following, we will deduce a series of energy estimates. Firstly, in order to get the initial energy estimates, we define the following normalized entropy η(v, u, s; v∞ , 0, s∞ ) around (v∞ , 0, s∞ ): η(v, u, s; v∞ , 0, s∞ ) = (e(v, θ)+

u2 )−e(v∞ , s∞ )−ev (v∞ , s∞ )(v −v∞ )−es (v∞ , s∞ )(s−s∞ ), (3.12) 2

and deduce the following entropy identity: ηt (v, u, s; v∞ , 0, s∞ ) + (ψu)x −

 μuux v

2

+ μ uvθx = x



ωx2 vθ

+

vω 2 θ

 φ.

(3.13)

With this in hand, we obtain the following lemma: Lemma 3.2. (ϕ, u, ξ)2L2

t + 0

R

 u2x dxdτ ≤ C(v∞ , s∞ ) (ϕ0 , u0 , ξ0 )2L2 + E 3 (T ) . vθ

(3.14)

Proof. Under the a priori assumption (3.4), we can easily deduce that η(v, u, s; v∞ , 0, s∞ ) = C(v∞ , s∞ )|(ϕ, u, ξ)|2 ,

(3.15)

based on which, we have by integrating entropy identity (3.13) with respect to t and x over [0, t] × R that    t 2 t  2 2 ω u vω x x 2 (ϕ, u, ξ)2 + μ dxdτ ≤ C(v∞ , s∞ ) (ϕ0 , u0 , ξ0 ) + + φdxdτ θ  0 R vθ  0 R vθ t  2 ≤ C(v∞ , s∞ ) (ϕ0 , u0 , ξ0 )2 + C|φ|L∞ ωx + ω 2 dxdτ t,x ≤ C(v∞ , s∞ ) ((ϕ0 , u0 , ξ0 )2 + E 3 (T )) .

0

R

(3.16) And we finish this proof. We complete the initial energy estimates by deducing the estimates of ω in the following lemma, which can be easily obtained by multiplying (2.2)5 with ω, and then integrating it with respect to t and x over [0, t] × R:

Lemma 3.3.

R

ω2 dx + 2

t  R

0

 ω02 ωx2 2 + vω dxdτ = dx v R 2

(3.17)

Next, we provide estimates on (ψx , ux , ξx , ωx ) in Lemma(3.4)-Lemma(3.6). First, for the estimates of (ψx , ux ), we have  t u2x μ 2 v 2 ψx2 + dx + uxx dxdτ 2 2γRθ R v 0 R   2 2 u0x v02 ψ0x + dx + C(v∞ , s∞ )E 3 (T ). ≤ 2 2γRθ 0 R 

Lemma 3.4.

(3.18)

2

ψx Proof. Applying ∂x to (3.12)1 and (2.2)2 , multiplying them by vγRθ , ux respectively, and then adding them together,integrating the final result with respect to t and x over [0, t] × R, we have    2 t t  ux μ 2 v 2 ψx2 ϕx u x ψ x φ x u x ψ x (γ + 1)vux ψx2 + dx + uxx dxdτ = +2 − dxdτ 2 2γRθ 2γRθ v θ R R v R 0 0  t  ω 2 ϕ x ψ x φx ω 2 ψ x v2 ωx ωxx ψx μu2x +(γ − 1) −2 x + +2 ωωx ψx − 2 ψx ϕx dxdτ 2 γRθ γRθv γRθ γRθ γRθv 0 R   2 2 u0x v02 ψ0x + dx + 2 2γRθ0 R   2 8  u20x v02 ψ0x = + dx, Ij + 2 2γRθ0 j=1 R (3.19) where Ij denotes the term on the right hand side of (3.19) which can be estimated one by one in the following: t

|I1 | ≤ C(v∞ , s∞ )|ux |L∞ t,x 

|I2 + I3 | ≤ C(v∞ , s∞ ) |ϕx |

L∞ t,x

+ |φx |

|I4 | ≤ C(v∞ , s∞ )|ψx |L∞ t,x

|I5 | ≤ C(v∞ , s∞ )|ϕx |

L∞ t,x

|ψx |

0

R

 t  0

t  0

t L∞ t,x

L∞ t,x

0

R

ψx2 dxdτ ≤ C(v∞ , s∞ )E 3 (T ),

R

vωx2

R

u2x + ψx2 dxdτ ≤ C(v∞ , s∞ )E 3 (T ),

ω2 + xx v

(3.20)

(3.21)

 dxdτ ≤ C(v∞ , s∞ )E 3 (T ),

ωx2 dxdτ ≤ C(v∞ , s∞ )E 4 (T ) ≤ C(v∞ , s∞ )E 3 (T ), v

since we assume in (3.17) that E(T ) ≤ 2 0 < 1. And t t μ 2 μ 2 2 2 uxx + C(v∞ , s∞ )(|ϕx |L∞ u dxdτ + |ψx |L∞ ) |I6 + I7 | ≤ t,x t,x 2v v x R R 0 0 t t μ 2 μ 2 2 ≤ uxx dxdτ + C(v∞ , s∞ )E (T ) ux dxdτ 4v 0 R v 0 t R μ 2 ≤ uxx dxdτ + C(v∞ , s∞ )E 4 (T ) 4v 0 t R μ 2 ≤ uxx dxdτ + C(v∞ , s∞ )E 3 (T ), 4v 0 R

(3.22)

(3.23)

(3.24)

|I8 | ≤ C(v∞ , s∞ )|ψx |L∞ |ϕx |L∞ t,x t,x

t 0

R

u2x dxdτ ≤ C(v∞ , s∞ )E 4 (T ) ≤ C(v∞ , s∞ )E 3 (T ). vθ

(3.25)

Substituting I1 − I8 back into (3.19), we finish this proof. As for ξx , we have

Lemma 3.5.

R

ξx2 dx

≤

R

2 ξ0x dx + C(v∞ , s∞ )E 3 (T ).

(3.26)

Proof. Multiplying ∂x (3.12)2 by ξx and then integrating it with respect to t and x over [0, t) × R, we have t  2ωx ωxx ξx ωx2 ϕx ξx ϕx ξx ω 2 2vωωx ξx vω 2 ξx φx ωx2 φx ξx 2 2 − + + − − ξx dx = ξ0x dx + 2θ vθ v θ θ θ vθ2 0 R R R  2 2φ ξ x x +2μ ux uvθxx ξx − μ uxvϕ2xθξx − μ uxvθ dxdτ 2 =

17  j=9

Ij ,

(3.27) where Ij denotes the term on the right hand side of (3.27) which can be estimated one by one in the following:  1  2 ωxx 2 (3.28) dxdτ ≤ C(v∞ , s∞ )E 3 (T ), vωx + |I9 | ≤ C(v∞ , s∞ )|ξx |L∞ t,x v 0 R |I10 + I11 | ≤ C(v∞ , s∞ )|ϕx |L∞ |ξx |L∞ t,x t,x

1

( 0

R

|I12 | ≤ C(v∞ , s∞ )|ξx |L∞ t,x

|I13 + I14 | ≤ C(v∞ , s∞ )|φx |

 |I15 | ≤ C(v∞ , s∞ ) |ξx |L∞ t,x

|ξx |

L∞ t,x

t 0

R

1

ωx2 + vω 2 )dxdτ ≤ C(v∞ , s∞ )E 4 (T ) ≤ C(v∞ , s∞ )E 3 (T ), v (3.29)

(

0

L∞ t,x



1

R

(

0

R

ωx2 + vω 2 )dxdτ ≤ C(v∞ , s∞ )E 3 (T ), v

(3.30)

ωx2 + vω 2 )dxdτ ≤ C(v∞ , s∞ )E 4 (T ) ≤ C(v∞ , s∞ )E 3 (T ), v (3.31)

μ 2 u dxdτ + |ξx |L∞ t,x 4v xx

t 0

R

μ 2 u dxdτ v x

 ≤ C(v∞ , s∞ )E 3 (T ), (3.32)

and 



|I16 + I17 | ≤ C(v∞ , s∞ ) |ϕx |L∞ + |ϕx |L∞ |ξx |L∞ t,x t,x t,x ≤ C(v∞ , s∞ )E 3 (T ).

t 0

R

Substituting I9 − I17 back into (3.27), we finish this proof.

μ 2 u dxdτ ≤ C(v∞ , s∞ )E 4 (T ) (3.33) v x

As for ωx , we have Lemma 3.6.  t  2 2 ωxx ωx2 ω0x 2 dx + + vωx dxdτ ≤ (x)dx + C(v∞ , s∞ )E 3 (T ). 2 v 2 0 R R R

(3.34)

Proof. Multiplying ∂x (2.2)4 by ωx and then integrating it with respect to t and x over [0, t] × R, we have  t  2 t   2 ωx ϕx ωxx ωx2 ω0x ωxx 2 dx + + vωx dxdt = (x)dx + − ϕx ωωx dxdt v v2 R 2 R R 2 R 0 0 19 2  ω0x (x)dx + Ij . = R 2 j=18 (3.35) Here the Ij denote the corresponding terms in the right hand side of (3.35). By Cauchy-Schwartz inequality, we deduce t 2 ωxx 1 t 1 t 2 2 ∞ 2 |I18 | ≤ dxdt+C(v∞ , s∞ )E 3 (T ), ωxx dxdt+C(v∞ , s∞ )|ϕx |Lt,x ωx dxdt ≤ 4 0 R 4 v R R 0 0 (3.36) and |I19 | ≤ |ϕx |L∞ t,x

t  0

R

 ωx2 2 + vω dxdt ≤ C(v∞ , s∞ )E 3 (T ). v

(3.37)

Substituting these back into (3.35), we finish this proof. After obtaining (ux , ψx , ξx ), next, we derive estimates on Lemma 3.7.

t 0

R

ψx2 dxdτ

+ R

t 0

R

ψx2 dxdτ and we have

ψx udx ≤ C(v∞ , s∞ )(E 2 (0) + E 3 (T )).

(3.38)

Proof. Multiplying (2.2)2 by ψx , integrating it with respect to t and x over [0, t] × R, we deduce that t 2 ψx dxdτ + ψx udx 0 R  Rt t  γRθ 2 u3x ωx2 ux 2 ≤ ux ω + 2 + 2 dxdτ ψ0x u0 dx + ux dxdτ + (γ − 1) 2 v v v R R R 0 0 t +C(v∞ , s∞ ) (uxx ψx + ϕx ux ψx ) dxdτ 0 R t  t  γRθ 2 μ 2 2 2 ∞ ≤ u ψ0x u0 dx + C(v∞ , s∞ ) u dxdτ + C(v , s )|u | + + ω vω ∞ ∞ x Lt,x x dxdτ v2 x vθ x R R R 0 0 t t 1 t 2 2 +C(v∞ , s∞ ) uxx dxdτ + ψ dxdτ + C(v∞ , s∞ )|ux |L∞ ϕx ψx dτ t,x 2 0 R x 0 0 R 1 t 3 ≤ ψ0x u0 dx + C(v∞ , s∞ )E (T ) + ψx2 dxdτ, 2 R R 0 (3.39) from which, (3.38) follows immediately.

In turn, we provide estimates on (ψxx , uxx , ξxx , ωxx ) in lemma3.8, lemma3.9 and lemma3.10. First, for (uxx , ψxx ), we have Lemma 3.8.    2 t 2  2 2 2 uxx u0xx uxxx v 2 ψxx v 2 ψ0xx +β dx + μ dxdτ ≤ +β dx + C(v∞ , s∞ )E 3 (T ). 2 2γRθ v 2 2γRθ 0 R R R (3.40) 2

ψxx Proof. Applying ∂x2 to (3.12)1 and (2.2)2 , multiplying them by β vγRθ , uxx respectively, and then adding them together, integrating the final equality with respect to t and x over [0, t] × R, we have    2 t 2  2 2 2 uxx u0xx uxxx v 2 ψxx v02 ψ0xx +β dx + μ dxdτ ≤ +β dx 2 2γRθ 2 2γRθ0 0 R R v R t  2 ωx ωxxx ψxx + ωxx +C(v∞ , s∞ ) ψxx + ϕx ωx ωxx ψxx + ωx2 ϕxx ψxx + ϕ2x ωx2 ψxx + u2xx ψxx + ux uxxx ψxx 0

R

+ϕx ux uxx ψxx + ωx2 ψxx + ωωxx ψxx + ux uxx ϕx ψxx + u2x ϕx ψxx + u2x ϕxx ψxx + φx ψxx uxx + ϕx ψxx uxx +ψxx φxx ux + φx ϕx ux ψxx + ϕ2x ux ψxx + ϕxx ux ψxx 2 2 2 2 +ωx2 ψxx + ω 2 ψxx + u2x ψxx + ux ψxx + ϕx uxx uxxx + ux ϕxx uxxx + ux ϕ2x uxxx | dxdτ   2 45 2  u0xx v 2 ψ0xx = +β dx + Ij , 2 2γRθ R j=20

(3.41) where Ij denotes the corresponding term on the right hand side of (3.41). For simplicity, we only estimate some typical terms in the following: t |I20 | ≤ C(v∞ , s∞ ) |ωx |L∞ ωxxx ψxx dτ 0 t 1 1 ≤ C(v∞ , s∞ )E(T ) ωx  2 ωxx  2 ωxxx dτ 0 t  ≤ C(v∞ , s∞ )E(T ) ωx 2 + ωxx 2 + ωxxx 2 dτ ≤ C(v∞ , s∞ )E(T )3

0



|I21 | ≤ C(v∞ , s∞ ) ≤ C(v∞ , s∞ )E(T ) ≤ C(v∞ , s∞ )E(T )

t

0

|ωxx |L∞ ωxx ψxx dτ

t

0 t 0

3

1

ωxx  2 ωxxx  2 dτ 

ωxx 2 + ωxxx 

(3.43) 2

dτ

≤ C(v∞ , s∞ )E(T )3 , t |ϕx |L∞ |ωx |L∞ ωxx ψxx dτ |I22 | ≤ C(v∞ , s∞ ) 0 t 1 1 1 1 ≤ C(v∞ , s∞ )E(T )2 ϕx  2 ϕxx  2 ωx  2 ωxx  2 dτ 0 t  ≤ C(v∞ , s∞ )E(T )2 ϕx 2 + ϕxx 2 + ωx 2 + ωxx 2 dτ 0

(3.42)

≤ C(v∞ , s∞ )E(T )4 ≤ C(v∞ , s∞ )E(T )3 ,

(3.44)



t

|ωx |2L∞ ϕxx ψxx dτ |I23 | ≤ C(v∞ , s∞ )E(T ) 0 t 2 ≤ C(v∞ , s∞ )E(T ) ωx ωxx dτ 0 t  ≤ C(v∞ , s∞ )E(T )2 ωx 2 + ωxx 2 dτ 4

(3.45)

0

≤ C(v∞ , s∞ )E(T )4 ≤ C(v∞ , s∞ )E(T )3 , and

|I24 | ≤ C(v∞ , s∞ )

t

0

≤ C(v∞ , s∞ )E(T )

|ϕx |2L∞ |ωx |L∞ ωx ψxx dτ

t

0

≤ C(v∞ , s∞ )E(T )3

3

t 0

1

ϕx ϕxx ωx  2 ωxx  2 dτ

(3.46)



ωx 2 + ωxx 2 dτ

≤ C(v∞ , s∞ )E(T )5 ≤ C(v∞ , s∞ )E(T )3 . The other terms can be estimated similarly and hence the proof is completed. Similarly, multiply ∂x2 (3.12)2 by ξxx and integrating with respect to x and t over [0, t] × R, we have

Lemma 3.9.

R

2 ξxx dx

≤

R

2 ξ0xx dx + C(v∞ , s∞ )E(T )3 .

(3.47)

For ωxx , we have Lemma 3.10. R

2 ωxx dx + 2

t  R

0

 2 2 ω0xx ωxxx 2 + vωx dxdτ ≤ dx + C(v∞ , s∞ )E 3 (T ). v 2 R

Proof. Multiplying ∂x2 (2.2)4 with ωxx and integrating on QT , we have  t  2 2 ωxxx ωxx 2 dx + + vωx dxdτ v 0 R 2  R t ωxx ωxxx ϕx ωx ωxx ϕxx ωx ϕ2x ωxxx = + + v2 v2 v3 0 R 2 + −ϕx ωxx − ϕxx ωx ωxx − ϕx ωx2 ) dxdτ =

30  j=25

(3.48)

(3.49)

Ij ,

where Ij denotes the terms on the right hand side of (3.49) and can be estimated in the following: 1 |I25 | ≤ 8 1 |I26 | ≤ 8

t 0

R

t 0

R

2 ωxxx dxdτ + C(v∞ , s∞ )|ϕx |2L∞ t,x v 2 ωxxx dxdτ + C(v∞ , s∞ )|ωx |2L∞ t,x v

t 0

R

2 vωxx dxdτ,

(3.50)

ϕ2xx dxdτ,

(3.51)

t 0

R

1 |I27 | ≤ 8

1 |I29 | ≤ 8

t

t 2 ωxxx 2 dxdτ + C(v∞ , s∞ )|ϕx |L∞ vωx2 dxdτ, t,x R v R 0 0 t 2 vωxx dxdτ, |I28 | ≤ C(v∞ , s∞ )|ϕx |L∞ t,x 0

t 0

R

2 vωxx dxdτ

+

R

C(v∞ , s∞ )|ωx |2L∞ t,x

and I30 ≤ C(v∞ , s∞ )|ϕx |L∞ t,x

t 0

t 0

R

ϕ2xx dxdτ,

ωx2 dxdτ

R

(3.52) (3.53) (3.54)

(3.55)

Substituting I25 − I30 back into (3.49), we finish this proof. t 2 As for the term ψxx dxdτ , we have 0

R

Lemma 3.11. t 2 ψxx dxdτ + β ψxx ux dx ≤ β ψ0xx u0x dx + C(v∞ , s∞ )E 3 (T ). β 0

R

R

(3.56)

R

Proof. Multiplying ∂x (2.2)2 with βψxx and integrating it with respect to t and x over [0, t] × R, we have t 2 ψxx + β ux ψxx dx β R 0 R t  ≤β ψ0xx u0x dx + C(v∞ , s∞ ) uxx ωx ωxx + ϕx ωx2 + uuxx + ωωx + u2x ϕx − ux x − φx ux + ϕx ux dxdτ 0 R R 42  ψ0xx u0x dx + Ij , =β R

j=31

(3.57) where Ij denotes the terms on the right hand side of (3.57) and can be estimated in the following: |I31 | ≤ C(v∞ , s∞ )|ωx |L∞ t,x

t 0

ωx ωxx dτ ≤ E(T )

|I32 | ≤ C(v∞ , s∞ )|ϕx |

L∞ t,x

|ωx |

L∞ t,x

t 0

t 0

(ωx 2 + ωxx 2 )dτ ≤≤ C(v∞ , s∞ )E 3 (T ), (3.58)

2

ωx uxx dτ ≤ E (T )

|I33 | ≤ C(v∞ , s∞ )|ϕx |L∞ |ux |L∞ t,x t,x |I34 | ≤ C(v∞ , s∞ )|ω|L∞ t,x

t 0

|ux |L∞ |I35 | ≤ C(v∞ , s∞ )|ϕx |L∞ t,x t,x

t 0



t 0

(ωx 2 +uxx 2 )dτ ≤ C(v∞ , s∞ )E 3 (T ),

uxx 2 dτ ≤ C(v∞ , s∞ )E 3 (T ),

uxx ωx dτ ≤ C(v∞ , s∞ )E 3 (T ),

t 0

ux uxx dτ ≤ C(v∞ , s∞ )E 3 (T ),

(3.59) (3.60) (3.61) (3.62)

|I36 | ≤ C(v∞ , s∞ )

t R

0

u2xx dτ ≤ C(v∞ , s∞ )E 3 (T ),

+ |φx |L∞ ) |I37 + I38 | ≤ (|ϕx |L∞ t,x t,x t



t 0

(3.63)

ux uxx dτ ≤ E 3 (T ),

2 ψxx dxdτ + 4 2

(3.64)

t

u3xxx dxdτ |I39 | ≤ C(v∞ , s∞ ) 0 0 R R t 2 ψxx ≤ dxdτ + E 2 (0) + C(v∞ , s∞ )E 3 (T ), R 2 0 t |I40 | ≤ C(v∞ , s∞ )|ϕx |L∞ uxx ψxx dτ ≤ C(v∞ , s∞ )E 3 (T ), t,x

(3.65)

(3.66)

0

|I41 | ≤



C(v∞ , s∞ )|ϕx |2L∞ t,x

|I42 | ≤ C(v∞ , s∞ ) ≤ C(v∞ , s∞ )E(T )

t 0

t

0 0

ux ψxx dτ ≤ C(v∞ , s∞ )E 4 (T ) ≤ C(v∞ , s∞ )E 3 (T ),

|ux |L∞ ϕxx ψxx dτ ≤ E(T )

t



t 0

1

(3.67)

1

ux  2 uxx  2 ψxx 

ux 2 + uxx 2 + ψxx 2 dτ ≤ C(v∞ , s∞ )E 3 (T ).

(3.68)

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