Research article

Large-time behavior of cylindrically symmetric Navier-Stokes equations with temperature-dependent viscosity and heat conductivity

  • Received: 27 November 2023 Revised: 09 January 2024 Accepted: 17 July 2024 Published: 21 August 2024
  • 35Q35, 76N10

  • In this study, the initial-boundary value problem for cylindrically symmetric Navier-Stokes equations was considered with temperature-dependent viscosity and heat conductivity. Firstly, we established the existence and uniqueness of a strong solution when the viscosity and heat conductivity were both power functions of temperature. Moreover, the large-time behavior of the strong solution was obtained with large initial data, since all of the estimates in this paper were independent of time. It is worth noting that we identified the relationship between the initial data and the power of the temperature in the viscosity for the first time.

    Citation: Dandan Song, Xiaokui Zhao. Large-time behavior of cylindrically symmetric Navier-Stokes equations with temperature-dependent viscosity and heat conductivity[J]. Communications in Analysis and Mechanics, 2024, 16(3): 599-632. doi: 10.3934/cam.2024028

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  • In this study, the initial-boundary value problem for cylindrically symmetric Navier-Stokes equations was considered with temperature-dependent viscosity and heat conductivity. Firstly, we established the existence and uniqueness of a strong solution when the viscosity and heat conductivity were both power functions of temperature. Moreover, the large-time behavior of the strong solution was obtained with large initial data, since all of the estimates in this paper were independent of time. It is worth noting that we identified the relationship between the initial data and the power of the temperature in the viscosity for the first time.



    As it is well-known that the cylindrically symmetric Navier-Stokes equations take the form

    ρt+(rρu)rr=0, (1.1)
    ρ(ut+uur)ρv2r+Pr=(λ(ru)rr)r2uμrr, (1.2)
    ρ(vt+uvr)+ρuvr=(μvr)r+2μvrr(μv)rruvr2, (1.3)
    ρ(wt+uwr)=(μwr)r+μwrr, (1.4)
    ρ(e+uer)+P(ru)rr=(κrθr)rr+Q, (1.5)

    where ρ(r,t) is the density, u(r,t), v(r,t), and w(r,t) are velocities in different directions, θ(r,t) is the temperature, the pressure P and the internal energy e are related with the density and temperature

    P=P(ρ,θ)=Rρθande=e(ρ,θ)=cvθ, (1.6)

    the specific gas constant R and the specific heat at constant volume cv are positive constants, respectively; the symbol Q denotes

    Q=λ(ru)2rr24μuurr+μw2r+μ(vrvr)2, (1.7)

    μ and λ are viscosity coefficients, and κ is the heat conductivity coefficient.

    Without loss of generality, we shall consider the system (1.1) with the following initial-boundary data:

    {(ρ,u,v,w,θ)|t=0=(ρ0,u0,v0,w0,θ0)(r),0<arb<,(u,v,w,rθ)|r=a=(u,v,w,rθ)|r=b=0,t0. (1.8)

    Our main goal is to show the large-time behavior of global solutions to the initial-boundary value problem (1.1)–(1.8) with large initial data. For this purpose, it is convenient to transform the initial-boundary value problem (1.1)–(1.8) into Lagrangian coordinates. We introduce the Lagrangian coordinates (t,x) and denote (˜ρ,˜u,˜v,˜w,˜θ)(t,x)=(ρ,u,v,w,θ)(t,r), where

    r=r(t,x)=r0(x)+t0u(s,r(s,x))ds, (1.9)

    and

    r0(x):=f1(x),f(r):=rayρ0(y)dz.

    Note that the function f is invertible on [a,b] provided that ρ0(y)>0 for each y[a,b] (which will be assumed in Theorem 1.1). Due to (1.1)1 and (1.8), we see

    tr(t,x)ayρ(t,y)dy=0.

    Then it is easy to check

    r(t,x)ayρ(t,y)dy=f(r0(x))=xandr(t,1)byρ(t,y)dy=0, (1.10)

    which translates the domain [0,T]×[a,b] into [0,T]×[0,1]. Hereafter, we denote (˜ρ,˜u,˜v,˜w,˜θ) by (ρ,u,v,w,θ) for simplicity. The identities (1.9) and (1.10) imply

    rt(t,x)=u(t,x),rx(t,x)=r1τ(t,x), (1.11)

    where τ:=ρ1 is the specific volume. By means of identities (1.11), system (1.1)–(1.8) is changed to

    τt=(ru)x, (1.12)
    utv2r+rPx=r(λ(ru)xτ)x2uμx, (1.13)
    vtuvr=r(μrvxτ)x+2μvx(μv)xμτvr2, (1.14)
    wt=r(μrwxτ)x+μwx, (1.15)
    et+P(ru)x=(κr2θxτ)x+Q, (1.16)

    where t>0, xΩ=(0,1), P=Rθτ, e=Rθγ1, and Q=λ(ru)2xτ4μuux+μr2w2xτ+μτ(rvxτvr)2.

    Throughout this paper, we assume that μ,λ, and κ are power functions of absolute temperature as follows:

    μ=˜μθα,λ=˜λθα,κ=˜κθβ, (1.17)

    where constants ˜μ,˜λ,˜κ,α, and β are positive constants.

    The objective of this paper is to study the global existence and stability of the solutions to an initial-boundary value problem of (1.12)–(1.16) with the initial data:

    (τ,u,v,w,θ)(x,0)=(τ0,u0,v0,w0,θ0),x(0,1), (1.18)

    and the boundary conditions:

    (u,v,w,θx)(0,t)=(u,v,w,θx)(1,t)=0,t0. (1.19)

    Using Navier-Stokes equations as a model for describing fluid motion has been widely accepted by the physics community. In recent years, some significant progress has been made in the study of Navier-Stokes equations with constant viscosity coefficients. When the initial value has a certain small property and vacuum state does not exist, the global existence, uniqueness, and large-time behavior of the solutions can be easily calculated [1,2,3,4,5,6,7,8]. However, solving the problem of large initial values is very challenging, and the first significant breakthrough was achieved by Lions [9]. Besides, by assuming that the initial value is only sufficiently small in the energy space, Hoff [10,11] confirmed the existence of global weak solutions. In the process of studying fluid motion, a vacuum state is often involved, which makes calculations far more complex. The results in [12,13] indicate the Cauchy problem of Navier-Stokes equations with constant coefficients containing vacuum state is not appropriate. This uncertainty is reflected by the fact that the solutions of the system have no continuous dependence on the initial values. Based on physical considerations, Liu-Xin-Yang [12] studied the Cauchy problem of the Navier-Stokes equations with density dependent viscosity, and proved its local suitability. However, only when the temperature and density change within a suitable range, real fluids can be considered as ideal fluids (viscosity coefficients are constants). In the case of large changes in temperature or density, the viscosity of the real fluid will vary greatly [14].

    On the other hand, Navier-Stokes equations can be developed using the Chapman-Enskog expansion of the microscopic particle collision model Boltzman equation. Consequently, it can be determined that the viscosity depends on the temperature. However, compared to the abundant research using classical models, the studies on the physical case using the temperature-dependent viscosity model are lacking. Because the viscosity and heat conductivity are both temperature-dependent, degeneracy and strong non-linearity may appear. Pan-Zhang [15] and Huang-Shi [16] obtained global strong solutions and large-time behavior in bounded domains for one-dimensional Navier-Stokes equations, when α=0 and 0<β<1. The studies of Liu-Yang-Zhao-Zhou [17] and Wan-Wang [18] also acquired global solutions of Navier-Stokes equations in one dimensional and cylindrically symmetrical cases, respectively, with the requirement that |γ1| was small enough. Wang-Zhao [19] removed the smallness condition of |γ1|, and established global classical solutions to Navier-Stokes equations in the one-dimensional whole space when μ and κ satisfy:

    μ=˜μh(τ)θα,κ=˜κh(τ)θα,

    where α is small enough. In their calculations, the viscosity and heat-conductivity were dependent on temperature and density, and to overcome the difficulties caused by density, the following conditions could not be removed:

    h(τ)1τ1L(Ω)+h(τ)1τL(Ω)C.

    This means that estimate of τxL2(Ω) can be directly obtained without the upper and lower bounds of density, as long as the coefficient μ1 or κ1 appears. However, if h(τ) is constant, then the constants l1=l2=0 and the result of this case cannot be established using the model in [19]. Recently, Sun-Zhang-Zhao [20] considered an initial-boundary value problem of the compressible Navier-Stokes equations for one-dimensional viscous and heat-conducting ideal polytropic fluids with temperature-dependent transport coefficients, and discovered the global-in-time existence of strong solutions. In that paper, the initial data could be large if α0 is small and the growth exponent β0 is arbitrarily large. It is worth mentioning that the smallness of α>0 depends on the size of the initial data. However, unfortunately the study did not provide a specific relationship between α and the initial data in [20]. Our main results are concluded as follows.

    Theorem 1.1. For given positive constants M0,V0>0, assume that

    (τ0,u0,v0,w0,θ0)H2(Ω)M0,infx(0,1){τ0,θ0}V0. (1.20)

    Then there exist ϵ0>0 and C0 which depend only on β, M0, and V0, such that the initial-boundary value problem (1.12)–(1.19) with 0αϵ0:=min{|α1|,|α2|} and β>0 admit a unique global-in-time strong solution (τ,u,v,w,θ) on [0,1]×[0,+) satisfying

    C10τ(x,t)C0,C11θ(x,t)C1,

    and

    (τˉτ,u,v,w,θE0)C([0,+);H2(Ω)),

    where α1,α2 defined in what follows are dependent only on β, M0, and V0 (see details in (3.2), (3.5), and (3.6)). Moreover, for any t>0, the exponential decay rate is

    (τˉτ,u,v,w,θE0)2H1+rˉr2H2Ceγ0t, (1.21)

    where

    ˉτ=10τdx,E0=10(θ0+u20+v20+w202cv)dx,ˉr=[a2+2ˉτx]12.

    A few remarks are in order.

    Remark 1. For k=1,2 and 1p, we adopt the simplified nations for the standard Sobolev space as follows:

    :=L2(Ω),k:=Hk(Ω),f:=maxxΩ|f(x)|,Lp:=Lp(Ω).

    Remark 2. We remark here that the growth exponent β(0,+) can be arbitrarily large, and the choice of ϵ0>0 depends only on β, V0, and the H2norm of the initial data. An outline of this paper is as follows. We devote Section 2 to a discussion of a number of a priori estimates independent of time, which are needed to extend the local solution to all time. Based on the previous estimates, the main results, Theorem 1.1 are proved in Section 3.

    Remark 3. In this paper, the positive c, C, and Ci(i=0,1,,16) are some positive constants which depend only on β, M0, and V0, but not on the time t. Furthermore, c and C are different from line to line.

    First of all, define

    X(t1,t2;m1,m2;N):={(τ,u,v,w,θ)C([t1,t2];H2(Ω)),τxL2(t1,t2;H1(Ω))(ux,vx,wx,θx)L2(t1,t2;H2(Ω)),τtC([t1,t2];H1(Ω))L2(t1,t2;H1(Ω)),(ut,vt,wt,θt)C([t1,t2];L2(Ω))L2(t1,t2;H1(Ω)),τm1,θm2,E(t1,t2)N2,(x,t)[0,1]×[t1,t2]},

    where N, m1, m2, and t1,t2(t2>t1) are constants and

    E(t1,t2):=supt1tt2(τx,ux,θx)21+θt2+t2t1θt2dt

    with

    θt|t=t1:=1cv[P(ru)x+(κr2θxτ)x+Q]|t=t1,
    θxt|t=t1:=1cv[P(ru)x+(κr2θxτ)x+Q]x|t=t1.

    The main purpose of this section is to derive the global t-independent estimates of the solutions (τ,u,v,w,θ)X(0,T;m1,m2,N).

    We start with the following basic energy estimate.

    Lemma 2.1. Assume that the conditions listed in Theorem 1.1 hold. Then there exists a constant 0<ϵ11 depending only on M0 and V0, such that if

    mα22,Nα2,αH(m1,m2,N)ϵ1, (2.1)

    where

    H(m1,m2,N):=(m1+m2+N+1)5,

    then for T0,

    10ηˆθ(τ,u,v,w,θ)(x,t)dx+T010[τu2θ+u2x+w2xτθ+θβθ2xτθ2+τθ(rvxτvr)2]dxdsC, (2.2)

    where

    ηˆθ(τ,u,v,w,θ):=ˆθϕ(τˉτ)+u2+v2+w22+cvˆθϕ(θˆθ),ϕ(z):=zlogz1.

    Proof. Multiplying (1.12)–(1.16) by Rˆθ(ˉτ1τ1), u, v, w, and (1ˆθθ1), respectively, integrating over [0,1], and adding them together, one obtains

    ddt10ηˆθ(τ,u,v,w,θ)dx+10[˜κr2θβθ2xτθ2+Qθ]dx=0, (2.3)

    where Q=λ(ru)2xτ4μuux+μr2w2xτ+μτ(rvxτvr)2.

    Apparently, by means of λ=2μ+λ, one has

    λ(ru)2x4τμuux=(2μ+λ)r2u2x+(2μ+λ)τ2u2r2+2λτuux=2μr2u2x+2μτ2u2r2+2μ+3λ3[rux+τur]22μ3[rux+τur]2=23μ(r2u2x+τ2u2r2)+2μ+3λ3[rux+τur]2+2μ3[ruxτur]2 23μ(r2u2x+τ2u2r2).

    Thus, one has

    QCu2xτ+Cτu2+Cw2xτ+Cτ(rvxτvr)2,

    which combined with (2.1) and (2.3) yields

    ddt10ηˆθ(τ,u,v,w,θ)(t,x)dx+c10(τu2θ+u2x+w2xτθ+θβθ2xτθ2+τθ(rvxτvr)2)dx0. (2.4)

    Integrating (2.4) over (0,T), we can obtain (2.2) by the initial conditions (τ0,u0,v0,θ0).

    Next, by means of Lemma 2.1, we derive the upper and lower bounds of τ.

    Lemma 2.2. Assume that the conditions of Lemma 2.1 hold. Then for (x,t)Ω×[0,),

    C10τ(x,t)C0.

    Proof. The proof is divided into three steps.

    Step 1 (Representation of the formula for τ).

    It follows from (1.13) that

    (ur)t+u2v2r2+2uμxr+Px=(λ(lnτ)t)x=λ(lnτ)xt+λx(ru)xτ.

    that is

    (uλr)t+g+(λ1P)x=(lnτ)xt, (2.5)

    where

    g:=u2v2λr2+2uμxλr(λ1)xPλx(ru)xλτ(λ1)tur.

    Integrating (2.5) over [0,t]×[x1(t),x], we have

    xx1(t)(uλru0λ0r0)dξ+t0xx1(t)gdξds+t0λ1P(x)λ1P(x1)ds=lnτ(x,t)lnτ(x1(t),t)[lnτ0(x)lnτ(x1(t),0)], (2.6)

    where x1(t)[0,1] is determined by the following progresses. Next, for convenience, we define

    F:=(ru)xτλ1Px0g(ξ)dξ,φ:=t0F(x,s)ds+x0u0λ0r0dξ.

    It follows from the definitions above that

    φx=uλr,φt=F. (2.7)

    By the definition of F and (1.12), one has

    t0[λ1P(x1(t),s)+x1(t)0g(ξ,s)dξ]ds=t0((ru)xτF)(x1(t),s)ds=lnτ(x1(t),t)lnτ(x1(t),0)t0F(x1(t),s)ds. (2.8)

    Due to (1.12) and (2.7), we have

    (τφ)t(ruφ)x=τφtruφx=τFu2λ=(ru)xτPλτx0g(ξ)dξu2λ. (2.9)

    Integrating (2.9) over [0,t]×Ω, one has

    10φτdx=10τ0x0(u0λ0r0)(ξ)dξdxt010[τλP+τx0gdξ+u2λ]dxds. (2.10)

    Hence, by virtue of the mean value theorem, there exits x1(t)[0,1] such that φ(x1(t),t)=10φτdx. By the definition of φ, (2.8), and (2.10), one obtains

    t0F(x1(t),s)ds=φ(x1(t),t)x1(t)0u0λ0r0(ξ)dξ=10τ0x0u0λ0r0(ξ)dξdxt010(τλP+τx0gdξ+u2λ)dxdsx1(t)0u0λ0r0(ξ)dξ. (2.11)

    Putting (2.11) into (2.8), it follows that

    t0(Pλ(x1(t),s)+x1(t)0g(ξ,s)dξ)ds=lnτ(x1(t),t)lnτ(x1(t),0)10τ0x0u0λ0r0(ξ)dξdx+x1(t)0u0λ0r0(ξ)dξ+t010(τλP+τx0gdξ+u2λ)dxds. (2.12)

    Inserting (2.12) into (2.6), we derive

    t0Pλds+t0x0gdξdst010(τλP+τx0gdξ+u2λ)dxds+xx1(t)(uλru0λ0r0)dξ+10τ0x0u0λ0r0dξdxx1(t)0u0λ0r0dξ=lnτlnτ0. (2.13)

    Let

    g=u2v2λr2+g1,

    where

    g1:=2uμxλr(λ1)xPλx(ru)xλτ(λ1)tur.

    It follows from (2.13) that

    τ=B1AD, (2.14)

    where

    A:=exp{t0[Pλ(x,s)+x0(g1(ξ,s)+u2λr2)dξ+10τx0(v2λr2g1)dξdx]ds},B:=exp{t0[10(τλP+τx0u2λr2(ξ)dξ+u2λ)dx+x0v2λr2dξ]ds},D:=τ0exp{10τ0x0u0λ0r0dξdxx1(t)0u0λ0r0dξ+xx1(t)(uλru0λ0r0)(ξ)dξ}.

    By (2.14), one has

    τD1B=A. (2.15)

    Define that

    J:=Pλ(x,s)+x0(g1(ξ,s)+u2λr2)dξ+10τx0(v2λr2g1)dξdx.

    Then, multiplying (2.15) by J gives

    τD1BJ=ddtA.

    Since A(0)=1, integrating the above equality over (0,t) about time, one has

    τ=DB1+1λt0B(s)B(t)D(t)D(s)τ[Pλ(x,s)+x0(g1(ξ,s)+u2λr2)dξ+10τx0(v2λr2g1)dξdx]ds. (2.16)

    Step 2 (Lower bound for τ). First of all, by means of (2.1) and (2.2), one has

    C1DC. (2.17)

    Next, we estimate B. Employing Jensen's inequality to the convex function ϕ, we have

    10zdxlog10zdx110ϕ(z)dx. (2.18)

    By (2.18) and Lemma 2.1, one obtains

    C110τdx,ˉθ:=10θdxC, (2.19)

    which means that

    C110τλPdxC. (2.20)

    Hence, by means of the definition of B and (2.20), choosing ε1 suitably small, there exist two constants C1 and C2, such that

    ec1tB(t)ec2t. (2.21)

    That is,

    ec1(ts)B(s)B(t)ec2(ts). (2.22)

    Apparently, by means of (2.1) and (2.19), we deduce

    |τ10τx0g1dξdx|C|α|τ2(θ1θxu+θατ1θx+θ1τ1θxu1+θ1τ1θtu)Cε1. (2.23)

    Similarly, one also has

    x0g1dξCε1. (2.24)

    Thus, for tt0<,

    τDB1Cε1t0ec2(ts)ds=DB1Cε1c2(1ec2t)Cect0ε2(1ec2t0).

    For a enough large t, we have

    infxΩτ(x,t)Ct0B(s)B(t)θdsε2(1ec2t). (2.25)

    So, we need the estimates of θ and B(s)B(t). By the mean value theorem and (2.19), there exits x2(t)[0,1], such that

    C1θ(x2(t),t)C. (2.26)

    By Cauchy-Schwarz's inequality and (2.19), one has

    |[ln(θ+1)]β2+1[ln(θ(x2(t),t)+1)]β2+1|=|xx2(ln(θ+1))β2θxτ(θ+1)τ(ξ)dξ|(10(ln(θ+1))βθ2xτ(θ+1)2dx)12(10τdx)12C(10θβθ2xτθ2dx)1/2,

    which means that

    θCC10θβθ2xτθ2dx. (2.27)

    By (2.16)–(2.17), (2.23)–(2.24), (2.21), Lemma 2.1, and (2.19), one has

    10τdxCect+Ct0B(s)B(t)ds,

    that is

    t0B(s)B(t)dsCCect. (2.28)

    Putting (2.27) into (2.25), by (2.22), (2.28), and Lemma 2.1, for a enough large t, one has

    t0B(s)B(t)θdsCt0B(s)B(t)(110θβθ2xτθ2dx)dsCCectC(t/20+tt/2)B(s)B(t)10θβθ2xτθ2dxdsCCectCt/20ec(ts)10θβθ2xτθ2dxdsCtt/210θβθ2xτθ2dxdsCCectCect/2Ctt/210θβθ2xτθ2dxdsC. (2.29)

    Inserting (2.29) into (2.25), for a large enough time T0, when t>T0, it follows that

    infxΩτ(x,t)C.

    Step 3 (Upper bound for τ). By (2.17), (2.22)–(2.24), and Lemma 2.1, one obtains

    τC+Ct0ec2(ts)τ(10θβθ2xτθ2dx+1)ds, (2.30)

    where we have used the results

    {θC+Cτ10θβθ2xτθ2dxwhen0<β1,θC+C10θβθ2xτθ2dxwhen1<β<. (2.31)

    In fact, by Hölder's inequality, for 0<β1,

    |θ1/2(x,t)θ1/2(x2(t),t)|10θ12θxdxτ1/2(10θβθ2xτθ2dx)1/2(10θ1βdx)1/2τ1/2(10θβθ2xτθ2dx)1/2. (2.32)

    For 1<β<,

    |θβ/2(x,t)θβ/2(x2(t),t)|10θβ/2θxθdx(10θβθ2xτθ2dx)1/2(10τdx)1/2. (2.33)

    By means of (2.26) and (2.32)–(2.33), we can obtain (2.31).

    Thus, the inequality (2.30) combined with Gronwall's inequality and Lemma 2.1 yields that for any t0,

    supt0τ(x,t)C.

    However, we cannot get the time-space estimate of vx in Lemma 2.1. To obtain this estimate, we need the following result.

    Lemma 2.3. Assume that the conditions listed in Lemma 2.1 hold. Then for any p>0 and T0,

    10θ1pdx+T010(θβθ2xθp+1+θα(u2+u2x+w2x)θp+θατθp(rvxτvr)2)dxdsC. (2.34)

    Proof. By Lemma 2.1, the result of (2.34) has been established for p=1. In the following steps, we do the estimate for p>0 and p1. Multiplying (1.16) by θp, integrating over [0,1], and using integration by parts gives

    cvp1ddt10θ1pdx+p10˜κr2θβθ2xτθp+1dx+10Qθpdx=R10θ1pτ(ru)xdx=R10θ1pE0τ(ru)xdx+RE010(ru)xτdx. (2.35)

    Apparently, there exists constant C(p) depending on p such that

    |θ1pE0|C(p)|θ1/2E1/20|(E1/20+θ12p). (2.36)

    By means of (2.35), (2.36), Lemma 2.2, (1.13), and (1.12), we deduce

    cvp1ddt10θ1pdx+p10˜κr2θβθ2xτθp+1dx+10QθpdxC(p)θ1/2E1/2010(E1/20+θ12p)(|u|+|ux|)dx+RE0ddt10lnτdxC(p)θ1/2E1/20[(10u2+u2xθdx)12(10θdx)12+(10θ1pdx)12(10u2+u2xτθpdx)12]+RE0ddt10lnτdxC(p)θ1/2E1/202+C(p)10u2+u2xθdx+δ10u2+u2xτθpdx+C(δ,p)θ1/2E1/20210θ1pdx+RE0ddt10lnτdx. (2.37)

    Thus, employing the truth of

    t0θ1/2E1/202dsC, (2.38)

    we can conclude from (2.37), Grönwall's inequality, and Lemma 2.2 that (2.34) is correct. In fact,

    θ1/2E1/20θ1/2ˉθ1/2+ˉθ1/2E1/20. (2.39)

    By virtue of Lemmas 2.1–2.2 and (2.19), one has

    |ˉθζEζ0|=|10ddη{[10(θ+ηu2+v2+w22cv)dx]ζ}dη|=|ζ10[10(θ+ηu2+v2+w22cv)dx]ζ1dη10(u,v,w)22dx|C(u,v,w)(u,v,w)C10|(ux,(vr)x,wx)|dxC(10[u2xθ+1θ(rvxτvr)2+w2xθdx])1/2(10θdx)1/2C(10[u2xθ+1θ(rvxτvr)2+w2xθ]dx)1/2, (2.40)

    where we have used the fact that

    (vr)x=τr2(rvxτvr).

    For β<1, it follows from Lemma 2.1 and (2.19) that

    θ1/2ˉθ1/2C10θ12|θx|dxC(10θβθ2xθ2dx)12(10θ1βdx)12C(10θβθ2xθ2dx)12. (2.41)

    For 1β<,

    θ12ˉθ12Cθβ2ˉθβ2C10θβ21|θx|dxC(10θβθ2xθ2dx)12. (2.42)

    Hence, by (2.39)–(2.42) and Lemmas 2.1–2.2, we can derive (2.38). The proof of Lemma 2.3 is thus complete.

    According to Lemmas 2.1–2.3, we can conclude that the following results have been established.

    Corollary 2.1. Assume that the conditions listed in Lemma 2.1 hold. Then for <q<1, 0<p<, and T0,

    C1τC,C110τdxC,C110θdxC,10(|lnτ|+|lnθ|+θq+u2+v2+w2)dxC3,T010[(u2+u2x+v2+v2x+w2x+τ2t)(1+θp)+θβθ2xθ1+p]dxdsC. (2.43)

    Here, we have taken p=α in (2.34) to obtain the time-space estimates of v and vx.

    Using the result above, we establish the following estimate about τx.

    Lemma 2.4. Assume that the conditions listed in Lemma 2.1 hold. Then for T0,

    10τ2xdx+T010τ2x(1+θ)dxdsC2.

    Proof. According to the chain rule, one has

    (λτxτ)t=(λτtτ)x+λθτ(τxθtτtθx). (2.44)

    By means of (1.12), (1.13), and (2.44), we have

    (λτxτ)t=utr+Pxv2r2+2uμxr+λθτ(τxθtτtθx). (2.45)

    Multiplying (2.45) by λτxτ, integrating over [0,1] about x, and using (1.12) and (2.44), we obtain

    ddt10[12(λτxτ)2λuτxrτ]dx+10Rλθτ2xτ3dx=10(ur)xλτtτdx+10Rλτxθxτ2dx+10λτx(u2v2)τr2dx+102uμxλτxrτdx+10λθτ2(λτxr1uτ)(τxθtτtθx)dx:=5i=1Ii. (2.46)

    By Hölder's inequality, (2.1), (1.12), and Corollary 2.1, one has

    I1=10(uxrτur3)λτtτdxC(u,ux,τt)2C(u,ux)2. (2.47)

    Using Corollary 2.1 and taking p=β, one has

    T010θ2xθdxdsC. (2.48)

    Hence, we argue the term I2 as the following

    I2δ10τ2xθτ3dx+C(δ)10θ2xθdx. (2.49)

    By means of integration by parts, Corollary 2.1, and (2.1), one can derive

    I3=10logτ(λr2(u2v2))xdxClnτ10|(|α|θxu2,|α|θxv2,θαu2,θαv2,θαu2x,θαv2x)|dxC10(u2,v2,u2x,v2x)dx. (2.50)

    By virtue of (2.1), we derive

    I4C|α|m322N(10u2dx)1/2(10τ2xθτ3dx)1/2δ10τ2xθτ3dx+C(δ)10u2dx. (2.51)

    By means of (2.1), Corollary 2.1, and (1.16), one can deduce

    I5C10|α||θ1||(τ2xθt,uτxθt,τxuθx,u2θx,τxuxθx,uuxθx)|dxC|α|m22N10τ2xθτ3dx+C|α|m322N(10u2+u2xdx)1/2(10τ2xθτ2dx)1/2+C|α|m12N10u2+u2xdxε10τ2xθτ3dx+C(ε)10u2+u2xdx. (2.52)

    Inserting (2.47) and (2.49)–(2.52) into (2.46), and choosing ε suitable small, we obtain

    ddt10[12(λτxτ)2λuτxrτ]dx+c10θτ2xdxC(u,ux,θx/θ,v,vx)2. (2.53)

    Integrating (2.53) over [0,t], using Cauchy-Schwarz's inequality, (2.48), and Corollary 2.1, for any t0, one has

    10τ2xdx+t010τ2xθdxdsC. (2.54)

    By virtue of (2.54), we have

    ˉθ10τ2xdx=10τ2x(ˉθθ)dx+10τ2xθdxˉθ210τ2xdx+12ˉθθˉθ210τ2xdx+10τ2xθdxˉθ210τ2xdx+Cθˉθ2+10τ2xθdx. (2.55)

    It follows from (2.19) and (2.48) that

    T0θˉθ2dsCT010θ2xθdx10θdxdtC. (2.56)

    Thus, it follows from (2.55)–(2.56) that

    T010τ2xdxdtC. (2.57)

    The proof of Lemma 2.4 has been completed by (2.54) and (2.57).

    Next, based on the estimate of τx, we are devoted to derive the estimates on the first-order derivatives of wx.

    Lemma 2.5. Assume that the conditions listed in Lemma 2.1 hold. Then for T0,

    10w2xdx+T010w2xxdxdtC3.

    Proof. Multiplying (1.15) by wxx and integrating over [0,1] about x, we find from (2.1) and Lemma 2.4 that

    12ddtwx2+10μr2w2xxτdx=10rwxxwx(μrτ)xdx10μwxxwxdxC10|wxxwx|(|α|m12|θx|+1+|τx|)dxεwxx2+C(ε)wx2+C(ε)τx2wx2εwxx2+C(ε)wx2. (2.58)

    Taking ε suitably small in (2.58) finds

    12ddtwx2+c10w2xxdxCwx2. (2.59)

    The proof of Lemma 2.5 is complete by integrating (2.59) over (0,t) about time and choosing ε suitably small.

    Based on the above result, we have the following uniform first-order derivatives estimates on the velocity (u,v).

    Lemma 2.6. Assume that the conditions listed in Lemma 2.1 hold. Then for T0,

    10(u2x+v2x+τ2t)dx+T010(u2xx+v2xx+θ2x+u2t+v2t+w2t+τ2tx)dxdtC4.

    Proof. Multiplying (1.13) and (1.14) by uxx and vxx, respectively, and integrating over Ω about x, by integration by parts, one has

    12ddt10(u2x+v2x)dx+10r2τ(λu2xx+μv2xx)dx=10uxxrPxdx+10(vxxuvruxxv2r)dx10uxxr[(λ(ru)xτ)xλruxxτ]dx+210uμxuxxdx10vxx[rvx(μrτ)x+2μvx(μv)xμτvr2]dx:=5i=1IIi. (2.60)

    By Cauchy-Schwarz's inequality, one has

    II1εuxx2+C(ε)(θx,τx)2. (2.61)

    It follows from Sobolev's inequality, the boundary condition of v, and Corollary 2.1, that we have

    II2ε(uxx,vxx)2+C(ε)v2(u,v)2ε(uxx,vxx)2+C(ε)vx2. (2.62)

    Direct computation from (2.1) yields

    II3εuxx2+C(ε)10[τ2xu2x+(1+|α|m22θ2x)|(ux,uτx,u)|2]dxεuxx2+C(ε)(ux,u)2+C(ε)τx2(ux,u)22εuxx2+C(ε)(ux,u)2, (2.63)
    II4εuxx2+C(ε)|α|N2m22u2εuxx2+C(ε)u2, (2.64)

    and

    II5εvxx2+C(ε)10[v2x(1+|α|m22θ2x+τ2x)+v2]dx2εvxx2+C(ε)(vx,v)2. (2.65)

    Putting (2.61)–(2.65) into (2.60) and taking ε suitably small gives

    12ddt10(u2x+v2x)dx+c10(u2xx+v2xx)dxC(θx,τx,vx,ux,u,v)2. (2.66)

    Integrating (2.66) over (0,T) about time, and using Lemma 2.4 and Corollary 2.1, we find

    10(u2x+v2x)dx+T010(u2xx+v2xx)dxdtC+CT010θ2xdxdt. (2.67)

    For β>1, we take p=β1 in (2.43), and then

    T010θ2xdxdtC. (2.68)

    Substituting (2.68) into (2.67), it follows for β>1 that

    10(u2x+v2x)dx+T010(u2xx+v2xx+θ2x)dxdtC. (2.69)

    Next, we need to estimate the L2(Ω×(0,t))-norm of θx for 0<β1. We deduce from multiplying (1.16) by θ1β2 and integration by parts that

    2cv4βddt10θ2β2dx+2β210˜κr2θβ2θ2xτdx=R10θ2β2τ(ru)xdx+10θ1β2Qdx=R10ˉθ2β2θ2β2τ(ru)xdxRˉθ2β210(ru)xτdx+10θ1β2QdxC10|ˉθ2β2θ2β2||(u,ux)|dxRˉθ2β2ddt10lnτdx+10θ1β2Qdx. (2.70)

    Notice that

    10|ˉθ2β2θ2β2||(u,ux)|dxCˉθ1β4θ1β4(10(1+θ2β2)dx)1/2(10(u2+u2x)dx)1/2C(10θβ4|θx|dx)2+C10(1+θ2β2)dx10(u2+u2x)dxC10θ1β2dx10θ2xθdx+C10(1+θ2β2)dx10(u2+u2x)dxC10θ2xθdx+C10(1+θ2β2)dx10(u2+u2x)dx, (2.71)

    and

    10θ1β2QdxC(ˉθ1β2θ1β2+1)10(u2+u2x+v2+v2x+w2x)dxC10θβ2|θx|dx10(u2x+v2x+w2x)dx+C10(u2x+v2x+w2x)dxC10(θ12+θβ4)|θx|dx10(u2x+v2x+w2x)dx+C10(u2x+v2x+w2x)dxε10θβ2θ2xdx+C(ε)(10(u2x+v2x+w2x)dx)2+C10(θ2xθ+u2x+v2x+w2x)dx. (2.72)

    We can conclude from (2.70)–(2.72) that

    10θ2β2dx+T010θβ/2θ2xdxdtC+CT0(10(u2x+v2x+w2x)dx)2ds,

    which combined with Young's inequality and Corollary 2.1 yields

    T010θ2xdxdtCT010θβθ2xθ2dxds+CT010θβ/2θ2xdxdsC+CT0(10(u2x+v2x+w2x)dx)2dt. (2.73)

    By means of Lemma 2.5, (2.67), and (2.73), we find for 0<β1,

    10(u2x+v2x)dx+T010(u2xx+v2xx+θ2x)dxdtC. (2.74)

    By virtue of (1.12)–(1.16), (2.1), Corollary 2.1, Lemma 2.4, (2.69), and (2.74), it follows that

    10τ2tdx+T010(u2t+v2t+w2t+τ2tx)dxdsC. (2.75)

    To obtain the first-order derivative estimate of the temperature, we need to first establish the uniform upper and lower bounds of θ.

    Lemma 2.7. Assume that the conditions listed in Lemma 2.1 hold. Then for T0,

    C11θC1.

    Proof. First of all, multiplying (1.16) by θ, and integrating over [0,1] about x, yields

    cv2ddt10θ2dx+10˜κr2θβθ2xτdx=10θQdxR10θ2(ru)xτdxC(u,ux,v,vx,wx)210θdx+ux210θ2dx. (2.76)

    Applying Gronwall's inequality to (2.76), we can obtain

    10θ2dx+T010θβθ2xdxdtC. (2.77)

    Based on the estimate above, we can get the bound of 10θβθ2xdx which will be used to obtain the upper bound of θ. Multiplying (1.16) by θβθt and integrating over (0,1) about x, it follows that

    cv10θβθ2tdx+R10θβ+1θt(ru)xτdx10θβθtQdx=10(˜κr2θβθxτ)xθβθtdx. (2.78)

    By integration by parts, one has

    10(˜κr2θβθxτ)xθβθtdx=10˜κr2θβθxτ(θβθx)tdx=˜κ2ddt10r2τ(θβθx)2dx+˜κ210(2ruτruτr3uxτ2)(θβθx)2dx. (2.79)

    Inserting (2.79) into (2.78), we can deduce that

    ˜κ2ddt10r2τ(θβθx)2dx+cv10θβθ2tdx=R10θβ+1θt(ru)xτdx+10θβθtQdx+˜κ210(ruτr3uxτ2)(θβθx)2dxcv210θβθ2tdx+C10θβ+2(u2+u2x)dx+C10θβ(u4+u4x+v4+v4x+w4x)dx+C(u,ux)10(θβθx)2dxcv210θβθ2tdx+C(u2,u2x,u4,u4x,v4,v4x,w4x)+C(10θβθ2tdx)2. (2.80)

    By Sobolev's inequality, Corollary 2.1, and Lemmas 2.5–2.6, one can find that

    T0(u2,u2x,u4,u4x,v4,v4x,w4x)dsC. (2.81)

    By virtue of (2.80), Grönwall's inequality, and (2.81), we can obtain

    10(θβθx)2dx+T010θβθ2tdxdsC. (2.82)

    Thanks to (2.82), it follows that

    θβ+1ˉθβ+1C(10(θβθx)2dx)12C. (2.83)

    That is, for t0,

    θC. (2.84)

    Thanks to (2.77) and (2.84), one has

    T010(θβ+1ˉθβ+1)2dxdtT010θ2βθ2xdxdtCsup0tTθβT010θβθ2xdxdtC. (2.85)

    Combining (2.83) and (2.84), one has

    T0|ddt10(θβ+1ˉθβ+1)2dx|dtCT010(θβ+1ˉθβ+1)2dxdt+CT0θβθt2dtCsup0tTθβT010θβθ2xdxdtC. (2.86)

    So, from (2.83), (2.85), and (2.86), one has

    limt+10(θβ+1ˉθβ+1)2dx=0.

    From (2.83), when t+,

    (θβ+1ˉθβ+1)2C(θβ+1ˉθβ+1)θβθx0,

    and we can obtain that there exists some time T01 such that when t>T0,

    θ(x,t)γ12. (2.87)

    Fixing T0 in (2.87), multiplying (1.16) by θp, p>2, and integrating over [0,1] about x yield

    cvp1ddtθ1p1p1+p10˜κr2θβθ2xτθp+1dx+10Qθpdx=R10θτθp(ru)xdx1210u2+u2xτθpdx+Cθ1p2Lp1.

    Hence,

    ddtθ1Lp1C,

    where C is a generic positive constant independent of p. Thus, integrating the above inequality over (0,t) and letting p, we arrive that

    θ1(x,t)C(T0+1)θ(x,t)[C(T0+1)]1,(x,t)[0,1]×[T0,+).

    The proof of Lemma 2.7 is complete.

    Lemma 2.8. Assume that the conditions listed in Lemma 2.1 hold. Then for T0,

    10θ2xdx+T010(θ2xx+θ2t)dxdsC5.

    Proof. Multiplying (1.16) by θxx, integrating over [0,1] on x, and by Hölder's, Poincaré's, and Cauchy-Schwarz's inequalities, Corollary 2.1, Lemma 2.4, and Lemma 2.7, we have

    cv2ddt10θ2xdx+10κr2θ2xxτdx=10θxx[Rθτ(ru)xθx(κr2τ)xQ]dxε10θ2xxdx+C(ε)10[θ2(ru)2xθ2x(κr2τ)2xQ2]dxε10θ2xxdx+C(ε)ux2θ2+C(ε)θx2+C(ε)θx2τx2+C(ε)10(u4+v4+u4x+v4x+w4x)dxεθxx2+C(ε)(ux2+θx2+θxθxx+u2u2+v2v2)+C(ε)(ux2ux2+vx2vx2+wx2wx2)εθxx2+C(ε)(ux,vx,wx,uxx,vxx,wxx)2+C(ε)θx2. (2.88)

    Choosing ε suitably small in (2.88) gives

    cv2ddt10θ2xdx+c10θ2xxdxC(ux,vx,wx)21+Cθx2. (2.89)

    Integrating (2.89) and using Lemmas 2.5–2.6, one has

    θx(t)2+T0θxx2dsC. (2.90)

    Hence, similar to (2.75), by means of (1.16), Corollary 2.1, Lemmas 2.4–2.7, and (2.90), one can deduce that

    T010θ2tdxdtC.

    Next, we derive the second-order derivatives estimates of (τ,u,v,w,θ).

    Lemma 2.9. Assume that the conditions listed in Lemma 2.1 hold. Then for T0,

    10(u2t+v2t+w2t+θ2t+u2xx+v2xx+w2xx+θ2xx+τ2xt)dx+T010(u2xt+τ2tt+v2xt+w2xt+θ2xt)dxdsC6.

    Proof. Applying t to (1.13) and multiplying by ut in L2, one has

    12ddt10u2tdx+10˜λr2θαu2xtτdx=10ruxt[Pt(λτ)t(ru)xλτ((ru)xtruxt)]dx10rxut[λ(ru)xτ]tdx+10ut[(v2r)trtPx+rxPt+rt(λ(ru)xτ)x2(uμx)t]dx:=3i=1IIIi. (2.91)

    Applying t to (1.14) and multiplying by vt in L2, one has

    12ddt10v2tdx+10˜μr2θαv2xtτdx=10{vt[2(μvx)trx(μrvxτ)t(μv)xt]rvxtvx[μrτ]t}dx+10vt[rt(μrvxτ)x(uvr)t(μτvr2)t]dx:=5i=4IIIi. (2.92)

    Applying t to (1.15) and multiplying by wt in L2, one has

    12ddt10w2tdx+10˜μr2θαw2xtτdx=10wtrt(μrwxτ)xdx10{rwxwxt(μrτ)t+wt[rx(μrwxτ)t(μwx)t]}dx:=7i=6IIIi. (2.93)

    Adding (2.91)–(2.93) together, we get

    12ddt10(u2t+v2t+w2t)dx+10˜λr2θαu2xt+˜μr2θαv2xt+˜μr2θαw2xtτdx=7i=1IIIi. (2.94)

    Before the computations of III1 to III7, we need to keep in mind the following facts:

    (u,v,w)C,arb,rx=r1τ,rt=u,rtx=ux,C1τC,C1θC,|(ru)x|C|(u,ux)|,|(ru)xtruxt|C|(u2,ut,uux)|,|(ru)xt|C|(u2,ut,uux,uxt)|.

    Then, by Hölder's, Sobolev's, and Cauchy-Schwarz's inequalities, one has

    III1C10|uxt||(θt,τt,θtux,τtux,u,ut,ux)|dxεuxt2+C(ε)(θt,τt,u,ut)2+C(ε)(θt,τt)2ux2εuxt2+δθxt2+C(ε,δ)(θt,τt,u,ut,τxt)2, (2.95)

    and

    III2C10|ut||(θt,θtux,u2,ut,ux,uxt,τt,τ2t)|dxεuxt2+C(ε)(ut,θt,u,ux,τt)2+εθt2ux2+C(ε)τt2τt2εuxt2+δθxt2+C(ε,δ)(ut,θt,u,ux,τt,τxt)2. (2.96)

    By virtue of (1.13), one has

    |(λ(ru)xτ)x|C|(ut,v2,θx,τx)|.

    Thus, it follows from Hölder's, Sobolev's, and Cauchy-Schwarz's inequalities that

    III3C10|ut||(vt,v2,θx,τx,τt,θt,ut,utθx,θxθt,θxt)|dxεθxt2+C(ε)(vt,ut,v,θx,θt,θt,τx,τt)2+ε(ut,θt)2θx2ε(uxt,θxt)2+C(ε)(vt,ut,θt,τt,v,θx,τx)2, (2.97)

    and

    III4C10|vt||(θtvx,vx,vxt,vxτt,θxθt,θxt,θxvt)|dx+C10|vxtvx||(θt,v,τt)|dxε2(vxt,θxt)2+C(ε)(vt,vx)2+C(ε)(θt,τt,vt)2(vx,θx)2ε(vxt,θxt)2+C(ε)(vt,vx,θt,τt,τtx)2. (2.98)

    It follows from (1.14) that

    |(μrvxτ)x|C|(vt,v,vx,θx)|.

    Then

    III5C10|vt||(vt,v,θx,vx,ut,θt,τt)|dxC(vt,v,θx,vx,ut,θt,τt)2. (2.99)

    By virtue of (1.15), we can obtain

    III6C10|wt||u||(μrwxτ)x|dxC10|wt||(wt,wx)|dxC(wt,wx)2, (2.100)

    and

    III7C10|wxt||(wxθt,wxτt,wx)|+|wt||(θtwx,τtwx,wx,wxt)|dxεwxt2+C(ε)(wx,wt)2+C(ε)(θt,τt)2wx2ε(wxt,θxt)2+C(ε)(wx,wt,θt,τt,τxt)2. (2.101)

    Putting (2.95)–(2.101) into (2.94) gives

    12ddt(ut,vt,wt)2+c(uxt,vxt,wxt)2ε(uxt,vxt,wxt,θxt)2+C(ε)(ut,vt,wt,θt,wx,τx,θx)2+C(ε)(τt,u,v)21. (2.102)

    Applying t to (1.16) and multiplying by θt in L2, it follows that

    cv2ddt10θ2tdx+10˜κθβr2θ2xtτdx=10θt[Qt(P(ru)x)t]θxθxt(κr2τ)tdx. (2.103)

    First of all, by means of the definition of Q, one has

    |θtQt|C|θt||(u,τt,ut,ux,uxt,uxτt,uxut,u2x,uxuxt)|+C|θt||(θt,θtu2x,τtu2x,θtw2x,w2x,τtw2x,wxwxt,θtv2x,τtv2x)|+C|θt||(v2x,vxt,τtvx,vt,v,vx,vxvxt,vxvt)|C(ε)|(θt,u,τt,ut,ux,wx,vx,vt,v,vx)|2+ε|(uxt,wxt,vxt)|2+C(ε)|(τt,ut,ux)|2|(ux,wx,vx)|2+C(ε)|θt|2|(ux,vx,wx)|2. (2.104)

    Using (2.104) and Sobolev's inequality, we can derive from (2.103) that

    cv2ddt10θ2tdx+10˜κθβr2θ2xtτdxC10(|θt||(Qt,θt,τt,θtux,τtux,u,ut,ux,uxt)|+|θx||(θxtθt,θxt,θxtτt)|)dxC(ε)(θt,u,τt,ut,ux,θx,wx,vx,vt,v,vx)2+ε(uxt,wxt,vxt,θxt)2+C(ε)(τt,θt,ut,ux)2(ux,wx,vx,θx)2+C(ε)τt2θt2C(ε)(θt,u,τt,ut,ux,wx,vx,vt,v,τtx,uxx)2+ε(uxt,wxt,vxt,θxt)2+C(ε)(ux,vx,wx)21θt2+C(ε)τt21θt2. (2.105)

    Adding (2.102) to (2.105) and choosing ε>0 suitably small, it follows that

    12ddt(cvθt,ut,vt,wt)2+c(uxt,vxt,wxt,θxt)2C(ut,vt,wt,θt,wx,τx,θx)2+C(ux,τt,v)21+C(ux,vx,wx)21θt2+Cτt21θt2. (2.106)

    By means of (2.106) and Grönwall's inequality, we deduce

    (ut,vt,wt,θt)2+T0(uxt,vxt,wxt,θxt)2dsC. (2.107)

    According to (1.13), one has

    λr2uxxτ=utv2r+rPx+2uμxr[(λ(ru)xτ)xλruxxτ],

    which means that

    |uxx|C|(ut,v,θx,τx,θxux,τxux,u,ux)|.

    Hence, by means of (2.107), we obtain

    uxx2C.

    Similarly, use the equations (1.12)–(1.16), we also can derive

    (vxx,wxx,θxx,τtx)2+T0τtt2dsC7. (2.108)

    Here, we omit the details of (2.108). The proof of Lemma 2.9 is complete.

    Lemma 2.10. Assume that the conditions listed in Lemma 2.1 hold. Then for T0,

    10τ2xxdx+T010(τ2xx+τ2xxt+u2xxx+v2xxx+w2xxx+θ2xxx)dxdsC7.

    Proof. Apply x to (2.45) and multiply by (λτx/τ)x in L2 to get

    12ddt10(λτxτ)2xdx+10Rθλτ(λτxτ)2xdx=10(λτxτ)x[λθτ(τxθtτtθx)]xdx+10(λτxτ)x(utrv2r2+2uμxr)xdxR10(λτxτ)x[2θxτxτ2θxxτ2θτ2xτ3θτxλτ(λτ)x]dxC(ε)10[|(τx,θx)|2|(τxθt,τtθx)|2+|(τxxθt,τxθxt,τtxθx,τtθxx)|2]dx+C(ε)10[|(uxt,ut,vx,v,uxθx,θxx,θ2x,θx)|2+|(θxx,τ2x,θxτx)|2]dx+ε10(λτxτ)2xdx:=9i=8IIIi+ε10(λτxτ)2xdx, (2.109)

    where the following fact has been used:

    (θτ)xx=θxxτ2θxτxτ2+2θτ2xτ3θτxxτ2=θxxτ2θxτxτ2+2θτ2xτ3θλτ[(λτxτ)xλxτxτ+λτ2xτ2].

    By Sobolev's inequality and Lemmas 2.6–2.9, we have

    III8C(ε)((τx,θx)4(τt,θt)2+θt2τxx2+τx2θxt2+τt2θxx2+θx2τxt2)C(ε)((τx,θx)4+(τx,θx)2(τxx,θxx)2+θt21τxx2+θxt2τx2+τt21+θx21)C(ε)((τx,θx,τt)2+θt2τxx2), (2.110)

    and

    III9C(ε)(uxt,ut,vx,v,θxx,θx)2+C(ε)(ux,θx,τx)2(θx,τx)2ετxx2+C(ε)(uxt,ut,vx,v,θxx,θx,τx)2. (2.111)

    Noting that

    |τxx|C|(λτxτ)x|+C|(θxτx,τ2x)|,

    we can derive from Sobolev's inequality and Lemma 2.4 that

    τxx2C(λτxτ)x2+Cθx2τx2+Cτx44C(λτxτ)x2+Cθx21+Cτx4+Cτx3τxx.

    So, it follows from Cauchy-Schwarz's inequality and Lemma 2.4 that

    τxx2Cθx21+Cτx2+C(λτxτ)x2. (2.112)

    Taking ε suitably small, putting (2.110)–(2.111) into (2.109), and using Lemmas 2.4, 2.8, and 2.9, we find

    12ddt10(λτxτ)2xdx+c10(λτxτ)2xdxC(τt,θt,θx,ut,v)21+Cτx2+Cθt21(λτxτ)x2. (2.113)

    By (2.113), Grönwall's inequality, Corollary 2.1, and Lemmas 2.6 and 2.8–2.9, one obtains

    10(λτxτ)2xdx+T010(λτxτ)2xdxdsC. (2.114)

    It follows from (2.112) and (2.114) that

    τxx2+T0τxx2dsC. (2.115)

    Letting x act on (1.13) gives

    ˜λθ2r2uxxxτ+rx(λτtτ)x+r[(λτ)xτt]x+r(λτ)x(ru)xx=uxt(v2r)x+(rPx)x+2(uμx)xrλτ[(ru)xxxruxxx]. (2.116)

    It follows from (2.115) and (2.116) that

    T0uxxx2dsCT0(θxτt,τxt,τtτx)2ds+CT0(θ2xτt,θxxτt,θxτtx,θxτtτx)2ds+CT0(τxxτt,τxτtx,τ2xτt)2ds+CT0(θx,θxτx,θxux,θxuxx)2ds+CT0(uxt,vx,v,θx,τx,θxx,θxτx,τxx)2ds+CT0(uxθx,θ2x,θxx)2ds+CT0(u,τx,τxx,ux,τxux,uxx)2dsCT0(τt,τxx,τtx,θx,ux,uxt,vx,v,θxx,u,τx,uxx)2dsC,

    where the following fact has been used:

    (θx,τx,θ2x,τt,θxτt,τ2x)C+C(θx,τt,τx)21C.

    Similarly, using (1.14)–(1.15), we also have

    T0(vxxx,wxxx)2dsC.

    Letting x act on (1.16) gives

    ˜κθβr2θxxxτ=cvθxt+(Pτt)x(κr2τ)xxθx2(κr2τ)xθxxQx. (2.117)

    It follows from (2.114) and (2.117) that

    T0θxxx2dsCT0(θxt,θxτt,τxτt,τtx)2ds+CT0(θ3x,θxxθx,θ2x,θ2xτx,θxτx,θxτ2x,θxτxx)2ds+CT0(θxx,τxθxx)2+Qx2ds. (2.118)

    By the definition of Q, one has

    T0Qx2dsCT0(θx,θxux,u,τx,ux,uxx,uxτx,u2x,uxuxx)2ds+CT0(θxw2x,w2x,wxwxx,w2xτx)2ds+CT0(θxv2x,τxv2x,v2x,vxvxx,v2xτx,vx,vxx,vxτx,v)2ds. (2.119)

    Since the following estimates have been obtained:

    (θx,τx,ux,wx,vx)C(θx,τx,ux,wx,vx)1C,

    putting (2.119) into (2.118) yields

    T0θxxx2dsCT0(θxt,τt,τxt,θx,θxx,τx,τxx,ux,u,uxx,wx,wxx,vx,vxx,v)2dsC.

    The proof of Lemma 2.10 is complete.

    With all a priori estimates from Section 2 at hand, we can complete the proof of Theorem 1.1. For this purpose, it will be shown that the existence and uniqueness of local solutions to the initial-boundary value problem (1.12)–(1.19) can be obtained by using the Banach theorem and the contractivity of the operator defined by the linearization of the problem on a small time interval.

    Lemma 3.1. Letting (1.20) hold, then there exists T0=T0(V0,V0,M0)>0, depending only on β, V0, and M0, such that the initial boundary value problem (1.12)–(1.19) has a unique solution (τ,u,v,w,θ)X(0,T0;12V0,12V0,2M0).

    Proof of Theorem 1.1: First, to prove Theorem 1.1, according to (1.20), one has

    τ0V0,θ0V0,xΩ,(τ0,u0,v0,w0,θ0)H2M0.

    Combined with Lemma 3.1, there exists t1=T0(V0,V0,M0) such that (τ,u,v,w,θ)X(0,t1;12V0,12V0,2M0).

    We find the positive constant |α|α1, where α1 satisfies

    (12V0)|α1|2,(2M0)|α1|2,|α1|H(12V0,12V0,2M0)ϵ1, (3.1)

    where ϵ1 is chosen in Lemma 2.1. That means that one can choose

    |α1|:=min{ln2|ln2lnV0|,ln2|ln2+lnM0|,ϵ1H1(12V0,12V0,2M0)}. (3.2)

    One deduces from Lemmas 2.1–2.10 with T=t1 that for each t[0,t1], the local solution (τ,u,v,w,θ) satisfies

    C10v(x,t)C0,C11θ(x,t)C1,x(0,1), (3.3)

    and

    sup0tt1(τ,u,v,w,θ)22+t10θt2dtC28, (3.4)

    where Ci(i=2,,7) is chosen in Section 2 and C28:=7i=2Ci. It follows from Lemma 2.9 and Lemma 2.10 that (τ,u,v,w,θ)C([0,T);H2). If one takes (τ,u,v,w,θ)(,t1) as the initial data and applies Lemma 3.1 again, the local solution (τ,u,v,w,θ) can be extended to the time interval [t1,t1+t2] with t2(C0,C1,C8) such that (τ,u,v,w,θ)X(t1,t1+t2;12C0,12C1,12C8). Moreover, for all (x,t)[0,1]×[0,t1+t2], one gets

    12C0v(x,t),12C1θ(x,t),

    and

    supt1tt1+t2(τ,u,v,w,θ)22+t1+t2t1θt2dt4C28,

    which combined with (3.3) and (3.4) implies that for all t[0,t1+t2],

    12C0v(x,t),12C1θ(x,t),
    sup0tt1+t2(τ,u,v,w,θ)22+t1+t20θt2dt5C28.

    Take αmin{α1,α2}, where αi(i=1,2) are positive constants satisfying (3.1) and

    (12C0)α22,(5C8)α22,α2H(12C0,12C1,5C8)ϵ1,

    where the value of ϵ1 is chosen in Lemma 2.1. That means that we can choose

    |α2|:=min{ln2|ln2lnC0|,ln2|ln5+lnC8|,ϵ1H1(12C0,12C1,5C8)}. (3.5)

    Then one can employ Lemmas 2.1–2.10 with T=t1+t2 to infer the local solution (τ,u,v,w,θ) satisfying (3.3) and (3.4).

    Choosing

    ϵ0=min{α1,α2}, (3.6)

    and repeating the above procedure, one can extend the solution (τ,u,v,w,θ) step-by-step to a global one provided that |α|ϵ0. Furthermore,

    (τ,u,v,w,θ)2H2++0[(ux,vx,wx,θx)2+τ2]dtC29,

    from which we derive that the solution (τ,u,v,w,θ)X(0,+;C0,C1,C9).

    The large-time behavior (1.21) follows from Lemmas 2.4–2.10 by using a standard argument [21].

    First, thanks to (1.15), (2.1), (2.43), (2.55), (2.62), (2.73), Corollary 2.1, and Lemmas 2.4–2.10, taking ˆθ=E0, one has

    ddt10ηE0(τ,u,v,w,θ)dx+c1(u,v)21+c1(wx,θx)20, (3.7)
    ddt10[12(λτxτ)2λuτxrτ]dx+c2τx2C10(u,ux,θx,v,vx)2, (3.8)
    ddt(ux,vx,wx)2+c3(uxx,vxx,wxx)2C11(θx,τx,vx,ux,u,v,wx)2, (3.9)
    ddtθx2+c4θxx2C12(ux,vx,wx)21+C12θx2. (3.10)

    By Cauchy-Schwarz's inequality, one has

    |λuτxrτ|14(λτxτ)2+Cu2. (3.11)

    Hence, by means of (3.11), Poincaré's inequalities, Corollary 2.1, and Lemma 2.7, one can deduce

    cτx2C13u210[12(λτxτ)2λuτxrτ]dxC(τx,ux)2.

    Multiplying (3.7)–(3.10) by C14, C15, and C16, respectively, and adding them together with (3.10), one has

    ddtA+c(ux,vx,wx,θx)2H1+cτx20, (3.12)

    where we have defined

    A:=10C14ηE0(τ,u,v,w,θ)+C15[12(λτxτ)2λuτxrτ]dx+C16(ux,vx,wx)2+θx2,

    and chosen constants C14>C15>C16>0 suitably large such that

    c1C14C10C15C11C16C12>0,
    c2C15C11C16C12>0,
    c3C16C12>0.

    Taking C142>C13 and using Poincaré's inequality gives

    c(τˉτ,u,v,w,θE0)2AC(ux,vx,wx,θx)21+Cτx2, (3.13)

    where we have used the facts

    θE02C10|θˉθ|2dx+C(u,v,w)2C(θx,ux,vx,wx)2.

    By means of (3.12) and (3.13), we can derive that

    (τˉτ,u,v,w,θE0)(t)2H1(Ω)Cect. (3.14)

    By means of ˉr, one has

    r2ˉr2=2x0τˉτdξ. (3.15)

    By means of (3.14) and (3.15), we have

    rˉr22Cect.

    The proof is thus complete.

    Dandan Song: Writing-original draft, Writing-review & editing, Supervision, Formal Analysis; Xiaokui Zhao: Writing-review & editing, Methodology, Supervision.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors are grateful to the referees for their helpful suggestions and comments on the manuscript. This work was supported by the NNSFC (Grant No. 12101200), the Doctoral Scientific Research Foundation of Henan Polytechnic University (No. B2021-53) and the China Postdoctoral Science Foundation (Grant No. 2022M721035).

    The authors declare there is no conflict of interest.



    [1] Y. Cho, H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91–129. https://doi.org/10.1007/s00229-006-0637-y doi: 10.1007/s00229-006-0637-y
    [2] H. J. Choe, H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differential Equations, 190 (2003), 504–523. https://doi.org/10.1016/S0022-0396(03)00015-9 doi: 10.1016/S0022-0396(03)00015-9
    [3] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579–614. https://doi.org/10.1007/s002220000078 doi: 10.1007/s002220000078
    [4] E. Feireisl, H. Petzeltová, Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77–96. https://doi.org/10.1007/s002050050181 doi: 10.1007/s002050050181
    [5] D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc., 303 (1987), 169–181. https://doi.org/10.2307/2000785 doi: 10.2307/2000785
    [6] X. D. Huang, Z. P. Xin, A blow-up criterion for classical solutions to the compressible Navier-Stokes equations, Sci. China Math., 53 (2010), 671–686. https://doi.org/10.1007/s11425-010-0042-6 doi: 10.1007/s11425-010-0042-6
    [7] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337–342. https://doi.org/10.3792/pjaa.55.337 doi: 10.3792/pjaa.55.337
    [8] O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations, J. Differential Equations, 245 (2008), 1762–1774. https://doi.org/10.1016/j.jde.2008.07.007 doi: 10.1016/j.jde.2008.07.007
    [9] P. L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Compressible models, Oxford University Press, New York, 1998.
    [10] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215–254. https://doi.org/10.1006/jdeq.1995.1111 doi: 10.1006/jdeq.1995.1111
    [11] D. Hoff, M. M. Santos, Lagrangean structure and propagation of singularities in multidimensional compressible flow, Arch. Ration. Mech. Anal., 188 (2008), 509–543. https://doi.org/10.1007/s00205-007-0099-8 doi: 10.1007/s00205-007-0099-8
    [12] T. P. Liu, Z. P. Xin, T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Systems, 4 (1998), 1–32. https://doi.org/10.3934/dcds.1998.4.1 doi: 10.3934/dcds.1998.4.1
    [13] Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229–240. https://doi.org/10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C
    [14] Y. B. Zel'dovich, Y. P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena. Vol. II, Academic Press, New York, 1967.
    [15] R. H. Pan, W. Z. Zhang, Compressible Navier-Stokes equations with temperature dependent heat conductivity, Commun. Math. Sci., 13 (2015), 401–425. https://doi.org/10.4310/CMS.2015.v13.n2.a7 doi: 10.4310/CMS.2015.v13.n2.a7
    [16] B. Huang, X. D. Shi, Nonlinearly exponential stability of compressible Navier-Stokes system with degenerate heat-conductivity, J. Differential Equations, 268 (2020), 2464–2490. https://doi.org/10.1016/j.jde.2019.09.006 doi: 10.1016/j.jde.2019.09.006
    [17] H. X. Liu, T. Yang, H. J. Zhao, Q. Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data, SIAM J. Math. Anal., 46 (2014), 2185–2228. https://doi.org/10.1137/130920617 doi: 10.1137/130920617
    [18] L. Wan, T. Wang, Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients, J. Differential Equations, 262 (2017), 5939–5977. https://doi.org/10.1016/j.jde.2017.02.022 doi: 10.1016/j.jde.2017.02.022
    [19] T. Wang, H. J. Zhao, One-dimensional compressible heat-conducting gas with temperature-dependent viscosity, Math. Models Methods Appl. Sci., 26 (2016), 2237–2275. https://doi.org/10.1142/S0218202516500524 doi: 10.1142/S0218202516500524
    [20] Y. Sun, J. W. Zhang, X. K. Zhao, Nonlinearly exponential stability for the compressible Navier-Stokes equations with temperature-dependent transport coefficients, J. Differential Equations, 286 (2021), 676–709. https://doi.org/10.1016/j.jde.2021.03.044 doi: 10.1016/j.jde.2021.03.044
    [21] M. Okada, S. Kawashima, On the equations of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ., 23 (1983), 55–71. https://doi.org/10.1215/kjm/1250521610 doi: 10.1215/kjm/1250521610
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