Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients
Ling Wan Tao Wang











 Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients
    
Ling Wan

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Tao WangCorresponding author. Email addresses: ling.wan@whu.edu.cn (L. Wan), tao.wang@whu.edu.cn (T. Wang).

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

DICATAM, Mathematical Division, University of Brescia, Via Valotti 9, 25133 Brescia, Italy

    
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We consider the Navier–Stokes equations for compressible heat-conducting ideal polytropic gases in a bounded annular domain when the viscosity and thermal conductivity coefficients are general smooth functions of temperature.
A global-in-time, spherically or cylindrically symmetric, classical solution to the initial boundary value problem is shown to exist uniquely and converge exponentially to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ.
The initial data can be large if γ is sufficiently close to 1.
These results are of Nishida–Smoller type and extend the work [Liu et al., SIAM J. Math. Anal. 46 (2014), 2185–2228] restricted to the one-dimensional flows.




Keywords: Compressible Navier–Stokes equations;
  Temperature dependent viscosity coefficients;
  Global symmetric solutions; 
  Nishida–Smoller type result;
  Exponential stability



Mathematics Subject Classification:
  35Q35 (35B40, 76N10)







§  INTRODUCTION


The motion of a compressible viscous and heat-conducting fluid in Ω⊂ℝ^d can be described by the Navier–Stokes equations in Eulerian coordinates:

 
 
    ∂_tρ+div(ρu)=0,

    ∂_t(ρu)+div(ρu⊗u)
 +∇P=div 𝕊,

    ∂_t(ρE)+div(ρuE+uP)
 =div(κ∇θ+𝕊·u).
 

Here  the primary dependent variables are the density ρ, the velocity field u∈ℝ^d, and the temperature θ.
The symbol E=e+12|u|^2 is the specific total energy.
For ideal polytropic gases, the pressure P and the specific internal energy e are related with ρ and θ by equations of state:

    P=Rρθ,   e=c_vθ,

where R>0,  c_v=R/(γ-1), and γ>1 are the specific gas constant, the specific heat at constant volume, and the adiabatic exponent, respectively. 
The fluid is assumed to be Newtonian so that the viscous stress tensor 𝕊 is of form

    𝕊=μ(∇u+(∇u)^𝖳)
    +λ  div u 𝕀,

where  (∇u)^𝖳 is the transpose matrix of ∇u and 𝕀 is the d× d identity matrix.
The viscosity coefficients μ, λ, and the thermal conductivity coefficient κ are prescribed through constitutive relations as functions of the density and temperature satisfying μ>0, κ>0, and 2μ+dλ>0 (see <cit.>).


In this paper we establish the existence and exponential decay rate of global-in-time,  spherically or cylindrically symmetric, classical solutions to (<ref>)–(<ref>) in the bounded annular domain Ω={x=(x_1,⋯,x_d)∈ℝ^d: a<r<b} for 0<a<b<∞ with large initial data. Here we set d∈ℕ_+ and  r=|x| in the spherically symmetric case, while d=3 and r=√(x^2_1+x^2_2) in the cylindrically symmetric case. 
The system (<ref>)–(<ref>) is
supplemented with  the spherically or cylindrically symmetric initial data:

    (ρ,u,θ)(0,x)
    =(ρ_0,u_0,θ_0)(x)    for  x∈Ω,

and the boundary conditions:

    u=0,  ∂θ/∂n=0   on  ∂Ω,

where n denotes the unit outward normal vector to ∂Ω. 


We are interested in the 
case where the transport coefficients μ, λ, and  κ are smooth functions of the temperature.
More specifically, we suppose that

    μ,λ,κ∈ C^3(0,∞),  μ(θ)>0,  κ(θ)>0, 2μ(θ)+dλ(θ)>0    for all  θ>0.

Our main motivation is provided by the kinetic theory of gases.
By virtue of the Chapman–Enskog expansion, the compressible Navier–Stokes system (<ref>) is the first order approximation of the Boltzmann equation, and the transport coefficients μ, λ, and κ depend solely on the temperature (see <cit.> and <cit.>). 
In particular, if the intermolecular potential varies as r^-a with r being the molecule distance, then μ, λ, and κ satisfy

    μ=μ̅θ^a+4/2a,  λ=λ̅θ^a+4/2a,  κ=κ̅θ^a+4/2a   for  a> 0,

where μ̅, λ̅, and κ̅  are constants.
In spite of this obvious physical relevance, there is no global existence result currently available beyond the small data <cit.> for the Navier–Stokes system (<ref>)–(<ref>)  with  general adiabatic exponent γ and transport coefficients (<ref>).




Let us mention some related results about the global existence and large-time behavior for the full compressible Navier–Stokes equations (<ref>) with large data. 
The global existence and uniqueness of smooth solutions to (<ref>)–(<ref>) in one-dimensional bounded domains are proved in the seminal work by <cit.> for constant transport coefficients and large initial data.
The crucial step in <cit.> is to obtain the positive upper and lower bounds of the specific volume τ (i.e. the reciprocal of density ρ) and temperature θ, which is achieved by means of a decent representation for  τ and the maximum principle.

The results in <cit.> have been generalized to cover the spherically and cylindrically symmetric flows.
In the case of spherical symmetry, <cit.> (resp. <cit.>) showed the  existence of global-in-time (generalized) solutions in bounded annular domains (resp. in exterior domains), while <cit.> investigated the flows between a static solid core and a free boundary connected to a surrounding vacuum state.
In the cylindrically symmetric case, <cit.> established the global solvability with large data in a bounded annular domain.
Later, <cit.> proved the global existence of a spherically or cylindrically symmetric weak solution with large discontinuous data in a ball.
As for the large-time behavior of global solutions, see <cit.> (resp. <cit.>) for spherically symmetric flows in bounded domains (resp. in exterior domains), and <cit.> for cylindrically symmetric flows in bounded domains.
In all of these works the transport coefficients μ, λ, and κ are supposed to be constants.

The argument in <cit.> can be applied to the case of constant viscosity and temperature dependent thermal conductivity; 
see <cit.> for one-dimensional flows and <cit.> for cylindrically symmetric flows. 
Under certain assumptions on  temperature dependent thermal conductivity κ,  pressure P,  and internal energy e,
<cit.> show the global existence of spherically or cylindrically symmetric, classical and strong solutions for (<ref>) with constant viscosity coefficients μ and λ, which generalizes the work by <cit.> for one-dimensional flows.
The assumption of constant viscosity coefficients μ and λ is essential in  <cit.>.


Temperature dependence of the viscosity coefficients μ and λ turns out to have a strong influence on the solution and lead to difficulty in mathematical analysis for global solvability with large data. 
<cit.> recently obtain the first result on global well-posedness of smooth solutions to the one-dimensional Navier–Stokes system (<ref>)–(<ref>) with general adiabatic exponent γ,  temperature dependent viscosity, and large initial data.
Unfortunately, this result do not cover the models satisfying (<ref>).
As far as we are aware, the only global solvability result currently available for (<ref>)–(<ref>) with constitutive relations (<ref>) and large data
is due to  <cit.> for one-dimensional flows.
In <cit.> the initial density and velocity can be large, but γ-1 and the H^3-norm of θ_0-1 with θ_0 being the initial temperature have to be small. 
In other words, this is a Nishida–Smoller type global solvability result with large data.
The original Nishida–Smoller type global solvability result is about the one-dimensional ideal polytropic isentropic compressible Euler system in <cit.> (see also <cit.> for the  nonisentropic case). 

It is a  natural and interesting problem to establish a Nishida–Smoller type global existence result for the spherically or cylindrically symmetric solutions of (<ref>)–(<ref>) with temperature dependent transport coefficients (<ref>).
Our main objective is to study this problem  and to show the exponential stability of solutions toward the constant equilibrium states as well.  
For this purpose, we reduce the initial boundary value problem (<ref>)–(<ref>) to the corresponding problem in either spherical or cylindrical coordinates.

We let (u, v, w) be the velocity components in either spherical or cylindrical coordinates.
Thus

    u(t,x)=ũ(t,r)/rx,   
    v=w≡ 0

 in the spherically symmetric case, while

    u(t,x)
    =ũ(t,r)/r(x_1,x_2,0)
    +ṽ(t,r)/r(-x_2,x_1,0)+w̃(t,r)(0,0,1)

in the cylindrically symmetric case. 
In both cases, the density and temperature depend only on the time t and the radius r, i.e. (ρ,θ)(t,x)=(ρ̃,θ̃)(t,r).
The system for (ρ̃,ũ, ṽ,w̃,θ̃) then takes the form



   ρ̃_t+(r^m ρ̃ũ)_r/r^m=0, 


   ρ̃(ũ_t+ũũ_r)-ρ̃ṽ^2/r+P̃_r=
[ν̃(r^m ũ)_r/r^m]_r-2m ũμ̃_r/r,  


   ρ̃(ṽ_t+ũṽ_r)+ρ̃ũṽ/r=
(μ̃ṽ_r)_r+2μ̃ ṽ_r/r^m-m(μ̃r^m-1ṽ)_r/r^m-μ̃ṽ/r^2m, 


   ρ̃(w̃_t+ũw̃_r)=
(μ̃w̃_r)_r+mμ̃ w̃_r/r, 


   ρ̃(ẽ_t+ũẽ_r)+P̃(r^m ũ)_r/r^m=
(κ̃r^mθ̃_r)_r/r^m+𝒬̃,
 


where μ̃=μ(θ̃), ν̃=2μ(θ̃)+λ(θ̃), P̃=Rρ̃θ̃, ẽ=c_vθ̃, and

    𝒬̃=
    ν̃(r^m ũ)_r^2/r^2m-2mμ̃(r^m-1ũ^2)_r/r^m+μ̃w̃_r^2+μ̃[ṽ_r-ṽ/r^m]^2.

In spherically symmetric case, d∈ℕ_+, m=d-1, and ṽ=w̃=0, whereas in cylindrically symmetric case, d=3 and m=1. 
The initial and boundary conditions (<ref>) and (<ref>) are reduced to


(ρ̃,ũ,ṽ,w̃,θ̃)(0,r)   =
(ρ̃_0,ũ_0,ṽ_0,w̃_0,θ̃_0)(r),            a≤r≤b,


(ũ,ṽ,w̃,θ̃_r)(t,a)   =(ũ,ṽ,w̃,θ̃_r)(t,b)=0,           t≥0.

These boundary conditions are supposed to be compatible with the initial data.





To establish the global existence, it is convenient to transform the initial boundary value problem (<ref>)–(<ref>) into that in Lagrangian coordinates.
We introduce the Lagrangian coordinates (t,x) and denote 
    (ρ,u,v,w,θ)(t,x) =(ρ̃,ũ,ṽ,w̃,θ̃)(t,r),
 
where

    r=r(t,x)=r_0(x)+∫_0^t ũ(s,r(s,x))ds,

and 

    r_0(x):=h^-1(x),   
    h(r):=∫_a^r z^m ρ̃_0(z)dz.

Notice that  the function h is invertible on [a,b] provided that ρ̃_0(z)>0 for each z∈[a,b] (which will be assumed in Theorem <ref>). 
Here, without loss of generality,
we set that h(b)=1.
Due to (<ref>), (<ref>), and (<ref>), we see 

    ∂/∂ t∫_a^r(t,x) z^m ρ̃(t,z)dz
    =
    ∂/∂ t∫_b^r(t,x) z^m ρ̃(t,z)dz=0.

Then it is easy to check that

    ∫_a^r(t,x) z^m ρ̃(t,z)dz=h(r_0(x))=x
       and  ∫_b^r(t,1) z^m ρ̃(t,z)dz=0.

Hence r(t,0)=a, r(t,1)=b for t≥ 0, and the region {(t,r): t≥ 0, a≤ r≤ b} under consideration is transformed into {(t,x):t≥ 0,0≤ x≤ 1}.
The identities (<ref>) and (<ref>) imply

    r_t(t,x)=u(t,x),   
    r_x(t,x)=r^-mτ(t,x),

where τ:=1/ρ is the specific volume.
By virtue of (<ref>),  the system (<ref>) is reformulated to that for (τ,u,v,w,θ)(t,x) as
 

   τ_t=(r^m u)_x,
 
   u_t-v^2/r+r^mP_x=
r^m[ν(r^m u)_x/τ]_x-2mr^m-1uμ_x,

 
   v_t+uv/r=
r^m[μr^m v_x/τ]_x+2μv_x-m( μr^m-1  v)_x-μτv/r^2m,

  
   w_t=
r^m[μr^m w_x/τ]_x+mμr^m-1 w_x,

  
   e_t+P(r^m u)_x=
[κr^2m θ_x/τ]_x+𝒬, 


where t>0, x∈ℐ:=(0,1), and

    P   =R θ/τ,  e=c_v θ,  c_v=R/γ-1,
    𝒬   =
    ν(r^m u)_x^2/τ-2mμ(r^m-1 u^2)_x+μ r^2m w_x^2/τ +μτ[r^m v_x/τ-v/r^m]^2.

The initial and boundary conditions are 
 
(τ,u,v,w,θ)(0,x)   =
(τ_0,u_0,v_0,w_0,θ_0)(x),
           x∈ℐ,
               
(u,v,w,θ_x)(t,0)   =(u,v,w,θ_x)(t,1)=0,           t≥0,

where (τ_0,u_0,v_0,w_0,θ_0):=
(1/ρ̃_0,ũ_0,ṽ_0,w̃_0,θ̃_0)∘ r_0, the symbol ∘ denotes composition, and r_0 is defined by (<ref>).





We now state our main results in the following theorem.
 
Suppose that the transport coefficients μ, λ, and κ satisfy (<ref>). Let the initial data (τ_0,u_0,v_0,w_0,θ_0) be compatible with the boundary conditions (<ref>) and satisfy

    (τ_0,u_0,v_0,w_0)_H^3(ℐ)+
     (√(c_v)(θ_0-1),θ_0xx)_H^1(ℐ)≤Π_0,
       V_0^-1≤τ_0(x)≤ V_0,    θ_0(x)≥ V_0^-1     for all x∈ℐ,

where  Π_0  and V_0 are positive constants independent of γ-1.
Then there exist constants ϵ_0>0 and C_1>0, which depend only on Π_0  and V_0, such that if γ-1≤ϵ_0, then the initial boundary value problem (<ref>)–(<ref>) has a unique global solution  (τ,u,v,w,θ)∈ C([0,∞),H^3(ℐ)) satisfying






    C_1^-1≤τ(t,x)≤ C_1,  12≤θ(t,x)≤ 2
         for all (t,x)∈[0,∞)×ℐ,

and the exponential decay rate

    (τ-τ̅,u,v,w,θ-θ̅)(t)_H^1(ℐ)+r(t)-r̅_H^2(ℐ)≤ C_γe^-c_γt     for all t∈[0,∞),
 
where C_γ and c_γ are positive constants depending on γ, and

    τ̅=∫_ℐτ_0dx,  θ̅=∫_ℐ[θ_0+1/2c_v(u_0^2+v_0^2+w_0^2)]dx,  r̅=[a^m+1+(m+1)τ̅x]^1/m+1.
 



 The techniques in this paper can be applied to obtain analogous results for the initial boundary value problem (<ref>)–(<ref>) with the following boundary conditions
 
    (u,v,w)|_x=0,1=0,  θ|_x=0,1=1,     t≥ 0.
  


 We deduce from (<ref>) and (<ref>) that no vacuum will be developed if the initial data do not contain a vacuum.
 It follows from 
 Sobolev's imbedding theorem that the unique solution constructed in Theorem <ref> is a globally smooth non-vacuum solution with large initial data. 
 Moreover, this result in Lagrangian coordinates can easily be converted to an equivalent statement for the corresponding problem in Eulerian coordinates.









Now we outline the main ideas to deduce Theorem <ref>.
As shown in <cit.>,
the crucial step to construct the global solutions for the initial boundary value problem (<ref>)–(<ref>) with large initial data is to obtain the positive upper and lower bounds of the specific volume τ and the temperature θ.
In the case of constant viscosity coefficients μ and λ, the pointwise bounds for the specific volume τ and the upper bound for the temperature θ can be obtained by modifying the argument in <cit.>. 
The positive lower bound for the temperature then follows from the standard maximum principle.
However, we find that this methodology does not work in the case of density/temperature dependent viscosity. 
 

For ideal polytropic gases (<ref>)–(<ref>), the temperature θ satisfies

    c_vθ_t+Rθ(r^m u)_x/τ=
    [κ r^2mθ_x/τ]_x+𝒬.

It can be expected to get the uniform-in-time bounds for √(c_v)(θ-1,θ_t)(t)_H^1(ℐ) by very careful energy estimates even when the viscosity and thermal conductivity coefficients are functions of the temperature.
Recalling that c_v=R/(γ-1), under the a priori assumption that γ-1 is sufficiently small, we can use the smallness of (θ-1,θ_t)(t)_H^1(ℐ) to handle the possible growth of solutions induced by the temperature dependence of the viscosity. 
The bounds for the specific volume τ from below and above can be established by developing the argument by <cit.> (see Lemma <ref>).

There are three main differences between our results for the symmetric flows (<ref>) and the work in <cit.> restricted to the one-dimensional case.
  
  		
  		
  		
  		
  		
  	
  	
  1.
First, the system (<ref>) under consideration here is more complicated than the one-dimensional compressible Navier–Stokes system (i.e. the system (<ref>) with m=v=w=0) which makes the form and treatment of equations simpler.

  2.
The second difference is that the analysis in <cit.> takes place in the whole space without boundary. Thus the energy estimates for the two-order and three-order derivatives of the solutions (u_xx(t)_L^2(ℝ) etc.) can be deduced directly from integrating by parts (see, for instance, <cit.> for the estimate of u_xx(t)_L^2(ℝ)).
Due to the presence of the boundary conditions (<ref>) in our case, we cannot use the methods in <cit.> to estimate the two-order and three-order derivatives of the solutions (τ,u,v,w,θ). 
To overcome such a difficulty, we make the estimates for (u_t,v_t,w_t,θ_t) and (u_xt,v_xt,w_xt,θ_xt), which yield the  bounds for (τ_xx,u_xx,v_xx,w_xx,θ_xx)(t)_H^1(ℐ).

  3.
A third difference concerns the assumptions on the initial data. 
In <cit.>, √(c_v)(θ_0-1)_H^3(ℝ) is required to be bounded by some (γ-1)-independent positive constant. 
According to this assumption, the H^1-norm of θ_0xx has to be small because of the smallness of γ-1. 
In Theorem <ref>, θ_0xx_H^1(ℐ) is assumed in (<ref>) to be bounded by some (γ-1)-independent positive constant. Thus θ_0xx_H^1(ℐ) can be large even when the adiabatic exponent γ goes to 1. 


 
The rest of this paper is organized as follows. 
First, in Section <ref>, we derive a number of desired a priori estimates. More specifically, the basic energy estimate is obtained in Subsection <ref>;
the uniform-in-time pointwise bounds of the specific volume τ is shown in Subsection <ref> by applying the argument developed by Kanel'; the estimates on first-order, second-order, and third-order derivatives of the solution (τ,u,v,w,θ) will be deduced in Subsections <ref>, <ref>, and <ref>, respectively. Finally, in Section 3, by combining the a priori estimates and the continuation argument, we prove the existence, uniqueness and exponential decay rate of global-in-time solutions for the problem (<ref>)–(<ref>).
 
 
 Notations.
Throughout this paper, we use ℐ:=(0,1). 
 For  1≤ q≤∞ and k∈ℕ, we denote by L^q(ℐ) the usual Lebesgue space on ℐ equipped with the norm ·_L^q(ℐ) and by H^k(ℐ) the standard Sobolev space in the L^2 sense equipped with the norm ·_H^k(ℐ).
 For notational simplicity, we shall use
 
    ·:=·_L^2(ℐ),  ·_k:=·_H^k(ℐ),  ·_L^q:=·_L^q(ℐ)
     .

 If I⊂ℝ and X is a Banach space, then
 we denote by C(I; X) the space of continuous functions on I with values in X, by L^q(I; X) the space of L^q-functions on I with values in X, and by ·_L^q(I; X)  the norm of the space L^q(I; X).
To simplify the presentation, we employ C, c, and C_i (i∈ℕ) to denote various positive constants, depending only on  Π_0  and V_0, where  Π_0  and V_0 are determined by (<ref>)–(<ref>). 
Hence C, c, and C_i are independent of γ-1 and t.
The symbol A≲ B (or B≳ A) means that A≤ C B holds uniformly for some (γ-1)-independent constant C.
We also use C_γ, c_γ, and C_i(γ) (i∈ℕ_+) to denote positive constants depending on γ.




§ A PRIORI ESTIMATES
 
This section is devoted to deriving certain a priori estimates on the solutions (τ, u, v, w, θ)∈ X(0,T;M,N) to the initial boundary value problem (<ref>)–(<ref>) with temperature dependent transport coefficients (<ref>) for T>0, M≥ 1, and  N≥ 1.
Here we define the set

    X(t_1,t_2 ;M,N):={
    (τ,u,v,w,θ)∈ C([t_1,t_2];H^3(ℐ)):
           τ_x∈ L^2(t_1,t_2;H^2(ℐ)),τ_t∈ C([t_1,t_2];H^2(ℐ))∩ L^2(t_1,t_2;H^2(ℐ)),
              (u_t,v_t,w_t,θ_t)∈ C([t_1,t_2];H^1(ℐ))∩ L^2(t_1,t_2;H^2(ℐ)),ℰ_t_1(t_2)≤ N^2,
            (u_x,v_x,w_x,θ_x)∈ L^2(t_1,t_2;H^3(ℐ)),τ(t,x)≥ M^-1 ∀ (t,x)∈[t_1,t_2 ]×ℐ},

for constants M, N, t_1, and t_2 (t_1≤ t_2),  where

    ℰ_t_1(t_2):=sup_t∈[t_1,t_2]{(u,√(c_v)(θ-1),√(c_v)θ_t)(t)_1^2
    +θ_xx(t)^2}
    +∫_t_1^t_2√(c_v)θ_t(s)_1^2ds,

with

    θ_t|_t=t_1   :=.1/c_v[-P(r^m u)_x+[κ r^2mθ_x/τ]_x+𝒬]|_t=t_1,
    θ_xt|_t=t_1   :=.1/c_v[-P(r^m u)_x+[κ r^2mθ_x/τ]_x+𝒬]_x|_t=t_1.


We set without loss of generality that R=1 and hence c_v=1/(γ-1).
For the sake of simplicity, we will use the following abbreviation:

    ·:=·_L^∞([0,T];L^∞( ℐ)).
 
Since (τ, u, v, w, θ)∈ X(0,T;M,N), it follows from  Sobolev's inequality that

    M^-1≤τ(t,x)≲ N,  (θ-1,θ_t)(t)_1≲  (γ-1)^1/2N  ∀ (t,x)∈[0,T]×ℐ,
    
       θ_x ≲ (γ-1)^1/4N,   (θ-1,θ_t) ≲ (γ-1)^1/2N,  ∫_0^Tθ_t(t)_1^2dt≲ (γ-1)N^2.


We shall make repeated use of the following estimate:
 
    a≤ r(t,x)≤ b  ∀ (t,x)∈[0,T]×ℐ,

 which follows from (<ref>) and (<ref>). 




 §.§ Basic energy estimate
 
In the following lemma we show the basic energy estimate and the pointwise bounds for θ.
 
 Assume that the conditions listed in Theorem <ref> hold.
 Then there exists a positive constant ϵ_1 depending only on  Π_0  and V_0, such that if
 
    (γ-1)^1/4 M^2 N^2≤ϵ_1,

 then 
 
    12≤θ(t,x)≤ 2    ∀ (t,x)∈[0,T]×ℐ,
       sup_t∈[0,T](√(ϕ(τ/τ̅)),u,v,w,√(c_v)(θ-1))(t)^2≲ 1,
       ∫_0^T∫_ℐ[mτ u^2+τ v^2+
     τ_t^2+u_x^2+v_x^2+w_x^2+θ_x^2/τ]dxd t ≲ 1,

 where ϕ(z):=z-ln z -1 and τ̅ is given by (<ref>).


The estimate (<ref>) follows immediately from (<ref>).
 In order to prove (<ref>) and (<ref>), we deduce an entropy-type energy estimate for the initial boundary value problem (<ref>)–(<ref>).
 Let θ̂>0 be an arbitrary but fixed constant.
 Multiplying (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) by θ̂(τ̅^-1-τ^-1), u, v, w, and (1-θ̂θ^-1), respectively, we have
 
    η_θ̂(τ,u,v,w,θ)_t
     +θ̂κ r^2mθ_x^2/τθ^2
     +θ̂𝒬/θ=ℛ_x,

 where 
 
    η_θ̂(τ,u,v,w,θ):=θ̂ϕ(τ/τ̅)+12(u^2+v^2+w^2)+c_vθ̂ϕ(θ/θ̂)

 is the relative entropy (see, for instance, <cit.>), 𝒬 is given by (<ref>), and
 
    ℛ:=    ν r^m u(r^m u)_x/τ+
     [1-θ̂/θ]
     κ r^2mθ_x/τ
     + r^m u[θ̂/τ̅-θ/τ]
       -2mμ r^m-1 u^2
     +μ r^2m (vv_x+ww_x)/τ-m μ r^m-1 v^2.

 Thanks to the boundary conditions (<ref>), we integrate (<ref>) over ℐ to obtain
  
    d/dt∫_ℐη_θ̂(τ,u,v,w,θ)
      +∫_ℐθ̂[κ r^2mθ_x^2/τθ^2
      +𝒬/θ]=0.

  By virtue of (<ref>), we infer that, for σ>0, 
  
    ν(r^m u)_x^2-
      2mμτ(r^m-1 u^2)_x =ν r^2m u_x^2+(2mμ+
      m^2 λ)τ^2 u^2/r^2+2m λ r^m-1τ u u_x
          =[mλ r^m u_x/√(σ)+
      √(σ)τ u/r]^2+
      [ν-m^2λ^2/σ]r^2mu_x^2+
      [2mμ+m^2λ -σ]τ^2u^2/r^2
         ≥[ν-m^2λ^2/σ]r^2mu_x^2
      +[2mμ+m^2λ -σ]τ^2u^2/r^2.

In light of (<ref>) and (<ref>), 
 we can choose σ =m(mλ^2+2μ^2+(m+1)μλ)/ν>0 so that

    ν(r^m u)_x^2-2mμτ(r^m-1 u^2)_x/τθ≥β/τ[r^2mu_x^2+m τ^2 u^2/r^2],

where β is given by 

    β:=2μ+(m+1)λ/θmin{μν/mλ^2+2μ^2+(m+1)μλ,μ/ν}.

Using (<ref>), (<ref>), (<ref>), and (<ref>), we integrate by parts to find

    2∫_ℐμ(θ)/θvv_x
    =-∫_0^T∫_ℐd/dθ[μ(θ)/θ]θ_xv^2
    ≲ (γ-1)^1/4NM∫_ℐτ v^2,
 
which combined with (<ref>) and (<ref>) implies

    ∫_ℐμτ/θ[r^m v_x/τ-v/r^m]^2≳∫_ℐv_x^2/τ+(1-ϵ_1)∫_ℐτ v^2.

Plug (<ref>) and (<ref>) into (<ref>), 
and use (<ref>) to derive that if (<ref>) holds for some sufficiently small ϵ_1>0, then

    d/dt∫_ℐη_θ̂(τ,u,v,w,θ)
    +c∫_ℐθ̂[mτ u^2+τ v^2+
    u_x^2+v_x^2+w_x^2+θ_x^2/τ]
    ≤ 0.
 
It follows from (<ref>), (<ref>), and (<ref>) that 

    ϕ(θ)≳ (θ-1)^2   and  τ_t^2/τ≲ mτ u^2+u_x^2/τ.

If we take θ̂=1, by virtue of the conditions assumed in Theorem <ref>, we infer

    ∫_ℐη_1(τ_0,u_0,v_0,w_0,θ_0)≲ 1.

Integrating (<ref>) with θ̂=1 over (0,t) yields the estimates (<ref>) and (<ref>).

 


 §.§ Pointwise bounds for the specific volume
 
In this subsection we employ the argument developed by Kanel' <cit.> to obtain the uniform bounds for the specific volume τ.
To this end, we first make the estimate for τ_x/τ  in the following lemma.  
 
If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then 

    sup_t∈[0,T]τ_x/τ(t)^2
    +∫_0^T∫_ℐτ_x^2/τ^3≲ 1+lnτ.



According to the chain rule, we have

    [ντ_x/τ]_t=
    [ντ_t/τ]_x+
    ν'(θ)/τ(τ_xθ_t-τ_tθ_x),
 
which combined with (<ref>) and (<ref>) implies

    [ντ_x/τ]_t-u_t/r^m-P_x
    
    =-v^2/r^m+1+2m uμ_x/r+
    ν'(θ)/τ(τ_xθ_t-τ_tθ_x).

Multiply (<ref>) by ντ_x/τ and use (<ref>), (<ref>) to deduce

    [1/2(ντ_x/τ)^2-
    u/r^mντ_x/τ]_t
    +νθτ_x^2/τ^3
    +[u/r^mντ_t/τ]_x-(u/r^m)_x ντ_t/τ
    -ντ_xθ_x/τ^2
          =ντ_x/τm u^2-v^2/r^m+1
    +2m uμ_x/rντ_x/τ
    +ν'(θ)/τ^2(ντ_x-r^-muτ)
    (τ_xθ_t-τ_tθ_x).

Integrating the last identity, we get

    d/dt∫_ℐ[1/2(ντ_x/τ)^2-
    u/r^mντ_x/τ]
    +∫_ℐνθτ_x^2/τ^3
    =∑_q=1^7𝒦_q,

where each term 𝒦_q in the decomposition will be defined and estimated below. First we consider the term

    𝒦_1:=
    ∫_ℐ(u/r^m)_xντ_t/τ.

According to (<ref>) and (<ref>), we infer

    𝒦_1=
    ∫_ℐντ_t/τ(u_x/r^m-mτ u/r^2m+1)≲∫_ℐ[u_x^2/τ+
    τ_t^2/τ+mτ u^2].

The second term
is defined and estimated as

    𝒦_2:=∫_ℐντ_xθ_x/τ^2≤ϵ∫_ℐτ_x^2/τ^3
    +C(ϵ) ∫_ℐθ_x^2/τ.

For the term 

    𝒦_3:=
    ∫_ℐντ_x/τm u^2-v^2/r^m+1,

by virtue of the boundary conditions (<ref>),
we integrate by parts and use (<ref>), (<ref>) to derive

    𝒦_3=-∫_ℐlnτ[ν/r^m+1(m u^2-v^2)]_x≲lnτ∫_ℐ|(mθ_x u^2,θ_x v^2,
    mτ u^2,τ v^2,mu u_x,vv_x)|.

In view of (<ref>) and (<ref>), we have

    𝒦_3≲lnτ[1+
    (γ-1)^1/4NM]
    ∫_ℐ[mτ u^2+τ v^2
    +u_x^2+v_x^2/τ].

The term 

    𝒦_4=
    ∫_ℐν'(θ)/τr^-mu τ_tθ_x

can be treated by using (<ref>) as

    𝒦_4   ≲u^1/2u_x^1/2τ_t/√(τ)θ_x/√(τ)
    
       ≲τ_t/√(τ)^2+
    (γ-1)^1/2N^4M^2
    u_x/√(τ)θ_x/√(τ)
       ≲[1+
    (γ-1)^1/2N^4M^2]
    ∫_ℐτ_t^2+u_x^2+θ_x^2/τ.

We have from (<ref>) and (<ref>) that

    𝒦_5:=∫_ℐν'(θ)ν/τ^2θ_tτ_x^2≲ 
    Nθ_t∫_ℐτ_x^2/τ^3≲ϵ_1 ∫_ℐτ_x^2/τ^3.

Using (<ref>), we get

    𝒦_6:=   ∫_ℐτ_x/τθ_x (2m r^-1uμ'(θ)ν 
    -νν'(θ)τ^-1τ_t)
    ≲   ϵ∫_ℐτ_x^2/τ^3
    +C(ϵ)(γ-1)^1/2N^2M^2∫_ℐ[
    mτ u^2+τ_t^2/τ].

For the last term

    𝒦_7:=-∫_ℐν'(θ)/τ r^-mu τ_xθ_t ,

we have from (<ref>) that

    𝒦_7
    ≲ϵ∫_ℐτ_x^2/τ^3
    +C(ϵ) u^2 Nθ_t^2
    ≲ϵ∫_ℐτ_x^2/τ^3
    +C(ϵ) N^3  θ_t^2.

Plugging (<ref>)–(<ref>) into (<ref>) and using (<ref>), (<ref>) yield

    d/dt∫_ℐ[1/2(ντ_x/τ)^2-
    u/r^mντ_x/τ]
    +∫_ℐτ_x^2/τ^3
         ≲ N^3  θ_t^2+
    [1+ lnτ]
    ∫_ℐ[mτ u^2+v^2+
    τ_t^2+u_x^2+v_x^2+θ_x^2/τ].

We integrate (<ref>) over (0,t), apply Cauchy's inequality, and use (<ref>), (<ref>), (<ref>) to conclude (<ref>).


Now we establish the uniform bounds for the specific volume, which are essential for the proof of the main theorem.
 
If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then 

    C_1^-1≤τ(t,x)≤ C_1     for all (t,x)∈[0,T]×ℐ.

 

Define

    Φ(τ):=∫_1^τ/τ̅√(ϕ(z))/zdz,

where τ̅ is defined by (<ref>).
We infer that for suitably large constant C and τ≥ C,

    Φ(τ) ≳∫_C^τ/τ̅√(ϕ(z))/zdz≳∫_C^τ/τ̅ z^-1/2dz.

Hence 

    τ^1/2≲ 1+|Φ(v)|    ∀ τ∈(0,∞).

Similarly, it follows that

    |lnτ|^3/2≲ 1+|Φ(τ)|    ∀ τ∈(0,∞).

Thus, we have

    τ^1/2+lnτ^3/2≲
     1+sup_(t,x)∈[0,T]×ℐ|Φ(τ(t,x))|.

By virtue of (<ref>) and (<ref>), we obtain

    ∫_ℐτ(t,x)dx=τ̅    ∀ t∈[0,T],

which implies that for each t∈[0,T], there exists y(t)∈ℐ such that τ(t,y(t))=τ̅. Hence

    |Φ(τ)(t,x)|=
    |∫_y(t)^x∂/∂ xΦ(τ(t,y))dy|
         ≤∫_ℐ√(ϕ(τ/τ̅(t,y)))|τ_x/τ(t,y)|dy
    ≤√(ϕ(τ/τ̅))(t)τ_x/τ(t),

which combined with Lemmas <ref>–<ref>  yields

    |Φ(τ)(t,x)| ≲ 1+lnτ^1/2    ∀ (t,x)∈[0,T]×ℐ.

Combine (<ref>) with (<ref>) to deduce (<ref>).


We plug (<ref>) into (<ref>), (<ref>), and (<ref>) to obtain the following corollary.
 
If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then 

    sup_t∈[0,T](τ,τ_x,u,v,w,√(c_v)(θ-1))(t)^2
    +∫_0^T(τ_x,τ_t,√(m) u, u_x,v, v_x, w_x,θ_x )(t)^2d t ≤ C_2.
 




 §.§ Estimates on first-order derivatives


This part is devoted to deducing the bounds for  (u_x,v_x,w_x,√(c_v)θ_x)(t) uniformly in time t.
We first make the estimate for w_x in the next lemma.
 
If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then 

    sup_t∈[0,T]w_x(t)^2 +∫_0^Tw_xx(t)^2dt≤ C_3.



Multiply (<ref>) by w_xx, integrate the resulting identity over ℐ, and use 
    w_t(t,0)=w_t(t,1)=0
 to find

    d/dtw_x(t)^2
    +2∫_ℐμ r^2mw_xx^2/τ
    =-2∫_ℐw_xxr^mw_x[μ r^m/τ]_x
    -2m∫_ℐw_xxμ  r^m-1w_x.

In view of (<ref>), (<ref>) and (<ref>),
we have

    |[μ r^m/τ]_x|≲|(1,θ_x,τ_x)|.

By virtue of (<ref>), (<ref>) and (<ref>),  we have

    sup_t∈[0,T](θ-1,θ_t)(t)_1+(θ_x,θ_t)+
    ∫_0^Tθ_t(t)_1^2dt≲ 1.

Then using Cauchy's and Sobolev's inequalities yields

    d/dtw_x(t)^2
    + w_xx(t)^2≲∫_ℐ w_x^2(1+θ_x^2+τ_x^2)
         ≲[1+θ_x^2]w_x(t)^2
    +sup_s∈[0,T]τ_x(s)^2w_x(t)_L^∞^2
         ≲[1+C(ϵ)sup_s∈[0,T]τ_x(s)^4]w_x(t)^2+ϵw_xx(t)^2,

which combined with (<ref>) implies

    d/dtw_x(t)^2+ w_xx(t)^2≲w_x(t)^2.

We conclude (<ref>) by integrating (<ref>) over (0,t). 




In the following lemma we obtain the bounds for u_x(t) and v_x(t) uniformly in time t. 

If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then 

    sup_t∈[0,T](u_x,v_x)(t)^2 +∫_0^T(u_xx,v_xx)(t)^2dt≤ C_4.



 Multiply (<ref>) and (<ref>) by u_xx and v_xx, respectively, and then add the resulting identities to get
 
    (u_x^2+v_x^2)_t-2(u_xu_t+v_xv_t)_x
        +2r^2m/τ(ν u_xx^2+μ v_xx^2)
        -2u_xxr^m P_x
         =2[-u_xxv^2/r+v_xxuv/r]
        -2u_xx r^m[(ν (r^m u)_x/τ)_x
        -ν r^m u_xx/τ]+4m r^m-1u μ_x u_xx
          -2v_xx[r^m v_x (μ r^m/τ)_x+2μ r^m-1 v_x    -m(μ r^m-1 v)_x-μτ v /r^2m].

 Integrating the last identity and using the boundary conditions
 
    u_t|_∂ℐ=v_t|_∂ℐ=0,

 we have
 
    d/dt(u_x,v_x)(t)^2
     +2∫_ℐr^2m/τ(ν u_xx^2+μ v_xx^2)=∑_q=8^12𝒦_q,

 where each term 𝒦_q in the decomposition will be defined and estimated below.
 First, we get
 
    𝒦_8:=2∫_ℐ u_xxr^m P_x
     ≤ϵu_xx(t)^2
     +C(ϵ) (θ_x,τ_x)(t)^2.

 For the term
 
    𝒦_9:=
     2∫_ℐ[-u_xxv^2/r+v_xxuv/r],

 we have from Sobolev's inequality and (<ref>) that
 
    𝒦_9   ≤ϵ(u_xx,v_xx)(t)^2
     +C(ϵ)|(u,v)(t)_L^∞^2v(t)^2
       ≤ϵ(u_xx,v_xx)(t)^2
     +C(ϵ)(u_x,v_x)(t)v(t)^2.
 
 To estimate the term
 
    𝒦_10:=-2∫_ℐ
     u_xx r^m[(ν (r^m u)_x/τ)_x-
     ν r^m u_xx/τ],

we first compute from (<ref>), (<ref>) and (<ref>) that

    |(ν (r^m u)_x/τ)_x-ν r^m u_xx/τ| 
    
    ≲|(τ_x,θ_x)||(u_x,mu)|+|(u_x,muτ_x,mu)|.

It follows from (<ref>), (<ref>), and (<ref>) that 

    𝒦_10   ≤ϵu_xx(t)^2+C(ϵ)
    ∫_ℐ[τ_x^2 u_x^2+(1+θ_x^2)|(u_x,muτ_x,mu)|^2]
       ≤ϵu_xx(t)^2+C(ϵ)
    [1+θ_x^2][
    (u_x,mu)(t)^2+τ_x(t)^2(u_x,mu)(t)_L^∞^2]
       ≤ϵu_xx(t)^2+C(ϵ)
    [(u_x,mu)(t)^2+(u_x,mu)(t)(u_xx,u_x)(t)] 
       ≤ 2ϵu_xx(t)^2+C(ϵ)(u_x,mu)(t)^2.

  The term
  
    𝒦_11
      := 4m∫_ℐ  r^m-1u μ_x u_xx

  can be easily estimated as
  
    𝒦_11≤ϵu_xx(t)^2+C(ϵ)θ_x^2 mu(t)^2
      ≤ϵu_xx^2+C(ϵ)mu(t)^2.

  The last term is 
  
    𝒦_12:=
      -2∫_ℐ
       v_xx[r^m v_x (μ r^m/τ)_x+2μ r^m-1 v_x   -m(μ r^m-1 v)_x-μτ v /r^2m].

  Similar to the derivation of (<ref>), we can get
  
    𝒦_12   ≤ϵv_xx(t)^2
      +C(ϵ)∫_ℐ[
      v_x^2(1+θ_x^2+τ_x^2)+v^2]
    
    
         ≤ 2ϵv_xx(t)^2+C(ϵ)(v_x,v)(t)^2  .

  Plugging (<ref>)–(<ref>) into 
  (<ref>) and taking ϵ sufficiently small yield

    d/dt(u_x,v_x)(t)^2+( u_xx,v_xx)(t)^2
    ≲(θ_x,τ_x,u_x,mu,v_x,v)(t)^2+(u_x,v_x)(t)v(t)^2.
  
Integrating (<ref>) over (0,t), we deduce from (<ref>) that

    (u_x,v_x)(t)^2+∫_0^t( u_xx,v_xx)(s)^2ds
    
    ≲ 1+sup_s∈[0,t](u_x,v_x)(s),
 
from which we can conclude (<ref>).


The following lemma is the estimate on √(c_v)θ_x(t).

If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then 

    sup_t∈[0,T]√(c_v)θ_x(t)^2 +∫_0^Tθ_xx(t)^2dt≤ C_5.



 Multiply (<ref>) by θ_xx, use the boundary condition for θ, and integrate the resulting identity to get
 
    d/dt√(c_v)θ_x(t)^2
     +2∫_ℐκ r^2mθ_xx^2/τ
     =∑_q=13^15𝒦_q,

 where the term 𝒦_q will be given below.
 For the term
 
    𝒦_13:=∫_ℐ
     2θ_xx[θ(r^m u)_x/τ
     -ν(r^m u)_x^2/τ+2mμ(r^m-1 u^2)_x],

  it follows from (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) that

    𝒦_13   ≤ϵθ_xx(t)^2
    +C(ϵ)∫_ℐ|(u_x,mu,1)|^2|(u_x,mu)|^2
       ≤ϵθ_xx(t)^2
    +C(ϵ)(u_x,mu,1)(t)_L^∞^2(u_x,mu)^2
       ≤ϵθ_xx(t)^2
    +C(ϵ)(u_x,mu,1)(t)^2(u_x,mu,u_xx)(t)^2 
       ≤ϵθ_xx(t)^2+C(ϵ)(u_x,mu,u_xx)(t)^2.

Defining

    𝒦_14:=∫_ℐ2θ_xx[
    -μ r^2m w_x^2/τ -μτ(r^m v_x/τ-v/r^m)^2],

 we have from (<ref>), (<ref>) and (<ref>) that

    𝒦_14   ≤ϵθ_xx^2
    +C(ϵ)∫_ℐ|(w_x,v_x,v)|^4
       ≤ϵθ_xx^2
    +C(ϵ)(w_x,v_x,v)(t)_L^∞^2
    (w_x,v_x,v)(t)^2
       ≤ϵθ_xx^2+C(ϵ)(w_xx,v_xx,w_x,v_x,v)(t)^2

The term

    𝒦_15:=∫_ℐ2 θ_xx[κ r^2mθ_xx/τ-(κ r^2mθ_x/τ)_x]

can be bounded by using (<ref>) as

    𝒦_15   ≤ϵθ_xx(t)^2
    +C(ϵ)∫_ℐθ_x^2[1+θ_x^2+τ_x^2]
       ≤ϵθ_xx(t)^2+C(ϵ)[1+θ_x^2] (θ_x,τ_x)(t)^2
       ≤ϵθ_xx(t)^2+C(ϵ)(θ_x,τ_x)(t)^2.

We insert (<ref>)–(<ref>) into (<ref>) to obtain

    d/dt√(c_v)θ_x(t)^2+θ_xx(t)^2≲(mu,v,τ_x,u_x,v_x,w_x,θ_x,u_xx,v_xx,w_xx)(t)^2,

which combined with (<ref>), (<ref>), (<ref>) and (<ref>) implies (<ref>).


Using the system (<ref>), we can deduce the next lemma from Corollary <ref>, Lemmas <ref>–<ref>. 

If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then 
 
    sup_t∈[0,T]τ_t(t)^2 +∫_0^T(τ_xt,
     u_t,v_t,w_t,c_vθ_t)(t)^2dt≲ 1.



 We have from (<ref>), (<ref>), (<ref>), and (<ref>) that
 
    |τ_t|≲ |(mu,u_x)|,  
     |τ_xt|≲ |(u_xx,u_x,mu,muτ_x)|,

 which combined with (<ref>) and (<ref>) yields 
 
    sup_t∈[0,T]τ_t(t)^2 +∫_0^Tτ_xt(t)^2dt≲ 1.

 By virtue of (<ref>), (<ref>) and (<ref>), we can deduce from (<ref>)  that
 
    |c_vθ_t|≲|(mu,u_x,θ_xx,θ_x^2,τ_xθ_x,θ_x,mu^2,u_x^2,w_x^2,v_x^2,v^2)|,

 which combined with (<ref>), (<ref>), (<ref>), and (<ref>) implies
 
    c_vθ_t(t)^2   ≲(mu,u_x,θ_xx,θ_x,τ_x)(t)^2
     +∫_ℐ|(mu,u_x,w_x,v_x,v)|^4
       ≲(mu,u_x,θ_xx,θ_x,τ_x)(t)^2
     +(mu,u_x,w_x,v_x,v)^2(mu,u_x,w_x,v_x,v)_1^2
       ≲(mu,u_x,θ_xx,θ_x,τ_x,u_xx,w_x,w_xx,v_xx,v_x,v)(t)^2.

 Then it follows from Corollary <ref>, Lemmas <ref>–<ref> that 
 
    ∫_0^Tc_vθ_t(t)^2dt≲ 1.


By virtue of (<ref>) and (<ref>), we have 

    c_vθ_t(0)≤ C,

for some (γ-1)-independent positive constant C.

 The other estimates in (<ref>) can be proved by a similar computation.



 §.§ Estimates on second-order derivatives


In this subsection, we aim to derive the uniform bounds for (τ_xx,u_xx,v_xx,w_xx,θ_xx)(t).  
For this purpose, we make the estimates for (u_t,v_t,w_t) in the next lemma. 
 
If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then
 
    sup_t∈[0,T](u_t,v_t,w_t)(t)^2 +
     ∫_0^T(u_xt,τ_tt,v_xt,w_xt)(t)^2dt≲ 1.


 The proof is divided into three steps.
	
Step 1. 
 We let ∂_t act on (<ref>), and multiply the resulting identity by u_t to yield
 
    [1/2u_t^2]_t+ν r^2mu_xt^2/τ
     +[u_tr^m P_t-u_t r^m (ν (r^m u)_x/τ)_t]_x
     =J_1+J_2+J_3,

 with 
 
    J_1   :=
      -(r^m)_xu_t[ν(r^m u)_x/τ]_t,
    
     J_2   :=
     r^m u_xt[P_t-(ν/τ)_t(r^m u)_x
     -ν/τ((r^m u)_xt-r^m u_xt)],
    
      J_3   :=u_t[(v^2/r)_t-(r^m)_tP_x+(r^m)_xP_t+
      (r^m)_t(ν (r^mu)_x/τ)_x-2m(r^m-1u μ_x)_t].

 It follows  from (<ref>), (<ref>) and (<ref>) that
 
    (u,v,w)≲ 1.

  To estimate J_1 and J_2, we first get from (<ref>) that
  
    |(r^m u)_xt-r^m u_xt|
         ≲|((r^m)_txu,(r^m)_xu_t,(r^m)_tu_x)|
      ≲ |(mu^2,u_t,uu_x)|,
    |(r^m u)_xt|
         ≲|(mu^2,u_t,uu_x,u_xt)|.

  Then we have from (<ref>) and (<ref>) that
  
    J_1≲ 
      |u_t|
      |(θ_tτ_t,τ_t^2,mu,u_t,u_x,u_xt)|
      ≲ϵ u_xt^2+C(ϵ)
      |(τ_t,τ_t^2,mu,u_t,u_x)|^2,

 and
  
    J_2≲ 
      |u_xt|
      |(θ_t,τ_t,θ_tτ_t,τ_t^2,mu^2,u_t,uu_x)|
      ≲ϵ u_xt^2+C(ϵ)
      |(θ_t,τ_t,τ_t^2,mu,u_t,u_x)|^2.
 
 According to (<ref>), we have from (<ref>),  (<ref>), (<ref>), and (<ref>) that
 
    |[ν (r^m u)_x/τ]_x|
     ≲|(u_t,v^2,θ_t,τ_t,θ_x)|,

and 
 
    J_3
     
     ≲ |u_t||(v_t,v,θ_x,τ_x,θ_t,τ_t,u_t,θ_xt)|≲|(u_t,v_t,v,θ_x,τ_x,θ_t,τ_t,θ_xt)|^2.

 Integrating (<ref>) over (0,t)×ℐ, using (<ref>)–(<ref>), (<ref>)–(<ref>),  (<ref>), and (<ref>) yield
 
    u_t(t)^2+∫_0^tu_xt^2
    
    
     ≲ 1+∫_0^tθ_xt^2+ ∫_0^tτ_t_L^∞^2τ_t^2 ≲ 1+∫_0^tτ_t_1^2≲ 1.


By virtue of (<ref>), τ_tt=(r^m u)_xt, and hence 

    |τ_tt|≲ |u_xt|+|(u_t,uu_x)|+|mu||(u_x,1)|.
 
Combining this with (<ref>),(<ref>), (<ref>),  and (<ref>) implies

    ∫_0^Tτ_tt(t)^2dt≲ 1.
 
 
Step 2. 
 Letting ∂_t act on (<ref>), and  multiplying the resulting identity by v_t  yield
 
    [1/2v_t^2]_t+μ r^2mv_xt^2/τ -[v_t r^m (μ r^m v_x/τ)_t]_x =J_4+J_5,

 with 
 
    J_4   :=v_t[-(uv/r)_t+(r^m)_t(μ r^m v_x/τ)_x -(μτ v/r^2m)_t],
    
     J_5   := v_t [-(r^m)_x(μ r^m v_x/τ)_t+2(μ r^m-1v_x)_t-m(μ r^m-1v)_xt]-r^m v_xtv_x[μ r^m /τ]_t.

 By virtue of (<ref>), (<ref>) and (<ref>),  we derive from (<ref>) that
 
    |[μ r^m v_x/τ]_x|
     ≲|(v_t,v,θ_x,v_x)|,
 
 and
 
    J_4≲ |v_t||(u_t,v_t,v,v_x,θ_x,τ_t,θ_t)|.

It follows from (<ref>) and (<ref>) that
 
    J_5   ≲
     |v_t||(v_xt,
     τ_t v_x,v_x,v_t,θ_xt,θ_x,θ_t,u_x,mu)|
     +|v_xtv_x||(θ_t,τ_t,u)|
       ≲ϵ v_xt^2+C(ϵ)
     |(v_t, τ_t v_x,v_x,θ_xt,θ_x,θ_t,u_x,mu)|^2.
 
 Integrating (<ref>) over (0,t)×ℐ, using (<ref>)–(<ref>), (<ref>)–(<ref>), (<ref>), and (<ref>) yield
 
    v_t(t)^2+∫_0^tv_xt^2
     
     ≲ 1+∫_0^tθ_xt^2+
     ∫_0^tτ_t_L^∞^2v_x^2
     ≲ 1+∫_0^tτ_t_1^2≲ 1.

 

 
 
Step 3. 
 Let ∂_t act on (<ref>), and multiply the resulting identity by w_t to get
 
    [1/2w_t^2]_t+μ r^2mw_xt^2/τ
     -[w_t r^m (μ r^m w_x/τ)_t]_x
     =J_6+J_7,

    with
    
    J_6   :=w_t(r^m)_t[μ r^m w_x/τ]_x,
    
        J_7   : =-r^m w_x w_xt[μ r^m/τ]_t
        - w_t[
        (r^m)_x(μ r^m w_x/τ)_t
        -m(μ r^m-1w_x)_t].

According to (<ref>) and (<ref>), we infer

    J_6   ≲ |w_t||(w_t,w_x)|≲ |(w_t,w_x)|^2,
    
    J_7   ≲ 
    |w_xt||(θ_tw_x,τ_tw_x,w_x,w_t)|
    +|w_t||(θ_tw_x,τ_tw_x,w_x)|
       ≲ϵ w_xt^2+C(ϵ)
    |(w_t,θ_tw_x,w_x,τ_t w_x)|^2.

 Integrating (<ref>) over (0,t)×ℐ,
 using (<ref>)–(<ref>),
 (<ref>)–(<ref>), (<ref>), and (<ref>) yield
 
    w_t(t)^2+∫_0^tw_xt^2
     
     ≲ 1+
     ∫_0^tτ_x_L^∞^2w_x^2
     ≲ 1+∫_0^tτ_t_1^2≲ 1.

Combine the estimates (<ref>), (<ref>), (<ref>) and (<ref>) to
derive  (<ref>).




Noting that the equations for (u,v,w) are parabolic, we deduce the uniform  bounds for the L_x^2-norms of (u_xx,v_xx,w_xx) in the following lemma.
   If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then
 
    sup_t∈[0,T](u_xx,v_xx,w_xx,τ_xt)(t)≤ C_6.



It follows from (<ref>), (<ref>), and (<ref>) that

    |(r^m)_x|≲ m,  
    |(r^m)_xx|≲ |(τ_x,m)|,  
    |(r^m)_xxx|≲ |(τ_xx,τ_x,m)|,

and hence

    |(r^m u)_xx-r^m u_xx|≲ |(τ_x u,mu,u_x)|.

According to (<ref>), we get

    ν r^2m u_xx/τ
    = u_t-v^2/r+r^mP_x+2mr^m-1uμ_x
    -r^m[(ν(r^m u)_x/τ)_x-
    ν r^m u_xx/τ].

Using (<ref>),  (<ref>) and (<ref>), we obtain

    |u_xx|≲|(u_t,v,θ_x,τ_x)|+
    |(θ_xτ_t,τ_xτ_t)|+
    |(τ_x u,mu,u_x)|,

which combined
with Corollary <ref>, Lemmas <ref>–<ref>
yields

    sup_t∈[0,T]u_xx(t)≲ 1.

Since the proof of the other estimates in (<ref>) can be shown in a similar way, we omit it.



The following lemma concerns the bounds for  the L^2_x-norms of c_vθ_t and θ_xx.

   If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then
 
    sup_t∈[0,T](c_vθ_t,θ_xx)(t)^2 +∫_0^T√(c_v)θ_xt(t)^2dt≤ C_7.



Let ∂_t act on (<ref>) and multiply the resulting identity with c_vθ_t to discover
 
    [(c_vθ_t)^2/2]_t+c_v κ r^2mθ_xt^2/τ
     -[c_vθ_t(κ r^2mθ_x/τ)_t]_x
         =c_vθ_t𝒬_t-c_vθ_t(P(r^m u)_x)_t-
     c_vθ_xθ_xt[κ r^2m/τ]_t.

By virtue of the boundary conditions (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>), we integrate the last identity to get

    c_vθ_t(t)^2+∫_0^t√(c_v)θ_xt^2
         ≲c_vθ_t(0)^2+∫_0^t(c_vθ_t,𝒬_t,
    (P(r^m u)_x)_t)^2
    +∫_0^t∫_ℐc_vθ_x^2|(θ_t,τ_t,mu)|^2
         ≲ 1+
    ∫_0^t(𝒬_t,(P(r^m u)_x)_t)^2
    +sup_s∈[0,T]√(c_v)θ_x(s)^2∫_0^t(θ_t,τ_t,mu)_1^2
         ≲ 1+∫_0^t(𝒬_t,
    θ_tτ_t,τ_t^2,τ_tt)^2.

Using (<ref>),   (<ref>) and (<ref>),
after some elementary calculations, we have

    |𝒬_t|≲    
    |(θ_t,τ_t)|τ_t^2+|τ_tτ_tt|+
    |θ_t||(u_x,mu)|+|(u_x,mu,u_t,u_xt,u_xu_t)|+|w_xw_xt|
       +w_x^2|(θ_t,τ_t,mu)|
    +|(θ_t,τ_t)||(v_x,v)|^2
    +|(v_x,v)||(v_xt,v_t,v_x,v_xτ_t,v)|,

and

    𝒬_t^2≲    
    |(θ_t,τ_t,mu,w_x,v_x,v)|^6+τ_t^2τ_tt^2+w_x^2w_xt^2
    +v_x^2v_xt^2
       +
    |(θ_t,u_x,τ_t,u_t,v_x,v_t)|^4+
    |(θ_t,u_x,mu,u_t,u_xt,v_xt,v_t,v_x,v)|^2.

Employing Sobolev's inequality, (<ref>), Corollary <ref>, and Lemmas <ref>–<ref> yields

    ∫_0^t∫_ℐ |(θ_t,τ_t,mu,w_x,v_x,v)|^6
         ≲sup_[0,t](θ_t,τ_t,mu,w_x,v_x,v)^4
    ∫_0^t(θ_t,τ_t,mu,w_x,v_x,v)_1^2
    ≲ 1,
       ∫_0^t∫_ℐ|(θ_t,θ_x,u_x,τ_t,u_t,v_x,v_t)|^4
         ≲sup_[0,t](θ_t,θ_x,u_x,τ_t,u_t,v_x,v_t)^2
    ∫_0^t(θ_t,θ_x,u_x,τ_t,u_t,v_x,v_t)_1^2
    ≲ 1,

and 

    ∫_0^t∫_ℐ[τ_t^2τ_tt^2+w_x^2w_xt^2
    +v_x^2v_xt^2]≲sup_[0,t](τ_t,w_x,v_x)_1^2∫_0^t(τ_tt,w_xt,v_xt)^2
    ≲ 1.

Combining (<ref>)–(<ref>) implies
 
    sup_t∈[0,T]c_vθ_t(t)^2 +∫_0^T√(c_v)θ_xt(t)^2dt≲ 1.


By virtue of the equation (<ref>), we have

    κ r^2mθ_xx/τ
    =c_vθ_t+Pτ_t
    -(κ r^2m/τ)_x θ_x-𝒬,

which implies

    |θ_xx|≲ |(c_vθ_t,τ_t)|+|θ_x||(θ_x,τ_x,m)|
    +|(τ_t^2,u_x,mu,w_x^2,v_x^2,v^2)|.

We use this estimate and (<ref>), (<ref>) to have
 
    sup_t∈[0,T]θ_xx(t)^2 ≲ 1.

The estimate (<ref>) then follows from (<ref>) and (<ref>).




In the following lemma we make an estimate for ((u_xxx,v_xxx,w_xxx,θ_xxx) in L^2(0,T;L^2(ℐ)).
 
 If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then
 
    ∫_0^T(u_xxx,v_xxx,w_xxx,
     θ_xxx/√(c_v))(t)^2
     dt≲ 1+∫_0^Tτ_xx(t)^2dt.



Differentiating (<ref>) with respect to x, we have
 
    r^2mν u_xxx/τ+(r^m)_x[ντ_t/τ]_x+r^m[(ν/τ)_xτ_t]_x+r^m(ν/τ)_x(r^m u)_xx
           =
     u_xt-[v^2/r]_x+(r^m P_x)_x+2m(r^m-1u μ_x)_x-r^m ν/τ[(r^m u)_xxx-r^m u_xxx].

 By a direct computation, applying (<ref>), (<ref>), (<ref>), and (<ref>) gives

    |u_xxx|
    ≲    |u_xt|+|(v_x,v,θ_x,τ_x,θ_xx,τ_xx,τ_x^2,u_x)|
    +|u||(τ_xx,τ_x,m)|+|u_x||(τ_x,1)|+|u_xx|
       + |(τ_tx,θ_xτ_t,τ_xτ_t)|
    +|(θ_xxτ_t,τ_xxτ_t,τ_t,τ_x^2τ_t,θ_xτ_xt,τ_xτ_xt)|
    +|τ_tx||(θ_x,τ_x)|.
 
Applying (<ref>), Corollary <ref> and Lemmas <ref>–<ref>, we infer

    ∫_0^Tu_xxx^2
    ≲ 1+∫_0^Tτ_xx^2
    +∫_0^T∫_ℐ[τ_x^4+θ_xx^2τ_t^2+
    τ_xx^2τ_t^2+τ_x^4τ_t^2+τ_x^2τ_xt^2].

The last term on the right can be estimated by employing the Sobolev's inequality as

    ∫_0^T∫_ℐ[θ_xx^2τ_t^2+
    τ_xx^2τ_t^2+τ_x^4τ_t^2+τ_x^4+τ_x^2τ_xt^2]
         ≲sup_t∈[0,T]τ_t_L^∞^2∫_0^T[(θ_xx,τ_xx)^2+τ_xxτ_x^3]
    +∫_0^Tτ_xτ_xx(τ_x,τ_tx)^2
         ≲
    1+∫_0^Tτ_xx^2.

Here we have used (<ref>), (<ref>) and (<ref>). Hence we get

    ∫_0^Tu_xxx^2
    ≲ 1+∫_0^Tτ_xx^2.


The estimate for (v,w) in (<ref>) can be obtained similarly.



We next show the estimate of θ_xxx in (<ref>).
Differentiate (<ref>) with respect to x to get

    κ r^2mθ_xxx/τ
    =c_vθ_xt+(Pτ_t)_x-
    [κ r^2m/τ]_xxθ_x
    -2[κ r^2m/τ]_xθ_xx-𝒬_x,

which combined with (<ref>) yields

    |θ_xxx|/√(c_v)   ≲|√(c_v)θ_xt|
    +|(θ_xτ_t,τ_xτ_t,τ_xt)|
    +|(θ_xx,τ_xx,τ_x^2,θ_x,τ_x)|
    +|(θ_xx,τ_xθ_xx,𝒬_x)|,
    |√(c_v)θ_xt|   ≲|θ_xxx|/√(c_v)
    +|(θ_xτ_t,τ_xτ_t,τ_xt)|
    +|(θ_xx,τ_xx,τ_x^2,θ_x,τ_x)|
    +|(θ_xx,τ_xθ_xx,𝒬_x)|.

Applying (<ref>), Corollary <ref> and Lemmas <ref>–<ref>, we infer

    ∫_0^Tθ_xxx/√(c_v)^2
       ≲ 1+∫_0^Tτ_xx^2
    +∫_0^T∫_ℐ[θ_xx^2τ_x^2+𝒬_x^2]
       ≲ 1+∫_0^Tτ_xx^2
    +sup_[0,T]θ_xx^2∫_0^Tτ_xxτ_x
    +∫_0^T𝒬_x^2
       ≲ 1+∫_0^Tτ_xx^2
    +∫_0^T𝒬_x^2.

To conclude (<ref>), it remains to prove

    ∫_0^T𝒬_x^2≲ 1.

According to the definition of 𝒬, we can compute that

    |𝒬_x|≲    τ_t^2|(θ_x,τ_x)|+|τ_tτ_xt|+
    |θ_x||(mu,u_x)|+|(mu,τ_x,u_x,u_xx,u_x^2)|
       +w_x^2|(θ_x,τ_x,1)|+|w_xw_xx|+
    |(θ_x,τ_x)||(v_x,v)|^2+|(v_x,v)||(v_xx,v_x)|,

which yields

    𝒬_x^2≲    
    |(θ_x,τ_x,τ_t,w_x,v,v_x)|^6
    +|(θ_x,mu,u_x,w_x,v,v_x)|^4
       +|(mu,τ_x,u_x)|^2+τ_t^2τ_xt^2+w_x^2w_xx^2+v_x^2v_xx^2+u_xx^2+v_xx^2.

Applying (<ref>)–(<ref>), 
Corollary <ref>, and Lemmas <ref>–<ref>, we
can deduce (<ref>) and therefore complete the proof of this lemma.





 
If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then
 
    sup_t∈[0,T]τ_xx(t)^2 +
     ∫_0^T(τ_xx,u_xxx,τ_xxt,v_xxx,w_xxx,
     θ_xxx/√(c_v))(t)^2
    dt≤ C_8.



 Let ∂_x act on (<ref>) and multiply the resulting identity
 by (ντ_x/τ)_x to find

    [1/2(ντ_x/τ)_x^2]_t
    +θ/ντ[ντ_x/τ]_x^2
    -[ντ_x/τ]_x
    [ν'(θ)/τ(τ_xθ_t-τ_tθ_x)]_x
       =-[ντ_x/τ]_x
    [-θ_xx/τ-2θτ_x^2/τ^3
    +2θ_xτ_x/τ^2-θτ_x/ντ(ν/τ)_x]
          +[ντ_x/τ]_x[
    u_t/r^m-v^2/r^m+1
    +2m uμ_x/r]_x.

Integrating the last identity and using Cauchy's inequality yield

    (ντ_x/τ)_x(t)^2
    +∫_0^t(ντ_x/τ)_x^2
         ≲.(ντ_x/τ)_x|_t=0^2+∫_0^t∫_ℐ[|(θ_xx,τ_x^2,θ_xτ_x)|^2+
    |(u_xt,u_t,v_x,v,u_xθ_x,θ_xx,θ_x)|^2]
            +
    ∫_0^t∫_ℐ[|(τ_xθ_tx,τ_tθ_xx)|^2+
    |(τ_xxθ_t,τ_txθ_x)|^2+
    |(τ_xθ_t,τ_tθ_x)|^2
    |(τ_x,θ_x)|^2].

We only show the estimates for the last three terms on the right-hand side of (<ref>), since the other terms can be easily treated by using Corollary <ref> and Lemmas <ref>–<ref>.
The term

    ∫_0^t∫_ℐ|(τ_xθ_t,τ_tθ_x)|^2
    |(τ_x,θ_x)|^2

can be estimated by using (<ref>) and (<ref>) as

    ∫_0^t∫_ℐ|(τ_xθ_t,τ_tθ_x)|^2
    |(τ_x,θ_x)|^2
    ≲∫_0^t∫_ℐ
    |(τ_x,τ_t,θ_x)|^4
    ≲ϵ∫_0^tτ_xx^2+C(ϵ).

It follows from (<ref>), (<ref>), (<ref>), and (<ref>) that

    ∫_0^t∫_ℐ|(τ_xθ_tx,τ_tθ_xx)|^2
    ≲∫_0^t(τ_x,τ_t)(τ_xx,τ_xt)(θ_xt,θ_xx)^2
         ≲ϵ∫_0^t(τ_xx,τ_xt)^2+C(ϵ)
    ∫_0^t(θ_xt,θ_xx)^2
    ≲ϵ∫_0^tτ_xx^2+C(ϵ).

In view of (<ref>) and (<ref>), we have

    ∫_0^t∫_ℐ|(τ_xxθ_t,τ_txθ_x)|^2
       ≲ (θ_t,θ_x)^2 ∫_0^t(τ_xx,τ_xt)^2≲ 1+
    (γ-1)^1/2  N^2∫_0^tτ_xx^2.

Thanks to (<ref>)–(<ref>) and (<ref>), we derive

    (ντ_x/τ)_x(t)^2
    +∫_0^t(ντ_x/τ)_x^2
    ≲ C(ϵ)
    +(ϵ+ ϵ_1^2)∫_0^tτ_xx^2.

Noting that

    |τ_xx|
    ≲|(ντ_x/τ)_x|
    +|(θ_xτ_x,τ_x^2)|,

 taking ϵ and ϵ_1 sufficiently small, we get

    τ_xx(t)^2+∫_0^tτ_xx^2
       ≲ 1+∫_ℐτ_x^2(θ_x^2+τ_x^2)+
    ∫_0^t∫_ℐτ_x^2(θ_x^2+τ_x^2)
       ≲ 1+sup_s∈[0,t]τ_x(s)^2_L^∞≲ 1+sup_s∈[0,t]τ_xx(s).

Apply Cauchy's inequality to (<ref>) to get

    sup_t∈[0,T]τ_xx(t)^2
    +∫_0^Tτ_xx^2≲ 1,

which combined with (<ref>) gives (<ref>).








 §.§ Estimates on the third-order derivatives


This subsection is devoted to deriving the estimates for (τ_xxx,u_xxx,v_xxx,w_xxx,θ_xxx).  
To do this, we first make the estimates for (u_xt,v_xt,w_xt) in the following lemma. 
 
If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then
 
    sup_t∈[0,T](u_xt,v_xt,w_xt)(t)^2 +∫_0^T(u_xxt,v_xxt,w_xxt)(t)^2dt≲ 1,
       sup_t∈[0,T](u_xxx,τ_tt,τ_xxt,v_xxx,w_xxx)(t)≤ C_9.



 Letting ∂_t act on (<ref>) and multiplying the resulting identity by u_xxt yields
 
    [1/2u_xt^2]_t-(u_xtu_tt)_x+ν r^2mu_xxt^2/τ=u_xxtR_u

 with 
 
    R_u:=    -(v^2/r)_t+(r^mP_x)_t+2m(r^m-1uμ_x)_t-(r^m)_t[ν (r^m u)_x/τ]_x
       -r^m[(ν/τ)_x(r^mu)_x]_t-r^m(ν/τ)_t(r^mu)_xx-r^mν/τ[(r^mu)_xxt-r^m u_xxt].

 Thanks to (<ref>), we integrate (<ref>) and use Cauchy's inequality to have
 
    u_xt(t)^2+∫_0^tu_xxt^2≲u_xt|_t=0^2
     +∫_0^t∫_ℐR_u^2.

 By virtue of the chain rule, (<ref>), and (<ref>), for a general smooth function f(τ,θ), we have
 
    f(τ,θ)_xt≲    |(τ_xt,θ_xt)|+|(τ_x,τ_t,θ_x,θ_t)|^2,
    |(r^mu)_xxt-r^m u_xxt| ≲    |((r^m)_xxtu,(r^m)_xtu_x,(r^m)_xxu_t,(r^m)_xu_xt, (r^m)_tu_xx)|
    ≲    |(u_xx,u_x,mu,τ_x)| +|(u_x^2,u_x)|+|(τ_x u_t,u_t)|+|(u_xt,u_xx)|,

 and
 
    |R_u| ≲     |(v_t,v)|+|(θ_x,τ_x)|+|(τ_xt,θ_xt)|+
     |(τ_x,τ_t,θ_x,θ_t)|^2+|u_tθ_x|
       + |(u_t,v,θ_x,τ_x)|+|(θ_x,τ_x)τ_tt|
     +|(τ_xt,θ_xt)||τ_t|+
     |(τ_x,τ_t,θ_x,θ_t)|^2|τ_t|
       +|(θ_t,τ_t)τ_xt|+
     |(u_xx,u_x,mu,τ_x,u_x^2,τ_x u_t,u_t,u_xt)|.

 Applying Sobolev's inequality, we get from
 (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) that
 
    (τ_t,τ_x,u_x,v_x,w_x)≲ 1.

Using (<ref>), (<ref>), Corollary <ref>, and Lemmas <ref>–<ref>, we derive

    ∫_0^t∫_ℐR_u^2≲ 1.

Insert the last estimate into (<ref>) to get

    u_xt(t)^2+∫_0^tu_xxt^2≲ 1.
 
 The estimates for v and w in (<ref>) can be obtained in a similar way. 

 We next show the estimate for u_xxx in (<ref>).
 In light of (<ref>), we have
   
    u_xxx(t)^2
       ≲ 1+θ_xxτ_t^2+τ_xxτ_t^2
       +τ_txτ_x^2≲  1.

The other estimates in (<ref>) can be proved similarly by using the system (<ref>).  
The proof of this lemma is completed.




In the following lemma, we deduce the L^∞(0,T;L^2(ℐ))-norm for √(c_v)θ_xt,
and we can have the bound for θ_xxx/√(c_v) in L^∞(0,T;L^2(ℐ)) due to the equation (<ref>). 
 
If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then
 
    sup_t∈[0,T](√(c_v)θ_xt,θ_xxx/√(c_v))(t)
     ^2 +∫_0^Tθ_xxt(t)^2dt≤ C_10.



 Let ∂_t act on (<ref>) and multiply the resulting identity by θ_xxt to get
 
    [c_v/2θ_xt^2]_t
     -(c_vθ_ttθ_xt)_x+κ r^2mθ_xxt^2/τ= θ_xxtR_θ,

 with 
 
    R_θ:=[P(r^m u)_x]_t
     -[(κ r^2mθ_x/τ)_xt
     -κ r^2mθ_xxt/τ]-𝒬_t.

 Integrating the last identity and using the boundary condition for θ yield
 
    √(c_v)θ_xt(t)^2+∫_0^tθ_xxt(s)^2ds
     ≲√(c_v)θ_xt|_t=0^2
     +∫_0^t∫_ℐR_θ^2.

By virtue of (<ref>) and (<ref>), we have

    √(c_v)θ_xt|_t=0^2≲ 1.

To estimate the last term on the right, we compute

    |(P(r^m u)_x)_t|≲     |τ_t(θ_t,τ_t)|+|τ_tt|,
    
    |𝒬_t|≲     |(θ_t,τ_t)τ_t^2|+|τ_tτ_tt|+
    |θ_t(r^m-1u^2)_x|+|(r^m-1u^2)_xt|+|w_xw_xt|
       + |(μ r^2m/τ)_tw_x^2|
    +|(θ_t,τ_t)||(v_x,v)|^2+|(v_x,v)||(v_xt,v_x,v_t,v)|,

and
 
    |[(κ r^2mθ_x/τ)_xt
     -κ r^2mθ_xxt/τ]|
    
       ≲  |(θ_x,τ_x,1)θ_xt|+ |(θ_t,τ_t,1)θ_xx|+
     |θ_x||(mu,u_x)|
          +|θ_x|[|(θ_x,τ_x)|+|(θ_t,τ_t)|
     +|(θ_xt,τ_xt)|+
     |(θ_x,θ_t,τ_x,τ_t)|^2].

 Plug (<ref>)–(<ref>) into (<ref>) and use 
 (<ref>), (<ref>), (<ref>), Corollary <ref>,  and Lemmas <ref>–<ref> to conclude 
  
    sup_t∈[0,T]√(c_v)θ_xt(t)
      ^2 +∫_0^Tθ_xxt(t)^2dt≲ 1.

The estimate (<ref>) follows by using (<ref>), (<ref>) and (<ref>).


By using the system (<ref>), we can get the following estimates for (u_xxxx,v_xxxx,w_xxxx,θ_xxxx).
The proof is similar to that of Lemma <ref> and hence we omit the details for brevity.
 If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then
 
    ∫_0^T(u_xxxx,v_xxxx,w_xxxx,
     θ_xxxx/c_v)(t)^2
     dt≲ 1
     +∫_0^Tτ_xxx(t)^2dt.



The following lemma is to establish the bound for L_x^2-norm of τ_xx(t,x). We note that, unlike the above estimates, the bound in (<ref>) depends on γ-1. 
 
 If the conditions listed in Lemma <ref> hold for a sufficiently small ϵ_1, then
  
    sup_t∈[0,T]τ_xxx(t)^2 +
      ∫_0^T(τ_xxx,u_xxxx,v_xxxx,w_xxxx,
      θ_xxxx/√(c_v))(t)^2
      dt≲ c_v.



Let ∂_x^2 act on (<ref>) and multiply the resulting identity by (ντ_x/τ)_xx to get






    [1/2(ντ_x/τ)_xx^2]_t
    +θ/ντ[ντ_x/τ]_xx^2
    =[ντ_x/τ]_xxR_τ,

with

    R_τ=[u_t/r^m-v^2/r^m+1
    +2m uμ_x/r+ν'(θ)/τ(τ_xθ_t-τ_tθ_x)+(θ/τ)_x]_xx+θ/ντ(ντ_x/τ)_xx.

Integrating (<ref>) and using Cauchy's inequality yield

    (ντ_x/τ)_xx(t)^2
    +∫_0^t(ντ_x/τ)_xx(s)^2ds
    ≲ 1+∫_0^t∫_ℐR_τ^2.
	
From (<ref>) and (<ref>), we have

    (τ_xt,u_xx,v_xx,w_xx)≲ 1.

Using (<ref>) and (<ref>), 
after a direct calculation, we have

    R_τ^2≲    
    |(u_t,u_xt,u_xxt)|^2+|(v,v_x,v_xx)|^2+|(θ_x,θ_xx,θ_xxx)|^2
       +|(τ_x,θ_x,τ_xx,θ_xx,θ_xxx)|^2
    +|(τ_xxxθ_t,τ_xxt,τ_xxθ_xt,θ_xxt)|^2.

Plug (<ref>) into (<ref>) and use 
Corollary <ref>, Lemmas <ref>–<ref> to get

    (ντ_x/τ)_xx(t)^2
    +∫_0^t(ντ_x/τ)_xx(s)^2ds
         ≲.(ντ_x/τ)_xx|_t=0^2
    +∫_0^tθ_xxx^2+
    θ_t^2∫_0^tτ_xxx^2+
    ∫_0^t∫_ℐτ_xx^2θ_xt^2.

By virtue of (<ref>), (<ref>) and (<ref>), we have

    (ντ_x/τ)_xx(t)^2
    +∫_0^t(ντ_x/τ)_xx(s)^2ds
    ≲ c_v+
    ϵ_1[∫_0^tτ_xxx^2+sup_s∈[0,t]τ_xx(s)^2].

Employ (<ref>) and (<ref>) to have

    |ντ_xxx/τ|
    ≲|(ντ_x/τ)_xx|+
    |(τ_xx,θ_xx,τ_x,θ_x)|,

which combined with (<ref>) and (<ref>) implies (<ref>) if ϵ_1 is sufficiently small.






§ PROOF OF THEOREM <REF>


In this section we complete the proof of Theorem <ref> by combining the a priori bounds obtained in Section <ref>, the continuation argument and Poincaré's inequality. 
We divide this section into two parts. 
The first is devoted to proving the existence and uniqueness of global solutions to the problem (<ref>)–(<ref>), and the second one  aims at showing the convergence decay rate of the solutions toward constant states.



 §.§ Global solvability

We first present the local solvability result to the initial boundary value problem (<ref>)–(<ref>) in the following proposition, which can be proved by the standard iteration method (see, for instance, <cit.>).

 
Let the initial data (τ_0,u_0,v_0,w_0,θ_0)∈ H^3(ℐ) satisfy that

    τ_0(x)≥λ_1^-1,  θ_0(x)≥λ_2^-1  ∀ x∈ℐ,
       u_0_1+√(c_v)(θ_0-1,θ_t|_t=0)_1+θ_0xx≤Λ,
       (τ_0,u_0,v_0,w_0)_3+√(c_v)θ_0-1_1+(θ_0xx,√(γ-1)θ_0xxx)≤Π,

for some positive constants  λ_1, λ_2, Λ, and Π, where θ_t|_t=0 and θ_xt|_t=0 are defined by (<ref>) and (<ref>), respectively.
Then there exists a positive constant T_0=T_0(λ_1,λ_2,Λ,Π), which depends only on λ_1, λ_2, Λ, and Π, such that the initial boundary value problem (<ref>)–(<ref>) has a unique solution (τ, u, v, w, θ) ∈ X(0,T_0;2λ_1,2Λ).


According to (<ref>) and (<ref>), there exists a (γ-1)-independent positive constant C_0, such that
 
    τ_0(x)≥ V_0^-1,  θ_0(x)≥ V_0^-1  ∀ x∈ℐ,
       u_0_1+√(c_v)(θ_0-1,θ_t|_t=0)_1+θ_0xx≤ C_0,
       (τ_0,u_0,v_0,w_0)_3+√(c_v)θ_0-1_1+(θ_0xx,√(γ-1)θ_0xxx)≤Π_0.

Let ϵ_1, C_j (j=1,⋯,10) be chosen in Section <ref>.
We assume that γ-1≤ϵ_0 with

    ϵ_0:=min{δ_1,δ_2},

where

    δ_1:=[ϵ_1/(2C_0)^2(2V_0)^2]^4,  δ_2:=[ϵ_1/5C_11(2C_1)^2 ]^4,  
    C_11:=∑_j=1^10C_j.


Applying  Proposition <ref>, we can find a positive constant t_1=T_0(V_0,V_0,C_0,Π_0) such that there exists a unique solution  (τ, u, v, w, θ) ∈ X(0,t_1;2V_0,2C_0) to the initial boundary value problem (<ref>)–(<ref>).


Since γ-1≤δ_1,
we can apply  Corollary <ref> and Lemmas <ref>–<ref> with T=t_1 to deduce that there exists a positive constant C_1(γ), depending only on V_0, Π_0, and γ, such that for each t∈[0,t_1],  the local solution  (τ, u, v, w, θ) satisfies
 
    ℰ_0(t)≤ C_11,  τ(t,x)≥ C_1^-1,  θ(t,x)≥12  ∀ x∈ℐ,
       (τ,u,v,w)(t)_3+√(c_v)(θ_0-1)(t)_1+(θ_xx,√(γ-1)θ_xxx)(t)≤ C_1(γ).


If we take  (τ, u, v, w, θ) (t_1,·) as the initial data and apply Proposition <ref> again,
we can extend the local solution  (τ, u, v, w, θ) to the time interval [0,t_1+t_2] with
t_2=T_0(C_1,2,C_11,C_1(γ)) such that (τ, u, v, w, θ)∈ X(t_1,t_1+t_2;2C_1,2√(C_11)).
Hence the local solution (τ, u, v, w, θ)∈ X(0,t_1+t_2;2C_1,√(5C_11)). 
Noting that γ-1≤δ_2, we can apply  Corollary <ref> and Lemmas <ref>–<ref> with T=t_1+t_2 to deduce that (<ref>) and (<ref>) hold for each t∈[0,t_1+t_2].

Repeating the above procedure, we can then extend the solution (τ, u, v, w, θ) step by step to a global one provided that  γ-1≤ϵ_0 with ϵ_0 given by (<ref>). Furthermore,

    (τ,u,v,w,θ)(t)_3^2+∫_0^∞[(u_x,v_x,w_x,θ_x)_3^2+τ_x_2^2]dt≤ C_2(γ)
        ∀ t∈[0,∞),

where C_2(γ) is some positive constant depending on γ, Π_0, and V_0.






 §.§ Convergence decay rate

It follows from (<ref>) and Poincaré's inequality that

    (τ-τ̅)(t)^2=
    	∫_ℐ|τ(t,x)-∫_ℐτ(t,y)dy|^2dx≲τ_x(t)^2.

Thanks to the boundary conditions for (u,v,w), we apply Poincaré's inequality to get

    (u,v,w)(t)≲(u_x,v_x,w_x)(t).

Multiplying (<ref>), (<ref>), (<ref>) by u, v, w, respectively,  adding them with (<ref>), and integrating the resulting identity over [0,t]×ℐ yield







    ∫_ℐ[c_vθ+1/2(u^2+v^2+w^2)](t,x)dx=c_vθ̅    ∀ t∈[0,∞),

where θ̅ is given by (<ref>).
We use Poincaré's inequality and  (<ref>) to obtain

    (θ-θ̅)(t)^2≤∫_ℐ|θ(t,x)-∫_ℐθ(t,y)dy|^2dx+(u,v,w)(t)^2
    ≲(u_x,v_x,w_x,θ_x)(t)^2.

We have from  (<ref>) that for all (t,x)∈[0,∞)×ℐ,

    [r^m+1(t,x)]_x=(m+1)τ(t,x),  
    r^m+1(t,x)=a^m+1+(m+1)∫_0^xτ(t,y)dy.

Then 

    |r^m+1(t,x)-r̅^m+1|   =
    (m+1)|∫_0^x(τ(t,y)-τ̅)dy|,
    [r^m+1(t,x)-r̅^m+1]_x   =(m+1)[τ(t,x)-τ̅],

which combined with (<ref>) imply

    r(t)-r̅_2≲r^m+1(t)-r̅^m+1_2
    ≲τ(t)-τ̅_1.


To derive the exponential stability of the solution (τ,u,v,w,θ), we define the following energy functionals:

    ℋ_η(t)   :=∫_ℐη_θ̅(τ,u,v,w,θ)(t,x)dx,
      ℋ_τ(t)   :=∫_ℐτ_x^2(t,x)dx,  ℋ_θ(t):=∫_ℐθ_x^2(t,x)dx,
    ℋ_U(t)   :=∫_ℐ[u_x^2+v_x^2+w_x^2](t,x)dx,
      ℋ_0(t)   :=∫_ℐ[1/2(ντ_x/τ)^2-
    u/r^mντ_x/τ](t,x)dx,

where η_θ̅(τ,u,v,w,θ) is defined by (<ref>). 
Then we have from (<ref>) that

    ℋ_τ(t)-Cu(t)^2≲ℋ_0(t)≲ℋ_τ(t)+u(t)^2
    ≲ℋ_τ(t)+ ℋ_U(t).


It follows from (<ref>), (<ref>), and (<ref>) that

    d/dtℋ_U(t)+c(u_xx,v_xx,w_xx)(t)^2≲ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t).

In light of (<ref>) and (<ref>), we have

    c_vd/dtℋ_θ(t)+cθ_xx(t)^2≲ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t)+(u_xx,v_xx,w_xx)(t)^2.

Multiplying (<ref>) by a sufficiently small σ_1>0 and adding this with (<ref>) yield

    d/dt[ℋ_U(t)+c_vσ_1ℋ_θ(t)]+cσ_1(u_xx,v_xx,w_xx,θ_xx)(t)^2
    ≲ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t).

By virtue of (<ref>), (<ref>), and (<ref>),  we get

    d/dtℋ_0(t)+cℋ_τ(t)
    ≲    [1+(γ-1)^2][ℋ_U(t)+ℋ_θ(t)]
       +(γ-1)^2[
    ℋ_τ(t)+(u_xx,v_xx,w_xx,θ_xx)(t)^2].

Since c_v^-1=|γ-1|≤ϵ_0,
we multiply (<ref>) by a sufficiently small σ_2 and add this into (<ref>) to find

    d/dt[ℋ_0(t)+σ_2ℋ_U(t)+c_vσ_2σ_1ℋ_θ(t)]+cℋ_τ(t)+cσ_2σ_1(u_xx,v_xx,w_xx,θ_xx)(t)^2
    
         ≲[1+ϵ_0^2][ℋ_U(t)+ℋ_θ(t)]+ϵ_0^2[
    ℋ_τ(t)+(u_xx,v_xx,w_xx,θ_xx)(t)^2].

Take ϵ_0>0 suitably small to infer

    d/dt[ℋ_0(t)+σ_2ℋ_U(t)+c_vσ_2σ_1ℋ_θ(t)]+cℋ_τ(t)
    ≲ℋ_U(t)+ℋ_θ(t).

It follows from (<ref>) and that

    d/dtℋ_η(t)+c[ℋ_U(t)+ℋ_θ(t)]≤ 0.

Multiplying (<ref>) by a sufficiently small σ_3 and adding this into (<ref>) imply

    d/dtℋ(t)+c[ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t)]
    ≤ 0,

where 

    ℋ(t):=ℋ_η(t)+σ_3ℋ_0(t)+σ_3σ_2ℋ_U(t)+c_vσ_3σ_2σ_1ℋ_θ(t).

Using (<ref>), (<ref>), (<ref>), and (<ref>), we have

    [1/2-σ_3C]u(t)^2+cσ_3[ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t)]≤ℋ(t)≲ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t).

Take σ_3>0 with Cσ_3≤ 1/4 so that

    cσ_3[ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t)]≤ℋ(t)≲ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t),
 
which combined with (<ref>) yields

    ℋ_τ(t)+ℋ_U(t)+ℋ_θ(t)≲ℋ(t)≲e^-ct.

We combine (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) to conclude the exponential decay estimate (<ref>). Therefore, the proof of Theorem <ref> has been completed.







	Acknowledgement


Research of the authors was supported by the Fundamental Research Funds for the Central Universities, the Project funded by China Postdoctoral Science Foundation, and the grants from National Natural Science Foundation of China under contracts 11601398 and 11671309.
The authors express much gratitude to Huijiang Zhao for his support and advice.
Tao Wang would like to warmly thank Paolo Secchi, Alessandro Morando, and Paola Trebeschi for support and hospitality during his postdoctoral stay at University of Brescia. 




abbrvnat





