ASYMPTOTIC BEHAVIOR OF SPHERICALLY OR CYLINDRICALLY SYMMETRIC SOLUTIONS TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH LARGE INITIAL DATA

. In this paper, we study the asymptotic behavior of global spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations for the viscous heat conducting ideal polytropic gas ﬂow with large initial data in H 1 , when the heat conductivity coeﬃcient depends on the temperature, practically, κ ( θ ) = ˜ κ 1 + ˜ κ 2 θ q where constants ˜ κ 1 > 0, ˜ κ 2 > 0 and q > 0 (as to the case of ˜ κ 1 = 0, please refer to the Appendix). In addition, the exponential decay rate of solutions toward to the constant state as time tends to inﬁnity for the initial boundary value problem in bounded domain is obtained. The mass density and temperature are proved to be pointwise bounded from below and above, independent of time although strong nonlin-earity in heat diﬀusion. The analysis is based on some delicate uniform energy estimates independent of time.


XINHUA ZHAO AND ZILAI LI
We consider the initial and boundary conditions: and (u, v, w, θ r )| r=a,b = 0, t 0.
There are many classical literatures on the well-posedness and large-time behavior of solutions to (1)-(3) for constant viscosity and constant heat conductivity coefficients (κ 1 > 0,κ 2 = 0). Kazhikhov and Shelukhin [15] firstly obtained the global existence and uniqueness of classical solutions in one-dimensional bounded domain with arbitrarily large initial data. Then, significant progress has been made on the mathematical aspect of the initial and initial boundary value problems. For initial boundary value problem in bounded domains, the global existence and the large-time behavior of solutions have been established in [20,21]. In fact, the global solutions converge exponentially to constant states as time tends to infinity. This argument has been applied to the case of spherically symmetric solutions and cylindrically symmetric solutions with large initial data [10,25,26]. Recently, Cui and Yao [2] got the exponential decay for the global spherically or cylindrically symmetric solutions with large initial data for the compressible p-th power Newtonian fluid. For initial boundary value problem in unbounded domains and Cauchy problem, Jiang [12] got the asymptotic behavior of solutions to the compressible viscous ideal gas in one dimension. Recently, Li and Liang [18] studied the large-time behavior of solutions to the initial and initial boundary value problems in one-dimensional unbounded domains. They proved the temperature was bounded from below and above uniformly in both time and space. Wan and Wang [27] studied the cylindrically symmetric solutions of (1)-(3) in three-dimensional exterior domains.
For the cases of heat conductivity coefficient depending on temperature(κ 1 ≥ 0,κ 2 > 0), Jenssen and Karper [8] proved the global existence of a weak solution in one dimension with µ =μ(constant), κ(θ) =κ 2 θ b , b ∈ [0, 3 2 ). Later, Pan and Zhang [22] considered the existence and uniqueness of global strong solutions in one dimension under the milder assumption µ =μ, κ =κ 2 θ b , b ≥ 0. Hsiao and Luo [7] considered the large-time behavior of solutions for viscous one-dimensional real gas in one-dimensional bounded domain, when µ =μ and the heat conductivity coefficient κ(ρ, θ) satisfies In the case of the viscosity coefficients µ depending on ρ and κ(ρ, θ) satisfying (5), the global existence of solutions to the one-dimensional free boundary problem was studied by Kawohl [14]. For more about this case, see [4,9,11,17,23]. Then, Liu, Yang, Zhao and Zou [19] studied the existence and uniqueness of a global smooth non-vacuum solutions for one-dimensional Cauchy problem with µ, κ depending on temperature, provided that γ − 1 is sufficiently small. Next, when the viscosity and heat conductivity coefficients satisfy µ(ρ, θ) =μh(ρ)θ α , κ(ρ, θ) =κh(ρ)θ α with | α |≤ ε 0 for a positive constant ε 0 , Wang and Zhao [29] obtained the existence of global non-vacuum solutions to the Cauchy problem in one dimension. This result was generalized to the symmetric flows [28]. For other studies related to this topic such as the boundary layers, as well as other related models, we can refer to [13,24] and the references therein.
In this paper, we will consider spherically or cylindrically symmetric Navier-Stokes equations (1)-(3) under the assumptions that the viscosity coefficient µ, λ being constants and the heat conductivity coefficient κ(θ) satisfying We obtain the exponential decay of global smooth solutions without any restrictions on the size of initial data.
To state the main result, let us introduce the notations.
(i) For 1 p ∞, L p denotes the L p space on [0, L] with the norm · L p . Specially, we put · = · L 2 . For m ∈ N , W m,p denotes the Sobolev spaces on [0, L], whose norm is · W m,p , and H m = W m,2 .
(ii) Q T = [0, L] × [0, T ] for T > 0. As in [2,10], it is convenient to transfer the problem (1)-(3) into the equations in Lagrangian coordinates. Hence we consider the following initial-boundary problem in the Lagrangian coordinates: Here the new variables are ( w(x, t), and θ(x, t) are the specific volume, radial velocity, angular velocity, axial velocity and temperature of the flows, respectively. Moreover, the radius r(x, t) also depends on the Lagrangian mass coordinates (x, t), and

XINHUA ZHAO AND ZILAI LI
The initial conditions are The boundary conditions are It follows from the Lagrangian transformation that and Also we denote In what follows, we use C(and C i or c) to denote a generic positive constant depending only on the parameters of the system and the bounds of the initial data, but independent of t and T . For simplicity, , 1 , 2 , δ denote any small constants.
The following is the main result of this paper: . Assume that the initial data are compatible with boundary conditions, and the condition (6) holds. Then there exists a constant C > 0, such that the problem (7)-(10) admits a unique solution and for any t > 0, Moreover, where we definẽ Obviously,η,θ > 0.   [26]; The same conclusions as in Theorem 1.1 hold for the p-th power Newtonian fluid where the pressure P = Rρ p θ, see [2].
Compared with the previous result, there are two differences in this paper. On the one hand, we don't need the smallness of the initial data, that is, all initial data can be large. On the other hand, for the heat conductivity coefficient κ(θ) satisfying (6), in order to obtain the large-time behavior of solutions to the initial boundary problem (7)- (10), all the estimates should be independent of any length of time.
This will result in some mathematical difficulties. The first difficulty encountered here is to establish uniform point-wise positive upper bound of the temperature θ.
To overcome this difficulty, we use the iterative method of [3,22]. So we can mutually control two functionals Y and Z by choosing suitable α. The second difficulty is to establish uniform lower bound of the temperature. Compared with the constant heat conductivity coefficient, it is difficult to get d for any q > 0, see Lemma 2.17.

2.
Proof of Theorem 1.1. The global existence and uniqueness of solutions (η, u, v, w, θ) to the problem (7)-(10) in H 1 space has been obtained in [5,15,21]. Therefore, we are only concern about the large-time behavior of solutions in H 1 . For simplicity of presentation, we will takeκ 1 =κ 2 = 1.
As the constant heat conductivity coefficient, it is important to get the uniform upper bounds and lower bounds of the volume η(x, t) and temperature θ(x, t). In fact, the basic energy estimate, the uniform upper bound and lower bound of the volume η(x, t) could be obtained by the same method as the case of constant heat conductivity coefficient. Therefore, the proofs of Lemmas 2.1, 2.2 and 2.3 are similar to the arguments in [2,25], just with some slightly modifications, we omit the proofs here.
Next, by some simple calculations, we can get the lower bound of the temperature θ which depend on the fixed time T . Lemma 2.4. Under the conditions of Theorem 2.1, there exists two constants C 1 > 0 and C 2 > 0, independent of T , such that Proof. Let h = 1 θ . Choosing γ > 0 to satisfy 0 < 2µ β < γ < 1. Then multiplying (7) 5 by − 1 θ 2 , we have where we note that all the terms in the bracket {·} on the right hand side of the above resulting equation are nonnegative. Now multiplying the above inequality by 2ph 2p−1 with p ≥ 1 being an arbitrary integer, and integrating the result over [0, L] with respect to x, one yields Then, by the assumption of the initial data θ 0 , we have where C 1 > 0 and C 2 > 0 are independent of p, T . Letting p → +∞, we get This completes the proof.
Next two lemmas give H 1 estimates of v and w.
Proof. Multiplying (7) 3 by v, and integrating the resulting equality with respect to x over [0, L], we have It follows from Gronwall inequality, Lemma 2.2 and (21) that , Sobolev inequality and Lemma Next, multiplying (7) 3 by v t , integrating the resulting equality with respect to x over [0, L] and using (7) 1 , (11), one gets By Gronwall inequality, Poincaré inequality, and (21) This completes the proof of Lemma 2.5.
Analogously, we can also use the same method as that in Lemma 2.5 to prove the following lemma for the axial velocity w. Lemma 2.6. Under the conditions of Theorem 2.1, we have Lemma 2.7. Under the conditions of Theorem 2.1, for any α ∈ (0, 1], it holds that Proof. We only prove α ∈ (0, 1), the case of α = 1 is the direct result of Lemma 2.1. We use the idea as in [25]. Multiplying (7 , then integrating the result with respect to x over [0, L], using (7) 1 , we find that Thus, using (18), and integrating the result inequality with respect to s over [0, t], we have for any > 0 where we have used Lemma 2.1 and the following inequality, Finally, as (16), we find This completes the proof of Lemma 2.7.
As in [3,22], in order to get the first-order derivative estimate of the radial velocity u, and the uniform estimates of the temperature, we define the following two functionals Lemma 2.8. Under the conditions of Theorem 2.1, it holds that and max Proof. Applying (10) and Sobolev inequality, we have By interpolation inequality, we get Finally, for (25), it follows from (20) that Thus, the proof of Lemma 2.8 is finished.
Using the above lemma, we will get the relations between the first-order derivatives estimates of η, v, w, u and Z, Y . And the following four lemmas are key to obtain the large time behavior of the solution and the uniform upper bound of θ.
for any t ∈ [0, T ], and α will be determined later.
Proof. It follows from (7) 2 that Multiplying it by η x η , and integrating the result over [0, L], we have Hence by means of Cauchy inequality, Lemma 2.3 and Lemmas 2.7-2.8, we have Utilizing the boundary condition of u, it easy to see that On the one hand, it follows form Lemma 2.1 that On the other hand, using Lemmas 2.7, 2.8 and (21), one knows where α ∈ (0, 1) will be determined later. Finally, it is easy to see that Substituting (27)-(31) into (26) and using (21), Lemma 2.5, Gronwall inequality, we have The proof of Lemma 2.9 is completed. Proof. From (7) 3 and the integration by parts, we have After integrating the above inequality with respect to s over [0, t], we have Then utilizing (21), the boundary condition of v and Lemmas 2.5, 2.9, one gets Substituting (33) into (32) and applying Lemma 2.5, we derive This completes the proof of Lemma 2.10.
Next, we can also obtain the similar estimate for the axial velocity w.
Proof. We deduce from the integration by parts and (7) 2 that Then, from Hölder inequality, , the boundary condition of u, Lemmas 2.5, 2.7, 2.8, 2.9, and (21), one has After integrating the resulting inequality with respect to s over [0, t], and taking 0 < α < 1 2 , one gets We completed the proof of Lemma 2.12.
Next, we give the relation between Y and Z in the following lemma.
By means of these lemmas, we can obtain the estimates of Y and Z.
Proof. Multiplying (7) 2 by u t , then integrating the resulting equality with respect to x over [0, L], we have Since u 0 ∈ H 1 , we have Using integration by parts, (7) 1 , Lemmas 2.5, 2.9, 2.15 and (39), we know It follows form (39) that It is easy to see that and Then using Gronwall inequality, the above estimates and (21), we get This together with Lemma 2.15 prove the Lemma 2.16.
The following lemma gives the estimate of d dt L 0 (1+θ q ) 2 r 2m η θ 2 x dx . It is also useful for obtaining the uniform lower bound of θ.
Recalling (41), and using previous lemmas, we have the following useful result.
With all the a priori estimates in Lemmas 2.1-2.17 at hand, we will establish the asymptotic behavior and the convergence rates of η, u, v, w, θ and r.
Proof of Theorem 1.1 Define and ν = C vθ (logθ − 1) + Rθ logη. It follows from (7) that Hence, we obtain d dtZ It follows from Taylor expansion and Lemma 2.18 that Then by Poincaré inequality and Cauchy inequality, we infer On the other hand, it follows from (26) and Lemma 2.16 that Finally, by Lemmas 2.10-2.12, 2.17, we also have Adding (48) multiplied by to (50), one has d dt Adding (52) multiplied by to (51), using (49), we obtain d dt Hence we can get the asymptotic behavior of r(x, t). This completes the proof of Theorem 1.1.

3.
Appendix. Proof of Remark 1.2. For the case ofκ 1 = 0, the proof is similar to that ofκ 1 > 0 with some slight modifications. We only give some key steps of proof.
As the same as the proof of Lemma 2.9, for the inequality (27), we deal with as follow: where q < 1. Thus we get the estimate of (a). For the estimate of (b), by (33) and (a), we have Next, we can also obtain the similar estimate for w. By (53), (36) and (a), it follows where taking q < 1. Finally, as for the relation between Y and Z, the proof is similar to Lemma 2.13 with some slight modifications. For convenient reading, we give a simplified proof in the following. For I 1 , For I 2 , For I 3 + I 4 , For I 5 , For I 6 , using Poincaré inequality, Lemmas 2.1, 2.8, we obtain Thus for q > 1 2 , we have (1 + θ q )θ 2 s dxds + δZ.
Using the same method as the case of theκ 1 > 0, for 1 2 < q < 1, we can get Lemma 2.14 and Lemma 2.15. We omit the proof here.