Global Existence of the Cylindrically Symmetric Strong Solution to Compressible Navier-Stokes Equations

and Applied Analysis 3 Before stating the main result, we assume that (ρ 0 − ρ, u 0 , V 0 , w 0 ) ∈ H 1 ([0, 1]) , (13) with ρ 0 > 0 and ρ = (1/(b − a)) ∫b a ρ 0 r dr, and define

When  and  are density-dependent viscosity, there are many important results made on the compressible Navier-Stokes equations.For example, the existence of solution to 2D shallow water equations was investigated by Bresch and Desjardins [3,4].The well posedness of solution to the free boundary value problem with initial finite mass and the flow density being connected with the infinite vacuum either continuously or via jump discontinuity was considered by many authors; refer to [5] and the references therein.The global existence of classical solution for  ∈ (0, 1/2) was shown by Mellet and Vasseur [6].The qualitative behaviors of global solution and dynamical asymptotic of vacuum states were also addressed, such as the finite time vanishing of finite vacuum or asymptotical formation of vacuum in large time, the dynamical behaviors of vacuum boundary, the large time convergence to rarefaction wave with vacuum, and the stability of shock profile with large shock strength; refer to [7] and the references therein.
It should be mentioned here that important progress has been obtained on global existence and asymptotical behaviors of strong solution to compressible Navier-Stokes equations (1) with constant viscosity coefficients; the readers can refer to [8][9][10][11][12] and the references therein.When  is density-dependent viscosity, Lian et al. [13] proved that the strong solution exists globally in time and investigated the long time behaviors of the solution to one-dimensional case.However, for multidimensional case, there are a few results on the global existence and asymptotical behaviors of strong solution with density-dependent viscosity coefficients for compressible Navier-Stokes equations.
Recently, there have been some results on the existence of cylindrically symmetric solution to three-dimensional compressible Navier-Stokes equations.When viscosity coefficients are both constants, Frid and Shelukhin [14,15] proved the uniqueness of the weak solution under certain condition, Fan and Jiang in [16] showed the global existence of weak solution, and Jiang and Zhang obtained the existence of strong solution for the nonisentropic case in [17].When () =   , 0 ≤  ≤ , () =  (a positive constant), Yao et al. [18] showed the global existence for the three-dimensional compressible Navier-Stokes equations.However, when  and  are both density-dependent viscosity, there is no result made on the existence of cylindrically symmetric solution to three-dimensional compressible Navier-Stokes equations.
In the present paper, we consider the initial boundary value problem (IBVP) for the three-dimensional isentropic compressible Navier-Stokes equations and focus on the existence and time asymptotic behavior of the global strong solution.For simplicity, we deal with the case () =   , () = ( − 1)  with  > 1/2.We show that the unique global cylindrically symmetric strong solution to the IBVP (2)-( 4) exists and tends to the equilibrium state exponentially as time grows up for initial data satisfied in Theorem 1.
The rest part of the paper is arranged as follows.In Section 2, the main results about the global existence of strong solution to the compressible Navier-Stokes equations are stated in detail.Then, some important a priori estimates will be given in Section 3 and Theorem 1 is proven in Section 4.

Main Results
For simplicity, the viscosity terms are assumed to satisfy () =   and () =   () − () = ( − 1)  with  > 1/2 and D(U) = ∇U.Then (1) become Consider a flow between two circular coaxial cylinders and assume that the corresponding solution depends only on the radial variable  in Ω := { | 0 <  ≤  ≤ } and the time variable  ∈ [0, ].Then, the three-dimensional Navier-Stokes equations governing the flow reduce to the following system in the domain   := Ω × (0, ) of the form supplemented with the initial and boundary conditions The velocity vector  = (, V, ) is given by the radial, angular, and axial velocities.For simplicity, we take here V  () =   () ≡ 0 ( = 1, 2), since otherwise we can use to replace V and .
We are interested in the global existence of the initial boundary value problem for (3)-( 4).It is convenient to deal with the IBVP (3) in the Lagrangian coordinates.For simplicity, we assume that for  ∈ [, ] and  ∈ [0, ]; define the Lagrangian coordinates transformation which translates the domain and satisfies The initial boundary value problem (3)-( 4) is changed to for (, ) ∈ [0, 1] × [0, ], with the initial data and boundary conditions given by where  = (, ) is defined by Before stating the main result, we assume that with  0 > 0 and  = (1/( − )) ∫    0  d, and define Then, we have the following main results.

The A Priori Estimates
In this section, we establish the a priori estimates for any solution (, , V, ) with  > 0 to IBVP ( 9)- (12).Making use of similar arguments as in [19] with modifications, we can establish the following lemmas.

Lemma 6. Under the same assumptions as Lemma 4, it holds that
where  > 0 denotes a constant independent of time.
Lemma 7.Under the same assumptions as Lemma 4, it holds that In addition, it holds that where  > 0 denotes a constant independent of time.
Proof.Differentiating (9) 2 with respect to , multiplying the result by ()  , and integrating the result with respect to  over [0, 1], we have A complicated computation gives From (9) 1 , (9) 2 , ( 18), ( 21), ( 22), and Lemma 6, it holds that where  > 0 denotes a constant independent of time, and from (44), we can find With the same methods, we can obtain the following: where  1 > 0 and  2 > 0 denote two constants independent of time.