Global Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations with Density-Dependent Viscosity

We consider the exterior problem and the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coe ﬃ cient in this paper. For regular initial density, we show that there exists a unique global strong solution to the exterior problem or the initial boundary value problem, respectively. In particular, the strong solution tends to the equilibrium state as t → (cid:3) ∞ .


Introduction
The isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients read as follows: where  ∈ (0, +∞) is the time and x ∈   ,  is the spatial coordinate, and  > 0 and  denote the density and velocity, respectively.Pressure function is taken as () =   with  > 1, and is the strain tensor and ℎ(), () are the Lamé viscosity coefficients satisfying ℎ () > 0, ℎ () +  () ≥ 0. ( There is huge literature on the studies of the global existence of weak solutions and dynamical behaviors of jump discontinuity for the compressible Navier-Stokes equations with discontinuous initial data; for instance, if the viscosity coefficients ℎ() and () are both constants, for the case of one space dimension, Hoff investigated the global existence of discontinuous solutions of the Navier-Stokes equations [1][2][3].
Hoff derived the construction of global spherically symmetric weak solutions of compressible Navier-Stokes equations for isothermal flow with large and discontinuous initial data [4]; therein, Hoff also showed that the discontinuities persist for all time, convecting along particle trajectories, and decaying at a rate inversely proportional to the viscosity coefficient.Hoff also obtained the global existence theorems for the multidimensional Navier-Stokes equations of isothermal compressible flows with the polytropic equation of state () =   ( ≥ 1) [5,6].The global existence of weak solutions was proved for the Navier-Stokes equations for compressible, heat-conducting flow in one space dimension with large, discontinuous initial data by Chen et al. in [7].
Hoff showed the global existence of weak solutions of the Navier-Stokes equations for compressible, heat-conducting fluids in two and three space dimensions when the initial data may be discontinuous across a hypersurface of   [8].The global existence of solutions of the Navier-Stokes equations for compressible, barotropic flow in two space dimensions which exhibit convecting singularity curves, was proved by Hoff in [9].If the viscosity coefficients ℎ() =   , () = 0, for the case of one space dimension, Fang and Zhang proved the global existence of unique piecewise smooth solution to the free boundary value problem for (1) with 0 <  < 1, where the initial density is piecewise smooth with possibly large jump discontinuities [10].Lian et al. addressed the initial boundary value problem for (1) with 0 <  ≤ 1 subject to piecewise regular initial data with initial vacuum state included [11], where they obtained the global existence of unique piecewise regular solution and the finite time vanishing of vacuum state.In particular, they got that the jump discontinuity of density decays exponentially but never vanishes in any finite time and the piecewise regular solution tends to the equilibrium state exponentially as  → +∞.
In this present paper, we consider the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and discontinuous initial data and focus on the regularities and dynamical behaviors of global weak solution and so forth.As  > 1, we show that the initial boundary value problem with piecewise regular initial data admits a unique global piecewise regular solution, where the discontinuity in piecewise regular initial density is bounded.In particular, the jump discontinuity of density decays exponentially and the piecewise regular solution tends to the equilibrium state exponentially as  → +∞.
There are also many significant progresses achieved recently on the compressible Navier-Stokes equations with density-dependent viscosity coefficients.For instance, the mathematical derivations are obtained in the simulation of flow surface in shallow region [12,13].The prototype model is the viscous Saint-Venant equation (corresponding to (1) with () =  2 , ℎ() = , and () = 0).Many authors considered the well-posedness of solutions 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; refer to [14][15][16][17][18][19][20][21][22] and references therein.The global existence of classical solutions is shown by Mellet and Vasseur in [23].The qualitative behaviors of global solutions and dynamical asymptotics of vacuum states are also made, such as the finite time vanishing of finite vacuum or asymptotical formation of vacuum in long time, the dynamical behaviors of vacuum boundary, the long time convergence to rarefaction wave with vacuum, and the stability of shock profile with large shock strength; refer to [24][25][26] and references therein.
The rest part of the paper is arranged as follows.In Section 2, the main results about the existence and dynamical behaviors of global piecewise regular solution for compressible Navier-Stokes equations are stated.Then, some important a priori estimates will be given in Section 3. Finally, the theorem is proved in Section 4.
We will investigate the spherically symmetric solutions of the system (4) in a spherically symmetric domain Ω and denote that which gives the following system for  > 0: and the initial data and boundary condition become For simplicity, we consider the initial data with one discontinuous point  0 ∈ ( − ,  + ); namely, where  − and  > 0 are positive constants, and  > 0 is bounded.Then, we can give the main results as follows.

The a Priori Estimates
According to the analysis made in [27], there is a curve  = () defined by  ()  =  ( () , ) , (0) =  0 ,  > 0, (16) along which the Rankine-Hugoniot conditions hold where [((), )] := (() + 0, ) − (() − 0, ).It is convenient to make use of the Lagrange coordinates so as to establish the uniformly a priori estimates and take the Lagrange coordinates transform By ( 18) and the conservation of mass for (, ) the Lagrange coordinates transform ( 18) maps (, ) ∈ [ − ,  + ] ×  + into (, ) ∈ [0, 1] ×  + .The curve  = () in Eulerian coordinates is changed to a line  =  0 in Lagrangian coordinates, where and the jump conditions become The relations between Lagrangian and Eulerian coordinates are satisfied as and the initial boundary value problem ( 7)-( 8) is reformulated to where the initial data satisfies Next, we will deduce the a priori estimates for the solution (, ) to the initial boundary value problem (23).To prove the a priori estimates, we assume a priori that there are constants Lemma 4. Let  > 0. Under the conditions in Theorem 1, it holds for any solution (, ) to the initial boundary value problem (23) that where − ∫ Proof.Multiplying (23) 2 by  2  and integrating the result with respect to  over [0, 1], it holds from ( 21) and ( 23) and integrating (27) with respect to , we have Using ( 21) and ( 23) 1 , we deduce which gives It holds from (25) that From ( 24), (30), and (31), we have where  is a positive constant independent of time and we assume that From ( 28) and (32), Lemma 4 can be proved.
Lemma 5. Let  > 0. Under the conditions in Theorem 1, it holds for any solution (, ) to the initial boundary value problem (23) that where  is a positive constant independent of time.
Proof.Differentiating (23) 1 with respect to , we get Summing ( 35) and ( 23) 2 , it holds Note that and so which together with (36) implies Multiplying (39) by (+ 2   ) 2 and integrating the result with respect to  and , it holds that From ( 24), ( 26), (30), and (31), we have that where Ĉ is a positive constant independent of time and we assume that The proof of (34) is completed.Proof.It follows from ( 19) and (34) that which yields Thus, we can choose  + =  * + 1 to have for any positive integer  ∈ , where () is a positive constant dependent of time.
Proof.Multiplying (23) 2 with  2  2−1 , integrating by parts over [0, 1], we have Since it holds that it follows from (48) that which together with (43) and Young's inequality yields and applying the Gronwall's inequality to (51), we can obtain where () is a positive constant dependent of time.It holds from ( 24), ( 26), (30), and (31) that where C is a positive constant independent of time and we assume that From ( 52) and (53), we can deduce (47).
Lemma 8. Let  > 0, for  ∈ , and  > 1/2( − 1).Under the conditions in Theorem 1, it holds for any solution (, ) to the initial boundary value problem (23) that where () is a positive constant dependent of time.
Next, we find that Similarly, we can obtain where we have used By Gagliardo-Nirenberg-Sobolev inequality, (62), and (65), we have Thus, there is a  0 > 0 and a constant  1 > 0 such that For  ∈ [0,  0 ], denote and then from (23), we can obtain and multiplying (70) by 4V 3 and integrating the result over From ( 56), it holds that It holds from ( 24), ( 26), (30), and (31) that where Č is a positive constant independent of time and we assume that Then, it follows from (72) and (73) that where ( 0 ) is a positive constant independent of time  0 .By Gronwall's inequality, (75) leads to It holds for Therefore, we can choose to get where  0 ,  1 are positive constants independent of time.
where  > 0 denotes a constant independent of time.

Lemma 7 .
Let  > 0. Under the conditions in Theorem 1, it holds for any solution (, ) to the initial boundary value problem (23) that

Lemma 11 .
Let  > 0. Under the conditions in Theorem 1, it holds for any solution (, ) to the initial boundary value problem (