Open Access

Global solution to the exterior problem for spherically symmetric compressible Navier-Stokes equations with density-dependent viscosity and discontinuous initial data

Boundary Value Problems20132013:90

https://doi.org/10.1186/1687-2770-2013-90

Received: 5 November 2012

Accepted: 1 April 2013

Published: 15 April 2013

Abstract

In this paper, we consider the exterior problem for spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and discontinuous initial data. We prove that there exists a unique global piecewise regular solution for piecewise regular initial density with bounded jump discontinuity. In particular, the jump of density decays exponentially in time and the piecewise regular solution tends to the equilibrium state exponentially as t + .

Keywords

spherically symmetric Navier-Stokes equationsexterior problemdiscontinuous initial data

1 Introduction

In the present paper, we consider the exterior problem to N-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients. In general, the N-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients reads
{ ρ t + div ( ρ U ) = 0 , ( ρ U ) t + div ( ρ U U ) + P ( ρ ) div ( μ ( ρ ) D ( U ) ) ( λ ( ρ ) div U ) = 0 ,
(1.1)
where t ( 0 , + ) is the time and x R N , N is the spatial coordinate, ρ > 0 and u denote the density and velocity, respectively. Pressure function is taken as P ( ρ ) = ρ γ with γ > 1 , and
D ( U ) = ( U ) + T ( U ) 2
(1.2)
is the strain tensor and μ ( ρ ) , λ ( ρ ) are the Lamé viscosity coefficients satisfying
μ ( ρ ) > 0 , μ ( ρ ) + N λ ( ρ ) 0 .
(1.3)

There are many significant progresses achieved on the global existence of weak solutions and dynamical behaviors of jump discontinuity for the compressible Navier-Stokes equations with discontinuous initial data, for example, as the viscosity coefficients μ ( ρ ) and λ ( ρ ) are both constants, Hoff investigated the global existence of discontinuous solutions of one-dimension Navier-Stokes equations [13]. The construction of global spherically symmetric weak solutions of compressible Navier-Stokes equations for isothermal flow with large and discontinuous initial data was derived by Hoff [4], therein it is also showed that the discontinuities in the density and pressure persist for all time, convecting along particle trajectories, and decaying at a rate inversely proportional to the viscosity coefficient. Hoff also proved the global existence theorems for the multidimensional Navier-Stokes equations of isothermal compressible flows with the polytropic equation of state p ( ρ ) = ρ γ ( γ 1 ) [5, 6]. The global existence of weak solutions was obtained for the Navier-Stokes equations for compressible, heat-conducting flow in one space dimension with large, discontinuous initial data by Chen-Hoff-Trivisa [7]. The global existence of weak solutions of the Navier-Stokes equations for compressible, heat-conducting fluids in two- and three-space dimensions was proved by Hoff, when the initial data may be discontinuous across a hypersurface of R n [8]. Hoff also showed the global existence of solutions of the Navier-Stokes equations for compressible, barotropic flow in two space dimensions which exhibit convecting singularity curves [9].

If the viscosity coefficients μ ( ρ ) = ρ α , λ ( ρ ) = 0 , for the case of one space dimension, the global existence of unique piecewise smooth solution to the free boundary value problem was obtained by Fang-Zhang for (1.1) with 0 < α < 1 , where the initial density is piecewise smooth with possibly large jump discontinuities [10]. Lian-Guo-Li addressed the initial boundary value problem for (1.1) with 0 < α 1 subject to piecewise regular initial data with initial vacuum state included [11], Lian-Liu-Li-Xiao also consider the Cauchy problem for one-dimensional isentropic compressible Navier-Stokes equations with adensity-dependent viscosity coefficient [12]; in these two cases above, they proved the global existence of unique piecewise regular solution and proved the finite time vanishing of vacuum state in [11]. In particular, they got that the jump discontinuity of density decays exponentially, but never vanish in any finite time and the piecewise regular solution tends to the equilibrium state as t + .

Recently, there is also huge literature on the studies of 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 [13, 14]. The prototype model is the viscous Saint-Venant (corresponding to (1.1) with P ( ρ ) = ρ 2 , μ ( ρ ) = ρ and λ ( ρ ) = 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 [1522] and references therein. The global existence of classical solutions is shown by Mellet and Vasseur [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 [2427] and references therein.

In this present paper, we consider the exterior problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and discontinuous initial data and focus on the existence, regularity and dynamical behaviors of global weak solutions, etc. We show that the exterior 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 as t + .

The rest part of the paper is arranged as follows. In Section 2, the main results about the existence, regularity 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.

2 Notations and main results

For simplicity, we will take μ ( ρ ) = ρ α and λ ( ρ ) = ( α 1 ) ρ α and D ( U ) = U in (1.1). The isentropic compressible Navier-Stokes equations become
{ ρ t + div ( ρ U ) = 0 , ( ρ U ) t + div ( ρ U U ) + P ( ρ ) div ( ρ α U ) ( α 1 ) ( ρ α div U ) = 0 .
(2.1)
The initial data and boundary conditions of (2.1) are imposed as:
{ ( ρ , U ) ( x , 0 ) = ( ρ 0 , U 0 ) ( x ) , x Ω , U = 0 , on  Ω , lim | x | + ( ρ , u ) ( x , t ) = ( ρ ¯ , 0 ) , t [ 0 , T ] ,
(2.2)

where Ω : = R 3 / Ω r , Ω r is a ball of radius r centered at the origin in R 3 , and ρ ¯ > 0 is a constant.

We are concerned with the spherically symmetric solutions of the system (2.1) in the spherically symmetric exterior domain Ω. To this end, we denote that
| x | = r , ρ ( x , t ) = ρ ( r , t ) , U ( x , t ) = u ( r , t ) x r ,
(2.3)
which leads to the following system of equations for r > 0 :
{ ρ t + ( ρ u ) r + 2 ρ u r = 0 , ( ρ u ) t + ( ρ u 2 + ρ γ ) r + 2 ρ u 2 r α ( ρ α u r ) r α ρ α ( 2 u r ) r 2 ( α 1 ) ( ρ α ) r u r = 0 ,
(2.4)
with the initial data and boundary conditions
{ ( ρ , u ) ( r , 0 ) = ( ρ 0 , u 0 ) ( r ) , r [ r , + ) , u ( r , t ) = 0 , lim r + ( ρ , u ) ( r , t ) = ( ρ ¯ , 0 ) , t [ 0 , T ] .
(2.5)

Next, we give the definition of weak solution to the exterior problem (2.1)-(2.2).

Definition 2.1 (weak solution)

For any T > 0 , ( ρ , u ) is said to be a weak solution of the exterior problem (2.1)-(2.2), if ( ρ , u ) has the following regularities:
{ ρ 0 a.e. , 0 < ρ ρ ¯ L ( [ 0 , T ] ; L 1 ( Ω ) L 2 ( Ω ) ) , ρ U L ( [ 0 , T ] ; L 2 ( Ω ) ) , ( ρ α 1 2 ) L ( [ 0 , T ] ; L 2 ( Ω ) ) , ρ α U L 2 ( [ 0 , T ] ; L 2 ( Ω ) ) ,
(2.6)
and the equations (2.1) are satisfied in the sense of distribution. Namely, it holds for all φ C 0 ( Ω ¯ × [ 0 , T ] ) that
Ω ρ 0 φ ( x , 0 ) d x + 0 T Ω ρ φ t d x d t + 0 T Ω ρ U φ d x d t = 0 ,
(2.7)
and for all ψ = ( ψ 1 , ψ 2 , ψ 3 ) C 0 ( Ω ¯ × [ 0 , T ] ) that
Ω ρ 0 U 0 ψ ( x , 0 ) d x + 0 T Ω ( ρ ( ρ U ) ψ t + ρ U ρ U : ψ ) d x d t + 0 T Ω ρ γ div ψ d x d t ρ α U , ψ = 0 ,
(2.8)
where the diffusion term makes sense as
ρ α U , ψ = 0 T Ω ρ α 1 2 ( ρ U ) ψ d x d t 2 α 2 α 1 0 T Ω ( ρ U ) ( ( ρ α 1 2 ) ) ψ d x d t .
(2.9)
For simplicity, we consider the initial data in exterior problem (2.4)-(2.5) with one discontinuous point ξ 0 ( r , + ) , namely,
{ ρ 0 ρ ¯ L 1 L 2 ( [ r , ξ 0 ) ( ξ 0 , + ) ) , inf x [ r , ξ 0 ) ( ξ 0 , + ) ρ 0 > ρ > 0 , ρ 0 ( ξ 0 0 ) ρ 0 ( ξ 0 + 0 ) , | ρ 0 ( ξ 0 + 0 ) ρ 0 ( ξ 0 0 ) | < δ , r ( ρ 0 α 1 2 ) r L 2 ( [ r , ξ 0 ) ( ξ 0 , + ) ) , r 2 u 0 H 2 ( [ r , ξ 0 ) ( ξ 0 , + ) ) ,
(2.10)
where ρ and δ > 0 are positive constants, and δ > 0 is bounded. Next, we define that
(2.11)
(2.12)

and denote that in this paper. Then we can give the main results as follows.

Theorem 2.1 Let α > 2 3 and γ > 1 . Assume that the initial data satisfies (2.10) and E 0 + E 1 < ν α r 2 ρ ¯ γ 2 + α 1 2 , ν is a positive constant. Then there exist two positive constants ρ , ρ and a unique global piecewise regular solution ( ρ , u ) to the exterior problem (2.4)-(2.5), namely, satisfying
{ 0 < ρ ρ ( r , t ) ρ , ( r , t ) [ r , ξ ( t ) ) ( ξ ( t ) , + ) × [ 0 , T ] , ρ L ( [ 0 , T ] ; H 1 ( [ r , ξ ( t ) ) ( ξ ( t ) , + ) ) ) , u L ( [ 0 , T ] ; H 1 ( [ r , ξ ( t ) ) ( ξ ( t ) , + ) ) ) , u r L 2 ( [ 0 , T ] ; H 1 ( [ r , ξ ( t ) ) ( ξ ( t ) , + ) ) ) ,
(2.13)
where r = ξ ( t ) is a curve defined by
d ξ ( t ) d t = u ( ξ ( t ) , t ) , ξ ( 0 ) = ξ 0 , t > 0 ,
(2.14)
along which the Rankine-Hugoniot conditions hold
[ u ( ξ ( t ) , t ) ] = 0 , [ ρ γ ( ξ ( t ) , t ) ] = [ α ( ρ α u r + 2 ρ α u r ) ( ξ ( t ) , t ) ] ,
(2.15)
where [ f ( ξ ( t ) , t ) ] : = f ( ξ ( t ) + 0 , t ) f ( ξ ( t ) 0 , t ) , and along the discontinuity r = ξ ( t ) the jump decays exponentially
| [ ρ 0 α ( ξ 0 ) ] | e C 0 t | [ ρ α ( ξ ( t ) , t ) ] | | [ ρ 0 α ( ξ 0 ) ] | e C 1 t ,
(2.16)
where C 0 , C 1 are positive constants independent of time. Furthermore, the solution tends to equilibrium state ( ρ ¯ , 0 )
( ρ ρ ¯ , u ) ( , t ) L ( [ r , ξ ( t ) ) ( ξ ( t ) , + ) ) 0 , t + .
(2.17)

Remark 2.1 Theorems 2.1 holds for the Saint-Venant model for shallow water, i.e., γ = 2 , α = 1 .

Remark 2.2 (2.16) shows that the discontinuity in the density decays exponentially in time.

Remark 2.3 For piecewise regular initial data, it can be shown in terms of the difference scheme made by Hoff [4], there is a (piecewise regular) weak solution to the exterior problem (2.4) and (2.5). In addition, there is a curve r = ξ ( t ) starting from ξ ( 0 ) = ξ 0 , so that the density is discontinuous cross r = ξ ( t ) , and the Rankine-Hugoniot conditions hold
[ u ( ξ ( t ) , t ) ] = 0 , [ ρ γ ( ξ ( t ) , t ) ] = [ α ( ρ α u r + 2 ρ α u r ) ( ξ ( t ) , t ) ] .
(2.18)

To extend the local solution globally in time, we need to establish the uniformly a-priori estimates.

3 The a-priori estimates

According to the analysis made in [28], there is a curve r = ξ ( t ) defined by
d ξ ( t ) d t = u ( ξ ( t ) , t ) , ξ ( 0 ) = ξ 0 , t > 0 ,
(3.1)
along which the Rankine-Hugoniot conditions hold
[ u ( ξ ( t ) , t ) ] = 0 , [ ρ γ ( ξ ( t ) , t ) ] = [ α ( ρ α u r + 2 ρ α u r ) ( ξ ( t ) , t ) ] ,
(3.2)
where [ f ( ξ ( t ) , t ) ] : = f ( ξ ( t ) + 0 , t ) f ( ξ ( t ) 0 , t ) . It is convenient to make use of the Lagrange coordinates in order to establish the uniformly a-priori estimates, denote the Lagrange coordinates transform
x = r r ρ ( r , t ) r 2 d r , τ = t ,
(3.3)
which maps ( r , t ) [ r , + ) × R + into ( x , τ ) [ 0 , + ) × R + . The relations between Lagrangian and Eulerian coordinates are satisfied as
x r = ρ r 2 , x t = ρ u r 2 .
(3.4)
Since the conservation of total mass holds, the curve r = ξ ( t ) in Eulerian coordinates is changed to a line x = x 0 in Lagrangian coordinates, where
x 0 = r ξ ( t ) ρ ( r , t ) r 2 d r = r ξ 0 ρ 0 ( r ) r 2 d r
(3.5)
in Lagrangian coordinates, and the jump conditions become
[ u ( x 0 , τ ) ] = 0 , [ ρ γ ( x 0 , τ ) ] = [ α ρ 1 + α ( r 2 u ) x ( x 0 , τ ) ] .
(3.6)
Meanwhile, the exterior problem (2.4)-(2.5) is reformulated to
{ ρ τ + ρ 2 ( r 2 u ) x = 0 , r 2 u τ + ( ρ γ ) x = α ( ρ 1 + α ( r 2 u ) x ) x 2 ( ρ α ) x u r , ( ρ , u ) ( x , 0 ) = ( ρ 0 , u 0 ) ( x ) , x [ 0 , + ) , u ( 0 , t ) = 0 , lim x + ( ρ , u ) = ( ρ ¯ , 0 ) , τ [ 0 , + ) ,
(3.7)
where the initial data satisfies
{ ρ 0 ρ ¯ L 1 L 2 ( [ 0 , x 0 ) ( x 0 , + ) ) , inf x [ 0 , x 0 ) ( x 0 , + ) ρ 0 > ρ > 0 , ρ 0 ( x 0 0 ) ρ 0 ( x 0 + 0 ) , | ρ 0 ( x 0 + 0 ) ρ 0 ( x 0 0 ) | < δ , r 2 ( ρ 0 α ) x L 2 ( [ 0 , x 0 ) ( x 0 , + ) ) , 1 r 2 ρ 0 ( r 2 u 0 ) L 2 ( [ 0 , x 0 ) ( x 0 , + ) ) , r 2 ρ 0 ( r 2 u 0 ) x L 2 ( [ 0 , x 0 ) ( x 0 , + ) ) , 1 r 2 ρ 0 ( r 2 ρ 0 ( r 2 ρ 0 ( r 2 u 0 ) x ) x ) L 2 ( [ 0 , x 0 ) ( x 0 , + ) ) .
(3.8)
First, we are ready to establish the a-priori estimates for the solution ( ρ , u ) to the exterior problem (3.7). To obtain the a-priori estimates, we assume a-priori that there are constants ρ ± > 0 so that
0 < ρ ρ ( x , τ ) ρ + , ( x , τ ) ( [ 0 , x 0 ) ( x 0 , + ) ) × [ 0 , T ] , T > 0 .
(3.9)
Lemma 3.1 Let T > 0 . Under the conditions in Theorem  2.1, it holds for any solution ( ρ , u ) to the exterior problem (3.7) that
(3.10)

where .

Proof Multiplying (3.7)2 by r 2 u and integrating the result with respect to x over [ 0 , + ) , making use of (3.6), (3.7)1 and (3.8), we have
(3.11)
integrating (3.11) with respect to τ, we obtain
(3.12)
Applying equations (3.6) and (3.7)1, it holds that
[ ρ α ] τ + α [ ρ γ ] = 0 ,
(3.13)
which implies
[ ρ α ] = [ ρ 0 α ] exp ( α 0 τ [ ρ γ ] [ ρ α ] d s ) .
(3.14)
From (3.9) and (3.14), we can find
γ α ρ γ α [ ρ γ ] [ ρ α ] γ α ρ + γ α .
(3.15)
It holds from (3.8), (3.14) and (3.15) that
(3.16)
where C ¯ is a positive constant independent of time and we assume that
0 < | [ ρ 0 α ] | min { γ 4 ( C ¯ + ρ 1 ) ρ γ α , α 2 ( C ¯ + ρ 1 ) ρ 1 + α r 4 } : = M 1 .
(3.17)

From (3.12) and (3.16), Lemma 3.1 can be obtained. □

Lemma 3.2 Let T > 0 . Under the conditions in Theorem  2.1, it holds for any solution ( ρ , u ) to the exterior problem (3.7) that
(3.18)
Proof Multiplying (3.7)1 by ρ α 1 and differentiating the equation with respect to x, we have
( ρ α ) x τ + α ( ρ 1 + α ( r 2 u ) x ) x = 0 .
(3.19)
Summing (3.19) and (3.7)2, we get
( r 2 u + ( ρ α ) x ) τ + ( ρ γ ) x = ( r 2 ) τ u 2 ( ρ α ) x u r .
(3.20)
Note that
r 3 ( x , τ ) = r 3 + 3 0 x 1 ρ ( z , τ ) d z ,
(3.21)
and so
r τ = 1 r 2 0 x ( 1 ρ ) t ( z , t ) d z = 1 r 2 0 x ( r 2 u ) z ( z , t ) d z = u ( x , τ ) ,
(3.22)
which together with (3.20) yields
( r 2 u + ( ρ α ) x ) τ + ( ρ γ ) x = 2 r 3 u 2 2 ( ρ α ) x u r .
(3.23)
Multiplying (3.23) by ( u + r 2 ( ρ α ) x ) r 2 , and integrating the result with respect to x and τ, we have
(3.24)
It holds from (3.8), (3.10), (3.14) and (3.15) that
(3.25)
where C ˆ is a positive constant independent of time and we assume that
0 < | [ ρ 0 α ] | < min { ρ γ α ρ + 2 α 2 γ ( C ˆ + ρ 2 ) E 1 , ρ γ α ( C ˆ + ρ 2 ) E 1 , ρ 1 + α r 4 ( C ˆ + ρ 2 ) E 1 , M 1 } : = δ .
(3.26)

The proof of (3.18) is completed. □

Lemma 3.3 Let T > 0 . Under the conditions in Theorem  2.1, there exist two constants 0 < ρ < ρ such that
0 < ρ ρ ( x , τ ) ρ , ( x , τ ) [ 0 , + ) × [ 0 , T ] .
(3.27)

Proof

Let
φ ( ρ ) : = 1 γ 1 ( ρ γ 1 ρ ¯ γ 1 ) + ρ ¯ γ ( ρ 1 ρ ¯ 1 ) ,
(3.28)
and
ψ ( ρ ) : = ρ ¯ ρ φ ( η ) 1 2 η α 1 d η .
(3.29)
It follows from (3.10) and (3.18) that
(3.30)

We can verify that

1 As ρ + , we have for some θ ( 0 , 1 ) that if 1 < γ 3 , it holds
(3.31)
and if γ > 3 , it holds
(3.32)
2 As ρ 0 , we have for some θ ( 0 , 1 ) that if 1 < γ 3 , it holds
(3.33)
where ν 1 > 0 is a positive constant, and if γ > 3 , it holds
(3.34)
where ν 2 > 0 is a positive constant, and we denote the positive constant ν : = min { ν 1 , ν 2 } . Applying (3.31)-(3.34) and E 0 + E 1 < ν α r 2 ρ ¯ γ 2 + α 1 2 , let
ρ = 1 2 ρ , ρ + = 2 ρ ,
(3.35)

we can prove (3.27). □

Lemma 3.4 Let T > 0 . Under the conditions in Theorem  2.1, it holds for any solution ( ρ , u ) to the exterior problem (3.7) that
| [ ρ 0 α ( x 0 ) ] | e C 0 τ | [ ρ α ( x 0 , τ ) ] | | [ ρ 0 α ( x 0 ) ] | e C 1 τ , τ [ 0 , T ] ,
(3.36)

where C 0 , C 1 are positive constants independent of time.

Proof From (3.14), (3.15) and (3.27), we can get (3.36). □

Lemma 3.5 Let T > 0 . Under the conditions in Theorem  2.1, it holds for any solution ( ρ , u ) to the exterior problem (3.7) that
(3.37)

where C > 0 denotes a constant independent of time.

Proof Multiplying (3.7)2 by ρ ( 1 + α ) ( r 2 u ) τ and integrating the result with respect to x over [ 0 , + ) , making use of (3.6) and (3.8), we obtain
(3.38)
which implies
(3.39)
From (3.7)2, (3.10), (3.18) and (3.27), we can deduce that
(3.40)
(3.41)
and
(3.42)
where C denotes a constant independent of time and ϵ ( 0 , 1 ) is a small constant which will be determined later. Use (3.39)-(3.42), and we can obtain that
(3.43)
Differentiating (3.7)2 with respect to τ, multiplying the result by ( r 2 u ) τ and integrating the result with respect to x over [ 0 , + ) , we have
(3.44)
A complicated computation gives
(3.45)
and by means of Gronwall’s inequality, (3.7)2, (3.10), (3.18), (3.27) and (3.43), it holds that
(3.46)

where C denotes a constant independent of time, choosing the constant ϵ small sufficiently, we can complete the proof of Lemma 3.5. □

Remark 3.1

By Lemmas 3.1-3.5, the following inequality holds:
(3.47)
Lemma 3.6 Under the conditions in Theorem  2.1, it holds for any solution ( ρ , u ) to the exterior problem (3.7) that
( ρ ρ ¯ , u ) ( , τ ) L ( [ 0 , x 0 ) ( x 0 , + ) ) 0 , τ + .
(3.48)

Proof

From Lemmas 3.1-3.5, we can obtain
0 + ( ρ ρ ¯ , u ) x L 2 ( [ 0 , x 0 ) ( x 0 , + ) ) 2 d τ r 4 0 + ( ρ ρ ¯ , u ) x r 2 L 2 ( [ 0 , x 0 ) ( x 0 , + ) ) 2 d τ C ,
(3.49)
and
(3.50)
which together with (3.49) implies
( ρ ρ ¯ , u ) x L 2 ( [ 0 , x 0 ) ( x 0 , + ) ) 2 W 1 , 1 ( R + ) .
(3.51)
It holds from Gagliardo-Nirenberg-Sobolev inequality that
( ρ ρ ¯ , u ) L ( [ 0 , x 0 ) ( x 0 , + ) ) ( ρ ρ ¯ , u ) L 2 ( [ 0 , x 0 ) ( x 0 , + ) ) 1 2 ( ρ ρ ¯ , u ) x L 2 ( [ 0 , x 0 ) ( x 0 , + ) ) 1 2 ,
(3.52)

which together with (3.10), (3.27) and (3.51) implies this lemma. □

4 Proof of the main results

Proof The global existence of unique piecewise regular solution to the exterior problem (2.4)-(2.5) can be established in terms of the short time existence carried out as in [2, 10], the uniform a-priori estimates and the analysis of regularities, which indeed follow from Lemmas 3.1-3.5. In addition, one can show that ( ρ , u ) is also global weak solution to the exterior problem (2.4)-(2.5) with initial data satisfying (2.10). We omit the details. The large time behaviors follow from Lemma 3.6 directly. The proof of Theorem 2.1 is completed. □

Declarations

Acknowledgements

The research of RX Lian is supported by NNSFC No. 11101145.

Authors’ Affiliations

(1)
College of Mathematics and Information Science, North China University of Water Resources and Electric Power

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© Lian and Yang; licensee Springer. 2013

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