Research Article | Open Access

Ruxu Lian, Lan Huang, Jian Liu, "Global Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations with Density-Dependent Viscosity", *Journal of Applied Mathematics*, vol. 2012, Article ID 395209, 22 pages, 2012. https://doi.org/10.1155/2012/395209

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

**Academic Editor:**Nazim I. Mahmudov

#### Abstract

We consider the exterior problem and the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient 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 .

#### 1. Introduction

The isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients read as follows: where is the time and , is the spatial coordinate, and denote the density and velocity, respectively. Pressure function is taken as with , and is the strain tensor and , are the Lamé viscosity coefficients satisfying

There are many important progress achieved recently on the compressible Navier-Stokes equations with density-dependent viscosity coefficient. For instance, the mathematical derivations are derived in the simulation of flow surface in shallow region [1–4]. The prototype model is the viscous Saint-Venant. 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 is investigated by many authors, refer to [5–12] and references therein. Mellet and Vasseur showed the global existence of strong solutions for [13]. 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 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 [14–17] and references therein.

In this present paper, we consider the exterior problem and the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and focus on the regularities and dynamical behaviors of global strong solution, and so forth. As , we show that the exterior problem and the initial boundary value problem with regular initial data both admit the unique global strong solution. In particular, the strong solution tends to the equilibrium state as .

The rest of the paper is arranged as follows. In Section 2, the main results about the dynamical behaviors of global strong solution for compressible Navier-Stokes equations are stated. Then, the theorems of the exterior problem and the initial boundary value problem are proved in Sections 3 and 4, respectively.

#### 2. Notations and Main Results

For simplicity, we will take and and in (1.1). The isentropic compressible Navier-Stokes equations become Firstly, we consider the exterior problem, namely, the initial data and boundary conditions of (2.1) are imposed as follows: where , is a ball of radius centered at the origin in , and is a constant.

We are concerned with the spherically symmetric solutions of system (2.1) in an spherically symmetric exterior domain . To this end, we denote that which leads to the following system of equations for , with the initial data and boundary conditions and the initial data satisfies for some constants and Then, we define that and give the main results as follows.

Theorem 2.1. *Let . Assume that the initial data satisfies (2.6) and , is a positive constant. Then, there exist two positive constants and and a unique global strong solution to the exterior problem (2.4) and (2.5), namely, satisfying
**
Furthermore, the solution tends to the equilibrium state *

Then investigates the initial boundary value problem, and the initial data and boundary conditions of (2.1) are assumed as follow: By (2.3), one considers (2.4) with the initial data and boundary conditions and the initial data satisfies for some constant

Then, can give the main results as follows.

Theorem 2.2. *Let . Assume that the initial data satisfies (2.12), there exist two positive constants and and a unique global strong solution to the initial boundary value problem (2.4) and (2.11), namely, satisfying
**
Furthermore, the solution tends to the equilibrium state exponentially
**
where and are positive constants independent of time and .*

*Remark 2.3. *Theorems 2.1 and 2.2 hold for one-dimensional Saint-Venant’s model for shallow water, that is, , .

#### 3. Proof of the Exterior Problem

##### 3.1. The A Priori Estimates

It is convenient to make use of the Lagrangian coordinates so as to establish the uniformly a priori estimates. Take the Lagrange coordinates transform which map into . The relation between Lagrangian and Eulerian coordinates are satisfied as The exterior problem (2.4) and (2.5) is reformulated to where the initial data satisfies First, we will establish the a-priori estimates for the solution to the exterior problem (3.3).

Lemma 3.1. *Let . Under the conditions in Theorem 2.1, it holds for any solution to the exterior problem (3.3) that
*

*Proof. *Multiplying (3.3) _{2} by and integrating the result with respect to over , making use of (3.3) _{1} and (3.4), we have
integrating (3.6) with respect to , we obtain
Lemma 3.1 can be obtained.

Lemma 3.2. *Let . Under the conditions in Theorem 2.1, it holds for any solution to the exterior problem (3.3) that
**
and there exist two constants such that
*

*Proof. *Differentiating (3.3) _{1} with respect to , we have
Summing (3.10) and (3.3) _{2}, we get
Note that
and so
which together with (3.11) yields
Multiplying (3.14) by and integrating the result with respect to and , we have

Let
It follows from (3.6) and (3.13) that
We can verify that(1) As , it holds for , where (2) As , it holds for , where
Applying (3.17)–(3.19) and , where is a positive constant, we can prove (3.9).

Lemma 3.3. *Let . Under the conditions in Theorem 2.1, it holds for any solution to the exterior problem (3.3) that
**
where denotes a constant independent of time.*

*Proof. *Multiplying (3.3)_{2} by and integrating the result with respect to over , making use of (3.4), we obtain
which implies
From (3.3) _{2}, (3.5), (3.8), and (3.9), we can deduce that for some small
using (3.22)–(3.24), we can obtain that

Differentiating (3.3) _{2} with respect to , multiplying the result by , and integrating the result with respect to over , we have
A complicated computation gives
integrating (3.27) with respect to , by means of (3.3) _{2}, (3.5), (3.8), (3.9), and (3.25), it holds that
choosing the constant small sufficiently, we can complete the proof of Lemma 3.3.

*Remark 3.4. *By Lemmas 3.1–3.3, the following inequality holds:

Lemma 3.5. *Under the conditions in Theorem 2.1, it holds for any solution to the exterior problem (3.3) that
**
where denotes a constant independent of time.*

*Proof. *From Lemmas 3.1–3.3, we can obtain
which together with (3.31) implies
It holds from Gagliardo-Nirenberg-Sobolev inequality that
which together with (3.5), (3.9), and (3.33) implies this lemma.

##### 3.2. Proof of Theorem 2.1

*Proof. *The global existence of unique strong solution to the exterior problem as (2.4) and (2.5) can be established in terms of the short-time existence carried out as in [6], the uniform a-priori estimates and the analysis of regularities, which indeed follow from Lemmas 3.1–3.3. We omit the details. The large time behaviors follow from Lemma 3.5 directly. The proof of Theorem 2.1 is completed.

#### 4. Proof of the Initial Boundary Value Problem

##### 4.1. The A-Priori Estimates

Take the Lagrange coordinates transform By (4.1) and the conservation of mass for the Lagrange coordinates transform (4.1) map into .

The relation between Lagrangian and Eulerian coordinates are satisfied as and the initial boundary value problem’s (2.4) and (2.11) are reformulated to where the initial data satisfies Then, we will establish the a-priori estimates for the solution to the initial boundary value problem (4.4).

Lemma 4.1. *Let . Under the conditions in Theorem 2.2, it holds for any solution to the initial boundary value problem (4.4) that
*

*Proof. *Multiplying (4.4)_{2} by and integrating the result with respect to over , using (4.4)_{1} and (4.5), we obtain
and integrating (4.7) with respect to , we obtain
Lemma 4.1 can be obtained.

Lemma 4.2. *Let . Under the conditions in Theorem 2.2, it holds for any solution to the initial boundary value problem (4.4) that
**
where is a positive constant independent of time.*

*Proof. *Differentiating (4.4)_{1} with respect to , we have
which together with (4.4)_{2} and gives
Multiplying (4.11) by , and integrating the result with respect to and , it holds that
The proof of (4.9) is completed.

Lemma 4.3. *Let . Under the conditions in Theorem 2.2, there exists a constant such that
*

*Proof. *It follows from (4.6) and (4.9) that

Lemma 4.4. *Let . Under the conditions in Theorem 2.2, it holds for any solution to the initial boundary value problem (4.4) that
**
for any positive integer , where is a positive constant dependent of time.*

*Proof. *Multiplying (4.4)_{2} with , integrating by parts over , we have
Since it holds that
it follows from (4.16) that
which together with (4.13) and Young’s inequality yields
and by applying the Gronwall’s inequality to (4.19), we can obtain (4.15).

Lemma 4.5. *Let , for , and . Under the conditions in Theorem 2.2, it holds for any solution to the initial boundary value problem (4.4) that
**
where is a positive constant dependent of time.*

*Proof. *By means of Sobolev imbedding theorem and Cauchy-Schwarz inequality, applying (4.6), (4.13), and (4.15), we get
Next, we find that
which together with (4.13), (4.15), and (4.22) gives
applying the Gronwall’s inequality to (4.24), we obtain (4.21).

Lemma 4.6. *Let . Under the conditions in Theorem 2.2, there exists a constant such that
*

*Proof. *It is easy to verify that
Indeed, it holds that
which together with (4.4), (4.9), and (4.13) gives
By Gagliardo-Nirenberg-Sobolev inequality and (4.27), it follows that
Thus, there is a and a constant such that

For , denote
then from (4.4), we can obtain
multiplying (4.33) by and integrating the result over , we have
From (4.21), it holds that
which yields to
where is a positive constant dependent of time . By Gronwall’s inequality, (4.36) leads to
It holds for that
namely,
Therefore, we can choose
to get

Lemma 4.7. *
where denotes a constant independent of time.*

*Proof. *The proof is similar to Lemma 3.3. We omit here.

*Remark 4.8. *By Lemmas 4.1–4.7, the following inequality holds:

Lemma 4.9. *Under the conditions in Theorem 2.2, it holds for any solution to the initial boundary value problems (2.4) and (2.11) that
**
where and denote the constants independent of time and .*

*Proof. *In a similar argument to show (4.8) and (4.12) with modification, we can obtain the following
It holds from (4.9), (4.13), and (4.25) that
Denote
By (4.45)–(4.48), a complicated computation gives rise to
which gives
By the fact that
where is a constant independent of time and Gagliardo-Nirenberg-Sobolev inequality
we obtain (4.44).

##### 4.2. Proof of Theorem 2.2

*Proof. *The proof of Theorem 2.2 is similar to Theorem 2.1. We omit the details.

#### Acknowledgments

The research of R. Lian is partially supported by NNSFC no. 11101145. The research of L. Huang is partially supported by NNSFC no. 11126323.

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