Abstract and Applied Analysis

Volume 2014 (2014), Article ID 132324, 12 pages

http://dx.doi.org/10.1155/2014/132324

Research Article

Global Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous Initial Data

Ruxu Lian,1,2 Jianwei Yang,¹ and Jian Liu³

¹College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

²Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China

³College of Teacher Education, Quzhou University, Quzhou 324000, China

Received 29 September 2013; Accepted 8 April 2014; Published 4 May 2014

Academic Editor: Adem Kılıçman

Copyright © 2014 Ruxu Lian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 in this paper. For piecewise regular initial density with bounded jump discontinuity, we show that there exists a unique global piecewise regular solution. In particular, the jump of density decays exponentially in time and the piecewise regular solution tends to the equilibrium state exponentially 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 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 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 and are both constants, for the case of one space dimension, Hoff investigated the global existence of discontinuous solutions of the Navier-Stokes equations [1–3]. 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 [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 , , 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 , 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 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 , 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 , , and ). 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 [14–22] 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 [24–26] 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.

2. Notations and Main Results

We will take and and in (1) for simplicity. The isentropic compressible Navier-Stokes equations read as

with the initial data and boundary condition

where and and are constants.

We will investigate the spherically symmetric solutions of the system (4) in a spherically symmetric domain and denote that

which gives the following system for :

and the initial data and boundary condition become

For simplicity, we consider the initial data with one discontinuous point ; namely,

where and are positive constants, and is bounded.

Then, we can give the main results as follows.

Theorem 1. Let . Assume that the initial data satisfies (9). Then, there exist two positive constants , and a unique global piecewise regular solution to the initial boundary value problem (7)-(8), namely, satisfying

where is a curve defined by

along which the Rankine-Hugoniot conditions hold

where , and along the discontinuity the jump decays exponentially

and furthermore, the solution tends to the equilibrium state exponentially

where , , , and are positive constants independent of time and , where .

Remark 2. Theorem 1 holds for the Saint-Venant model for shallow water; that is, .

Remark 3. For the piecewise regular initial data, Hoff [4] has proved that there is a (piecewise regular) weak solution to the initial boundary value problem (7)-(8) in terms of the difference scheme. In addition, there is a curve starting from , such that the density is discontinuous cross the curve , and the Rankine-Hugoniot conditions hold

To extend the local solution globally in time, we have to prove the uniformly a priori estimates.

3. The a Priori Estimates

According to the analysis made in [27], there is a curve defined by

along which the Rankine-Hugoniot conditions hold

where . 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 . The curve in Eulerian coordinates is changed to a line 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 such that

Lemma 4. Let . 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 and integrating the result with respect to over , it holds from (21) and (23)₁ that

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 . 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 and integrating the result with respect to and , it holds that

From (24), (26), (30), and (31), we have that

where is a positive constant independent of time and we assume that

The proof of (34) is completed.

Lemma 6. Let . Under the conditions in Theorem 1, there exists a constant such that

Proof. It follows from (19) and (34) that

which yields

Thus, we can choose to have

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

for any positive integer , where is a positive constant dependent of time.

Proof. Multiplying (23)₂ with , integrating by parts over , 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 is a positive constant independent of time and we assume that

From (52) and (53), we can deduce (47).

Lemma 8. Let , for , and . 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.

Proof. By means of Sobolev imbedding theorem and Cauchy-Schwartz inequality, applying (34), (43), and (47), we get

Next, we find that

which together with (43), (47), and (57) gives

and applying the Gronwall’s inequality to (59), we obtain (56).

Lemma 9. Let . Under the conditions in Theorem 1, there exists a constant such that

Proof. It is easy to verify

Indeed, it holds that

which together with (23), (34), and (43) gives

Similarly, we can obtain

where we have used

By Gagliardo-Nirenberg-Sobolev inequality, (62), and (65), we have

Thus, there is a and a constant such that

For , denote

and then from (23), we can obtain

and multiplying (70) by and integrating the result over , we have

From (56), it holds that

It holds from (24), (26), (30), and (31) that

where is a positive constant independent of time and we assume that

Then, it follows from (72) and (73) that

where is a positive constant independent of time . By Gronwall’s inequality, (75) leads to

It holds for that

namely,

Therefore, we can choose

to get

Lemma 10. Let . Under the conditions in Theorem 1, it holds for any solution to the initial boundary value problem (23) that

where , are positive constants independent of time.

Proof. From (30), (31), (43), and (60), we can get (81).

Lemma 11. Let . Under the conditions in Theorem 1, it holds for any solution to the initial boundary value problem (23) that

where denotes a constant independent of time.

Proof. Multiplying (23)₂ by and integrating the result with respect to over , making use of (21) and (24), it holds that

which gives

It holds from (23)₂, (26), (34), and (43) that

where denotes a positive constant independent of time and is a small constant which will be chosen later. It holds from (84)–(87) that

Differentiating (23)₂ with respect to , multiplying the result by , and integrating the result with respect to over , we deduce

A complicated computation implies

by means of Gronwall’s inequality, (23)₂, (26), (34), (43), and (88), we have

where denotes a positive constant independent of time; choosing the constant small sufficiently, we complete the proof of Lemma 11.

Remark 12. By Lemmas 4–11, the following inequality holds:

Remark 13. In fact, there are several fundamental estimates, such as (26) in Lemma 4, (34) in Lemma 5, (43) in Lemma 6, (47) in Lemma 7, (60) in Lemma 9, and (82) in Lemma 11, which can be also found in [15, 20, 22, 26] and so forth.

Lemma 14. Under the conditions in Theorem 1, it holds for any solution to the initial boundary value problem (7)-(8) that

where and denote the constants independent of time and .

Proof. In a similar argument to show (28) and (40) with modification, we can obtain (94) and (95)

It holds from (34), (43), and (60) that

Denote

By (94)–(97), a complicated computation gives rise to

where is a positive constant and is the constant obtained in Lemma 10. From (97), we have

where we have used (26), (30), and (31). By the fact

where is a constant independent of time and Gagliardo-Nirenberg-Sobolev inequality

we obtain (93).

4. Proof of the Main Results

Proof. The global existence of unique piecewise regular solution to the initial boundary value problem (7)-(8) 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 4–11. In addition, one can show that is the global weak solution to the initial boundary value problem (7)-(8) with the initial data satisfying (9). We omit the details. The large time behaviors follow from Lemma 14 directly. The proof of Theorem 1 is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of R. X. Lian is supported by NNSFC no. 11101145, China Postdoctoral Science Foundation no. 2012M520360, Doctoral Foundation of North China University of Water Sources and Electric Power no. 201032, and Innovation Scientists and Technicians Troop Construction Projects of Henan Province.

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