Journal of Applied Mathematics

Volume 2013 (2013), Article ID 505108, 11 pages

http://dx.doi.org/10.1155/2013/505108

Research Article

Free Boundary Value Problem for the One-Dimensional Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous Initial Data

Ruxu Lian1,2 and Guojing Zhang³

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

³School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Received 9 February 2013; Accepted 6 June 2013

Academic Editor: Nazim Idrisoglu Mahmudov

Copyright © 2013 Ruxu Lian and Guojing Zhang. 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 study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and discontinuous initial data in this paper. For piecewise regular initial density, we show that there exists a unique global piecewise regular solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic time-rate, and the jump discontinuity of density also decays at an algebraic time-rate as the time tends to infinity.

1. Introduction

In the present paper, we consider the free boundary value problem to one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient for piecewise regular initial data connected with the infinite vacuum via jump discontinuity. In general, the one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient read

where and denote the flow density and velocity, respectively, the pressure-density function is taken as with , and the viscosity coefficient is with . Note here that the case and in (1) corresponds to the viscous Saint-Venant system.

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 example, as the viscosity coefficients are both constants, the global existence of discontinuous solutions of one-dimensional Navier-Stokes equations was derived by Hoff [1–3]. Hoff investigated the construction of global spherically symmetric weak solutions of compressible Navier-Stokes equations for isothermal flow with large and discontinuous initial data [4]; therein it is also proved 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. The global existence theorems for the multidimensional Navier-Stokes equations of isothermal compressible flows with the polytropic equation of state were also showed by Hoff [5, 6]. Chen et al. obtained the global existence of weak solutions for the Navier-Stokes equations for compressible, heat-conducting flow in one space dimension with large, discontinuous initial data [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 also proved by Hoff [9].

If the viscosity coefficients , , 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) with , where the initial density is piecewise smooth with possibly large jump discontinuities [10]. Lian et al. considered the initial boundary value problem for (1) with subject to piecewise regular initial data with initial vacuum state included in [11]. Lian et al. also addressed the Cauchy problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient [12]; in these two cases above, they proved the global existence of unique piecewise regular solution and the finite time vanishing of vacuum state was proved in [11]. 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 as .

Recently, there are also many significant progresses achieved 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 [13, 14]. The good-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 was considered by many authors; refer to [15–22] 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 [24–28] and references therein.

In this present paper, we consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes equations and focus on the existence, regularities, and dynamical behaviors of global piecewise regular solution, and so forth. As , , we show that the free boundary value problem with piecewise regular initial data admits a unique global piecewise regular solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic, and the jump discontinuity of density also decays at an algebraic time-rate as (refer to Theorem 2 for details).

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. Main Results

We are interested in the global existence and dynamics of the free boundary value problem for (1) with following initial data and boundary conditions:

where is the free boundary defined by

Next, we give the definition of weak solution to the free boundary problem (1) and (2).

Definition 1 (weak solution). For any , is said to be a weak solution of the free boundary problems (1) and (2), if has the following regularities:

and (1) is satisfied in the sense of distributions. Namely, it holds for all that

and for all that

For simplicity, we consider the initial data in FBVP (1) and (2) with one discontinuous point ; namely, for some constant

and the compatibility conditions between initial data and boundary conditions hold.

We will give the global existence and time-asymptotic behavior of piecewise regular solution as follows.

Theorem 2. Let and . Assume that the initial data satisfies (7). Then, there exists a unique global piecewise regular solution to the FBVP (1) and (2) satisfying for

where is a curve defined by

along which the Rankine-Hugoniot conditions hold

where , and along the discontinuity the jump satisfies

where is a positive constant independent of time.

If it further holds that , then satisfies

The domain expands outwards at an algebraic rate in time as

and the density decays pointwise to zero for any and as

where and are positive constants independent of time, and is a small constant.

Remark 3. Theorem 2 holds for the Saint-Venant model for shallow water; that is, , .

Remark 4. Fang-Zhang [10] obtained that

which show that the discontinuity in the density persists for all time. However, in this paper, from (15) we can shows that the discontinuity in the density decays at an algebraic rate in time; namely,

where is a positive constant independent of time, and is a small constant.

3. The A Priori Estimates

According to the analysis made in [29], there is a curve defined by

along which the Rankine-Hugoniot conditions hold

where .

It is convenient to make use of the Lagrange coordinates in order to establish the uniformly a-priori estimates. Let be a piecewise regular solution to the FBVP (1) and (2), and take the Lagrange coordinates transform

Since the conservation of total mass holds

the boundaries and are transformed into and , respectively, and the domain is transformed into , the curve in the Eulerian coordinates is changed to a line in the Lagrangian coordinates, where

and the jump conditions become

Meanwhile, the FBVP (1) and (2) is reformulated into

where the initial data satisfies

for some constant , and the consistencies between initial data and boundary value hold.

Next, we will give the a-priori estimates for the solution to the FBVP (24). Similarly to the arguments used in [10, 16, 20], we can establish the following a priori estimates and omit the details here.

Lemma 5. Let . Under the assumptions of Theorem 2, it holds for any piecewise regular solution to the FBVP (24) that

for any positive integer , denotes a constant independent of time and denotes a constant dependent on time, where .

Lemma 6. Let . Under the assumptions of Theorem 2, it holds for any piecewise regular solution to the FBVP (24) that

Proof. From (24)₁ and (24)₃, we have

which yields (30) similarly, because of (23) and (24)₁, it holds that

which together with (25) implies

Lemma 7. Let . Under the assumptions of Theorem 2, it holds for any piecewise regular solution to the FBVP (24) that

where satisfies and , satisfies and .

Proof. Multiplying (24)₁ by gives

which leads to

Summing (24)₂ and (37), we have

Multiplying (38) by and integrating the result over , we get

which together with the fact that

gives rise to (35).

Remark 8. The estimate (35) can be written in the following form in the Eulerian coordinates; that is to say, for all ,

Lemma 9. Let , for , and . Under the assumptions of Theorem 2, it holds for any piecewise regular solution to the FBVP (24) that

Proof. It follows from (24)1,2 that

By means of (25), (26), and (44), we have

where we have used

which can be deduced from (26) and (27). Making use of Gronwall's inequality to (46), we obtain (43).

Lemma 10. Let . Under the assumptions of Theorem 2, it holds for any piecewise regular solution to the FBVP (24) that

Proof. Denote

By (24)₁, we have

Multiplying (49) by , integrating the result over , and using (43), (44), we can obtain that for

where we use the fact that

Since it holds that

we have from (25) that

Substituting (53) in (50), we have

Using Gronwall's inequality, we get from (54) that

It follows from (29) and (55) that

as , ; for some large enough, we have

which implies (47).

We also have the regularity estimates for the solution to the FBVP (24) as follows.

Lemma 11. Let . Under the assumptions of Theorem 2, it holds for any piecewise regular solution to the FBVP (24) that

If it is also satisfied that

then the piecewise regular solution has the regularities

Proof. Multiplying (24)₂ by , integrating the result over , and making use of the boundary condition(24)₃, after a direct computation and recombination, we deduce

where we have used (26) and (27). On the other hand, integrating (24)₂ over and making use of (23) and the boundary conditions (24)₃, it holds that

which implies

It holds from (61) and (63) that

using Gronwall's inequality, (26), and (47), we have

where denotes a constant dependent of time.

Differentiating (24)₂ with respect to , we get

Taking inner product between (66) and , integrating the results over , and using the boundary conditions (24)₃, it holds that

The terms on the right-hand side of (67) can be bounded, respectively,as described below:

Summing (67) and (68) together and making use of (27) and (65), we obtain

Substituting (62) into (69), it follows from (27), (47), and (59) that

which together with Gronwall's inequality, (27), (47), and (65) yields

which implies , and it follows from the definition of that . The proof of this Lemma is completed.

Lemma 12. Let . Under the assumptions of Theorem 2, it holds for any piecewise regular solution to the FBVP (24) that

where is a positive constant independent of time.

Proof. From (27) and (34), we can obtain (72).

Finally, we will give the large time behaviors of the interface and decay rate of the density as follows.

Lemma 13. Let be any piecewise regular solution to the FBVP (1) and (2). Under the assumptions of Theorem 2, it holds for and time large enough that

and the density decays pointwise to zero for any and as

where and are positive constants independent of time and is a small constant.

Proof. We introduce the following functional in the Eulerian form as [22, 28]:

Differentiating (76) with respect to , using (1), (2), and , we have

Combining (77), we deduce

where we use the fact that as becomes large enough; it holds from (31) and

that

If , we have from (78) and the conservation of mass that

Hence, it holds that

From (30) and (35), we obtain

and then

which with (83) implies

If , we deduce from (76), (78), and the conservation of mass that

to which the application of Gronwall's inequality gives

We get from (84) that

which together with (89) yields

Also, it follows from the conservation of mass and Hölder's inequality that

which together with (84), (86), and (91) gives (73). Similarly, we have

which together with (35) implies that

Finally, it follows from (30) that

and as is large enough, it holds due to (35) that

which together with (86), (91), (95), and (96) gives (75).

4. Proof of the Main Results

Proof. The global existence of unique piecewise regular solution to the FBVP (1) and (2) can be established in terms of the short-time existence carried out as in [1, 2, 4], the uniform a-priori estimates, and the analysis of regularities, which indeed follow from Lemmas 5–11. We omit the details. The large time behaviors follow from Lemmas 12–13 directly. The proof of Theorem 2 is completed.

Acknowledgments

The authors thank the referee for the helpful comments and suggestions on the paper. 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. The research of G. J. Zhang is supported by NSF no. GFQQ2460101710.

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