Free Boundary Value Problem for the One-Dimensional Compressible Navier-Stokes Equations with a Nonconstant Exterior Pressure
We consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes (CNS) equations with density-dependent viscosity coefficient in the case that across the free surface stress tensor is balanced by a nonconstant exterior pressure. Under certain assumptions imposed on the initial data and exterior pressure, we prove that there exists a unique global strong solution which is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic time-rate.
We will investigate the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient for regular initial data in the case that across the free surface stress tensor is balanced by a nonconstant exterior pressure in the present paper. In general, the one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient read as where , , and stand for the flow density, velocity, and pressure, respectively, and the viscosity coefficient is with . We note here that as and in (1), the system corresponds to the viscous Saint-Venant system for shallow water.
There is huge literature on the studies of the compressible Navier-Stokes equations with density-dependent viscosity coefficients. For example, the mathematical derivations are achieved in the simulation of flow surface in shallow region [1, 2]. Bresch and Desjardins have investigated the existence of solutions to the 2D shallow water equations in [3, 4]. The global existence of classical solutions is proven by Mellet and Vasseur . The qualitative patterns of behavior of global solutions and dynamical asymptotics of vacuum states are also shown, such as the finite time vanishing of finite vacuum or asymptotical formation of vacuum in large time, the dynamical behavior of vacuum boundary, the large time convergence to rarefaction wave with vacuum, and the stability of shock profile with large shock strength; refer to [6–11] and references therein.
Recently, there is much significant progress achieved on the free boundary value problems; for instance, 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 considered by many authors; refer to [12–24] and references therein. In addition, the free boundary value problems for multidimensional compressible viscous Navier-Stokes equations with constant viscosity coefficients for either barotropic or heat-conducive fluids are investigated by many authors, such as in the case that across the free surface stress tensor is balanced by a constant exterior pressure and/or the surface tension; classical solutions with strictly positive densities in the fluid regions to FBVP for CNS (1) with constant viscosity coefficients are shown locally in time for either barotropic flows [25–27] or heat-conductive flows [28–30]. In the case that across the free surface the stress tensor is balanced by exterior pressure , surface tension , or both exterior pressure and surface tension , respectively, as the initial data is assumed to be near to a nonvacuum equilibrium state, the global existence of classical solutions with small amplitude and positive densities in fluid region to the FBVP for CNS (1) with constant viscosity coefficients is proved. Global existence of classical solutions to FBVP for compressible viscous and heat-conductive fluids is also obtained with the stress tensor balanced by the exterior pressure and/or surface tension across the free surface; refer to [33, 34] and references therein.
In the present paper, we focus on the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and a nonconstant exterior pressure, and the existence, regularities, and dynamical behavior of global strong solution will be addressed, and so forth. As , , we show that the free boundary value problem with regular initial data admits a unique global strong solution which is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic time-rate (refer to Theorem 1 for details).
The rest of the paper is arranged as follows. In Section 2, the main results about the existence and dynamical behavior of global strong solution to FBVP for compressible Navier-Stokes equations are stated. Then, some important a priori estimates will be given in Section 3 and the theorem is proven in Section 4.
2. Main Results
We will investigate the global existence and dynamics of the free boundary value problem for (1) with the following initial data and boundary conditions: where and are the free boundaries defined by and the function is the nonconstant exterior pressure.
Without the loss of generality, the total initial mass is renormalized to be one; that is, And we consider that the initial data satisfies where is a positive constant and ; note that the compatibility conditions between initial data and boundary conditions hold. Then, we have the global existence and time-asymptotical behavior of strong solution as follows.
Theorem 1 (FBVP). Let and . Assume that the initial data satisfies (5); ; is a constant; and uniformly for . Then, there exists a unique global strong solution to the FBVP (1) and (2) satisfying
with being a constant independent of time.
If it further holds that , then satisfies The domain expands outwards in time as where denotes a positive constant, and the density decays pointwise to zero for any and as where is a positive constant.
Remark 2. Theorem 1 holds for one-dimensional Saint-Venant model for shallow water; that is, , .
Remark 3. Equation (8) implies that as time goes to infinity, both the lower bound rate and the upper bound rate go to infinity.
Remark 4. In fact, we can choose the nonconstant exterior pressure like these or the linear combinations of these functions, and so forth.
Remark 5. In particular, let ; from (8), we have which implies that the upper bound rate of expands at an exponential rate; however, we proved that the upper bound rate of expands at an algebraic rate in  where we consider the free boundary value problem without the nonconstant exterior pressure.
3. The A Priori Estimates
Making use of the Lagrange coordinates, we can establish some a priori estimates. Define 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 FBVP (1) and (2) is reformulated into where the initial data satisfies and the consistencies between initial data and boundary conditions hold.
Next, we will deduce the a priori estimates for the solution to the FBVP (13).
Proof. From (13), we can find which imply that
Proof. Taking the product of (13)2 with , integrating on , and using boundary conditions, we have which leads to (20) after the integration with respect to , where we use the fact that, from (15), the domain expands as the time grows up, and it holds that
Proof. Multiplying (13)1 by gives which leads to Summing (13)1 and (25), we deduce Multiplying (26) by and integrating the result over , we obtain where we have the fact that which together with (15) and gives rise to (23).
Lemma 9. Let . Under the assumptions of Theorem 1, it holds that where is the positive constant independent of time.
Proof. Integrating (24) with respect to over , we know then integrating (13)2 over and using the boundary conditions, we have It holds from (31) and (32) that where denotes the positive constant independent of time.
Proof. Integrating (26) with respect to over , we have which together with (20) gives where we have used which can be deduced from (20), (23), (30), and (34). Making use of Gronwall’s inequality to (40), we obtain (37).
Proof. Define that By (13)1, we have Multiplying (43) by , where , integrating the result over , and using (37) and (38), we can find that Since it holds that which implies we have from (20) and (23) that Using the same method, we also have Substituting (47) and (48) into (44), we get which together with Gronwall’s inequality yields We have from (23), (46), and (50) that which implies
We also have the regularity estimates for the solution to the FBVP (13) as follows.
Proof. Multiplying (13)2 by , integrating the result over , and making use of the boundary conditions, after a direct computation and recombination, we find
Integrating (56) over , from (20), (23), (30), (41), , and , it is easily verified that
where denotes a constant dependent on time. From (13)2, (20), (23), (30), and (41), it holds that
Using (58), we can obtain that
which together with (13)2 implies
Differentiating (13)2 with respect to , we get Taking product between (61) and , integrating the results over , and using the boundary conditions (13)4,5, we have The terms on the right-hand side of (62) can be bounded, respectively, as described below: Summing (62)-(63) together and making use of (30) and (60), we have Integrating (64) over , it holds from (13)2, (20), and (60) that which gives which implies , and it follows from the definition of and that , . The proof of this lemma is completed.
Finally, we will give the large time behavior of the strong solution as follows.
Lemma 14. Let . Under the assumptions of Theorem 1, it holds for and time large enough that where denotes a positive constant, and the density decays pointwise to zero for any and as where is a positive constant.
Proof. From (13) we can find that
and, without loss of generality, we can renormalize to be zero; then, we denote
Applying (69), we can obtain
then the system (13) becomes
Multiplying (73)2 by and integrating the result over , after a straightforward calculation, we have
and as it holds from (71) that
which together with (75) leads to
multiplying (77) by for some to be determined later, we have
and we will prove the fact that as time is large enough, it holds that
where is a positive constant independent of time and we can find that (81) is true as we assume , where is some positive constant. Then, as , integrating (78) over , we have
As and , it holds from (75) that integrating (83) over and using we have
From (82) and (85), we know and it holds from the conservation of the mass that which gives Using (15), (17), and (23), we know and we can choose the positive constant such that as time is large enough, it holds that
Finally, we will prove the decay rate of the density, and it holds from the Gagliardo-Nirenberg-Sobolev inequality, (23), and (30) that which together with (86) gives (68).
4. Proof of the Main Results
Proof. The global existence of unique strong solution to the FBVP (1) and (2) can be established in terms of the short time existence carried out as in , the uniform a priori estimates, and the analysis of regularities, which indeed follow from Lemmas 6–13. We omit the details. The large time behavior follows from Lemma 14 directly. The proof of Theorem 1 is completed.
Conflict of Interests
The authors declare that they have no conflict of interests.
Both authors contributed to each part of this work equally.
The authors thank the referee for the helpful comments and suggestions on the paper. The research of Ruxu 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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