On the free boundary value problem for one-dimensional compressible Navier-Stokes equations with constant exterior pressure
© Lian and Liu; licensee Springer. 2014
Received: 8 January 2014
Accepted: 2 April 2014
Published: 6 May 2014
In this paper, we consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficient and constant exterior pressure. Under certain assumptions imposed on the initial data, the global existence and uniqueness of a strong solution to FBVP for CNS are established, in particular, the strong solution tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.
where , u and () stand for the flow density, velocity and pressure, respectively, and the viscosity coefficient is with . Note here that the case and in (1.1) corresponds to the viscous Saint-Venant system for shallow water.
Recently, there have been much significant progress 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 [1, 2]. The existence of solutions for the 2D shallow water equations is investigated by Bresch and Desjardins [3, 4]. 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 [5–15] and the references therein. The global existence of classical solutions is shown by Mellet and Vasseur . 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 the 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 [17–22] and the references therein.
In addition, some important progress has been made about free boundary value problems for multi-dimensional compressible viscous Navier-Stokes equations with constant viscosity coefficients for either barotropic or heat-conducive fluids by many authors; for example, 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 (1.1) with constant viscosity coefficients are proved locally in time for either barotropic flows [23–25] or heat-conductive flows [26–28]. 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 non-vacuum equilibrium state, the global existence of classical solutions with small amplitude and positive densities in fluid region to the FBVP for CNS (1.1) with constant viscosity coefficients is obtained. The global existence of classical solutions to FBVP for compressible viscous and heat-conductive fluids is also established with the stress tensor balanced by the exterior pressure and/or surface tension across the free surface; refer to [31, 32] and the references therein.
In this paper, we consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and constant exterior pressure, and we focus on the existence, regularities and dynamical behaviors of a global strong solution, etc. As we show that the free boundary value problem with regular initial data admits a unique global strong solution which tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity (refer to Theorem 2.1 for details).
The rest of the paper is arranged as follows. In Section 2, the main results about the existence and dynamical behaviors of a global strong solution to FBVP with two different initial data for compressible Navier-Stokes equations are stated. Then, some important a priori estimates are given in Section 3 and the theorem is proven in Section 4.
2 Main results
and the positive constant is the exterior pressure.
note that the compatibility conditions between the initial data and boundary conditions hold. Then we have the global existence and time-asymptotical behavior of a strong solution as follows.
Theorem 2.1 (FBVP)
with being a constant independent of time.
where and are positive constants independent of time.
Remark 2.1 Theorem 2.1 holds for the one-dimensional Saint-Venant model for shallow water, i.e., , .
Remark 2.2 The initial constraints (2.8) and (2.9) do not always require that the perturbation of the initial data around the equilibrium state is small. Indeed, it can be large provided that the state is large enough.
3 The a priori estimates
and the consistencies between the initial data and boundary conditions hold.
Next, we deduce the a priori estimates for the solution to FBVP (3.2).
which leads to (3.5) after the integration with respect to . □
where satisfies and , satisfies and .
From (3.12), (3.17) and (3.19), we have (3.7).
gives rise to (3.8). □
where and are positive constants independent of time.
The proof of this lemma is completed. □
We also have the regularity estimates for the solution to FBVP (3.2) as follows.
which implies , and it follows from the definition of and that . The proof of this lemma is completed. □
Finally, we give the large time behaviors of the strong solution as follows.
where and denote two positive constants independent of time.
where C is a positive constant independent of time.
we can deduce (3.56). □
4 Proof of the main results
Proof The global existence of a unique strong solution to FBVP (1.1) and (2.1) 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 3.1-3.4. We omit the details. The large time behaviors follow from Lemma 3.5 directly. The proof of Theorem 2.1 is completed. □
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, Innovation Scientists and Technicians Troop Construction Projects of Henan Province. The research of Jian Liu is supported by NNSFC No. 11326140, the Doctoral Starting up Foundation of QuZhou University No. BSYJ201314.
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