Abstract
In this paper, we consider the free boundary value problem (FBVP) for onedimensional isentropic compressible NavierStokes equations (CNS) with densitydependent 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 nonvacuum equilibrium state at an exponential timerate as the time tends to infinity.
Keywords:
NavierStokes equations; free boundary value problem; densitydependent viscosity coefficient; exterior pressure; strong solution1 Introduction
In the present paper, we consider the free boundary value problem to onedimensional isentropic compressible NavierStokes equations with densitydependent viscosity coefficient for regular initial data in the case that across the free surface the stress tensor is balanced by constant exterior pressure. In general, onedimensional isentropic compressible NavierStokes equations with densitydependent viscosity coefficient can be written as
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 SaintVenant system for shallow water.
Recently, there have been much significant progress achieved on the compressible NavierStokes equations with densitydependent 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 wellposedness 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 [515] and the references therein. The global existence of classical solutions is shown by Mellet and Vasseur [16]. 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 [1722] and the references therein.
In addition, some important progress has been made about free boundary value problems for multidimensional compressible viscous NavierStokes equations with constant viscosity coefficients for either barotropic or heatconducive 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 [2325] or heatconductive flows [2628]. In the case that across the free surface the stress tensor is balanced by exterior pressure [25], surface tension [29], or both exterior pressure and surface tension [30], 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 (1.1) with constant viscosity coefficients is obtained. The global existence of classical solutions to FBVP for compressible viscous and heatconductive 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 onedimensional isentropic compressible NavierStokes equations with densitydependent 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 nonvacuum equilibrium state at an exponential timerate 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 NavierStokes 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
We are interested in the global existence and dynamics of the free boundary value problem for (1.1) with the following initial data and boundary conditions:
where and are the free boundaries defined by
and the positive constant is the exterior pressure.
Without loss of generality, the total initial mass is renormalized to be one, i.e.,
Define
and
Firstly, we consider the initial data satisfying
where and are positive constants, δ is bounded, and we also consider the other initial data satisfying
note that the compatibility conditions between the initial data and boundary conditions hold. Then we have the global existence and timeasymptotical behavior of a strong solution as follows.
Theorem 2.1 (FBVP)
Let. Assume that the initial data satisfies (2.6) for, and satisfies (2.6) together with
foror the initial data satisfies (2.7) for, and satisfies (2.7) together with
for, whereνis a positive constant. Then there exists a unique global strong solutionto FBVP (1.1) and (2.1) satisfying, for,
withbeing a constant independent of time.
If it further holds that, thensatisfies
The solutiontends to the nonvacuum equilibrium state exponentially
whereandare positive constants independent of time.
Remark 2.1 Theorem 2.1 holds for the onedimensional SaintVenant 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
It is convenient to make use of the Lagrange coordinates in order to establish the a priori estimates. 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 . FBVP (1.1) and (2.1) is reformulated into
where the initial data satisfies
or
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).
Lemma 3.1Let. Under the assumptions of Theorem 2.1, it holds for any strong solutionto FBVP (3.2) that
Proof Taking the product of (3.2)_{2} with u, integrating on and using boundary conditions, we have
which leads to (3.5) after the integration with respect to . □
Lemma 3.2Let. Under the assumptions of Theorem 2.1, it holds for any strong solutionto FBVP (3.2) with the initial data satisfying (3.3) that
As the initial data satisfies (3.4), it holds
wheresatisfiesand, satisfiesand.
Proof Multiplying (3.2)_{1} by leads to
which gives
Summing (3.2)_{1} and (3.10), we deduce
Multiplying (3.11) by and integrating the result over , we have
Next, we prove (3.7) in the case that the initial data satisfies (3.3) firstly, to obtain (3.7). We assume a priori that there are constants so that
By means of (3.2)_{1} and the boundary condition, we have
which yields
From (3.13), we can obtain
It holds from (3.5), (3.13), (3.15) and (3.16) that
where is a positive constant independent of time, and we assume that
Using the same method we can obtain
From (3.12), (3.17) and (3.19), we have (3.7).
Then, we prove (3.8) as the initial data satisfies (3.4), where we have the fact
Applying equation (3.2)_{1} and the boundary condition, we have that
and
which together with
and
gives rise to (3.8). □
Lemma 3.3Let. Under the assumptions of Theorem 2.1, it holds
whereandare positive constants independent of time.
Proof Define
and
It is easy to verify that and . In addition, it holds as that
As the initial data satisfies (3.3), it follows from (3.5) and (3.7) that
where and ν are positive constants independent of time, and we assume that
As and as , from the condition
we can find that there are two positive constants and independent of time and choose
such that
As the initial data satisfies (3.4), it follows from (3.5) and (3.8) that
where ν is a positive constant independent of time, and we use the fact that
which together with Young’s inequality gives
where C is a positive constant independent of time. As and as , by means of the condition
we can obtain two positive constants and independent of time such that
The proof of this lemma is completed. □
We also have the regularity estimates for the solution to FBVP (3.2) as follows.
Lemma 3.4Let. Under the assumptions of Theorem 2.1, it holds for any strong solutionto FBVP (3.2) that
If it is also satisfied that
then the strong solutionhas the regularities
Proof Multiplying (3.2)_{2} by , integrating the result over and making use of the boundary conditions, after a direct computation and recombination, we have
Integrating equation (3.44) over , from (3.5), (3.7), (3.8) and (3.26), it is easily verified that
where C denotes a constant independent of time. From (3.2)_{2}, (3.5), (3.7), (3.8) and (3.26), we deduce
Using (3.46), we can obtain that
which together with (3.2)_{2} implies
Differentiating (3.2)_{2} with respect to τ, we get
Taking product between (3.49) and , integrating the results over and using the boundary conditions (3.2)_{4}, we have
The terms on the righthand side of (3.50) can be bounded respectively as follows:
and
Summing (3.50)(3.52) together and making use of (3.26) and (3.48), we obtain
Integrating equation (3.53) over , we get from (3.2)_{2}, (3.5) and (3.26) that
which gives
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.
Lemma 3.5Let. Under the assumptions of Theorem 2.1, it holds for any strong solutionto FBVP (3.2) that
whereanddenote two positive constants independent of time.
Proof Applying (3.6) and (3.12) with modification, we can obtain
and
We have from (3.5), (3.7), (3.26), the GagliardoNirenbergSobolev inequality and Young’s inequality that
and define
As the initial data satisfies (3.3), we have
where C and are positive constants independent of time. By (3.57)(3.62), a complicated computation gives rise to
where is a positive constant independent of time. From (3.63), we have
As the initial data satisfies (3.4), from (3.22)(3.24), we obtain
which implies that the domain expands as t grows up, so that we have
Then it holds from (3.65) and (3.66) that
Using (3.58), after a complicated computation, we have
where is a positive constant independent of time, and we have
where C is a positive constant independent of time.
By the fact
where is a constant independent of time, and the GagliardoNirenbergSobolev inequality
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 [7], the uniform a priori estimates and the analysis of regularities, which indeed follow from Lemmas 3.13.4. We omit the details. The large time behaviors follow from Lemma 3.5 directly. The proof of Theorem 2.1 is completed. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to each part of this work equally. All authors read and approved the final manuscript.
Acknowledgements
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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