On the free boundary value problem for one-dimensional compressible Navier-Stokes equations with constant exterior pressure

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.

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 regular initial data in the case that across the free surface the stress tensor is balanced by constant exterior pressure. In general, one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient can be written as

where \rho >0, u and P(\rho )={\rho }^{\gamma } (\gamma >1) stand for the flow density, velocity and pressure, respectively, and the viscosity coefficient is \mu (\rho )={\rho }^{\alpha } with \alpha >0. Note here that the case \gamma =2 and \alpha =1 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 [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 [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 [25], surface tension [29], or both exterior pressure and surface tension [30], 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 \gamma >1 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.

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 x=a(t) and x=b(t) are the free boundaries defined by

and the positive constant p_e is the exterior pressure.

Without loss of generality, the total initial mass is renormalized to be one, i.e.,

Define

and

where \overline{\rho }:=p_e^{\frac{1}{\gamma }}>0.

Firstly, we consider the initial data satisfying

where \underline{\rho } and \delta >0 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 time-asymptotical behavior of a strong solution as follows.

Theorem 2.1 (FBVP)

Let \gamma >1. Assume that the initial data satisfies (2.6) for , and satisfies (2.6) together with

for \alpha >\frac{1}{2} or the initial data satisfies (2.7) for , and satisfies (2.7) together with

for , where ν is a positive constant. Then there exists a unique global strong solution (\rho ,u,a,b) to FBVP (1.1) and (2.1) satisfying, for T>0,

with c>0 being a constant independent of time.

If it further holds that u_0\in H^2([a_0,b_0]), then (\rho ,u,a,b) satisfies

The solution (\rho ,u) tends to the non-vacuum equilibrium state exponentially

where C_1 and C_2 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., \gamma =2, \alpha =1.

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 (\overline{\rho },0) is small. Indeed, it can be large provided that the state \overline{\rho }>0 is large enough.

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 x=a(t) and x=b(t) are transformed into \xi =0 and \xi =1, respectively, and the domain [a(t),b(t)] is transformed into [0,1]. 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 (\rho ,u) to FBVP (3.2).

Lemma 3.1 Let T>0. Under the assumptions of Theorem  2.1, it holds for any strong solution (\rho ,u) to FBVP (3.2) that

Proof Taking the product of (3.2)₂ with u, integrating on [0,1] and using boundary conditions, we have

which leads to (3.5) after the integration with respect to \tau \in [0,T]. □

Lemma 3.2 Let T>0. Under the assumptions of Theorem  2.1, it holds for any strong solution (\rho ,u) to FBVP (3.2) with the initial data satisfying (3.3) that

As the initial data satisfies (3.4), it holds

where a(\tau ) satisfies a^{\prime }(\tau )=u(0,\tau ) and a(0)=a_0, b(\tau ) satisfies b^{\prime }(\tau )=u(1,\tau ) and b(0)=b_0.

Proof Multiplying (3.2)₁ by {\rho }^{\alpha -1} leads to

which gives

Summing (3.2)₁ and (3.10), we deduce

Multiplying (3.11) by (u+{({\rho }^{\alpha })}_{\xi }/\alpha ) and integrating the result over [0,1]\times [0,\tau ], 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 {\rho }_{\pm }>0 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 \hat{C} 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 3.3 Let T>0. Under the assumptions of Theorem  2.1, it holds

where {\rho }_{\ast } and {\rho }^{\ast } are positive constants independent of time.

Proof Define

and

It is easy to verify that \phi (\rho )\ge 0 and {\psi }^{\prime }(\rho )\ge 0. In addition, it holds as \rho \to +\mathrm{\infty } that

and as \rho \to 0 that

As the initial data satisfies (3.3), it follows from (3.5) and (3.7) that

where \tilde{C} and ν are positive constants independent of time, and we assume that

As and as \alpha >\frac{1}{2}, from the condition

we can find that there are two positive constants {\rho }_{\ast } and {\rho }^{\ast } 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 {\rho }_{\ast } and {\rho }^{\ast } independent of time such that

The proof of this lemma is completed. □

We also have the regularity estimates for the solution (\rho ,u) to FBVP (3.2) as follows.

Lemma 3.4 Let T>0. Under the assumptions of Theorem  2.1, it holds for any strong solution (\rho ,u) to FBVP (3.2) that

If it is also satisfied that

then the strong solution (\rho ,u) has the regularities

Proof Multiplying (3.2)₂ by {\rho }^{-(1+\alpha )}{u}_{\tau }, integrating the result over [0,1] and making use of the boundary conditions, after a direct computation and recombination, we have

Integrating equation (3.44) over [0,\tau ], 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 u_{\tau }, integrating the results over [0,1] and using the boundary conditions (3.2)₄, we have

The terms on the right-hand 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