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.


Introduction
We will investigate the free boundary value problem for onedimensional 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 densitydependent viscosity coefficient read as   + ()  = 0, ()  + ( 2 +  ())  = ( ()   )  , (, ) ∈  × [0, ] , where  > 0, , and () =   ( > 1) stand for the flow density, velocity, and pressure, respectively, and the viscosity coefficient is () =   with  > 0. We note here that as  = 2 and  = 1 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 [5].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][7][8][9][10][11] and references therein.
Recently, there is much significant progress achieved on the free boundary value problems; for instance, 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 [12][13][14][15][16][17][18][19][20][21][22][23][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 2 Journal of Applied Mathematics 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][26][27] or heat-conductive flows [28][29][30].In the case that across the free surface the stress tensor is balanced by exterior pressure [27], surface tension [31], or both exterior pressure and surface tension [32], 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  > 1, 0 <  ≤ 1, 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.

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   () > 0 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 inf where  is a positive constant and  0 :=   (0); note that the compatibility conditions between initial data and boundary conditions hold.Then, we have the global existence and timeasymptotical behavior of strong solution as follows.
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 [16] where we consider the free boundary value problem without the nonconstant exterior pressure.

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  = 0 and  = 1, respectively, and the domain [(), ()] is transformed into [0, 1].The FBVP (1) and ( 2) is reformulated into where the initial data satisfies inf and the consistencies between initial data and boundary conditions hold.
Lemma 9. Let  > 0. Under the assumptions of Theorem 1, it holds that where  is the positive constant independent of time.
Proof.Integrating (24) with respect to  over [0, ], we know then integrating (13) 2 over [0, ] × [0, ] and using the boundary conditions, we have It holds from (31) where  denotes the positive constant independent of time.
Lemma 10.Let  > 0. Under the assumptions of Theorem 1, it holds for any strong solution (, ) to the FBVP (13) that for any positive integer  ∈ , and () > 0 denotes a constant dependent on time.
Finally, we will give the large time behavior of the strong solution as follows.
Lemma 14.Let  > 0. Under the assumptions of Theorem 1, it holds for  ∈ (0, 1] and time  large enough that where 0 <  ≤ −1 denotes a positive constant, and the density decays pointwise to zero for any  ∈ [(), ()] and  > 0 as where  > 0 is a positive constant.
Proof.From ( 13) we can find that and, without loss of generality, we can renormalize ∫ 1 0  0 () 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 [0, 1], after a straightforward calculation, we have which implies where  > 0 is a positive constant independent of time and we can find that (81) is true as we assume   () = ((1 + ) ] ), where ] is some positive constant.Then, as 0 <  ≤ min{ − 1, }, integrating (78) over [0, ], we have