^{1,2}

^{3}

^{1}

^{2}

^{3}

We study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and discontinuous initial data in this paper. For piecewise regular initial density, we show that there exists a unique global piecewise regular solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic time-rate, and the jump discontinuity of density also decays at an algebraic 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 piecewise regular initial data connected with the infinite vacuum via jump discontinuity. In general, the one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient read

There is huge literature on the studies of the global existence of weak solutions and dynamical behaviors of jump discontinuity for the compressible Navier-Stokes equations with discontinuous initial data; for example, as the viscosity coefficients are both constants, the global existence of discontinuous solutions of one-dimensional Navier-Stokes equations was derived by Hoff [

If the viscosity coefficients

Recently, there are also many significant progresses 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 [

In this present paper, we consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes equations and focus on the existence, regularities, and dynamical behaviors of global piecewise regular solution, and so forth. As

The rest part of the paper is arranged as follows. In Section

We are interested in the global existence and dynamics of the free boundary value problem for (

Next, we give the definition of weak solution to the free boundary problem (

For any

For simplicity, we consider the initial data in FBVP (

We will give the global existence and time-asymptotic behavior of piecewise regular solution as follows.

Let

If it further holds that

Theorem

Fang-Zhang [

According to the analysis made in [

It is convenient to make use of the Lagrange coordinates in order to establish the uniformly a-priori estimates. Let

Meanwhile, the FBVP (

Next, we will give the a-priori estimates for the solution

Let

Let

From (_{1} and (_{3}, we have
_{1}, it holds that

Let

Multiplying (_{1} by _{2} and (

The estimate (

Let

It follows from (_{1,2} that

Let

Denote
_{1}, we have

We also have the regularity estimates for the solution

Let

If it is also satisfied that

Multiplying (_{2} by _{3}, after a direct computation and recombination, we deduce
_{2} over _{3}, it holds that

Differentiating (_{2} with respect to _{3}, it holds that

Let

From (

Finally, we will give the large time behaviors of the interface and decay rate of the density as follows.

Let

We introduce the following functional

If

If

Also, it follows from the conservation of mass and Hölder's inequality that

Finally, it follows from (

The global existence of unique piecewise regular solution to the FBVP (

The authors thank the referee for the helpful comments and suggestions on the paper. The research of R. X. 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. The research of G. J. Zhang is supported by NSF no. GFQQ2460101710.