Free boundary value problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity

Abstract

We study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in this paper. Under certain assumptions imposed on the initial data, we show that there exists a unique global strong 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 also at an algebraic time-rate as the time tends to infinity.

Keywords:

• Navier-Stokes equations
• free boundary value problem
• density- dependent viscosity coefficient
• strong solution

AMS Subject Classifications:

• 35Q35
• 76D03

Acknowledgement

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.