Free Boundary Value Problem for the One-Dimensional Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous Initial Data

Ruxu Lian,Guojing Zhang

doi:10.1155/2013/505108

Abstract

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.

Highlights

We consider the free boundary value problem to one-dimensional isentropic compressible NavierStokes equations with density-dependent viscosity coefficient for piecewise regular initial data connected with the infinite vacuum via jump discontinuity
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 [1,2,3]
As γ > 1, 0 < α ≤ 1, we show that the free boundary value problem with piecewise regular initial data admits 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, and the jump discontinuity of density decays at an algebraic time-rate as t → +∞

Summary

Introduction

We consider the free boundary value problem to one-dimensional isentropic compressible NavierStokes equations with density-dependent viscosity coefficient for piecewise regular initial data connected with the infinite vacuum via jump discontinuity. The qualitative behaviors of global solutions and dynamical asymptotics of vacuum states are made, such as the finite time vanishing of finite vacuum or asymptotical formation of vacuum in longtime, the dynamical behaviors of vacuum boundary, the longtime convergence to rarefaction wave with vacuum, and the stability of shock profile with large shock strength; refer to [24,25,26,27,28] and references therein 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.

Main Results

The A Priori Estimates

Proof of the Main Results