One-dimensional Compressible Navier-stokes Equations Posedness

We consider the initial-boundary-value problems for 1D compressible isentropicNavier-Stokes system with density-dependent viscosity coefficients and dispersion(surface tension):Whereρ(x,t), u(x,t) and P(ρ) =ρ^γ(γ＞1) stand for the density , velocity and pressure respectively. For simplicity, we setν= 1 , and the viscosity coefficientμ(ρ) is assumed to beμ(p) =ρ^α, with 0＜α＜(?).First , if the initial data (ρ₀,u₀) satisfiesρ₀∈H²([0,1]), u₀∈H¹([0, l]), the existence and uniqueness of the global solutions to IBVP (*1) are proved. As the viscosity coefficient depends the density, the main difficulty in studying the problem is related to obtaining a priori estimates for density from below, thus we need a lot of accurate estimates. Furthermore, for smooth initial data,there is a global smooth solution. Then , the behavier of the strong solutions at an infinitely growing time is also considered, we can show that the strong solution tends to the non-vacuum equilibrium state exponentially in time.