Navier-Stokes Equations With Degenerate Viscosity, Vacuum And Gravitational Force

In this paper, we study the one-dimensional compressible Navier-Stokes equations for isentropic flow with gravitational force and fixed boundary condition. We are interested in the case that the gas is in contact with the vacuum continuously when viscosity coefficient depends on the density. Precisely, the viscosity coefficientμis proportional toρ^θand 0 <θ<1/2. By giving a series of a priori estimates on the solution coping with the degeneracy of vacuum, gravitational force and boundary effect, we give global existence and uniqueness.The main result of this paper is:Theorem 2.2 (Existence). Under the conditions (A1)-(A3), the free boundary problem (2.1)-(2.4) has a weak solution (p(x, t),u(x, t)) which satisfies Definition 2.1 and p{x,t) satisfieswhere k₂ =( k₁)/2+ v.Theorem 2.4 (Uniqueness). Under the conditions (A1)-(A3) and k₂ <(20/21) (1-2θ)(α/θ)^1/2, let (ρ₁,u₁)(x,t) and (ρ2,u2)(x,t) be two weak solutions to the initial-boundary value problem (2.1)-(2.4) in 0≤t≤T as described in Definition 2.1. Then (ρ1,u1)(x,t) = (ρ2,u2)(x,t) a.e. in (x,t)∈[0,1]×[0,T].