Global solutions to 1D compressible Navier-Stokes/Allen-Cahn system with density-dependent viscosity and free-boundary

Abstract

This paper is concerned with the Navier-Stokes/Allen-Cahn system, which is used to model the dynamics of immiscible two-phase flows. We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form of η(ρ) = ρ^α. The existence of unique global H^2m-solutions (m ∈ ℕ) to the free boundary problem is proven for when \(0 < \alpha < {1 \over 4}\). Furthermore, we obtain the global C^∞-solutions if the initial data is smooth.