Free boundary value problem of one dimensional two-phase liquid-gas model

In this paper, we study a free boundary value problem for two-phase liquid-gas model with mass-dependent viscosity coefficient when both the initial liquid and gas masses connect to vacuum continuously. The gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid. We give the proof of the global existence and uniqueness of weak solutions when β ∈ (0, 1), which have improved the result of Evje and Karlsen, and we obtain the regularity of the solutions by energy method.