Global strong solutions of the nonisentropic Navier-Stokes/Allen-Cahn system with density-dependent viscosity

In this paper, we study a diffuse interface model for the compressible immiscible two-phase flows. The model is a nonlinear coupling of the compressible non-isentropic Navier-Stokes equations and the modified Allen-Cahn equations. We are interested in the case of density-dependent viscosity $ \eta(\rho) $ and temperature-dependent heat conductivity $ \kappa(\theta) $. The global existences of unique strong solutions to 1D initial boundary problem are obtained, when $ \eta = \rho^\alpha $, $ \kappa = \theta^\beta $ with $ 0\le\alpha<\frac14 $, $ \beta\ge1 $, and $ \eta = 1+\rho^\alpha $, $ \kappa = \theta^\beta $ with $ \alpha\ge0 $, $ \beta>0 $, respectively.