Global strong/classical solutions to the one-dimensional compressible Navier-Stokes-Allen-Cahn system with density-dependent viscosity

This paper is concerned with a one-dimensional isentropic compressible Navier-Stokes-Allen-Cahn system with density-dependent viscosity, which models the motion of a mixture of two viscous compressible fluids. The case when the pressure p () = ^, the viscosity (, ) = ^, the interface thickness () = ^ and the relaxation time function a (, , ᵧ) = ^ is considered, where and are the density and the phase variable, respectively, and, , , are parameters. Under some suitable assumptions on the parameters, , , and the initial data, we prove the global existence and large-time behavior of nonvacuum strong and classical solutions to its Cauchy problem with large initial data. This appears to be the first global existence result on the Cauchy problem of the compressible Navier-Stokes-Allen-Cahn system with density-dependent viscosity and large data.