The global classical solution to a 1D two-fluid model with density-dependent viscosity and vacuum

In this paper, we consider the initial-boundary problem for a 1D two-fluid model with density-dependent viscosity and vacuum. The pressure depends on two variables but the viscosity only depends on one of the densities. We prove the global existence and uniqueness of the classical solution in the one-dimensional space with large initial data and vacuum. We use a new Helmholtz free energy function and the material derivative of the velocity field to deal with the general pressure with two variables, without the equivalence condition. We also develop a new argument to handle the general viscosity.