The Cauchy Problem for a Non-conservative Compressible Two-Fluid Model with Far Field Vacuum in Three Dimensions

In this paper, we study the wellposedness of the Cauchy problem for a non-conservative compressible two-fluid model with density-dependent viscosity coefficients vanishing at far field in three dimensions. The non-conservative pressure term (an implicit function) and the degenerate viscosity coefficients due to the vanishing of the volume fractions and the densities are the main issues. To overcome the difficulties, we construct iteration sequences in terms of the average densities and the velocities, and explore some new connections between the pressure term (including its gradients) and some other terms of the average densities. Those estimates are uniform for the positive lower bound of the average densities, and they are not trivial in particular when the adiabatic indexes are close to 1. Moreover, to get the strong convergence for the full sequences, one can not use the mean value theorem in the pressure term to get the desired estimates of the difference between the average densities due to the possible vanishing of the densities. Instead, we introduce some equations in terms of some new quantities associated with the volume fractions, the densities, and the average densities. Compared with the existing results on the same model, this work can be viewed as the first result on the wellposedness of regular solutions that allow the volume fraction and the density to vanish.