Incompressible inviscid limit of the viscous two-fluid model with general initial data

In this paper, we study the incompressible inviscid limit of the viscous two-fluid model in the whole space $R^3$ with general initial data in the framework of weak solutions. By applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of the densities and the velocity, we prove rigorously that the weak solutions of the compressible two-fluid model converge to the strong solution of the incompressible Euler equations in the time interval provided that the latter exists. Moreover, thanks to the Strichartz's estimates of linear wave equations, we also obtain the convergence rates. The main ingredient of this paper is that our wave equations include the oscillations caused by the two different densities and the velocity and we also give an detailed analysis on the effect of the oscillations on the evolution of the solutions.