Low Mach number limit for the compressible Euler-Navier-Stokes two-phase flow model in R3

This paper is concerned with the two-phase flow model consisting of the compressible isothermal Euler equations coupled with the compressible isentropic Navier-Stokes equations through a drag forcing term.For the 3-D Cauchy problem,we rigorously justify the low Mach number limit, which means that the solutions converge to that of a two-phase flow model coupled with the compressible Euler equations and the incom pressible Navier-Stokes equations locally and globally in time as Mach number goes to zero. MSC 2020:35Q30,35B35,35L6576D3374J40