Large-friction and incompressible limits for pressureless Euler/isentropic Navier-Stokes flows

We investigate the large-friction and incompressible limits for a two-phase flow (Euler-NS) system which couples the pressureless Euler equations and the isentropic compressible Navier-Stokes equations through a drag force term with the friction coefficient 1τ>0 in Rᵈ (d≥2). We establish the uniform regularity estimates with respect to $τ$ so that the solution of the Cauchy problem for the Euler-NS system exists globally in time, provided that the initial data are uniformly close to the equilibrium state in a critical Besov space. These uniform estimates allow us to rigorously justify the strong convergence of the Euler-NS system to a one-velocity two-phase drift-flux (DF) model as τ→ 0, with an explicit convergence rate of order √τ. We also study the large-time asymptotic behavior of solutions for the Euler-NS system, uniformly with respect to $τ$. Moreover, when the Mach number ε>0 is considered, we prove the incompressible limit of the DF model toward the Transport-Navier-Stokes (TNS) system as ε→ 0, and justify the combined large-friction and incompressible limit for the Euler-NS system toward the TNS system in the regime τ=ε→ 0. Each singular limit process is globally valid in time for {ill-prepared} initial data.