In this paper, we consider a two-phase problem of inhomogeneous incompressible viscous fluids in the N-dimensional Euclidean space \(\textbf{R}^N\) for \(N\ge 3\) . One fluid occupies an upper half-space-like domain \(\Omega _+(t)\) , while another fluid occupies \(\Omega _-(t)=\textbf{R}^N\setminus \overline{\Omega _+(t)}\) for time \(t\ge 0\) . The two fluids are thus separated from one another by a sharp interface for \(t\ge 0\) , and the time-dependent domains \(\Omega _\pm (t)\) need to be determined as part of the problem. In this situation, it is known that local existence theorems hold for several two-phase flows such as homogeneous or inhomogeneous incompressible two-phase flows, compressible two-phase flows, and compressible-incompressible two-phase flows. On the other hand, our aim of this paper is to construct global-in-time solutions for small initial data and to show large time decay of solutions. Furthermore, this paper provides a new tool to prove global existence theorems for two-phase problems in unbounded domains on the basis of maximal regularity and time decay estimates of the two-phase Stokes semigroup in an \(L_p\) -in-time and \(L_q\) -in-space setting.
