The Green's function of the pressureless Euler-Navier-Stokes system in R³

In this paper, we investigate the space-time pointwise behaviors of global solutions to the pressureless Euler-Navier-Stokes system in three dimensions, which characterizes the motion of a two-phase fluid. First, we construct the Green's function for the linearized system, which consists of the Huygens wave, the diffusion wave, the Riesz wave, and the singular part containing the stationary delta wave arising from the pressureless structure. Then, under the assumptions on the initial data with appropriate spatial decay rates, we obtain the space-time pointwise estimates of global solutions to the Cauchy problem for the linearized system.