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In this paper, we present a refined framework for the global-in-timewell-posedness theory for the pressureless Euler--Navier--Stokes system and theoptimal temporal decay rates of certain norms of solutions. Here the couplingof two systems, pressureless Euler system and incompressible Navier--Stokesystem, is through the drag force. We construct the global-in-time existenceand uniqueness of regular solutions for the pressureless Euler--Navier--Stokessystem without using a priori large time behavior estimates. Moreover, we seekfor the optimal Sobolev regularity for the solutions. Concerning the temporaldecay for solutions, we show that the fluid velocities exhibit the same decayrate as that of the heat equations. In particular, our result provides that thetemporal decay rate of difference between two velocities, which is faster thanthe fluid velocities themselves, are at least the same as the second-orderderivatives of fluid velocities.

A revisit to the pressureless Euler--Navier--Stokes system in the whole space and its optimal temporal decay