An elementary approach to the pressureless Euler-Navier-Stokes system

TL;DR

This paper proves the global existence of strong solutions to the pressureless Euler-Navier-Stokes system in the whole space without small initial data assumptions, using elementary energy methods and providing decay estimates.

Contribution

It establishes the global existence of solutions without smallness or regularity assumptions on initial density, using a simple energy approach.

Findings

Global existence of strong solutions proved

Optimal decay estimates obtained

Long-time behavior of density described

Abstract

The pressureless Euler-Navier-Stokes system can be obtained formally from the Vlasov-Navier-Stokes system, under the assumption that the distribution function describing the density of particles is monokinetic. Its study has been the subject of several recent papers, which have established the global existence of solutions with high enough regularity, for small initial data. In this work, we demonstrate the global existence of strong solutions in the whole space case, without assuming the initial density to be small and regular: it suffices for it to be bounded and for the total mass to be finite. In passing, we obtain optimal decay estimates for the energy and dissipation functionals. As a corollary, we get a long-time description of the density. All these results are based on an elementary energy method, with no need of sophisticated Fourier analysis tools.