Optimal well-posedness for the pressureless Euler–Navier–Stokes system

Abstract

In this work, we investigate the Cauchy problem for the pressureless Euler–Navier–Stokes system in R3. We first establish the global small solutions of this system with critical regularity and then obtain the optimal time decay rate of the solutions by a suitable energy argument (independent of the spectral analysis). The proof crucially depends on non-standard product estimates and interpolations. In comparison with previous studies about time-decay by Choi and Jung [J. Math. Fluid Mech. 23, 99 (2021); arXiv:2112.14449], the smallness requirement of the low frequencies of initial data could be removed.