Global existence and time decay rates of the two-phase fluid system in R3

We are concerned with a two-phase fluid model in R 3 . This model was first derived by Choi (SIAM J. Math. Anal. 48: 3090–3122, 2016) by taking the hydrodynamic limit from the Vlasov–Fokker–Planck/isentropic Navier–Stokes equations with strong local alignment forces. Under the assumption that the H 3 norm of the initial data is small but its higher-order Sobolev norm can be arbitrarily large, the global existence and uniqueness of classical solutions are obtained by an energy method. Moreover, if in addition, the initial data norm of the H ˙ - s ( 0 ≤ s < 3 2 ) or B ˙ 2 , ∞ - s ( 0 < s ≤ 3 2 ) is small, we also obtain the optimal time decay rates of solutions.