Pointwise space-time estimates of two-phase fluid model in dimension three

Abstract

We studied the pointwise space-time behavior of the classical solution to the Cauchy problem of two-phase fluid model derived by Choi (SIAM J Math Anal 48:3090–3122, 2016) when the initial data is sufficiently small and regular. This model is the compressible damped Euler system coupled with the compressible Naiver–Stokes system via a drag force. As we know, Liu and Wang (Commun Math Phys 196:145–173, 1998) verified that the solution of the compressible Naiver–Stokes system obeys the generalized Huygens’ principle, while Wang and Yang (J Differ Equ 173:410–450, 2001) verified the solution of the compressible Euler system does not obey the generalized Huygens’ principle due to the damped mechanism. In this paper, we proved that both of two densities and two momentums for the two-phase fluid model obey the generalized Huygens’ principle as that in Liu and Wang (Commun Math Phys 196:145–173, 1998). The main contribution is to overcome the difficulty of the non-conservation arising from the damped mechanism of the system. As a byproduct, we also extended \(L^2\)-estimate in Wu et al. (SIAM J Math Anal 52(6):5748–5774, 2020) to \(L^p\)-estimate with \(p>1\).