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

In this paper, we investigate the pointwise space-time behavior of two-phase fluid model derived by Choi \cite{Choi} [SIAM J. Math. Anal., 48(2016), pp. 3090-3122], which is the compressible damped Euler equations coupled with compressible Naiver-Stokes equations. Based on Green's function method together with frequency analysis and nonlinear coupling of different wave patterns, it shows that both of two densities and momentums obey the generalized Huygens' principle as the compressible Navier-Stokes equations \cite{LW}, however, it is different from the compressible damped Euler equations \cite{Wang2}. The main contributions include seeking suitable combinations to avoid the singularity from the Hodge decomposition in the low frequency part of the Green's function, overcoming the difficulty of the non-conservation arising from the damped mechanism of the system, and developing the detailed description of the singularities in the high frequency part of the Green's function. Finally, as a byproduct, we extend $L^2$-estimate in \cite{Wugc} [SIAM J. Math. Anal., 52(2020), pp. 5748-5774] to $L^p$-estimate with $p>1$.