We investigate the global existence and optimal time decay rate of solution to the one dimensional (1D) two-phase flow described by compressible Euler equations coupled with compressible Navier-Stokes equations through the relaxation drag force on the momentum equations (Euler-Navier-Stokes system). First, we prove the global existence of strong solution and the stability of the constant equilibrium state to 1D Cauchy problem of compressible Euler-Navier-Stokes system by using the standard continuity argument for small $H^{1}$ data while its second order derivative can be large. Then we derive the optimal time decay rate to the constant equilibrium state. Compared with multi-dimensional case, it is much harder to get optimal time decay rate by direct spectrum method due to a slower convergence rate of the fundamental solution in 1D case. To overcome this main difficulty, we need to first carry out time-weighted energy estimates (not optimal) for higher order derivatives, and based on these time-weighted estimates, we can close a priori assumptions and get the optimal time decay rate by spectrum analysis method. Moreover, due to non-conserved form and insufficient decay rate of the coupled drag force terms between the two-phase flows, we essentially need to use momentum variables $(m= \rho u, M=n\omega)$, rather than velocity variables $(u, \omega)$ in the spectrum analysis, to fully cancel out those non-conserved and insufficiently time-decay drag force terms. Finally, we study the singularity formation of the two-phase flow. We consider the blow-up of Euler equations in Euler-Navier-Stokes system. For Euler equations, we use Riemann invariants to construct decoupled Riccati type ordinary differential equations for smooth solutions and provide some sufficient conditions under which the classical solutions must break down in finite time.
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