Global existence and optimal time decay rate to one-dimensional two-phase flow model

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 a 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 the multi-dimensional case, it is much harder to get the optimal time decay rate by the direct spectrum method due to a slower convergence rate of the fundamental solution in the 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 spectral analysis method. Moreover, due to the 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 insufficient time decay drag force terms.