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

TL;DR

This paper proves the global existence and determines the optimal decay rate of solutions for a 1D two-phase flow model described by coupled Euler and Navier-Stokes equations, overcoming unique 1D challenges.

Contribution

It establishes the global strong solution and optimal decay rate for the 1D Euler-Navier-Stokes system using novel energy estimates and spectral analysis tailored for 1D difficulties.

Findings

Proved global existence of strong solutions for small initial data.

Derived the optimal time decay rate for solutions to the 1D two-phase flow model.

Developed a spectral analysis method using momentum variables to handle non-conserved drag force terms.

Abstract

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.