Abstract:

This paper is mainly concerned with the time-asymptotic stability of stationary solutions to the outflow problem of the non-isentropic compressible Navier-Stokes-Allen-Cahn equations in the half space $\mathbb{R}^+$. The models can be used to describe the motion of a mixture of macroscopically immiscible two viscous compressible fluids. When the adiabatic exponent $\gamma$ is sufficiently close to $1$, we prove that the one-dimensional non-isentropic compressible Navier-Stokes-Allen-Cahn equations admits a unique global solution, which tends to the non-degenerate stationary solution as time goes to infinity. In this paper, the initial perturbations of temperature function and the phase field variable, and the strength of the stationary solution are required to be sufficiently small, but the initial perturbations of the density and velocity functions can be arbitrarily large. Our analysis is based on the basic $L^2$-energy method and some new techniques, which take into account the effect the phase field variable and the stationary solution.

Key words: compressible Navier-Stokes-Allen-Cahn equations, outflow problem, stability of the non-degenerate stationary solution