Existence and nonlinear stability of stationary solutions to the outflow problem of the one-dimensional full compressible Navier-Stokes-Allen-Cahn system

This paper is concerned with the large-time behavior of solutions toward a stationary solution for the outflow problem of the full compressible Navier-Stokes-Allen-Cahn system in the half-space $R^{+}$. The model can be used to describe the motion of a mixture of two viscous compressible fluids. First, we give some sufficient conditions for the existence of stationary solution via the manifold theory and the center manifold theory. Second, by using the elementary $L^2$-energy method, it is shown that the stationary solution is time-asymptotically stable provided that the initial perturbation and the strength of the stationary solution are sufficiently small. Finally, the convergence rates of solutions towards the stationary one are also established by employing a time and space weighted energy method. Our analysis is based on some new techniques which take into account the effect of the phase field variable.