Asymptotic Stability of Rarefaction Wave to the 1D Compressible Navier–Stokes–Allen–Cahn System With Density and Phase‐Field Variable–Dependent Viscosity in the Half Space

ABSTRACT In this paper, we study the time‐asymptotic behavior of solutions toward rarefaction wave for the impermeable wall problems of the one‐dimensional isentropic compressible Navier–Stokes–Allen–Cahn system with density and phase‐field variable–dependent viscosity on the half line , which models the motion of a mixture of two viscous compressible fluids. We consider the case when the viscosity coefficient and the pressure , where is the density of the fluid, is the phase‐field variable, and are parameters. Under some suitable assumptions on the boundary data and spatial‐asymptotic state of the velocity field, the time‐asymptotic profile of these problems is the 2‐rarefaction wave of the resulting Euler system. By employing the elementary ‐energy method, it is shown that this 2‐rarefaction wave is time‐asymptotically stable provided that the parameters and the initial perturbation satisfy some conditions. Here, the initial perturbation for the phase‐field variable is small, but the initial perturbations for the density and velocity of the fluid and the strength of the rarefaction wave can be arbitrarily large.