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 R + = ( 0 , + ∞ ) , which models the motion of a mixture of two
viscous compressible fluids. We consider the case when the viscosity
coefficient ν ( ρ , χ ) = ρ α χ β and the pressure p ( ρ ) = ρ γ , where
ρ= ρ( t,x) is the density of the fluid, χ=
χ( t,x) is the phase field variable, and α,β,γ∈R
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 L 2 -energy method, it is
shown that this 2-rarefaction wave is time-asymptotically stable
provided that the parameters α,β,γ and the initial perturbation
satisfy some conditions. Here the initial perturbation for the phase
field variable is small, but the initial perturbation for the density
and velocity of the fluid, and the strength of the rarefaction wave can
be arbitrarily large.