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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
  • Dan Lei,
  • Zhengzheng Chen
Dan Lei
Anhui University
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Zhengzheng Chen
Anhui University

Corresponding Author:chenzz@ahu.edu.cn

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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.
25 Nov 2024Submitted to Mathematical Methods in the Applied Sciences
26 Nov 2024Submission Checks Completed
26 Nov 2024Assigned to Editor
04 Dec 2024Review(s) Completed, Editorial Evaluation Pending
15 Dec 2024Reviewer(s) Assigned