Stability of the composite wave for compressible Navier-Stokes/Allen-Cahn system. (English) Zbl 1442.35348
Summary: This paper is devoted to the study of the nonlinear stability of the composite wave consisting of two rarefaction waves and a viscous contact wave for the Cauchy problem to a one-dimensional compressible non-isentropic Navier-Stokes/Allen-Cahn system which is a combination of the classical Navier-Stokes system with an Allen-Cahn phase field description. We first construct the composite wave through Euler equations under the assumption of \(\chi(x, t) \equiv 1\) for the large time behavior, and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding Cauchy problem of the non-isentropic Navier-Stokes/Allen-Cahn system. The proof is mainly based on a basic energy method.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35M10 | PDEs of mixed type |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B35 | Stability in context of PDEs |
| 76N15 | Gas dynamics (general theory) |
| 35Q30 | Navier-Stokes equations |
Keywords:
Navier-Stokes/Allen-Cahn system; contact wave; rarefaction wave; composite wave; asymptotic stabilityReferences:
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