Communications in Mathematical Sciences

Volume 21 (2023)

Number 6

Sharp interface limit for compressible Navier–Stokes/Allen–Cahn system with shock wave

Pages: 1487 – 1503

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n6.a2

Authors

Yunkun Chen (School of Mathematics and Computer Science, Anshun University, Anshun, China)

Bin Huang (College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, China)

Xiaoding Shi (College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, China)

Abstract

In this paper, the sharp interface limit for the diffusion interface model system of immiscible two-phase flow called compressible Navier–Stokes/Allen–Cahn system is studied in one dimension. The results show that, for the initial perturbations with small energy but possibly large oscillations of shock wave solutions, and the strength of initial phase field is allowed to vary arbitrarily within its physical meaning, then the sharp interface limit of the compressible Navier–Stokes/Allen–Cahn system is the standard two-phase compressible Navier–Stokes equations.

Keywords

compressible Navier–Stokes equations, Allen–Cahn equation, shock wave, stability, sharp interface limit

2010 Mathematics Subject Classification

35B40, 35B65, 35C20, 35M10, 35Q35, 76N10, 76T30

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This paper is supported by the National Natural Science Foundation of China (Nos. 12171024, 11971217, 11971020), Natural Science Foundation of Jiangxi Province (No. 20202ACBL211002), Academic and Technical Leaders Training Plan of Jiangxi Province (No. 20212BCJ23027).

Received 24 July 2022

Received revised 24 July 2022

Accepted 9 November 2022

Published 22 September 2023