Communications in Mathematical Sciences

Volume 18 (2020)

Number 4

Global solutions of a diffuse interface model for the two-phase flow of compressible viscous fluids in 1D

Pages: 1055 – 1086

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n4.a8

Authors

Shijin Ding (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Yinghua Li (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Abstract

This paper is concerned with a coupled Navier–Stokes/Cahn–Hilliard system describing a diffuse interface model for the two-phase flow of compressible viscous fluids in a bounded domain in one dimension. We prove the existence and uniqueness of global classical solutions for $\rho_0 \in C^{3,\alpha} (I)$. Moreover, we also obtain the global existence of weak solutions and unique strong solutions for $\rho_0 \in H^1 (I)$ and $\rho_0 \in H^2 (I)$, respectively. In these cases, the initial density function $\rho_0$ has a positive lower bound.

Keywords

compressible, Navier–Stokes, Cahn–Hilliard, global solutions

2010 Mathematics Subject Classification

35A01, 35A02, 35Q35

Full Text (PDF format)

Ding’s research is supported by the National Natural Science Foundation of China (Nos. 11371152, 11571117, 11771155, 11871005), and by the Guangdong Provincial Natural Science Foundation (No. 2017A030313003).

Li’s research is supported by the National Natural Science Foundation of China (Nos. 11671155, 11771156, 11971179), by the Guangdong Provincial Natural Science Foundation (Nos. 2017A030313024, 2019A1515010993), and by the Guangzhou Natural Science Foundation (No. 201707010136).

Received 3 May 2018

Accepted 19 January 2020

Published 28 July 2020