Communications in Mathematical Sciences

Volume 17 (2019)

Number 6

Existence of weak solutions to the steady two-phase flow

Pages: 1699 – 1712

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n6.a9

Authors

Senming Chen (School of Mathematics, South China University of Technology, Guangzhou, China)

Changjiang Zhu (School of Mathematics, South China University of Technology, Guangzhou, China)

Abstract

In this paper, we prove the existence of weak solutions to the steady two-phase flow. The result holds in three dimensions on the condition that the adiabatic constants $\gamma , \theta \gt 1$ and $\gamma \gt \frac{7}{3}, \theta = 1$. By constructing a special example, we show that the weak solutions are non-unique. It turns out that the uniform approximation scheme restricts the type of weak solutions, which leads to some open problems.

Keywords

two-phase model, weak solutions, non-uniqueness

2010 Mathematics Subject Classification

35D30, 35Q30, 76T10

Received 19 March 2019

Accepted 10 June 2019

Published 26 December 2019