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# 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

#### 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

The research was supported by the National Natural Science Foundation of China #11771150, 11831003.

Received 19 March 2019

Accepted 10 June 2019

Published 26 December 2019