Communications in Mathematical Sciences

Volume 18 (2020)

Number 5

Low Mach number limit of steady Euler flows in multi-dimensional nozzles

Pages: 1191 – 1220



Mingjie Li (College of Science, Minzu University of China, Beijing, China)

Tian-Yi Wang (Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan, Hubei, China; Gran Sasso Science Institute, L’Aquila, Italy)

Wei Xiang (City University of Hong Kong, Kowloon Tong, Hong Kong)


In this paper, we consider the steady irrotational Euler flows in multidimensional nozzles. The first rigorous proof on the existence and uniqueness of the incompressible flow is provided. Then, we justify the corresponding low Mach number limit, which is the first result of the low Mach number limit on the steady Euler flows. We establish several uniform estimates, which does not depend on the Mach number, to validate the convergence of the compressible flow with the conservative extra force to the corresponding incompressible flow, which is free from the conservative extra force effect, as the Mach number goes to zero. The limit is on the Hölder space and is unique. Moreover, the convergence rate is of order $\varepsilon^2$, which is higher than the ones in the previous results on the low Mach number limit for the unsteady flow.


multidimensional nozzles, low Mach number limit, steady flow, homentropic Euler equations, convergence rate

2010 Mathematics Subject Classification

35B40, 35L65, 35Q31, 76N15

The research of Mingjie Li is supported by the NSFC Grant No. 11671412.

The research of Tian-Yi Wang was supported in part by NSFC Grant No. 11601401 and 11971024.

The research ofWei Xiang was supported in part by the Grants Council of the HKSAR, China (Project No. CityU 11332916, 11304817, Project No. CityU 11303518 and Project No. CityU 11304820).

Received 7 November 2019

Accepted 3 February 2020

Published 23 September 2020