Methods and Applications of Analysis

Volume 28 (2021)

Number 3

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part II

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

Existence of radially symmetric stationary solutions for the compressible Navier–Stokes equation

Pages: 299 – 312



Itsuko Hashimoto (Kanazawa University, Osaka City University, Oksaka, Japan)

Akitaka Matsumura (Osaka University, Osaka, Japan)


The present paper is concerned with the existence of radially symmetric stationary solutions for exterior problems in $\mathbb{R}^n (n \geq 2)$ to the compressible Navier–Stokes equation, describing the motion of viscous barotropic gas without external forces, where boundary and far field data are prescribed. For both inflow and outflow problems, the existence of a unique radially stationary solution is shown in a suitably small neighborhood of the far field state. The estimates of algebraic decay rate toward the far field state are also obtained. Furthermore, it is shown that the boundary layer of the density appears as the velocity data tend to zero in the inflow problem, but not in the outflow problem.


compressible Navier–Stokes equation, stationary solution, radially symmetric solution

2010 Mathematics Subject Classification

Primary 35Q30. Secondary 76N10.

Itsuko Hashimoto was supported by JSPS Grant No. 17K14227.

Received 27 April 2020

Accepted 15 July 2020

Published 10 June 2022