Methods and Applications of Analysis

Volume 29 (2022)

Number 1

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

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)

Global existence and exponential decay of strong solutions for the inhomogeneous incompressible Navier–Stokes equations with vacuum

Pages: 57 – 94



Dehua Wang (Department of Mathematics, University of Pittsburgh, Pennsylvania, U.S.A.)

Zhuan Ye (Department of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, China)


The inhomogeneous incompressible Navier–Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solutions with vacuum are established for large initial data. The exponential decay-in-time of the strong solution is also obtained, which is different from the homogeneous case. The initial density may have vacuum and even compact support.


Navier–Stokes equations, vacuum, inhomogeneous, incompressible, exponential decay, global strong solution

2010 Mathematics Subject Classification

35B65, 35Q35, 76D05, 76N10

D. Wang’s research was supported in part by the National Science Foundation under grants DMS-1613213 and DMS-1907519.

Z. Ye was supported by the National Natural Science Foundation of China (No. 11701232), by the Natural Science Foundation of Jiangsu Province (No. BK20170224), and by the Qing Lan Project of Jiangsu Province.

Received 17 September 2020

Accepted 15 January 2021

Published 10 June 2022