Methods and Applications of Analysis

Volume 28 (2021)

Number 4

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

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 well-posedness of 3D incompressible inhomogeneous Navier–Stokes equations

Pages: 507 – 546

DOI: https://dx.doi.org/10.4310/MAA.2021.v28.n4.a6

Authors

Chenyin Qian (Department of Mathematics, Zhejiang Normal University, Jinhua, China)

Ping Zhang (Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China)

Abstract

In this paper, we prove the global well-posedness of 3D inhomogeneous incompressible Navier–Stokes equations with initial velocity to be sufficiently small in the critical Besov space, $\dot{B}^{3/p-1}_{p,1}$ for $1 \lt p \lt 6$ and with initial density in the critical Besov space and bounded away from vacuum. The key ingredient used in the proof lies in a new estimate to the pressure term. In particular, our result improves the previous ones by Abidi et al. (2013) [3], Zhai and Yin (2017) [32], Burtea (2017) [6] and so on.

Keywords

inhomogeneous Navier–Stokes systems, Littlewood–Paley theory, well-posedness, Besov spaces

2010 Mathematics Subject Classification

35Q30, 35Q35, 76D03

Full Text (PDF format)

P. Zhang is partially supported by NSF of China under Grants 11371347 and 11688101, and by an innovation grant from the National Center for Mathematics and Interdisciplinary Sciences of the Chinese Academy of Sciences.

Received 20 April 2020

Accepted 28 August 2020

Published 10 June 2022