Communications in Mathematical Sciences

Volume 18 (2020)

Number 7

Local well-posedness of isentropic compressible Navier–Stokes equations with vacuum

Pages: 1891 – 1909

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n7.a4

Authors

Huajun Gong (College of Mathematics and Statistics, Shenzhen University, Shenzhen, China)

Jinkai Li (South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou, China)

Xian-Gao Liu (School of Mathematical Sciences, Fudan University, Shanghai, China)

Xiaotao Zhang (South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou, China)

Abstract

In this paper, the local well-posedness of strong solutions to the Cauchy problem of the isentropic compressible Navier–Stokes equations is proved with the initial data being allowed to have vacuum. The main contribution of this paper is that the well-posedness is established without assuming any compatibility condition on the initial data, which was widely used before in many works concerning the well-posedness of compressible Navier–Stokes equations in the presence of vacuum.

Keywords

isentropic Navier–Stokes equations, local well-posedness, vacuum, without compatibility condition

2010 Mathematics Subject Classification

35D35, 35Q30, 35Q35, 76N10

Full Text (PDF format)

H. Gong was supported in part by the National Natural Science Foundation of China grants 11601342, 11871345, and 61872429, and by the Natural Science Foundation of SZU grants 2019084 and 2019085. J. Li was supported in part by the National Natural Science Foundation of China grants 11971009, 11871005, and 11771156, by the Natural Science Foundation of Guangdong Province grant 2019A1515011621, by the Guangdong Province Basic and Applied Basic Research Fund 2020B1515310005, and by the Hong Kong RGC Grant CUHK-14302917. X. Liu was supported in part by the National Natural Science Foundation of China grant 11631011. X. Zhang was supported in part by the National Natural Science Foundation of China grant 11901209, by the Guangdong Province Basic and Applied Basic Research Fund 2020B1515310002, and by the National Postdoctoral Program for Innovative Talents of China grant BX20200135.

Received 19 May 2019

Accepted 17 April 2020

Published 11 December 2020