THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA∗

doi:10.1007/s10473-023-0415-x

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1675-1716.doi: 10.1007/s10473-023-0415-x

THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA^∗

Haitao WANG¹, Xiongtao ZHANG^2,†

1. Institute of Natural Sciences and School of Mathematical Sciences, LSC-MOE, Shanghai Jiao Tong University, Shanghai 200240, China;
2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

收稿日期:2021-12-09 发布日期:2023-08-08
通讯作者: †Xiongtao ZHANG, E-mail: zhangpanda1987@163.com
作者简介:Haitao WANG, E-mail: haitallica@sjtu.edu.cn
基金资助:
*National Key R&D Program of China (2022YFA1007300); Wang was supported by the NSFC (11901386, 12031013) and the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA25010403); Zhang was supported by the NSFC (11801194, 11971188) and the Hubei Key Laboratory of Engineering Modeling and Scientific Computing.

THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA^∗

Haitao WANG¹, Xiongtao ZHANG^2,†

1. Institute of Natural Sciences and School of Mathematical Sciences, LSC-MOE, Shanghai Jiao Tong University, Shanghai 200240, China;
2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received:2021-12-09 Published:2023-08-08
Contact: †Xiongtao ZHANG, E-mail: zhangpanda1987@163.com
About author:Haitao WANG, E-mail: haitallica@sjtu.edu.cn
Supported by:
*National Key R&D Program of China (2022YFA1007300); Wang was supported by the NSFC (11901386, 12031013) and the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA25010403); Zhang was supported by the NSFC (11801194, 11971188) and the Hubei Key Laboratory of Engineering Modeling and Scientific Computing.

摘要/Abstract

摘要： A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the $L^1\cap$ BV class was constructed in [1]. In the current paper, we will continue to study the uniqueness and regularity of the constructed solution. The key ingredients are the Hölder continuity estimates of the heat kernel in both spatial and time variables. With these finer estimates, we obtain higher order regularity of the constructed solution to Navier-Stokes equation, so that all of the derivatives in the equation of conservative form are in the strong sense. Moreover, this regularity also allows us to identify a function space such that the stability of the solutions can be established there, which eventually implies the uniqueness.

关键词: compressible Navier-Stokes equation, BV initial data, regularity, uniqueness

Abstract: A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the $L^1\cap$ BV class was constructed in [1]. In the current paper, we will continue to study the uniqueness and regularity of the constructed solution. The key ingredients are the Hölder continuity estimates of the heat kernel in both spatial and time variables. With these finer estimates, we obtain higher order regularity of the constructed solution to Navier-Stokes equation, so that all of the derivatives in the equation of conservative form are in the strong sense. Moreover, this regularity also allows us to identify a function space such that the stability of the solutions can be established there, which eventually implies the uniqueness.

Key words: compressible Navier-Stokes equation, BV initial data, regularity, uniqueness

Haitao WANG, Xiongtao ZHANG. THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1675-1716.

Haitao WANG, Xiongtao ZHANG. THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1675-1716.

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THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA^∗

THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA^∗

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