三维不可压缩广义Navier-Stokes方程组在Fourier-Triebel-Lizorkin空间中的适定性

doi:10.7523/j.ucas.2021.0065

中国科学院大学学报 ›› 2023, Vol. 40 ›› Issue (2): 145-154.DOI: 10.7523/j.ucas.2021.0065

• 数学与物理学 • 下一篇

三维不可压缩广义Navier-Stokes方程组在Fourier-Triebel-Lizorkin空间中的适定性

敏德载, 吴刚, 姚卓雅

中国科学院大学数学科学学院, 北京 100049

收稿日期:2021-03-08 修回日期:2021-04-21 发布日期:2021-10-13
通讯作者: 姚卓雅,E-mail:yaozhuoya19@mails.ucas.ac.cn
基金资助:
国家自然科学基金（11771423）资助

Well-posedness of 3D incompressible generalized Navier-Stokes system in Fourier-Triebel-Lizorkin spaces

MIN Dezai, WU Gang, YAO Zhuoya

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received:2021-03-08 Revised:2021-04-21 Published:2021-10-13

摘要/Abstract

摘要： 针对三维不可压缩广义Navier-Stokes方程组的柯西问题，研究其在临界Fourier-Triebel-Lizorkin空间$\widehat {\dot F}_{p,q}^{4 - \alpha - \frac{3}{p}}$($\mathbb{R}^3$)中的适定性问题。利用Fourier局部化方法和Banach不动点定理，证明当$p > \frac{3}{{5 - \alpha }}$，q≥1或者$p = \frac{3}{{5 - \alpha }}$，q∈[$\frac{3}{{5 - \alpha }}$，$\frac{6}{{5 - \alpha }}$]时，该方程组对适当小的初始值是整体适定的，对大初始值是局部适定的。

关键词: Navier-Stokes方程组, Fourier-Triebel-Lizorkin空间, 整体适定性, 局部适定性

Abstract: In this paper, we consider the Cauchy problem for the 3D incompressible generalized Navier-Stokes system and study the well-posedness in critical Fourier-Triebel-Lizorkin spaces $\widehat {\dot F}_{p,q}^{4 - \alpha - \frac{3}{p}}$($\mathbb{R}^3$). Making use of Fourier localization method and Banach fixed point theorem, we proved that if $p > \frac{3}{{5 - \alpha }}$, q ≥ 1, the system is locally well-posed for large initial data as well as global well-posed for small initial data. Also we established same result for $p = \frac{3}{{5 - \alpha }}$，q∈[$\frac{3}{{5 - \alpha }}$，$\frac{6}{{5 - \alpha }}$].

Key words: Navier-Stokes system, Fourier-Triebel-Lizorkin spaces, global well-posedness, local well-posedness

中图分类号:

O175.2

敏德载, 吴刚, 姚卓雅. 三维不可压缩广义Navier-Stokes方程组在Fourier-Triebel-Lizorkin空间中的适定性[J]. 中国科学院大学学报, 2023, 40(2): 145-154.

MIN Dezai, WU Gang, YAO Zhuoya. Well-posedness of 3D incompressible generalized Navier-Stokes system in Fourier-Triebel-Lizorkin spaces[J]. Journal of University of Chinese Academy of Sciences, 2023, 40(2): 145-154.

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三维不可压缩广义Navier-Stokes方程组在Fourier-Triebel-Lizorkin空间中的适定性

Well-posedness of 3D incompressible generalized Navier-Stokes system in Fourier-Triebel-Lizorkin spaces

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