数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1657-1670.

• 论文 • 上一篇    下一篇

半空间上Stokes半群的加权时空估计及其在非稳恒Navier-Stokes方程中的应用

张庆华1(),朱月萍2,*()   

  1. 1 南通大学理学院 江苏南通 226019
    2 南通师范高等专科学校 江苏南通 226010
  • 收稿日期:2020-05-19 出版日期:2021-12-26 发布日期:2021-12-02
  • 通讯作者: 朱月萍 E-mail:zhangqh@ntu.edu.cn;zhuyueping@ntu.edu.cn
  • 作者简介:张庆华, E-mail: zhangqh@ntu.edu.cn
  • 基金资助:
    国家自然科学基金(11771223)

Weighted Temporal-Spatial Estimates of the Stokes Semigroup with Applications to the Non-Stationary Navier-Stokes Equation in Half-Space

Qinghua Zhang1(),Yueping Zhu2,*()   

  1. 1 School of Sciences, Nantong University, Jiangsu Nantong 226019
    2 Department of Mathematics, Nantong Normal College, Jiangsu Nantong 226010
  • Received:2020-05-19 Online:2021-12-26 Published:2021-12-02
  • Contact: Yueping Zhu E-mail:zhangqh@ntu.edu.cn;zhuyueping@ntu.edu.cn
  • Supported by:
    the NSFC(11771223)

摘要:

该文研究半空间上Navier-Stokes方程的加权时空估计以及正则解的存在性. 利用半空间上Stokes半群的Ukai表达式以及分数幂积分的加权不等式, 首先导出Stokes流关于空间变量的$L^{r}$-$L^{q}$混合加权估计式. 然后在初始速度$u_{0}$属于一个带权重$w^{s-n}$ ($n\leq s<\infty$)的$L^{s}({\mathbb R}_{+}^{n})$空间的条件下, 借助于Hardy不等式、空间的内插以及弱$L^{s}$空间, 在带有时空权重的$L^{b}(0, T;L^{q}({\mathbb R}_{+}^{n}))$空间中考察了Navier-Stokes方程积分解的存在性. 该文还证明, 若$n=3$, $n\leq s\leq4$, 并且$u_{0}$还属于能量空间$L_{\sigma}^{2}({\mathbb R}_{+}^{n})$, 则这个积分解恰好是Navier-Stokes方程的正则解. 考虑到当$s>n$时, 带权重的空间$L_{w^{s-n}}^{s}({\mathbb R}_{+}^{n})$$L^{s}({\mathbb R}_{+}^{n})$并不一致, 该文所得的结果是对所列文献的有益补充.

关键词: 半空间, Navier-Stokes方程, 加权时空估计

Abstract:

This paper deals with the weighted temporal-spatial estimates and strong solvability of the Navier-Stokes equation in ${\mathbb R}_{+}^{n}$. With the aid of Ukai's representation of the Stokes semigroup, and weighted inequalities for the fractional integral operators, $L^{r}$-$L^{q}$ estimates with mixed spatial weights are made for the Stokes flow. Then by means of Hardy's inequality, and interpolation method for the weak $L^{s}$ space, existence of the integral solution in $L^{b}(0, T;L^{q}({\mathbb R}_{+}^{n}))$ with temporal and spatial weights for the Navier-Stoke equation, where the initial velocity $u_{0}$ belongs to $L^{s}({\mathbb R}_{+}^{n})$ with the weight $w^{s-n}$ for some $n\leq s<\infty$ is established. This solution is proved to be the regular one provided $n=3$, $n\leq s\leq4$, and $u_{0}$ also lies in $L_{\sigma}^{2}({\mathbb R}_{+}^{n})$. Considering that $L_{w^{s-n}}^{s}({\mathbb R}_{+}^{n})$ does not coincide with $L^{s}({\mathbb R}_{+}^{n})$ whenever $s>n$, results obtained here can be viewed as useful supplements to the literatures.

Key words: Half space, Navier-Stokes equation, Weighted temporal-spatial estimate

中图分类号: 

  • O175.24