外区域上非稳恒Navier-Stokes流的加权模估计

数学物理学报 ›› 2023, Vol. 43 ›› Issue (4): 1133-1148.

外区域上非稳恒Navier-Stokes流的加权模估计

张庆华()

南通大学理学院江苏南通 226019

收稿日期:2022-05-10 修回日期:2023-04-24 出版日期:2023-08-26 发布日期:2023-07-03
作者简介:张庆华, E-mail: zhangqh@ntu.edu.cn

On Weighted Estimates for the Nnonstationary 3D Navier-Stokes Flow in an Exterior Domain

Zhang Qinghua()

School of Science, Nantong University, Jiangsu Nantong 226019

Received:2022-05-10 Revised:2023-04-24 Online:2023-08-26 Published:2023-07-03

摘要/Abstract

摘要：

该文研究外区域上3维非稳恒Navier-Stokes流的加权模估计. 首先构造一个向量场去点乘Navier-Stokes方程的两边, 从而将Navier-Stokes流表示为一个积分方程的强解; 然后对积分方程的各项进行估计. 利用初始条件 $|x|^{\alpha}u_{0}\in L^{r}(\Omega)$ , $u_{0}\in L_{\sigma}^{3}(\Omega)$ , 其中 $\|u_{0}\|_{3}$ 充分小, 指数 $0<\alpha<3$ , 1＜r＜3且 $\max \left\{r, \frac{3}{2}\right\}＜q＜\infty$ 满足适当的条件, 该文推导出加权估计式 $\||x|^{\alpha}u(t)\|_{q}\leq C(1+t^{\frac{\alpha}{2}})t^{-\frac{3}{2}(\frac{1}{r}-\frac{1}{q})}$ , 其中 $t>0$ . 在情形 $0<\alpha\leq2$ 下, 该文的初始条件要比文献弱, 对指数 $q$ 的限制也比文献宽松.

关键词: 外区域, Navier-Stokes流, 加权模估计

Abstract:

This paper studies weighted estimates for the 3D Navier-Stokes flow in an exterior domain. By multiplying the Navier-Stokes equation with a well selected vector field, an integral equation is derived, from which we prove that $\||x|^{\alpha}u(t)\|_{q}\leq C(1+t^{\frac{\alpha}{2}})t^{-\frac{3}{2}(\frac{1}{r}-\frac{1}{q})}$ for all $t>0$ under the initial condition $|x|^{\alpha}u_{0}\in L^{r}(\Omega)$ and $u_{0}\in L_{\sigma}^{3}(\Omega)$ with sufficiently small norm $\|u_{0}\|_{3}$ , where $0<\alpha<3$ , $1 $0<\alpha\leq2$ and restriction on $q$ is weaker.

Key words: exterior domain, Navier-Stokes flow, weighted estimate

中图分类号:

O175.24

张庆华. 外区域上非稳恒Navier-Stokes流的加权模估计[J]. 数学物理学报, 2023, 43(4): 1133-1148.

Zhang Qinghua. On Weighted Estimates for the Nnonstationary 3D Navier-Stokes Flow in an Exterior Domain[J]. Acta mathematica scientia,Series A, 2023, 43(4): 1133-1148.

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