Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (6): 1657-1670.

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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;
  • Supported by:
    the NSFC(11771223)


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

CLC Number: 

  • O175.24