@Article{JPDE-38-251,
author = {Zhang , Qinghua},
title = {Estimates on Weighted $L^q$-Norms of the  Nonstationary 3D Navier-Stokes Flow in  an Exterior Domain},
journal = {Journal of Partial Differential Equations},
year = {2025},
volume = {38},
number = {3},
pages = {251--278},
abstract = {<p style="text-align: justify;">This paper is devoted to estimates on weighted $L^q$-norms of the nonstationary 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 establish the weighted estimate $∥|x|^αu(t)∥_q ≤ C(1+t^{\frac{α}{2} +ε} )t^{-\frac{3}{2}(1-\frac{1}{q})} ,$$t&gt;0,$ where $ 0&lt;α≤1$ and $\frac{3}{2}&lt;q&lt;∞,$ or $1&lt;α&lt;2$ and $\frac{3}{3−α }&lt;q&lt;∞,$ $0&lt;ε&lt;1$ is arbitrary, and $u_0∈L^3_σ(Ω),$ $|x|^αu_0 ∈L^1(Ω)$ with $∥u_0∥_3$ sufficiently small. With the aid of the representation of the flow, we also prove that if in addition $u_0∈D^{1−1/b,b} _a$&nbsp; for some $\frac{6}{5} \le a&lt;\frac{3}{2}$ and $1&lt;b&lt;2$ with $\frac{3}{a}+\frac{2}{b} =4,$ then the optimal estimate $∥|x|^αu(t)∥_q≤C(1+t^{\frac{\alpha}{2}})&nbsp; t^{-\frac{3}{2}(1-\frac{1}{q})},$ $t&gt;0$ holds, where $α&gt;0$ and $1&lt;q&lt;∞.$ Compared with the literature, here no extra restriction is laid on the range of the exponents $α$ and $q.$</p>},
issn = {2079-732X},
doi = {https://doi.org/10.4208/jpde.v38.n3.1},
url = {http://global-sci.org/intro/article_detail/jpde/24400.html}
}