发表机构
Zhengzhou University of Light Industry; Northwest University; University of Chinese Academy of Sciences; Henan Polytechnic University（郑州轻工业大学; 西北大学; 中国科学院大学; 河南理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对分数阶Navier-Stokes方程，建立了时空导数估计，推广了经典Navier-Stokes系统的先验界和空间导数估计，并推导了标准情形下的积分估计。
AI 中文摘要
本文研究分数阶Navier-Stokes方程解的时空导数估计。我们证明了$\Lambda^{n\alpha}u^{(m)}_{t} \in L^{\frac{2(6\alpha-5)}{4m\alpha+2n\alpha+4 \alpha-5}}~~~~~~~(0,T; L^{2}(\mathbb{R}^{3}))$和$ \Lambda^{n }u^{(m)}_{t} \in L^{\frac{2(6\alpha-5)}{4m\alpha+2n +4 \alpha-5}}~~~~~(0,T; L^{2}(\mathbb{R}^{3}))$。这推广了Duff在[7, Acta Math. 164, 1990]中关于经典Navier-Stokes系统的先验界以及Boutros和Gibbon在[1, Nonlinearity 37, 2024]中的空间导数估计。此外，我们推导出在标准Navier-Stokes方程中，$ u \in L^{\frac{q}{q-3}}~~(0,T;L^{q} (\mathbb{R}^{3}))$，其中$ 6\leq q\leq\infty $，以及$\Lambda^{k}u \in L^{\frac{q}{ q(k+1)-3}}~~~(0,T;L^{q} (\mathbb{R}^{3})) $，其中$k\geq1, 2\leq q\leq\infty $。
英文摘要
In this paper, we are concerned with space-time derivative estimates of solutions to the fractional Navier-Stokes equations. It is shown that $Λ^{nα}u^{(m)}_{t} \in L^{\frac{2(6α-5)}{4mα+2nα+4 α-5}}~~~~~~~(0,T; L^{2}(\mathbb{R}^{3}))$ and $ Λ^{n }u^{(m)}_{t} \in L^{\frac{2(6α-5)}{4mα+2n +4 α-5}}~~~~~(0,T; L^{2}(\mathbb{R}^{3}))$. This generalizes a priori bounds for the classical Navier-Stokes system by Duff in [7, Acta Math. 164, 1990] and Boutros and Gibbon's spatial derivative estimates in [1, Nonlinearity 37, 2024]. In addition, we derive that $ u \in L^{\frac{q}{q-3}}~~(0,T;L^{q} (\mathbb{R}^{3}))$ with $ 6\leq q\leq\infty $ and $Λ^{k}u \in L^{\frac{q}{ q(k+1)-3}}~~~(0,T;L^{q} (\mathbb{R}^{3})) $ with $k\geq1, 2\leq q\leq\infty $ in the standard Navier-Stokes equations.
Comments36 pages
