PreprintCC BY 4.0arXiv
On space-time derivative estimates for the fractional Navier-Stokes equations
Abstract
In this paper, we are concerned with space-time derivative estimates of solutions to the fractional Navier-Stokes equations. It is shown that Λnαu(m)t ∈ L(2(6α−5))/(4mα+2nα+4 α−5) (0,T; L2(ℝ3)) 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.
Related
- Sharp Basic Velocity Estimates for the Plane Steady Navier--Stokes Equations through Vorticity
- On solutions for damped fractional Navier-Stokes equations in a critical Lei-Lin-Gevrey space
- Global Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation
- Linear Corrector Estimates and Corrected Pressure Convergence for a Two-Parameter Hyperbolic Relaxation of the Incompressible Navier--Stokes Equations
- Weighted Lq-Lr estimates of a generalized Oseen evolution operator in exterior domains, with applications to the Navier-Stokes flow past a rigid body
BibTeX
@misc{wang2026spacetime,
title = {{On space-time derivative estimates for the fractional Navier-Stokes equations}},
author = {Yanqing Wang and Wei Wei and Gang Wu and Daoguo Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12864},
eprint = {2609.12864},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12864},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On space-time derivative estimates for the fractional Navier-Stokes equations
AU - Yanqing Wang
AU - Wei Wei
AU - Gang Wu
AU - Daoguo Zhou
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12864
UR - https://arxiv.org/abs/2609.12864
AB - In this paper, we are concerned with space-time derivative estimates of solutions to the fractional Navier-Stokes equations. It is shown that Λ^nαu^(m)_t ∈ L^(2(6α−5))/(4mα+2nα+4 α−5) (0,T; L^2(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.
ER -