A Critical Chemin--Lerner Regularity Criterion via One Velocity Component for the Three-Dimensional Navier--Stokes Equations
Abstract
We prove a scaling-critical regularity criterion involving only one velocity component for finite-energy suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations. Let 2<p<∞ and m=3p/(p−2), so that 2/p+3/m=1. We show that a singularity cannot occur provided ∑j∈ℤ |Δ̇j u3|Lp(0,T;Lm(ℝ3))<∞, that is, u3∈L̃p(0,T;Ḃ0m,1(ℝ3)). The assumption is a spatial-frequency ℓ1 refinement of the still unresolved critical condition u3∈ LptLmx and is complementary to the Lorentz-in-time refinement Lp,1tLmx obtained by Wang, Wu, and Zhang. By Bernstein embedding, the result extends to u3∈L̃ptḂsq,1 on the nonnegative-smoothness critical line s=−1+2/p+3/q≥0. The principal innovation is a frequency--scale matching scheme embedded in the local energy inequality. Each dyadic block of u3 is retained until it is paired with the vertical scale selected by a one-dimensional backward heat kernel. Low frequencies gain from the slab thickness, high frequencies from transferring the projection to the localized flux and applying an inverse Bernstein estimate, and spatially separated pressure sources from harmonic decay. These mechanisms generate a two-sided ℓ1 kernel, converting spatial-frequency summability into summability of physical-scale energy increments. Consequently, we obtain a uniform Type-I local energy bound without a Lorentz refinement in time; compactness and one-component rigidity then exclude singular blow-up limits whose third velocity component vanishes.
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BibTeX
@misc{guo2026critical,
title = {{A Critical Chemin--Lerner Regularity Criterion via One Velocity Component for the Three-Dimensional Navier--Stokes Equations}},
author = {Maotuo Guo and Wendong Wang and Shiyang Xiong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03877},
eprint = {2609.03877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03877},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Critical Chemin--Lerner Regularity Criterion via One Velocity Component for the Three-Dimensional Navier--Stokes Equations
AU - Maotuo Guo
AU - Wendong Wang
AU - Shiyang Xiong
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03877
UR - https://arxiv.org/abs/2609.03877
AB - We prove a scaling-critical regularity criterion involving only one velocity component for finite-energy suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations. Let 2<p<∞ and m=3p/(p−2), so that 2/p+3/m=1. We show that a singularity cannot occur provided ∑_j∈ℤ |Δ̇_j u^3|_L^p(0,T;L^m(ℝ^3))<∞, that is, u^3∈L̃^p(0,T;Ḃ^0_m,1(ℝ^3)). The assumption is a spatial-frequency ℓ^1 refinement of the still unresolved critical condition u^3∈ L^p_tL^m_x and is complementary to the Lorentz-in-time refinement L^p,1_tL^m_x obtained by Wang, Wu, and Zhang. By Bernstein embedding, the result extends to u^3∈L̃^p_tḂ^s_q,1 on the nonnegative-smoothness critical line s=−1+2/p+3/q≥0. The principal innovation is a frequency--scale matching scheme embedded in the local energy inequality. Each dyadic block of u^3 is retained until it is paired with the vertical scale selected by a one-dimensional backward heat kernel. Low frequencies gain from the slab thickness, high frequencies from transferring the projection to the localized flux and applying an inverse Bernstein estimate, and spatially separated pressure sources from harmonic decay. These mechanisms generate a two-sided ℓ^1 kernel, converting spatial-frequency summability into summability of physical-scale energy increments. Consequently, we obtain a uniform Type-I local energy bound without a Lorentz refinement in time; compactness and one-component rigidity then exclude singular blow-up limits whose third velocity component vanishes.
ER -