三维Navier–Stokes方程的仅通过一个速度分量的Chemin--Lerner正则性准则
A Critical Chemin--Lerner Regularity Criterion via One Velocity Component for the Three-Dimensional Navier--Stokes Equations
- School of Science, Harbin University of Science and Technology（哈尔滨理工大学理学院）
- School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）
- Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对三维不可压缩Navier–Stokes方程，提出仅用一个速度分量的尺度临界正则性准则，通过频率-尺度匹配方案等机制，得到统一I型局部能量界并排除奇异爆破解极限。
AI中文摘要：
我们针对三维不可压缩Navier–Stokes方程的有限能量合适弱解，证明了仅涉及一个速度分量的尺度临界正则性准则。设2<p<∞，且m=3p/(p-2)，满足2/p+3/m=1。我们证明，只要∑_{j∈ℤ} ||Δ̇_j u³||_{L^p(0,T;L^m(ℝ³))}<∞，即u³∈L̃^p(0,T;Ḃ^0_{m,1}(ℝ³))，就不会出现奇异性。该假设是对仍未解决的临界条件u³∈L^p_tL^m_x的空间频率ℓ¹细化，且与Wang、Wu和Zhang得到的洛伦兹时间细化L^{p,1}_tL^m_x互补。通过伯恩斯坦嵌入，该结果可推广到非负光滑度临界线s=-1+2/p+3/q≥0上的u³∈L̃^p_tḂ^s_{q,1}。主要创新是嵌入在局部能量不等式中的频率-尺度匹配方案：u³的每个二进块都会保留，直到与一维向后热核选定的垂直尺度配对；低频从平板厚度中获益，高频则通过将投影转移到局域通量并应用逆伯恩斯坦估计获益，空间分离的压力源则通过调和衰减获益。这些机制生成了一个双侧ℓ¹核，将空间频率可和性转化为物理尺度能量增量的可和性，因此我们得到了无需时间洛伦兹细化的统一I型局部能量界，随后紧性和单分量刚性排除了第三速度分量消失的奇异爆破解极限。
英文摘要：
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 $2<p<\infty$ and $m=3p/(p-2)$, so that $2/p+3/m=1$. We show that a singularity cannot occur provided \[ \sum_{j\in\mathbb Z} \|\dotΔ_j u^3\|_{L^p(0,T;L^m(\mathbb R^3))}<\infty, \] that is, $u^3\in\widetilde L^p(0,T;\dot B^0_{m,1}(\mathbb R^3))$. The assumption is a spatial-frequency $\ell^1$ refinement of the still unresolved critical condition $u^3\in L^p_tL^m_x$ 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 $u^3\in\widetilde L^p_t\dot B^s_{q,1}$ on the nonnegative-smoothness critical line $s=-1+2/p+3/q\ge0$. 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 $\ell^1$ 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.
