Critical strong-time one-component regularity criterion

Prove regularity for general suitable weak solutions of the three-dimensional incompressible Navier–Stokes equations under the critical one-component condition u^3\in L^p(0,T;L^m(\mathbb R^3)) with 2/p+3/m=1 and 3<m<\infty.

Background

The paper identifies the scaling-critical one-component condition controlling only the third velocity component in the ordinary mixed-norm space Lp_tLm_x. Earlier results established one-component criteria in the strict Serrin range or at the critical line with stronger time summability, such as Lorentz-in-time control. The paper proves regularity instead under a spatial-frequency \ell1 refinement in a critical Chemin–Lerner/Besov space, while leaving the ordinary strong-time endpoint unresolved.

References

Nevertheless, the natural endpoint assertion \begin{equation}\label{eq:open-strong-time} u3\in Lp(0,T;Lm(\mathbb R3)), \qquad \frac2p+\frac3m=1, \qquad 3<m<\infty, \end{equation} remains open for general suitable weak solutions.

eq:open-strong-time:

u3∈Lp(0,T;Lm(R3)),2p+3m=1,3<m<∞,u^3\in L^p(0,T;L^m(\mathbb R^3)), \qquad \frac2p+\frac3m=1, \qquad 3<m<\infty,

— A Critical Chemin--Lerner Regularity Criterion via One Velocity Component for the Three-Dimensional Navier--Stokes Equations  (2609.03877 - Guo et al., 3 Sep 2026) in Section 1, immediately before the revision paragraph introducing the spatial-frequency refinement

The required perturbed Besov energy estimates, particularly those in the $p>4$ anisotropic regime, are not established in the present paper.

— A Critical one-component regularity criteria for the three-dimensional Navier--Stokes equations  (2609.34637 - Maotuo et al., 28 Sep 2026) in Remark 1.1, Section 1, immediately after Theorem 1.2 (labelled Remark~\ref{rem:besov-extension} in the manuscript)