On the Critical One Components Regularity for the $3-D$ Navier-Stokes System in $L^p_T(\dot{B}^{\frac 1 2+\frac 2 p}_{2,\infty})$ spaces

Abstract: We consider the conditional regularity of the mild solution $v$ of the $3-D$ incompressible Navier-Stokes equations with initial data $v_0\in \dot{H}^{\frac 1 2}$ and vorticity $Ω<em>0\in L<sup>{r_0}$ for some $r_0\in (1,2)$. We prove that if the solution associated with initial data $v_0$ blows up at a finite time $T<sup>\ast$, then for any $2&lt;p&lt;\infty$, and any unit vectors $e$ in $\mathbb{R}<sup>3$, the integral $$\int_0^{{T^{\ast}\left\Vert}} (v(t)|e)</em>{\mathbb{R}^{3}\right\Vert_{\dot{B}^{\frac} 1 2+\frac 2 p}_{2,\infty}}^p{\rm d}t$$ blows up at $T<sup>\ast$. The conclusion improves the recent results in Chemin et al. (Arch Ration Mech Anal 224(3):871-905, 2017) and Han et al. (Arch. Rational Mech. Anal. 231:939-970, 2019).

• The paper establishes that controlling a single velocity component in critical Besov spaces ensures regularity of mild solutions, strictly improving previous criteria.
• It employs anisotropic decompositions, duality approaches, and refined Besov interpolation to manage nonlinearities beyond standard Sobolev embeddings.
• The work has significant implications for theoretical analysis and practical diagnostics in fluid dynamics, promoting componentwise regularity testing.

Critical One Component Regularity for 3D Navier-Stokes in $L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$ Spaces

Introduction and Motivation

The regularity theory of the incompressible 3D Navier-Stokes equations remains a central open problem in PDE and mathematical fluid mechanics. A major line of research investigates conditional regularity criteria: which finite (or infinite) space-time quantities are sufficient to guarantee regularity of weak solutions up to a possibly singular time $T^*$. Among the sharpest results are the so-called "one component" criteria: is it sufficient to control a single component of the velocity field (or its projection) in a critical function space to preclude singularity formation?

The work under discussion establishes a new, strictly improved blow-up criterion for the Navier-Stokes system in terms of one component (or directional projection) of the velocity, formulated in space-time critical Besov spaces $L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$, where $2 < p < \infty$.

Problem Setting and Regularity Framework

Let $v$ be a mild solution to the 3D incompressible Navier-Stokes equations on $\mathbb{R}^3$:

$$\begin{cases} \partial_t v + v \cdot \nabla v - \Delta v + \nabla P = 0, \ \operatorname{div} v = 0, \ v|_{t=0} = v_0. \end{cases}$$

with $v_0 \in \dot H^{1/2}$ and initial vorticity $\Omega_0 \in L^{r_0}$, $1 < r_0 < 2$.

The classical Prodi-Serrin-Ladyzhenskaya criteria assert that if $T^*$0 with $T^*$1, $T^*$2, then regularity up to $T^*$3 is guaranteed. More refined criteria concern critical spaces like $T^*$4 for velocity, or $T^*$5, and more recently, criteria that only require space-time control on a single velocity component or directional projection.

Main Result

Theorem:

If for some $T^*$6, and some unit vector $T^*$7,

$T^*$8

then the mild solution $T^*$9 is regular up to time $L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$0: that is,

$L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$1

and higher-order regularity follows for positive times.

Conversely, if $L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$2 develops a singularity at $L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$3, then for any unit vector $L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$4 and any $L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$5, the above integral must diverge. This gives a true one-component blow-up criterion in the critical space.

Strength of the claim: This criterion strictly improves earlier results in which the space $L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$6 was employed—indeed, the homogeneous Besov norm $L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$7 is strictly weaker than the corresponding Sobolev norm, as the embedding $L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$8 holds.

Method of Proof and Technical Innovations

The approach builds on, sharpens, and extends earlier works such as Chemin-Zhang [Chemin2016], [Zhang2017], Lei et al. [Lei2019], and Han et al. [Lei2019], which used anisotropic decompositions and fine harmonic analysis.

Anisotropic Biot–Savart Decomposition

The key is to decompose $L^p_T(\dot{B}^{\frac12+\frac2p}_{2,\infty})$9 into horizontal and vertical components and relate the horizontal velocities to 2D vorticity and the vertical component via anisotropic Biot–Savart laws. The equations governing the evolution of the vertical vorticity and $2 < p < \infty$0 (the third component) highlight the interaction mechanisms most relevant for one-component criteria.

Duality and Refined Besov-type Interpolation

To estimate nonlinear terms that cannot be handled by Sobolev embeddings alone, the authors develop a duality approach in the appropriate function spaces, exploiting interpolation inequalities, embedding relations between anisotropic and isotropic Besov spaces, and commutator/Bony paraproduct decompositions.

A key technical achievement is a refined estimate that allows control of quadratic nonlinearities (e.g., $2 < p < \infty$1) using only the one-component norm $2 < p < \infty$2 and suitable vorticity norms, even when these do not close under standard Sobolev machinery.

Complete Coverage of $2 < p < \infty$3

Unlike previous results, this analysis applies for all $2 < p < \infty$4 (excluding the endpoint $2 < p < \infty$5), regardless of whether $2 < p < \infty$6, and crucially, works with the endpoint-type Besov space parameter $2 < p < \infty$7.

Avoidance of Full Norm Closure Limitation

For $2 < p < \infty$8, where Hilbert-space interpolation fails to recover the critical norm, a sophisticated Bony decomposition and commutator estimate are needed, handling the lack of regularity in $2 < p < \infty$9 and yielding closure via anisotropic product and embedding theorems.

Comparison with Previous Work

• Chemin–Zhang [Chemin2016]: Proved results for $v$0 for the one-component $v$1 case by anisotropic methods.
• Lei et al. [Lei2019]: Extended to $v$2, but with stronger norm assumptions.
• This paper: Strictly generalizes all these results, replacing $v$3 norms by the larger (less regular) Besov spaces $v$4.

Furthermore, the methods here clarify and overcome the technical blockages previously encountered—particularly for $v$5 near $v$6, and for divergence control in strongly anisotropic settings.

Implications and Consequences

Theoretical Insights

• Demonstrates that regularity can be characterized by an arbitrarily chosen directional component of the velocity in a minimal (i.e., least regular, largest possible) critical norm.
• Suggests that true singularities in Navier-Stokes solutions are necessarily reflected in every direction/component, in the sense of divergence of the corresponding Besov space-time norm.

Potential for Future Generalizations

• Opens a pathway to accessing endpoint cases ($v$7), which would require control over even weaker norms.
• The techniques developed can be adapted to study componentwise or directionwise criteria in related active scalar or fluid equations, possibly even in higher geometric settings or non-Euclidean domains.
• The approach may give heuristics for constructing candidate singularity blow-up scenarios by controlling or failing to control only a single component.

Practical Analytic Effects

• Enables 'reductionist' regularity testing in numerical or analytic studies: only a single directional projection (not the full velocity field) need be estimated in the sharpest critical norms.
• Justifies certain model reductions or experimental diagnostics that focus on individual flow directions in turbulence or computational simulations.

Conclusion

This paper provides a significant advancement in the conditional regularity theory for the 3D Navier-Stokes equations, proving that control of a single component of the velocity in the minimized critical norm $v$8 suffices to preclude singularity formation. The work strictly strengthens prior results and employs new analytic techniques in anisotropic harmonic analysis, duality, and Besov interpolation. The findings sharpen both our understanding of nonlinear fluid dynamics regularity and the technical framework available for further attacks on the core regularity problem. Future work may aim to resolve the remaining endpoint or even more singular cases, and to extend these ideas to other nonlinear evolution equations.

Reference:

"On the Critical One Components Regularity for the $v$9 Navier-Stokes System in $\mathbb{R}^3$0 spaces" (2607.01587)