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<p>In this paper, we derive a refined anisotropic regularity criterion for the three-dimensional incompressible Navier–Stokes equations. The criterion is based on a sharper estimate associated with the strain tensor and a family of scale-invariant quantities measuring anisotropic deformation. We show that the boundedness of these quantities in a critical space-time class ensures that Leray–Hopf weak solutions remain regular on (0,&#xa0;<i>T</i>]. This establishes a hierarchy of anisotropic regularity conditions and refines several existing regularity criteria.</p>

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A refined scale-invariant anisotropic regularity criterion via \(\Psi _\alpha \) for the 3d Navier-Stokes equations

  • 期刊论文

  • 发表日期：

    2026年7月9日

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  • Fan Wu

摘要

In this paper, we derive a refined anisotropic regularity criterion for the three-dimensional incompressible Navier–Stokes equations. The criterion is based on a sharper estimate associated with the strain tensor and a family of scale-invariant quantities measuring anisotropic deformation. We show that the boundedness of these quantities in a critical space-time class ensures that Leray–Hopf weak solutions remain regular on (0, T]. This establishes a hierarchy of anisotropic regularity conditions and refines several existing regularity criteria.