Under a fixed scale-invariant bound on suitable weak solutions of 3D Navier-Stokes, smallness of the vertical velocity component yields a positive lower bound on the local regularity radius via harmonic pressure approximation.
Wolf, On the local pressure of the Navier–Stokes equations and related systems, Advances in Differential Equations22(2017), no
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Finite-Scale One-Component Regularity via Harmonic Pressure for the 3D Navier-Stokes Equations
Under a fixed scale-invariant bound on suitable weak solutions of 3D Navier-Stokes, smallness of the vertical velocity component yields a positive lower bound on the local regularity radius via harmonic pressure approximation.