Regularity for solutions to the Navier-Stokes equations with one velocity component regular

By Cheng He. 2002

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A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Gradient of One Velocity Component (2002, background): “Remark 1. In [ 6 ], He proved the same conclusion under a stronger condition ∇u3 ∈ L�, with 2/� + 3/ = 1.”
A Regularity Criterion for the Navier–Stokes Equations (2007, background): “Neustupa et al. (2002) considered the regularity in the sense of one component, and He (2002) followed.”
On the result of He concerning the smoothness of solutions to the Navier-Stokes equations (2003, methodology): “) In the recent paper [4], He followed similar aim and obtained the regularity of the Navier-Stokes system provided ∇u3 ∈ L(I;L), 2t + 3 s ≤ 1, 2 ≤ t ≤ ∞, 3 ≤ s ≤ ∞.”
Log Improvement of the Prodi-Serrin Criteria for Navier-Stokes Equations (2007, background): “Let us also cite a condition involving the lower bound of the pressure introduced by Seregin and Sverak in [19], and conditions involving only one of the component of u (see Penel and Pokorny [17], He [9], and Zhou [23, 26]).”