A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Gradient of One Velocity Component

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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 [15], and conditions involving only one of the component of u (see Penel and Pokorny [13], He [5], and Zhou [19]).”
Global Regularity Criterion for the 3D Navier–Stokes Equations Involving One Entry of the Velocity Gradient Tensor (2010, unknown): “conditions (see, for example, [4,15,18,26,27, 38 ]).”
Regularity Criteria for the Three-dimensional Navier-Stokes Equations (2008, background): “In addition, some other sufficient regularity conditions were established in terms of only one component of the velocity field of the 3D NSE on the whole space R 3 or under periodic boundary conditions (cf. e.g., [20], [23], [35], [46]).”
Regularity criterion for 3d navier-stokes equations in terms of the direction of the velocity (2007, background): “(see [14] for extension of this result).”
A new regularity criterion for weak solutions to the Navier–Stokes equations (2005, background): “3 Comparing with the previous regularity criterion [26] ∇u3 ∈ L with 2 α + 3 γ ≤ 3 2 , establishing a priori estimates here are much more difficult than those.”