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We present regularity conditions for a solution to the 3D Navier-Stokes equations, the 3D Euler equations and the 2D quasigeostrophic equations, considering the vorticity directions together with the vorticity magnitude. It is found that the regularity of the vorticity direction fields is most naturally measured in terms of norms of the Triebel-Lizorkin type. 1. The generalized Navier-Stokes equations. In this section we are concerned with the following ‘generalized’ Navier-Stokes equations: ∂v ∂t + (v · ∇)v = −∇p − νΛv, (1) div v = 0, (2) v(x, 0) = v0(x), (3) where v = (v, v, v), v = v(x, t), j = 1, 2, 3, is the velocity of the fluid flows, p = p(x, t) is the scalar pressure, v0(x) is a given initial velocity field satisfying div v0 = 0, and ν > 0 is the viscosity constant. We write Λ = (−∆) α 2 . We denote the system (1)–(3) by (NS)α. The case α = 2 corresponds to the usual Navier-Stokes equations, which we denote simply by (NS). Heuristically, the number α represents the ‘strength of dissipation’, and in this paper we are concerned with the case 0 < α ≤ 2. We note that the system (NS)α was first considered by J.-L. Lions in [26], and the global regularity α ≥ 52 is shown there. Taking curl of (1) we obtain the following vorticity equation: ∂ω ∂t + (v · ∇)ω = (ω · ∇)v − νΛω, (4) 2000 Mathematics Subject Classification: 35Q35, 76B03, 76D05.

On the conditional regularity of the Navier-Stokes and related equations