Regularity criterion of the 3D generalized Navier-Stokes equations in the Triebel-Lizorkin spaces

In this paper, we consider the regularity of weak solutions to the 3D incompressible generalized Navier-Stokes (GNS) equations with fractional dissipative operator (−∆)αu in the Triebel-Lizorkin spaces. Let u be a weak solution in R3×(0, T ), w = curlu, and w = (w1, w2, 0). Employing the Littlewood-Paley decomposition, we prove that u becomes a classical solution if w ∈ Lq(0, T ; Ḟ 0 r,σ) with 2α q + 3 r = 2α, 3 2α < r ≤ ∞, and σ ≤ 2αr 3 .