Regularity criteria for generalized Navier–Stokes equations via middle eigenvalue of strain tensor

We establish regularity criteria for unsteady 3D flows of generalized Newtonian fluids with the p-power constitutive law for 53<p≤ 2 in terms of the middle eigenvalue of the strain tensor in homogeneous Besov spaces. These criteria are also valid for the horizontal derivatives of two velocity components or two components of vorticity. Notably, the criterion via the space L1(0,T;B˙∞,∞0(R3)) gives a positive answer to the related open problem for the classical Navier–Stokes equations.