A refined regularity criterion for the Navier–Stokes equations involving one non-diagonal entry of the velocity gradient

Global existence of weak solution and regularity criteria for the 2D Bénard system with partial dissipation

“…In particular, if θ = Const., then system (1.1) reduces to the classical Navier-Stokes system which describes the motion of incompressible viscous fluid flows and has been extensively studied by many authors; see [18][19][20][21][22][23][24] and the references therein. In addition to this, the reader is referred to [25][26][27][28][29] to find more results about the related fluid flow equations.…”

A survey of geometric constraints on the blowup of solutions of the Navier--Stokes equation

“…There are no scale-critical regularity criteria in terms of just one entry of ∇u, but there are a number of results giving subcritical regularity criteria in terms of just one diagonal entry ∂ i u i or one non-diagonal entry ∂ i u j [4,11,17,18,39,41,47,48,50,51,53]. The regularity criteria in terms of one diagonal entry of ∇u are closer to being scale-critical than the regularity criteria in terms of just one non-diagonal entry.…”

The application of anisotropic Troisi inequalities to the conditional regularity for the Navier–Stokes equations

“…It is then described in detail in the proof of theorem 2 between (51) and ( 52). In its rudimentary stage it was used in [24], where the estimate of the term…”