The optimal regularity criterion for the Navier‐Stokes equations in terms of one directional derivative of the velocity

Abstract

We study the regularity criterion for the Navier‐Stokes equations in terms of one directional derivative of the velocity field. Using a suitable multiplicative Sobolev inequality we extend the results by Kukavica & Ziane, Cao and Zhang.

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

“…Cao extended [3] this result to the range of exponents 27 16 ≤ q ≤ 3, although the proof in this paper only covers the range 27 16 ≤ q ≤ 5 2 , with the rest of the range already proven by Kukavica and Ziane in [24]. Zhang extended [49] the range of exponents to include 3 √ 37 4 − 3 ≤ q ≤ 3, and Namlyeyeva and Skalák then extended the lower bound on this range further in [33], although still not to the endpoint case q = 3 2 . Finally, Skalák extended [44,46] this result to include the range 3 2 < q ≤ 19 6 .…”

Prodi–Serrin condition for 3D Navier–Stokes equations via one directional derivative of velocity

“…There are also many efforts to extend the range of q for ∂ 3 u, such as [3,16,12,21]. In particular, the first author of this paper and D. Fang and T. Zhang [7] proved that u is regular in (0, T ], if…”

Critical regularity criteria for Navier–Stokes equations in terms of one directional derivative of the velocity

“…Later on, Cao, 14 Namlyeyeva and Skalak, 15 and Zhang 16 extended the range of q to q ∈ [1.5620, 3].…”