An improved regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity field

Abstract

In this paper, we establish a new multiplicative Sobolev inequality. As applications, we refine and extend the results in Kukavica and Ziane (J Math Phys 48:065203, 2007) and Cao (Discrete Contin Dyn Syst 26:1141-1151, 2010) simultaneously.

Global Regularity Criterion for the 3D Incompressible Navier–Stokes Equations Involving the Velocity Partial Derivative

“…When ∇u, ∇u 3 , u 3 , and the like satisfy a certain integrable condition, the weak solution is regular, and a large number of results are obtained (for details, refer to [6][7][8][9][10][11][12][13][14][15][16]). And Penel and Pokorný [13], Kukavica and Ziane [14], Cao [15], and Zhang [16], respectively, proved that the weak solution is regular on (0, T] when the weak solution satisfies the following conditions:…”

“…We estimate the right side of (21). By using the Hölder inequality, the Young inequality, ( 14), (16), and (17), we get that…”

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

“…We recall the following three-dimensional Sobolev-Ladyzhenskaya inequalities. 14,16 Lemma 3. For any 1 ≤ q < ∞, there exists a constant C q such that for any ∈…”

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

Some remarks on the Navier-Stokes equations with regularity in one direction

“…One way is to consider regularity criteria involving only one velocity component, which were done in [3], [9], [10], [13], [25], [27], and the references therein. Another way is to study the possible components reduction of ∇u to ∇u 3 , see for instance [4], [5], [7], [10], [16], [20], [22], [24], [26], [27]; or to ∂ 3 u, see for example [2], [11], [15], [14], [23]. In [14], Penel and Pokorný showed that if (1.4)…”