On the interior regularity criteria and the number of singular points to the Navier-Stokes equations

Abstract

We establish some interior regularity criterions of suitable weak solutions for the 3-D Navier-Stokes equations, which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin's criterion and Escauriza-Seregin-Šverák's criterion. We also show that if weak solution u satisfiesfor some 3 < p < ∞, then the number of singular points is finite.

Remark on Wolf’s Condition for Boundary Regularity of the Navier–Stokes Equations

“…This type of theorems in the case of interior regularity appeared in [6], for further developments, see, for example, [3], [8], and [2].…”

Local Boundary Regularity for the Navier–Stokes Equations in Non-Endpoint Borderline Lorentz Spaces

“…This was also done near the curved part of the boundary in [18]. In the context of Lorentz spaces, the interior result is proven in [27].…”

“…Recently, a version is proven in [19] but with the additional restriction that p ∈ L 2 (−1, 0; L 1 (B)). (1.8) We will first present a different proof to [27] of Theorem 1.2 under the assumption that u ∈ L ∞ (−1, 0; L 3,q (B)). The contradiction argument has the same spirit as that used in [3], [19] and [27].…”

New Regularity Criteria Based on Pressure or Gradient of Velocity in Lorentz Spaces for the 3D Navier–Stokes Equations

“…Before formulating our results, we mention that, as (1.8), regularity criteria in terms of pressure Π or gradient of pressure ∇Π with only space direction belonging to Lorentz spaces can be found in [7,33]. As for other regularity criteria involving Lorentz spaces, see [5,10,13,21,22,31]. Now our first result is stated as follows.…”