New $$\varepsilon $$-Regularity Criteria of Suitable Weak Solutions of the 3D Navier–Stokes Equations at One Scale

Abstract

In this paper, by invoking the appropriate decomposition of pressure to exploit the energy hidden in pressure, we present some new ε-regularity criteria for suitable weak solutions of the 3D Navier-Stokes equations at one scale: for any p, q ∈ [1, ∞] satisfyingThis is an improvement of corresponding results recently proved by Guevara and Phuc in [7, Calc. Var. 56:68, 2017]. As an application of these ε-regularity criteria, we improve the known upper box dimension of the possible interior singular set of suitable weak solutions of the Navier-Stokes system from 975/758(≈ 1.286) [16] to 2400/1903(≈ 1.261).MSC (2000): 35B65, 35D30, 76D05

Leray's backward self-similar solutions to the 3D Navier-Stokes equations in Morrey spaces

“…Lemma 2.1. [12] Let Φ denote the standard normalized fundamental solution of Laplace equation in R 3 . For 0 < ξ < η, we consider smooth cut-off function…”

“…where α = 2/p + 3/q < 2. Additionally, we apply the pressure decomposition as [12] to bound the second term in the left hand side of (1.8). To sum up, we get the following energy bound…”

The role of the pressure in the regularity theory for the Navier-Stokes equations

“…In the developments of [12,14,8], the smallness assumption in the classical theorem has been relaxed; in particular, in [14,8], for any (r, m)…”

$\varepsilon$-regularity criteria in Lorentz spaces to the 3D Navier-Stokes equations

“…The difference between (1.6) and (1.7) is that the former requires only a radius (one scale) and the latter needs infinite radii (finitely many scales). Since then, there have been extensive mathematical investigations of regularity of suitable weak solutions and many regularity conditions are established (see [10,12,30,38,39,41] and references therein). However, almost all known results involving ε-regularity criteria are discussed in usual Lebesgues spaces and there is a little literature for investigating ε-regularity criteria in Lorentz spaces.…”

“…Thirdly, the structure of potential singular set S of solutions in (1.1) attracted extensive research such as upper bound of box dimension or upper bound for the number of singular points S( see e.g. [6,7,12,19,25,35,38,42]). We set S(t) = {(x, t) ∈ S}.…”