On Wolf’s Regularity Criterion of Suitable Weak Solutions to the Navier–Stokes Equations

Abstract

In this paper, we consider the local regularity of suitable weak solutions to the 3D incompressible Navier–Stokes equations. By means of the local pressure projection introduced by Wolf (in: Rannacher, Sequeira (eds) Advances in mathematical fluid mechanics, Springer, Berlin, 2010, Ann Univ Ferrara 61:149–171, 2015), we establish a Caccioppoli type inequality just in terms of velocity field for suitable weak solutions to this system

$$\begin{aligned} \Vert u\Vert ^{2}_{L^{\frac{20}{7},\frac{15}{4}}Q(\frac{1}{2})}+ \Vert \nabla u\Vert ^{2}_{L^{2}(Q(\frac{1}{2}))} \le C \Vert u\Vert ^{2}_{L^{\frac{20}{7}}(Q(1))}+C\Vert u\Vert ^{4}_{L^{\frac{20}{7}}(Q(1))}. \end{aligned}$$

This allows us to derive a new $\varepsilon$-regularity criterion: Let u be a suitable weak solution in the Navier–Stokes equations. There exists an absolute positive constant $\varepsilon$ such that if u satisfies

$$\begin{aligned} \iint _{Q(1)}|u|^{20/7}dxdt< \varepsilon , \end{aligned}$$

then u is bounded in some neighborhood of point (0, 0). This gives an improvement of previous corresponding results obtained in Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), in Guevara and Phuc (Calc Var 56:68, 2017) and Wolf (Ann Univ Ferrara 61:149–171, 2015).