On Wolf's Regularity Criterion of Suitable Weak Solutions to the Navier-Stokes Equations

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 2 L 20 7, 15 4 Q(12) + . u 2 L2(Q(12)) = C u 2 L 20 7 (Q(1)) + C u 4 L 20 7 (Q(1)). This allows us to derive a new e-regularity criterion: Let u be a suitable weak solution in the Navier-Stokes equations. There exists an absolute positive constant e such that if u satisfies Q(1) | u| 20/ 7dxdt e, 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).