On the box-counting dimension of potential singular set for suitable weak solutions to the 3D Navier-Stokes equations

Abstract

In this paper, we are concerned with the upper box-counting dimension of the set of possible singular points in space-time of suitable weak solutions to the 3D Navier-Stokes equations. By taking full advantage of the pressure $Î $ in terms of $\nabla Î $ in equations, we show that this upper box dimension is at most $135/104(\approx1.30)$, which improves the known upper box-counting dimension $95/63(\approx1.51)$ in Koh et al. [9, J. Differential Equations, 261: 3137--3148, 2016], $45/29(\approx1.55)$ in Kukavica et al. [11, Nonlinearity 25: 2775-2783, 2012] and $135/82(\approx1.65)$ in Kukavica [10, Nonlinearity 22: 2889-2900, 2009].

Thanks to referees' crucial comments, we improved the box dimension of potential singular set for suitable weak solutions from 180/131 to 135/104 in this version