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 $\Pi$ in terms of $\nabla \Pi$ in equations, we show that this upper box dimension is at most $135/104(\approx 1.30)$, which improves the known upper box-counting dimension $95/63(\approx 1.51)$ in Koh et al. [9, J. Differential Equations, 261: 3137--3148, 2016], $45/29(\approx 1.55)$ in Kukavica et al. [11, Nonlinearity 25: 2775-2783, 2012] and $135/82(\approx 1.65)$ 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

References in corpus (1)

• Partial regularity of suitable weak solutions to the multi-dimensional generalized magnetohydrodynamics equations

Cited by in corpus (2)

• New $\epsilon$-regularity criteria of suitable weak solutions of the 3D Navier-Stokes equations at one scale
• Improved bounds for box dimensions of potential singular points to the Navier--Stokes equations