New <i>ε</i>-regularity criteria and improved box-counting dimension of potential singular set for suitable weak solutions to the Navier-Stokes-Planck-Nernst-Poisson system

Abstract Since the seminal work of (Caffarelli et al 1982 Comm. Pure Appl. Math. 35 771—831) on the partial regularity theory for the three-dimensional incompressible Navier-Stokes equations, where they proved that the one-dimensional parabolic Hausdorff measure of the space-time singular set is zero, extensive research has been devoted to the analysis of the partial regularity and singularity for the Navier-Stokes equations. The current understanding suggests that the dimensional bound on the parabolic Hausdorff measure of space-time singular sets cannot be further sharpened with existing methods. Consequently, scholars turn to investigate an alternative fractal dimension–the box-counting dimension–with the aim of proving that the box-counting dimension of the potential space-time singular set S is at most one. This approach is motivated by the fact that the Hausdorff measure dim H ( S ) of S is always bounded below by its box-counting dimension dim B ( S ). Through a series of investigations, the initial bound <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>5</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:mfrac> </mml:math> established in (Robinson JC and Sadowski 2009 Nonlinearity 22 2093—9) has been progressively improved to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:msub> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>7</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>6</mml:mn> </mml:mrow> </mml:mfrac> </mml:math> in (Wang and Yang 2019 Nonlinearity 32 4817—33). Recently, (Gong et al 2021 SIAM J. Math. Anal. 53 3306—37) extended the foundational work of (Caffarelli et al 1982 Comm. Pure Appl. Math. 35 771—831) on the three-dimensional incompressible Navier-Stokes equations to the Navier-Stokes-Planck-Nernst-Poisson system and proved that the one-dimensional parabolic Hausdorff measure of the space-time singular set is also zero. However, the box-counting dimension of the singular set remains unexplored. In this paper, we investigate the box-counting dimension of the space-time singular set for suitable weak solutions to the Navier-Stokes-Planck-Nernst-Poisson system. By establishing a series of new ε -regularity criteria, we prove that the box-counting dimension of the space-time singular set is at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfrac> <mml:mrow> <mml:mn>7</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>6</mml:mn> </mml:mrow> </mml:mfrac> </mml:math> .