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

Since the seminal work of Caffarelli, Kohn and Nirenberg {CKNP} 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 dimH(S) of S is always bounded below by its box-counting dimension dimB(S). Through a series of investigations, the initial bound dimB(S)≤5/3 established in {RSA} has been progressively improved to dimB(S)≤7/6 in {WYI}. Recently, Gong, Wang and Zhang {GWZP} extended the foundational work of {CKNP} 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 7/6.