Regularity criteria for suitable weak solutions to the four dimensional incompressible magneto-hydrodynamic equations near boundary

Abstract

In this paper, we consider suitable weak solutions of the four dimensional incompressible magnetohydrodynamic equations. We give two different kind ε-regularity criteria. One only requires the smallness of scaling L p,q norm of u, another requires the smallness of scaling space time L 2 norm of ∇u and boundedness of scaling norm of H or ∇H . And as an application of the second kind criteria, we also prove that up to the boundary, the two-dimensional Hausdorff measure of the set of singular points is equal to zero.

Partially regular weak solutions of the Navier-Stokes equations in $\mathbb{R}^4 \times [0,\infty[$

“…Gu [9] obtained some partial regularity criteria for suitable weak solutions to (1.7) in space dimension 4 assuming that these solutions exist. It is likely that one can construct partially regular weak solutions of the incompressible magneto-hydrodynamic equations with our strategy, but we do not pursue it here.…”