Partial regularity of suitable weak solutions to the four-dimensional incompressible magneto-hydrodynamic equations

摘要

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto-hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier-Stokes equations. The so-called ε-regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense.