Partial regularity for Navier–Stokes and liquid crystals inequalities without maximum principle

In 1985, V. Scheffer discussed partial regularity results for what he called to the inequality. These maps essentially satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier-Stokes system of equations, but they are not required to satisfy the Navier-Stokes system itself. We extend this notion to a system considered by Fang-Hua Lin and Chun Liu in the mid 1990s related to models of the flow of nematic liquid crystals, which include the Navier-Stokes system when the director field $d$ is taken to be zero. In addition to an extended Navier-Stokes system, the Lin-Liu model includes a further parabolic system which implies a maximum principle for $d$ which they use to establish partial regularity of solutions. For the analogous inequality one loses this maximum principle, but here we establish certain partial regularity results nonetheless. Our results recover in particular the partial regularity results of Caffarelli-Kohn-Nirenberg for suitable weak solutions of the Navier-Stokes system, and we verify Scheffer's assertion that the same hold for of the weaker inequality as well.