Partial regularity of the Ericksen-Leslie system with Ginzburg-Landau approximations

In this talk, we will discuss the regularity of suitable weak solutions to the Ericksen-Leslie system with Ginzburg-Landau relaxation which allows variable length for modeling the hydrodynamics of liquid crystal molecules. The main difficulty of the mathematical analysis for the PDE system is the absence of the maximum principle. However, we show an improvement of flatness property for the suitable weak solutions with smallness imposed on some suitable localized scaling-invariant quantities. As a result, we construct a suitable weak solution that is smooth away from a closed set of parabolic Hausdorff dimension at most 15/7. This is joint work with Changyou Wang (Purdue).