Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbb R^2$

Abstract: In this paper, we first establish the regularity theorem for suitable weak solutions to the Ericksen-Leslie system in dimensions two. Building on such a regularity, we then establish the existence of a global weak solution to the Ericksen-Leslie system in $\mathbb R^2$ for any initial data in the energy space, under the physical constraint conditions on the Leslie coefficients ensuring the dissipation of energy of the system, which is smooth away from at most finitely many times. This extends earlier works by Lin,Lin, and Wang on a simplified nematic liquid crystal flow in dimensions two.