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In this paper we investigate the three dimensional general Ericksen-Leslie(E--L) system with Ginzburg-Landau type approximation modeling nematic liquidcrystal flows. First, by overcoming the difficulties from lack of maximumprinciple for the director equation and high order nonlinearities for thestress tensor, we prove existence of global-in-time weak solutions underphysically meaningful boundary conditions and suitable assumptions on theLeslie coefficients, which ensures that the total energy of the E--L system isdissipated. Moreover, for the E--L system with periodic boundary conditions, weprove the local well-posedness of classical solutions under the so-calledParodi's relation and establish a blow-up criterion in terms of the temporalintegral of both the maximum norm of the curl of the velocity field and themaximum norm of the gradient of the liquid crystal director field.

Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows