We deal with several models treating the time-evolution of nematic liquid crystals. We analyze the major theories such as the Ericksen-Leslie and the Beris-Edwards, for nematics living in the whole space. This allows us to successfully take advantage of Fourier analysis and paradifferential calculus tools when performing global-in-time well-posedness. The class affinity of the considered solutions is that of Besov spaces.
We finally take into consideration the formalism of Qian-Sheng, modeling the inertial contribution of twisting constitutive molecules. We perform the existence of the so-called twist-wave solutions.

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