The Landau-de Gennes theory on the study of nematic liquid crystals was initiated by the Physics Nobel Prize winner P.-G. de Gennes in the 1970's, and it began to attract the extensive attention of mathematicians in recent years. This theory is a phenomenological theory in which stable states of the liquid crystal material correspond to minimizers of a free energy. In its free energy form, there is however an unusual cubic term. In this talk we will discuss the dynamic effects induced by this cubic term by considering the $L^2$ gradient flow dynamics generated by the free energy. We work in both two and three dimensions and focus on understanding the relations between the physicality of the initial data and global well-posedness of the system.