Studies On Well-Posedness And Regularity Of Solutions For The Liquid Crystal Model

This dissertation is concerned with the well-posedness and regularity of solutions for the liquid crystal model. The study in this area is one of the most important and challenging fields in mathematics because of many unsolved mathematical problems. The thesis is organized as follows.The first chapter introduces the background and significance of our study. We review some basic results on liquid crystal model. Finally we recall several important lemmas.The second chapter focuses on well-posedness of solutions for the liquid crystal model. We consider the global regularity for the 2D generalized liquid crystal model with fractional diffusion terms-∧2αu for the velocity field and -∧2βd for the vector field. Global existence of smooth solutions is proved for the case a=0,β> 1.In Chapter 3, we establish various regularity criteria for the 2D generalized liquid crystal model to guarantee smoothness of solutions. It turns out that our regularity criteria recover the results in previous global existence results naturally.Problems unsolved but expected to have breakthroughs are collected in the fourth chapter.