Compactness, existence, and partial regularity in hydrodynamics of liquid crystals

This thesis mainly focuses on the PDE theories that arise from the study of hydrodynamics of nematic liquid crystals.

In Chapter 1, we give a brief introduction of the Ericksen--Leslie director theory and Beris--Edwards Q-tensor theory to the PDE modeling of dynamic continuum description of nematic liquid crystals. In the isothermal case, we derive the simplified Ericksen--Leslie equations with general targets via the energy variation approach. Following this, we introduce a simplified, non-isothermal Ericksen--Leslie system and justify its thermodynamic consistency.

In Chapter 2, we study the weak compactness property of solutions to the Ginzburg--Landau approximation of the simplified Ericksen--Leslie system. In 2-D, we apply the Pohozaev type argument to show a kind of concentration cancellation occurs in the weak sequence of Ginzburg--Landau system. Furthermore, we establish the same compactness for non-isothermal equations with approximated director fields staying on the upper semi-sphere in 3-D. These compactness results imply the global existence of weak solutions to the limit equations as the small parameter tends to zero.

In Chapter 3, we establish the global existence of a suitable weak solution to the co-rotational Beris–Edwards system for both the Landau–De Gennes and Ball–Majumdar bulk potentials in 3-D, and then study its partial regularity by proving that the 1-D parabolic Hausdorff measure of the singular set is 0.

In Chapter 4, motivated by the study of un-corotational Beris--Edwards system, we construct a suitable weak solution to the full Ericksen--Leslie system with Ginzburg--Landau potential in 3-D, and we show it enjoys a (slightly weaker) partial regularity, which asserts that it is smooth away from a closed set of parabolic Hausdorff dimension at most 15/7.