Collaborative Research: L-infinity variational problems and the Aronsson equation

This is a US-China Collaboration Mathematics Research proposal, as a supple- ment of the P1's NSF grant 0601162. The partners from China are Professor Ding, Shijin from South China Normal University and Professor Liu, Xiangao from Fudan University, both of whom are currently supported by NSFC (see their bibliography sketches). The proposal consists two parts. In Part I, we plan to continue our collabora- tion on the analysis of the Landau-Lifshitz equation coupled with time dependent Maxwell equation. such as existence of partially smooth suitable weak solutions, uniqueness problem for suitable weak solutions, existence of finite time singularity, and long time behavior of suitable weak solutions. In Part II, we plan to initiate a new collaboration on the analysis of the hydrodynamic flow of liquid crystal mate- rials, which models the flow phenomena of nematic liquid crystal and roughly is a strong coupling system between the macroscopic motion like Navier-Stokes equation and a "gradient-like" flow of the Ossen-Frank energy for the optical axis. We first plan to study the problem in dimension two, which includes to establish a small en- ergy regularity theorem and the existence of weak solutions that exhibit both local and global energy inequalities, and then the problem in dimension 3. In dimension 2, we also propose several natural, but difficult questions for the compressible flow.