| Abstract: |
| The Hunter-Saxton equation is an integrable equation, and can be used to study the nonlinear instability in the director field of a nematic liquid. In this talk, we will first introduce a new generalized framework for energy conservative solutions of the Hunter-Saxton equation, and then discuss how to apply this new framework to investigate the regularity structure and long-time behavior of these solutions. In particular, some new observations have been found and rigorously shown: (i) singularities for the energy measure may only appear at at most countably many times, and are completely determined by the absolutely continuous part of initial energy measure; (ii) the temporal and spatial locations of singularities are explicitly determined by the initial data; and (iii) the long-time behavior of energy conservative solution is given by a kink-wave that is determined by the total energy of the system only. |
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