  This article represents a first step towards understanding the well-posedness
for the dispersive Hunter-Saxton equation. This problem arises in the study of
nematic liquid crystals, and although the equation has formal similarities with
the KdV equation, the lack of $L^2$ control gives it a quasilinear character,
with only continuous dependence on initial data.
  Here, we prove the local and global well-posedness of the Cauchy problem
using a normal form approach to construct modified energies, and frequency
envelopes in order to prove the continuous dependence with respect to the
initial data.
