  We study an initial value problem for two-dimensional needle crystal growth
with anisotropic surface tension. The initial value problem is derived from the
so called one-sided model based on complex variables method. We then obtain the
existence and uniqueness of local solution of the needle crystal problem for
any initial interface. Furthermore, we obtain that, on average in time, the
solution gains 3/2 derivative of smoothness in spatial variable compared to the
initial data. The continuous dependence on the initial data of the solution map
is also established.
