  We present a rigorous derivation of the Ericksen-Leslie equation starting
from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann
equation, we first make the Hilbert expansion for the solution of the
Doi-Onsager equation. The existence of the Hilbert expansion is connected to an
open question whether the energy of the Ericksen-Leslie equation is dissipated.
We show that the energy is dissipated for the Ericksen-Leslie equation derived
from the Doi-Onsager equation. The most difficult step is to prove a uniform
bound for the remainder in the Hilbert expansion. This question is connected to
the spectral stability of the linearized Doi-Onsager operator around a critical
point. By introducing two important auxiliary operators, the detailed spectral
information is obtained for the linearized operator around all critical points.
However, these are not enough to justify the small Deborah number limit for the
inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity
equation is also strongly singular. For this, we need to establish a precise
lower bound for a bilinear form associated with the linearized operator. In the
bilinear form, the interactions between the part inside the kernel and the part
outside the kernel of the linearized operator are very complicated. We find a
coordinate transform and introduce a five dimensional space called the
Maier-Saupe space such that the interactions between two parts can been seen
explicitly by a delicate argument of completing the square. However, the lower
bound is very weak for the part inside the Maier-Saupe space. In order to apply
them to the error estimates, we have to analyze the structure of the singular
terms and introduce a suitable energy functional.
