  We study the small Deborah number limit of the Doi-Onsager equation for the
dynamics of nematic liquid crystals without hydrodynamics. This is a
Smoluchowski-type equation that characterizes the evolution of a number density
function, depending upon both particle position $x\in \mathbb{R}^d(d=2,3)$ and
orientation vector $m\in\mathbb{S}^2$ (the unit sphere). We prove that, when
the Deborah number tends to zero, the family of solutions with rough initial
data near local equilibria will converge strongly to a local equilibrium
distribution prescribed by a weak solution of the harmonic map heat flow into
$\mathbb{S}^2$. This flow is a special case of the gradient flow to the
Oseen-Frank energy functional for nematic liquid crystals. The key ingredient
is to show the strong compactness of the family of number density functions and
the proof relies on the strong compactness of the corresponding second moment
(or the $Q$-tensor), a spectral decomposition of the linearized operator near
the limit local equilibrium distribution, as well as the energy dissipation
estimate.
