  We prove global well-posedness for the microscopic FENE model under a sharp
boundary requirement. The well-posedness of the FENE model that consists of the
incompressible Navier-Stokes equation and the Fokker-Planck equation has been
studied intensively, mostly with the zero flux boundary condition. Recently it
was illustrated by C. Liu and H. Liu [2008, SIAM J. Appl. Math.,
68(5):1304--1315] that any preassigned boundary value of a weighted
distribution will become redundant once the non-dimensional parameter $b>2$. In
this article, we show that for the well-posedness of the microscopic FENE model
($b>2$) the least boundary requirement is that the distribution near boundary
needs to approach zero faster than the distance function. Under this condition,
it is shown that there exists a unique weak solution in a weighted Sobolev
space. Moreover, such a condition still ensures that the distribution is a
probability density. The sharpness of this boundary requirement is shown by a
construction of infinitely many solutions when the distribution approaches zero
as fast as the distance function.
