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

Global Well-Posedness for the Microscopic FENE Model with a Sharp Boundary Condition