Global Well-posedness for the Microscopic Fene Model with a Sharp Boundary Condition

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 natural 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. This condition is strictly weaker than the natural flux boundary condition. Under this condition it is shown that there exists a unique weak solution in a weighted Sobolev space. 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.