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 zero flux boundary condition. 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 no faster than the distance function.