2009/05/07 by Hailiang Liu, Liu, Hailiang, Jaemin Shin +1
Economics, Econometrics and Finance · Mathematics · #35K20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.0905.1142
openalex publication_date 2009/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.