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

TL;DR

This paper establishes the global well-posedness of the microscopic FENE model with a minimal boundary condition, showing that solutions exist uniquely when the distribution decays faster than the boundary distance, and demonstrating the condition's sharpness.

Contribution

It introduces a sharp boundary condition for the microscopic FENE model ensuring well-posedness, expanding understanding beyond zero flux boundary conditions.

Findings

Unique weak solutions exist under the new boundary condition.

The boundary decay condition is sharp, with infinitely many solutions if it is not met.

The distribution remains a probability density under the boundary requirement.

Abstract

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.