Local well-posedness of the compressible FENE dumbbell model of Warner type

TL;DR

This paper proves the local well-posedness of a coupled system modeling a dilute suspension of FENE dumbbells in a compressible fluid, demonstrating existence and uniqueness of smooth solutions in a mathematically rigorous framework.

Contribution

It establishes the first local well-posedness result for the compressible FENE dumbbell model coupled with Navier--Stokes equations, regardless of the inclusion of diffusion in the Fokker--Planck equation.

Findings

Existence of a unique local-in-time solution

Solution is smooth in spacetime variables

Result holds with or without center-of-mass diffusion

Abstract

We consider a dilute suspension of dumbbells joined by a finitely extendible nonlinear elastic (FENE) connector evolving under the classical Warner potential $U(s)=-\frac{b}{2} \log(1-\frac{2s}{b})$, $s\in[0,\frac{b}{2})$. The solvent under consideration is modelled by the compressible Navier--Stokes system defined on the torus $\mathbb{T}^d$ with $d=2,3$ coupled with the Fokker--Planck equation (Kolmogorov forward equation) for the probability density function of the dumbbell configuration. We prove the existence of a unique local-in-time solution to the coupled system where this solution is smooth in the spacetime variables and interpreted weakly in the elongation variable. Our result holds true independently of whether or not the centre-of-mass diffusion term is incorporated in the Fokker--Planck equation.