Incompressible limit for the 3D compressible FENE dumbbell model

TL;DR

This paper investigates the global-in-time transition from a compressible to an incompressible FENE dumbbell model in 3D, developing new estimates to handle the challenges posed by large volume viscosity.

Contribution

It introduces a novel momentum-based estimate that enables the derivation of uniform decay rates and convergence towards the incompressible limit in the FENE model.

Findings

Established decay rates for strong solutions.

Proved faster decay of the momentum's incompressible component.

Achieved a time-decreasing convergence rate to the incompressible limit.

Abstract

In this work, we study the global-in-time incompressible limit of the compressible FENE dumbbell model on the three-dimensional torus T^3, where the incompressible limit is driven by large volume viscosity. To establish this limit, we develop time-weighted a priori estimates that yield decay rates for strong solutions. A key challenge arises from the fact that increasing the volume viscosity suppresses the decay of high-frequency components, thereby weakening the dissipation of the density and complicating the derivation of uniform-in-time decay estimates. To overcome this difficulty, we introduce a novel momentum-based estimate and show that the incompressible component of the momentum decays faster in time than the velocity itself. Exploiting this enhanced decay, we successfully close the a priori estimates and establish a time-decreasing convergence rate toward the incompressible limit.