Incompressible limit for the 3D compressible FENE dumbbell model

Abstract The finite extensible nonlinear elastic (FENE) dumbbell model couples the Navier–Stokes equations governing fluid velocity with a Fokker–Planck equation describing the evolution of polymer distribution within the fluid. 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 bulk 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 bulk 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.