The FENE dumbbell model combines the Navier–Stokes equations of the fluid velocity with a Fokker-Planck equation for the dynamics of polymer distribution in the fluid medium. The FENE model admits a special equilibrium solution \((0,\psi _\infty )\) . This paper explores two types of enhanced dissipation associated with the system governing the perturbations near this steady state, one due to the equilibrium and one due to the coupling and interaction. Mathematically the linearized perturbation system admits a hidden wave structure. Making use of the smoothing and stabilizing effects in this wave structure, we are able to establish the global existence and stability of a 2D anisotropic FENE model in \(\mathbb {R}^2\) with the velocity equation involving only horizontal dissipation. Without the coupling, the corresponding 2D Navier–Stokes is not known to be stable. When the spatial domain is \({\mathbb {T}}\times \mathbb {R}\) , the FENE model with even less dissipation is shown to be stable, and the solution is shown to decay exponentially to its horizontal average. This result also relies on the above enhanced dissipation. The last part of our paper illustrates the importance of enhanced dissipation in the study of inviscid limit for a partially dissipated FENE system.
