Small-scale turbulence limit of Fokker–Planck equation for polymers in turbulent flow

We study the singular limit of Fokker–Planck equation of polymers density as the dominant time scale of small-scale component of turbulent flow goes to zero. Here, we complete the study of Flandoli–Tahraoui [Stretching of polymers and turbulence: Fokker Planck equation, special stochastic scaling limit and stationary law, J. Differential Equations 452 (2026) 113789] about scaling limit as the space scale of small-scale component of turbulent flow goes to zero by using stochastic modeling of turbulence. Depending on certain parameters related to turbulence modeling, the limit density has a generalized Cauchy distribution for the end-to-end vector. We also discuss the limit when we don’t have a probability density limit. Our approach is based on the derivation of appropriate estimates on [Formula: see text] with appropriate weight and investigation of the convergence.