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 [J. Differ. Equ. 452, 113789 (2026)] 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 generalized Cauchy distribution for the end-to-end vector. We discuss also the limit when we don’t have a probability density limit. Our approach is based on the derivation of an appropriate estimates on L2 with appropriate weight and investigate the convergence.