Stretching of Polymers and turbulence: Fokker Planck equation, special stochastic scaling limit and stationary law

TL;DR

This paper investigates the stretching behavior of dilute polymers in turbulent flows using stochastic models, analyzing their scaling limits, convergence to deterministic equations, and stationary solutions with power law decay.

Contribution

It introduces a new scaling limit for stochastic turbulence models acting on polymers, leading to a deterministic equation with a novel term and explicit stationary solutions.

Findings

Convergence of the polymer density SPDE to a deterministic limit.

Derivation of stationary solutions with power law decay.

Identification of a special stochastic scaling limit.

Abstract

The aim of this work is understanding the stretching mechanism of stochastic models of turbulence acting on a simple model of dilute polymers. We consider a turbulent model that is white noise in time and activates frequencies in a shell $N\leq |k|\leq 2N$ and investigate the scaling limit as $N\rightarrow \infty$, under suitable intensity assumption, such that the stretching term has a finite limit covariance. The polymer density equation, initially an SPDE, converges weakly to a limit deterministic equation with a new term. Stationary solutions can be computed and show power law decay in the polymer length.