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

Yassine Tahraoui

TL;DR

The paper analyzes the sharp, two-scale limit for the Fokker–Planck equation of dilute polymers in a turbulent flow decomposed into large-scale deterministic and small-scale stochastic components. Using stochastic diffusion limit techniques and kinetic-transport analysis, it derives a limit equation with diffusion in the end-to-end vector and identifies how the second, singular limit τ→0 yields either a generalized Cauchy density or a measure-valued limit depending on the tail parameter α. A key finding is that when α>d/2 the polymer distribution converges to a product of a macroscopic density and a generalized Cauchy density in end-to-end length, while for α≤d/2 the limit may concentrate as a Radon measure, capturing coil-stretch transition and potential infinite extension. The results provide a rigorous bridge between turbulence modeling, heavy-tail polymer statistics, and the onset of coil-stretch behavior, with implications for drag reduction theory and extensions to FENE-type models in future work.

Abstract

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[arXiv:2410.00520] 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 $L^2$ with appropriate weight and investigate the convergence.

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

TL;DR

The paper analyzes the sharp, two-scale limit for the Fokker–Planck equation of dilute polymers in a turbulent flow decomposed into large-scale deterministic and small-scale stochastic components. Using stochastic diffusion limit techniques and kinetic-transport analysis, it derives a limit equation with diffusion in the end-to-end vector and identifies how the second, singular limit τ→0 yields either a generalized Cauchy density or a measure-valued limit depending on the tail parameter α. A key finding is that when α>d/2 the polymer distribution converges to a product of a macroscopic density and a generalized Cauchy density in end-to-end length, while for α≤d/2 the limit may concentrate as a Radon measure, capturing coil-stretch transition and potential infinite extension. The results provide a rigorous bridge between turbulence modeling, heavy-tail polymer statistics, and the onset of coil-stretch behavior, with implications for drag reduction theory and extensions to FENE-type models in future work.

Abstract

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[arXiv:2410.00520] 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 with appropriate weight and investigate the convergence.

Paper Structure

This paper contains 29 sections, 19 theorems, 178 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and formulation of the problem
  3. Structure of the small scales
  4. Scaling limit results
  5. Notation and functional setting
  6. Scaling limit as $N\to +\infty$
  7. The derivation of \ref{['sys-I']}
  8. Noise specification and the parameter $a_\tau$
  9. Main results
  10. Statement of the main result
  11. Non-uniqueness of measure-valued solutions and selection principle ($0<\alpha\leq \dfrac{d}{2}$)
  12. On the Role of the Unbounded Potential
  13. Physical interpretation
  14. The coil-stretch operator
  15. Uniform estimates and continuity equation
  16. ...and 14 more sections

Key Result

Theorem 1

Under the assumption that the small scales have the given structure in assumption_noise-2D. There exists a new probability space, denoted by the same way (for simplicity) $(\Omega,\mathcal{F},P)$, $f_\tau\in L^2_{w-*}(\Omega ;L^\infty([0, T];H))), \nabla_rf_\tau\in L^2(\Omega ;L^2([0, T];H)))$ such Moreover $f_\tau$ is the unique solution of the following problem: $P$-a.s. for any $t\in [0,T]$:

Theorems & Definitions (42)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • proof
  • Remark 4
  • Theorem 5
  • proof
  • Remark 6
  • Proposition 7
  • proof
  • ...and 32 more