The zero center-of-mass diffusion limit for kinetic FENE fluids
Let $x \in \mathbb{T}^3$, $q \in B_1(0) \subset \mathbb{R}^3$, $\nu > 0$, $k > 2$, $U(q) = -k \log(1 - |q|^2)$ and $M = Z^{-1} e^{-U}$. For $\varepsilon > 0$ consider the FENE dumbbell model $$u_t + u \cdot \nabla_x u - \nu \Delta_x u + \nabla_x p = \nabla_x \cdot \tau, \qquad \nabla_x \cdot u = 0,$$ $$\psi_t + u \cdot \nabla_x \psi = \varepsilon \Delta_x \psi + \nabla_q \cdot \bigl[\nabla_q \psi + \psi \nabla_q U - (\nabla_x u) q \psi\bigr], \qquad \tau = \int_{B_1} q \otimes \nabla_q U \, \psi \, dq,$$ with zero configuration-space flux, where $\varepsilon$ is the centre-of-mass diffusion coefficient. Fix smooth divergence-free $u_0$ and $\psi_0 = M g_0$, where $g_0$ is smooth and bounded above and below by positive constants and $\int_{B_1} \psi_0(x, q) \, dq = 1$. Finite entropy means $\int \psi \log(\psi / M) < \infty$ together with the usual kinetic-fluid entropy inequality. Barrett and Süli proved global existence of large-data weak solutions for a general class of FENE-type models with centre-of-mass diffusion, and Masmoudi proved global existence of weak solutions for the FENE model without it.
Problem. Can global finite-entropy weak solutions $(u_\varepsilon, \psi_\varepsilon)$ be chosen so that, along some $\varepsilon_j \downarrow 0$, they converge to a global weak solution of exactly the same system with $\varepsilon = 0$? Require $u_{\varepsilon_j} \to u$ strongly in $L^2$ on finite time intervals, $\psi_{\varepsilon_j} \rightharpoonup \psi$ weakly in $L^1$ there, and convergence of the stress and transport fluxes to those computed from $(u, \psi)$ in distributions, with no additional defect.
Source: Dębiec and Süli (2025), Section 1, on the limit of vanishing centre-of-mass diffusion for weak solutions of the FENE model. Stated by AIM — Open Applied Problems for the periodic three-dimensional FENE dumbbell model.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, dilute polymeric fluids, FENE dumbbell model
References
@article{debiec2025,
author = {D{\k{e}}biec, Tomasz and S{\"u}li, Endre},
title = {On a class of generalised solutions to the kinetic {H}ookean dumbbell model for incompressible dilute polymeric fluids: existence and macroscopic closure},
journal = {Arch. Ration. Mech. Anal.},
volume = {249},
number = {4},
year = {2025},
pages = {43},
doi = {10.1007/s00205-025-02115-x},
eprint = {2306.16901},
archivePrefix = {arXiv},
}
@article{barrett2011,
author = {Barrett, John W. and S{\"u}li, Endre},
title = {Existence and equilibration of global weak solutions to kinetic models for dilute polymers {I}: finitely extensible nonlinear bead-spring chains},
journal = {Math. Models Methods Appl. Sci.},
volume = {21},
number = {6},
year = {2011},
pages = {1211--1289},
doi = {10.1142/S0218202511005313},
eprint = {1004.1432},
archivePrefix = {arXiv},
}
@article{masmoudi2013,
author = {Masmoudi, Nader},
title = {Global existence of weak solutions to the {FENE} dumbbell model of polymeric flows},
journal = {Invent. Math.},
volume = {191},
number = {2},
year = {2013},
pages = {427--500},
doi = {10.1007/s00222-012-0399-y},
eprint = {1004.4015},
archivePrefix = {arXiv},
}
@misc{aim-fene-vanishing-center-of-mass-diffusion,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The zero center-of-mass diffusion limit for kinetic FENE fluids}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/376-fene-vanishing-center-of-mass-diffusion.md},
note = {Entry 376 at commit 24edaad, 2026-10-15},
}
Resolution claims
We study the zero center-of-mass diffusion limit for the full kinetic FENE dumbbell model on the three-dimensional torus. For every solvent viscosity $\nu>0$, FENE exponent $k>2$, smooth divergence-free initial velocity, and smooth normalized initial density with bounded positive Maxwellian quotient, we select global positive-diffusion weak solutions converging to a global weak solution of the identical nondiffusive system. Velocity converges strongly …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00791, "The zero center-of-mass diffusion limit for kinetic FENE fluids", clopen.solutions.