Global regularity of the planar noncorotational FENE fluid
Let $B = \{q \in \mathbb{R}^2 : |q| < 1\}$, $k > 2$, $U(q) = -k \log(1 - |q|^2)$ and $M(q) = Z^{-1} e^{-U(q)}$, with $\int_B M = 1$. The FENE dumbbell model couples the Navier–Stokes equations for the velocity $u(t, x)$ to a Fokker–Planck equation for the polymer distribution $\psi(t, x, q)$; on $\mathbb{R}^2_x \times B_q$ it reads $$u_t + u \cdot \nabla_x u - \Delta_x u + \nabla_x p = \nabla_x \cdot \tau, \qquad \nabla_x \cdot u = 0,$$ $$\psi_t + u \cdot \nabla_x \psi = \nabla_q \cdot \bigl( \nabla_q \psi + \psi \nabla_q U - (\nabla_x u) q \psi \bigr), \qquad \tau = \int_B q \otimes \nabla_q U \, \psi \, dq,$$ with zero normal configuration flux on $\partial B$ and $\int_B \psi(t, x, q) \, dq = 1$. In the corotational model $\nabla_x u$ is replaced in the Fokker–Planck equation by its antisymmetric part. Masmoudi proved local well-posedness in Sobolev spaces for every $k > 0$, global existence for data close to the equilibrium $(0, M)$, and global existence in two dimensions for the corotational model, extending a result of Lin, Zhang and Zhang; he later proved global existence of weak solutions of the full model without smallness assumptions.
Problem. For every smooth compatible initial datum with $u_0$ divergence free, $\psi_0 = M g_0$, $g_0 > 0$, and $u_0$ and $g_0 - 1$ smooth and compactly supported in $x$, must the local classical solution extend to all positive times? On each finite time interval it is required that $u \in C_t H^s_x$ and $\psi - M \in C_t H^s_x(L^2_q(M^{-1} dq))$ for every $s$, with smoothness in $q \in B$. There is no smallness assumption and no centre-of-mass diffusion in $x$.
Source: Masmoudi (2013), §7, "Regularity in 2D". Stated in this form by AIM — Open Applied Problems, for k > 2 and data smooth and compactly supported in x.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, FENE dumbbell model, Fokker–Planck equations, polymeric fluids
References
@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},
}
@article{masmoudi2008,
author = {Masmoudi, Nader},
title = {Well-posedness for the {FENE} dumbbell model of polymeric flows},
journal = {Comm. Pure Appl. Math.},
volume = {61},
number = {12},
year = {2008},
pages = {1685--1714},
doi = {10.1002/cpa.20252},
}
@article{lin2008,
author = {Lin, Fanghua and Zhang, Ping and Zhang, Zhifei},
title = {On the global existence of smooth solution to the 2-{D} {FENE} dumbbell model},
journal = {Comm. Math. Phys.},
volume = {277},
number = {2},
year = {2008},
pages = {531--553},
doi = {10.1007/s00220-007-0385-1},
}
@misc{aim-fene-two-dimensional-regularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global regularity of the planar noncorotational FENE fluid}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/358-fene-two-dimensional-regularity.md},
note = {Entry 358 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00590, "Global regularity of the planar noncorotational FENE fluid", clopen.solutions.