Large-data global regularity for creeping-flow Oldroyd–B without stress diffusion
On the unit torus $\mathbb{T}^2$, consider the Oldroyd-B model in the creeping-flow regime without stress diffusion, with all material constants equal to one: $$-\Delta u + \nabla p = \nabla \cdot C, \qquad \nabla \cdot u = 0, \qquad \int_{\mathbb{T}^2} u \, dx = 0,$$ $$\partial_t C + u \cdot \nabla C - (\nabla u) C - C (\nabla u)^T = -(C - I), \qquad C(0) = C_0,$$ for a velocity $u$, a pressure $p$ of zero mean and a symmetric matrix field $C$. The velocity is determined by the elliptic Stokes equation at each time; there is no inertial term and no Laplacian acting on $C$. Kupferman, Mangoubi and Titi proved a Beale–Kato–Majda type breakdown criterion for the three-dimensional Cauchy problem: a solution with initial stress in $H^s$, $s > 5/2$, persists unless the time integral of the $L^\infty$ norm of the stress diverges (2008). For the Oldroyd-B system on $\mathbb{R}^d$ coupled to the Stokes equation, Constantin and Sun proved global existence and uniqueness of solutions in $C^\alpha \cap L^1$ for data with small $L^\infty$ norm, large Hölder seminorms being allowed, with exponential convergence to equilibrium (2012). With stress diffusion added, Constantin and Kliegl proved global existence and uniqueness of strong solutions for large data in the plane, with the Navier–Stokes equations in place of the Stokes equation (2012), and Wang, Zhang and Zhang proved global well-posedness of strong solutions to the initial-boundary value problem in two dimensions in the creeping-flow regime (2026).
Problem. Let $C_0$ be an arbitrary smooth symmetric positive-definite $2 \times 2$ matrix field on $\mathbb{T}^2$, with no smallness assumption. Does the system have a unique global smooth solution with $C$ positive definite?
Source: Stated by AIM — Open Applied Problems, after Wang, Zhang and Zhang (2026), abstract, who describe global well-posedness of the non-diffusive creeping-flow model as open even in two dimensions; see also Constantin and Sun (2012).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, global regularity, Oldroyd-B fluids, viscoelastic fluids
References
@misc{wang2026,
author = {Wang, Yinghui and Zhang, Shihao and Zhang, Zhuo},
title = {Global well-posedness of strong solutions to the initial-boundary value problem for a two-dimensional stress-diffusive {O}ldroyd-{B} model in the creeping flow regime},
year = {2026},
eprint = {2608.25333},
archivePrefix = {arXiv},
}
@article{constantin2012sun,
author = {Constantin, Peter and Sun, Weiran},
title = {Remarks on {O}ldroyd-{B} and related complex fluid models},
journal = {Commun. Math. Sci.},
volume = {10},
number = {1},
year = {2012},
pages = {33--73},
doi = {10.4310/CMS.2012.v10.n1.a3},
eprint = {1009.0249},
archivePrefix = {arXiv},
}
@article{kupferman2008,
author = {Kupferman, Raz and Mangoubi, Claude and Titi, Edriss S.},
title = {A {B}eale--{K}ato--{M}ajda breakdown criterion for an {O}ldroyd-{B} fluid in the creeping flow regime},
journal = {Commun. Math. Sci.},
volume = {6},
number = {1},
year = {2008},
pages = {235--256},
doi = {10.4310/CMS.2008.v6.n1.a12},
eprint = {0709.1455},
archivePrefix = {arXiv},
}
@article{constantin2012kliegl,
author = {Constantin, Peter and Kliegl, Markus},
title = {Note on global regularity for two-dimensional {O}ldroyd-{B} fluids with diffusive stress},
journal = {Arch. Ration. Mech. Anal.},
volume = {206},
number = {3},
year = {2012},
pages = {725--740},
doi = {10.1007/s00205-012-0537-0},
eprint = {1204.3503},
archivePrefix = {arXiv},
}
@misc{aim-non-diffusive-oldroyd-b,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Large-data global regularity for creeping-flow Oldroyd–B without stress diffusion}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/080-non-diffusive-oldroyd-b.md},
note = {Entry 080 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-00621, "Large-data global regularity for creeping-flow Oldroyd–B without stress diffusion", clopen.solutions.