Large-data weak solutions for three-dimensional stress-diffusive Oldroyd-B flow
Let $\Omega \subset \mathbb{R}^3$ be a smooth bounded domain and let $\nu, \mu > 0$. The Oldroyd-B model with stress diffusion, for the velocity $u$, the pressure $p$ and the symmetric conformation tensor $\sigma$, is $$u_t + u \cdot \nabla u - \nu \Delta u + \nabla p = \nabla \cdot \sigma, \qquad \nabla \cdot u = 0,$$ $$\sigma_t + u \cdot \nabla \sigma - (\nabla u)\sigma - \sigma (\nabla u)^T - \mu \Delta \sigma + 2(\sigma - I) = 0,$$ with $u = 0$ and $\partial_n \sigma = 0$ on $\partial\Omega$, where $(\nabla u)_{ij} = \partial_j u_i$. Smooth solutions satisfy the energy identity $$\frac{d}{dt} E(t) + \nu \int_\Omega |\nabla u|^2 + \int_\Omega (\operatorname{tr}\sigma - 3) = 0, \qquad E(t) = \frac12 \int_\Omega \bigl(|u|^2 + \operatorname{tr}\sigma\bigr).$$ Barrett and Boyaval proved global existence of weak solutions in two dimensions, on convex polygonal domains, and Constantin and Kliegl proved global regularity on $\mathbb{R}^2$. In three dimensions, Dębiec and Süli constructed global generalised solutions of the Hookean dumbbell model, whose formal macroscopic closure is this system, with a possibly nonzero defect measure in the momentum equation.
Problem. Given smooth compatible $u_0$ with $\nabla \cdot u_0 = 0$ and $u_0|_{\partial\Omega} = 0$, and a smooth symmetric uniformly positive-definite tensor $\sigma_0$ with $\partial_n \sigma_0 = 0$, must there be a global distributional solution with these boundary conditions and initial values such that, on each finite time interval, $u \in L^\infty_t L^2_x \cap L^2_t H^1_0$, $\sigma \in L^\infty_t L^2_x \cap L^2_t H^1_x$, $\sigma$ is positive semidefinite almost everywhere, and the integrated inequality $$E(t) + \nu \int_0^t\!\int_\Omega |\nabla u|^2 + \int_0^t\!\int_\Omega (\operatorname{tr}\sigma - 3) \le E(0)$$ holds? The momentum equation must contain exactly $\nabla \cdot \sigma$, with no additional stress-defect measure.
Source: Dębiec and Süli (2025), §1 and Remark 1.4, on the global existence of weak solutions to the stress-diffusive Oldroyd-B model in three dimensions; see also Barrett and Boyaval (2011). Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Oldroyd-B model, viscoelastic fluids, weak solutions
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 Boyaval, S{\'e}bastien},
title = {Existence and approximation of a (regularized) {O}ldroyd-{B} model},
journal = {Math. Models Methods Appl. Sci.},
volume = {21},
number = {9},
year = {2011},
pages = {1783--1837},
doi = {10.1142/S0218202511005581},
eprint = {0907.4066},
archivePrefix = {arXiv},
}
@article{constantin2012,
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-oldroyd-diffusive-three-dimensional-weak,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Large-data weak solutions for three-dimensional stress-diffusive Oldroyd-B flow}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/371-oldroyd-diffusive-three-dimensional-weak.md},
note = {Entry 371 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-00627, "Large-data weak solutions for three-dimensional stress-diffusive Oldroyd-B flow", clopen.solutions.