Global classical Maxwell–Stefan diffusion with arbitrary pair friction
Let $\Omega \subset \mathbb{R}^3$ be a bounded domain with smooth boundary and outer unit normal $\nu$, let $N \ge 4$, and fix constants $f_{ij} = f_{ji} > 0$ for $i \ne j$. The mole fractions $x_1, \dots, x_N$ and fluxes $J_1, \dots, J_N$ satisfy the Maxwell–Stefan system $$\partial_t x_i + \nabla \cdot J_i = 0, \qquad \nabla x_i = \sum_{j \ne i} f_{ij} (x_i J_j - x_j J_i), \qquad \sum_{i=1}^N x_i = 1, \qquad \sum_{i=1}^N J_i = 0,$$ with $J_i \cdot \nu = 0$ on $\partial\Omega$; there are no chemical reactions and no bulk advection. Bothe proved local-in-time well-posedness (2011), and Jüngel and Stelzer proved that bounded weak solutions exist globally in time (2013). Bothe proved that global strong solutions, classical for positive times, exist from uniformly positive data, without smallness, when the friction is additive, $f_{ij} = g_i + g_j$ with all $g_i > 0$ (2026 preprint); for $N \ge 4$ additivity imposes identities among the $f_{ij}$. For $N = 3$ Bothe proved global strong well-posedness for arbitrary positive constant frictions (2026 preprint).
Problem. For every smooth initial composition $x_0$ with $x_{i,0} \ge \delta > 0$ for some $\delta$, $\sum_i x_{i,0} = 1$ and the no-flux compatibility condition, does the unique local classical solution continue to every finite time, remaining classical and strictly positive on $\overline{\Omega} \times [0, T]$ for each $T < \infty$? The coefficients $f_{ij}$ need not be of the form $g_i + g_j$, and the initial data need not be close to a constant composition.
Source: Daus, Jüngel and Tang (2020), Remark 2, as the question of global classical solutions of Maxwell–Stefan systems. Stated by AIM — Open Applied Problems without reactions, for N ≥ 4 species in three dimensions and arbitrary symmetric friction coefficients.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, cross-diffusion systems, gas mixtures, Maxwell–Stefan equations
References
@article{daus2020,
author = {Daus, Esther S. and J{\"u}ngel, Ansgar and Tang, Bao Quoc},
title = {Exponential time decay of solutions to reaction-cross-diffusion systems of {M}axwell--{S}tefan type},
journal = {Arch. Ration. Mech. Anal.},
volume = {235},
number = {2},
year = {2020},
pages = {1059--1104},
doi = {10.1007/s00205-019-01439-9},
eprint = {1802.10274},
archivePrefix = {arXiv},
}
@incollection{bothe2011,
author = {Bothe, Dieter},
title = {On the {M}axwell--{S}tefan approach to multicomponent diffusion},
booktitle = {Parabolic Problems},
series = {Progr. Nonlinear Differential Equations Appl.},
volume = {80},
publisher = {Birkh{\"a}user/Springer Basel AG},
address = {Basel},
year = {2011},
pages = {81--93},
doi = {10.1007/978-3-0348-0075-4_5},
eprint = {1007.1775},
archivePrefix = {arXiv},
}
@article{jungel2013,
author = {J{\"u}ngel, Ansgar and Stelzer, Ines Viktoria},
title = {Existence analysis of {M}axwell--{S}tefan systems for multicomponent mixtures},
journal = {SIAM J. Math. Anal.},
volume = {45},
number = {4},
year = {2013},
pages = {2421--2440},
doi = {10.1137/120898164},
eprint = {1211.2394},
archivePrefix = {arXiv},
}
@misc{bothe2026,
author = {Bothe, Dieter},
title = {Global strong solutions for {M}axwell--{S}tefan diffusion with additive friction coefficients},
year = {2026},
eprint = {2609.06732},
archivePrefix = {arXiv},
}
@misc{bothe2026ternary,
author = {Bothe, Dieter},
title = {Global strong solutions for the ternary {M}axwell--{S}tefan diffusion system},
year = {2026},
eprint = {2610.00162},
archivePrefix = {arXiv},
}
@misc{aim-maxwell-stefan-arbitrary-friction-global-classical,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global classical Maxwell–Stefan diffusion with arbitrary pair friction}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/479-maxwell-stefan-arbitrary-friction-global-classical.md},
note = {Entry 479 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-00614, "Global classical Maxwell–Stefan diffusion with arbitrary pair friction", clopen.solutions.