Global strong electrodiffusion with unequal ionic diffusivities
Let $\Omega \subset \mathbb{R}^3$ be a bounded connected domain with smooth boundary, let $m \ge 3$, let $z_1, \dots, z_m \in \{-1, 1\}$ take both values, and let $D_1, \dots, D_m, \epsilon, \nu, K > 0$ be constants, the $D_i$ not necessarily equal. Consider the Nernst–Planck–Stokes system $$\begin{aligned} \partial_t c_i + u \cdot \nabla c_i &= D_i \nabla \cdot (\nabla c_i + z_i c_i \nabla \Phi), \qquad i = 1, \dots, m,\\ -\epsilon \Delta \Phi &= \rho := \sum_{i=1}^m z_i c_i,\\ \partial_t u - \nu \Delta u + \nabla p &= -K \rho \nabla \Phi, \qquad \nabla \cdot u = 0, \end{aligned}$$ with $c_i = \gamma_i$, $\Phi = W$ and $u = 0$ on $\partial\Omega$, where the boundary data $\gamma_i > 0$ and $W$ are smooth and time-independent. The initial concentrations $c_{i,0}$ are smooth and strictly positive with $c_{i,0} = \gamma_i$ on $\partial\Omega$, and the initial velocity $u_0$ is smooth, divergence-free and zero on $\partial\Omega$. With $V = \{v \in H^1_0(\Omega; \mathbb{R}^3) : \nabla \cdot v = 0\}$, a strong solution on $[0, T]$ has $c_i \in L^\infty(0, T; H^1(\Omega)) \cap L^2(0, T; H^2(\Omega))$ and $u \in L^\infty(0, T; V) \cap L^2(0, T; H^2(\Omega; \mathbb{R}^3) \cap V)$, satisfies the equations in the sense of distributions, the boundary conditions in the trace sense and the initial data as time traces. Constantin, Ignatova and Lee proved that a unique strong solution exists on a short time interval, that a strong solution on $[0, T_1)$ can be continued beyond $T_1$ when $\sup_{T < T_1} \int_0^T \|\rho(t)\|_{L^2}^4 \, dt < \infty$, and that strong solutions are global for two species with valences $1$ and $-1$ and any diffusivities, and for any number of species with valences $\pm 1$ and a common diffusivity (2021).
Problem. For every such domain, $m$, valences, constants, boundary data and initial data, does the unique local strong solution extend to every finite time, that is, is there for every $T < \infty$ a strong solution on $[0, T]$ with $c_i \ge 0$?
Source: Constantin, Ignatova and Lee (2021), Introduction (p. 2 of arXiv:2008.10462v1), who write that global existence of smooth solutions for Nernst–Planck equations coupled to Stokes flow remains open in its full generality. Stated by AIM — Open Applied Problems for at least three species of valence ±1 with arbitrary diffusivities.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, ionic electrodiffusion, Nernst–Planck equations, Stokes equations
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{constantin2021npns,
author = {Constantin, Peter and Ignatova, Mihaela and Lee, Fizay-Noah},
title = {{N}ernst--{P}lanck--{N}avier--{S}tokes systems far from equilibrium},
journal = {Arch. Ration. Mech. Anal.},
volume = {240},
number = {2},
year = {2021},
pages = {1147--1168},
doi = {10.1007/s00205-021-01630-x},
eprint = {2008.10462},
archivePrefix = {arXiv},
}
@misc{aim-global-strong-electrodiffusion-with-unequal-ionic-diffusivities,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global strong electrodiffusion with unequal ionic diffusivities}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/744-global-strong-electrodiffusion-with-unequal-ionic-diffusivities.md},
note = {Entry 744 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01154, "Global strong electrodiffusion with unequal ionic diffusivities", clopen.solutions.