Global positive classical evolution for the non-isothermal binary PNPF system
Fix $c > 1$. On $\mathbb{R}^3$ consider the non-isothermal Poisson–Nernst–Planck–Fourier system for positive ion concentrations $p, n$, absolute temperature $\theta$, ion velocities $v_p, v_n$ and electric potential $\phi$: $$\begin{aligned} p_t + \nabla \cdot (p v_p) &= 0, & p v_p &= -\nabla(p\theta) - p \nabla\phi,\\ n_t + \nabla \cdot (n v_n) &= 0, & n v_n &= -\nabla(n\theta) + n \nabla\phi,\\ -\Delta\phi &= p - n, & \phi(x, t) &\to 0 \quad (|x| \to \infty), \end{aligned}$$ $$c(p + n)\theta_t + c(p v_p + n v_n) \cdot \nabla\theta = \Delta\theta + p|v_p|^2 + n|v_n|^2 - p\theta\, \nabla \cdot v_p - n\theta\, \nabla \cdot v_n.$$ The initial data satisfy $p_0 - 1, n_0 - 1, \theta_0 - 1 \in C_c^\infty(\mathbb{R}^3)$ and $\min(p_0, n_0, \theta_0) \ge \eta > 0$, with no restriction on their size, and the initial potential is the decaying Newtonian solution of $-\Delta\phi_0 = p_0 - n_0$. Hsieh, Lin, Liu and Liu derived the system from an energetic variational approach and proved that it has a unique global solution when $\|(n_0 - 1, p_0 - 1, \theta_0 - 1)\|_{H^2} + \|\nabla\phi_0\|_{L^2}$ is small enough (2020).
Problem. For all such initial data, does the unique local positive classical solution extend for all time, that is, does $$\sup_{0 \le t \le T} \Bigl( \|(p - 1, n - 1, \theta - 1)(t)\|_{H^2} + \|\nabla\phi(t)\|_{L^2} \Bigr) < \infty \quad \text{and} \quad \inf_{\mathbb{R}^3 \times [0, T]} \min(p, n, \theta) > 0$$ hold for every finite $T$?
Source: Hsieh, Lin, Liu and Liu (2020), Section 6 (Concluding Remarks, p. 19 of arXiv:2001.08393v1), who call global well-posedness for arbitrary initial data challenging. Stated by AIM — Open Applied Problems for smooth, compactly supported, strictly positive perturbations of the constant state.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, non-isothermal models, Poisson–Nernst–Planck systems
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{hsieh2020pnpf,
author = {Hsieh, Chia-Yu and Lin, Tai-Chia and Liu, Chun and Liu, Pei},
title = {Global existence of the non-isothermal {P}oisson--{N}ernst--{P}lanck--{F}ourier system},
journal = {J. Differential Equations},
volume = {269},
number = {9},
year = {2020},
pages = {7287--7310},
doi = {10.1016/j.jde.2020.05.037},
eprint = {2001.08393},
archivePrefix = {arXiv},
}
@misc{aim-global-positive-classical-evolution-for-the-non-isothermal-binary-pnpf-system,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global positive classical evolution for the non-isothermal binary PNPF system}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/745-global-positive-classical-evolution-for-the-non-isothermal-binary-pnpf-system.md},
note = {Entry 745 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-01152, "Global positive classical evolution for the non-isothermal binary PNPF system", clopen.solutions.