Instantaneous strict separation for three-dimensional local Cahn–Hilliard
Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded, connected domain and $0 < \theta < \theta_0$. Consider the Cahn–Hilliard equation with constant mobility, the logarithmic (Flory–Huggins) potential and homogeneous Neumann conditions, $$\partial_t u = \Delta \mu, \qquad \mu = -\Delta u + \frac{\theta}{2} \log\frac{1+u}{1-u} - \theta_0 u, \qquad \partial_\nu u = \partial_\nu \mu = 0.$$ Take $u_0 \in H^1(\Omega)$ with $|u_0| \le 1$ almost everywhere and $|\Omega|^{-1} \int_\Omega u_0 \in (-1, 1)$, and let $u$ be the global energy weak solution: $u \in L^\infty(0,T; H^1(\Omega)) \cap H^1(0,T; H^1(\Omega)^*)$ and $\mu \in L^2(0,T; H^1(\Omega))$, the equations hold weakly, and the free energy decreases by $\int |\nabla \mu|^2$, its logarithmic density taken with the convention $0 \log 0 = 0$.
Miranville and Zelik proved that in two dimensions solutions are instantaneously strictly separated from the pure phases $\pm 1$. In three dimensions, Abels and Wilke proved that every weak solution converges to a single stationary state and is strictly separated from some time on, depending on the initial datum; Gal and Poiatti proved instantaneous strict separation for entropy densities sufficiently more singular than the logarithmic one at $\pm 1$; and Poiatti proved it for the nonlocal Cahn–Hilliard equation, in which $-\Delta u$ is replaced by a convolution with an interaction kernel.
Problem. For every $\tau > 0$, must there exist $\delta = \delta(\tau, u_0, \Omega, \theta, \theta_0) > 0$ such that $$\operatorname*{ess\,sup}_{(x,t) \in \Omega \times [\tau, \infty)} |u(x,t)| \le 1 - \delta?$$ No initial separation or small-energy assumption is made.
Source: Stated by AIM — Open Applied Problems, after Gal and Poiatti (2025), §6.1.1, who call instantaneous strict separation for the logarithmic potential in three dimensions an open problem; see also Poiatti (2025), §1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, Cahn–Hilliard equations, logarithmic potentials, strict separation
References
@article{gal2025,
author = {Gal, Ciprian G. and Poiatti, Andrea},
title = {Unified framework for the separation property in binary phase-segregation processes with singular entropy densities},
journal = {European J. Appl. Math.},
volume = {36},
number = {1},
year = {2025},
pages = {40--67},
doi = {10.1017/S0956792524000196},
}
@article{poiatti2025,
author = {Poiatti, Andrea},
title = {The {3D} strict separation property for the nonlocal {C}ahn--{H}illiard equation with singular potential},
journal = {Anal. PDE},
volume = {18},
number = {1},
year = {2025},
pages = {109--139},
doi = {10.2140/apde.2025.18.109},
eprint = {2303.07745},
archivePrefix = {arXiv},
}
@article{miranville2004,
author = {Miranville, Alain and Zelik, Sergey},
title = {Robust exponential attractors for {C}ahn--{H}illiard type equations with singular potentials},
journal = {Math. Methods Appl. Sci.},
volume = {27},
number = {5},
year = {2004},
pages = {545--582},
doi = {10.1002/mma.464},
}
@article{abels2007,
author = {Abels, Helmut and Wilke, Mathias},
title = {Convergence to equilibrium for the {C}ahn--{H}illiard equation with a logarithmic free energy},
journal = {Nonlinear Anal.},
volume = {67},
number = {11},
year = {2007},
pages = {3176--3193},
doi = {10.1016/j.na.2006.10.002},
}
@misc{aim-local-cahn-hilliard-three-dimensional-separation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Instantaneous strict separation for three-dimensional local Cahn–Hilliard}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/470-local-cahn-hilliard-three-dimensional-separation.md},
note = {Entry 470 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-00612, "Instantaneous strict separation for three-dimensional local Cahn–Hilliard", clopen.solutions.