Uniform control of Neumann advection–diffusion for non-gradient flows
Let $\Omega \subset \mathbb{R}^d$, $d \ge 2$, be a bounded connected smooth domain with outward unit normal $\nu$, let $\omega \Subset \Omega$ be a nonempty open set, and let $B \in W^{2,\infty}(\mathbb{R}^d; \mathbb{R}^d)$ satisfy $B \cdot \nu \ge c_0 > 0$ on $\partial\Omega$. Write $X(s; x)$ for the solution of $X'(s) = -B(X(s))$, $X(0) = x$, and assume that for some $T_0, r_0 > 0$ every $x \in \overline{\Omega}$ has an $s_x \in (0, T_0)$ with $X(s_x; \mathbb{B}(x, r_0)) \subset \omega$, where $\mathbb{B}(x, r_0)$ is the open ball of radius $r_0$ about $x$. For $\varepsilon > 0$ consider $$y_t - \varepsilon \Delta y + B \cdot \nabla y = 1_\omega h \ \text{ in } \Omega \times (0,T), \qquad \partial_\nu y = 0 \ \text{ on } \partial\Omega \times (0,T), \qquad y(0) = y_0 \in L^2(\Omega).$$ For each fixed $\varepsilon$ and $T$ this system is null controllable; what matters here is the size of the controls as $\varepsilon \to 0$. When $B = \nabla f$ is a gradient field, possibly time dependent, with $\partial_\nu f \ge c > 0$ on the boundary and under a flushing condition of this kind, Et-Tahri, Bárcena-Petisco, Boutaayamou and Maniar proved that for every sufficiently large control time the cost of null controllability stays bounded as $\varepsilon \to 0$. For general drifts with $B \cdot \nu \ge 0$ satisfying the flushing condition they proved the dissipation estimate of their argument; the gradient structure is used in their Carleman estimate, to remove the transport term.
Problem. Must there exist $T_* < \infty$ such that, for every $T > T_*$, some $C_T, \varepsilon_T > 0$ satisfy: every $0 < \varepsilon < \varepsilon_T$ and every $y_0 \in L^2(\Omega)$ admit $h \in L^2((0,T) \times \omega)$ with $y(T) = 0$ and $$\|h\|_{L^2((0,T) \times \omega)} \le C_T \|y_0\|_{L^2(\Omega)}?$$ No representation $B = \nabla f$ by a scalar potential is imposed.
Source: Et-Tahri, Bárcena-Petisco, Boutaayamou and Maniar (2025), Remark 1.6, for drifts that are not gradients; stated for autonomous drift with strictly outward flux by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, null controllability, transport-diffusion equations, vanishing viscosity
References
@article{ettahri2025,
author = {Et-Tahri, Fouad and B{\'a}rcena-Petisco, Jon Asier and Boutaayamou, Idriss and Maniar, Lahcen},
title = {On uniform null controllability of transport-diffusion equations with vanishing viscosity limit},
journal = {Math. Methods Appl. Sci.},
volume = {48},
number = {6},
year = {2025},
pages = {6531--6552},
doi = {10.1002/mma.10693},
eprint = {2412.16660},
archivePrefix = {arXiv},
}
@misc{aim-nongradient-neumann-uniform-control,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniform control of Neumann advection–diffusion for non-gradient flows}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/415-nongradient-neumann-uniform-control.md},
note = {Entry 415 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove uniform null controllability of an advection--diffusion equation with Neumann boundary conditions as the diffusion coefficient tends to zero. The drift is autonomous, belongs to $W^{2,\infty}$, points strictly outward on the boundary, and its backward flow carries a neighborhood of every point into the interior control region within a bounded time. For every sufficiently large control time, an $L^2$ …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00762, "Uniform control of Neumann advection–diffusion for non-gradient flows", clopen.solutions.