Localized Degenerate Control of Navier-Stokes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathbb T^d=\mathbb R^d/(2\pi\mathbb Z)^d$, $d=2$ or $3$, let $\nu>0$, and let $V$ be the space of $H^1$ divergence-free vector fields on $\mathbb T^d$. Consider \[\partial_tu+(u\cdot\nabla)u-\nu\Delta u+\nabla p=\eta,\qquad\nabla\cdot u=0.\] Fix a nonempty open subdomain $\mathcal D\subset\mathbb T^d$. A localized degenerate control space is a finite-dimensional linear subspace $E\subset\{v\in V:\operatorname{supp}v\subset\overline{\mathcal D}\}$, and controls $\eta(t)\in E$ are bounded measurable functions of time. Approximate controllability means: for every $u_0,u_1\in V$, $T>0$ and $\varepsilon>0$, a control and a solution with $u(0)=u_0$ satisfy $\|u(T)-u_1\|_{L^2}<\varepsilon$. Controllability in finite-dimensional projections means that, for every finite-dimensional $F\subset V$, one can instead require $P_Fu(T)=P_Fu_1$, where $P_F$ is the $L^2$ orthogonal projection.

Statement recorded in the supplied source.

Does such a finite-dimensional $E$ exist independently of $\nu$, and can it be effectively constructed so that this Navier--Stokes system is approximately controllable and/or controllable in every finite-dimensional projection? [S2]

Short English statement

Can one find and effectively construct finitely many forcing modes confined to any chosen region that control the fluid approximately or in all finite projections, with the modes independent of viscosity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6500007 (AMR-064-0007). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. A. Agrachev, Some open problems, arXiv:1304.2590v2 (2013), Section VII, pp. 8--9. https://arxiv.org/pdf/1304.2590.