Critical velocity control for a rigid body moving in a bounded viscous fluid
Let $\Omega \subset \mathbb{R}^3$ be a bounded connected smooth domain and $S$ a bounded connected smooth domain with centroid at the origin, $\overline{S} \subset \Omega$ and $\Omega \setminus \overline{S}$ connected. A rigid body of shape $S$ and constant density $\rho_S > 0$ moves in a viscous incompressible fluid of density and viscosity one filling the rest of $\Omega$, with no-slip conditions and no applied force. For a finite $T > 0$ let $h(t) \in \mathbb{R}^3$ and $Q(t) \in SO(3)$, with $h(0) = 0$ and $Q(0) = I$, and put $$S(t) = h(t) + Q(t) S, \qquad F(t) = \Omega \setminus \overline{S(t)}, \qquad \rho(t, x) = \mathbf{1}_{F(t)}(x) + \rho_S \mathbf{1}_{S(t)}(x).$$ The velocity $u$ is defined on all of $\Omega$, equal inside $S(t)$ to the rigid velocity $\ell(t) + \omega(t) \times (x - h(t))$ with $h' = \ell$ and $Q' Q^{\mathsf T} v = \omega \times v$. A weak solution has $(h, Q) \in W^{1,\infty}(0, T; \mathbb{R}^3 \times SO(3))$ and a divergence-free $u \in L^\infty(0, T; L^2(\Omega)^3) \cap L^2(0, T; H^1_0(\Omega)^3)$ of this form, and satisfies $$-\int_0^T \!\! \int_\Omega \rho u \cdot \bigl(\partial_t \varphi + (u \cdot \nabla) \varphi\bigr) + 2 \int_0^T \!\! \int_\Omega Du : D\varphi = \int_\Omega \rho(0, x)\, a(x) \cdot \varphi(0, x)$$ for every divergence-free $\varphi \in H^1(0, T; L^2(\Omega)^3) \cap L^\infty(0, T; H^1_0(\Omega)^3)$ with $D\varphi = 0$ in $S(t)$ and $\varphi(T) = 0$, where $Du = \frac12(\nabla u + \nabla u^{\mathsf T})$ and the initial velocity $a \in L^2(\Omega)^3$ is divergence-free, has zero normal trace on $\partial\Omega$ and is rigid on $S$. Let $E = \{(t, x) : 0 < t \le T,\ x \in \Omega \cap \overline{F(t)}\}$, which contains the fluid side of the body's boundary but not $\partial\Omega$ or the initial time.
Muha, Nečasová and Radošević proved that a weak solution keeping the body a positive distance from $\partial\Omega$ is smooth if $u \in L^r(0, T; L^s(F(t)))$ with $2/r + 3/s = 1$ and $3 < s < \infty$ and the body's acceleration is bounded. Maity and Takahashi proved that $h, Q \in C^\infty((0, T])$ and $u \in C^\infty(E)$ when $u \in L^r(0, T; L^q(F(t)))$ with $2/r + 3/q = 1$, $2 \le r < \infty$, and, in the critical case, when $\sup_t \|u(t)\|_{L^3(F(t))}$ is smaller than a constant $\varepsilon > 0$. For the Navier–Stokes equations in $\mathbb{R}^3$ without a body, Escauriaza, Seregin and Šverák proved regularity of $L^\infty(L^3)$ solutions without a smallness condition.
Problem. Let a weak solution satisfy $\operatorname{dist}(\overline{S(t)}, \partial\Omega) \ge \delta$ on $[0, T]$ for some $\delta > 0$. Does $$\operatorname*{ess\,sup}_{0 < t < T} \|u(t, \cdot)\|_{L^3(F(t))} < \infty$$ imply $h, Q \in C^\infty((0, T])$ and $u \in C^\infty(E)$, with derivatives at $T$ taken from the left? The norm may have any finite size, and no further assumption, such as bounded acceleration of the body or regularity of the pressure, is made.
Source: Stated by AIM — Open Applied Problems, after Maity and Takahashi (2025), who proved the conclusion when the critical norm is small enough (Theorem 1.4) and discuss the smallness condition in Remark 1.7.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, conditional regularity, fluid–structure interaction, Navier–Stokes equations
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{maity2025,
author = {Maity, Debayan and Takahashi, Tak{\'e}o},
title = {Regularity of weak solutions of a fluid-rigid body interaction system in a bounded domain under the {P}rodi--{S}errin condition},
journal = {J. Differential Equations},
volume = {424},
year = {2025},
pages = {726--759},
doi = {10.1016/j.jde.2025.01.073},
note = {Author manuscript: HAL hal-04599986},
}
@article{muha2023,
author = {Muha, Boris and Ne{\v{c}}asov{\'a}, {\v{S}}{\'a}rka and Rado{\v{s}}evi{\'c}, Ana},
title = {On the regularity of weak solutions to the fluid--rigid body interaction problem},
journal = {Math. Ann.},
volume = {389},
number = {2},
year = {2024},
pages = {1007--1052},
doi = {10.1007/s00208-023-02664-0},
eprint = {2211.03080},
archivePrefix = {arXiv},
}
@article{maity2024,
author = {Maity, Debayan and Takahashi, Tak{\'e}o},
title = {Uniqueness and regularity of weak solutions of a fluid-rigid body interaction system under the {P}rodi--{S}errin condition},
journal = {Discrete Contin. Dyn. Syst.},
volume = {44},
number = {3},
year = {2024},
pages = {718--742},
doi = {10.3934/dcds.2023123},
}
@article{escauriaza2003,
author = {Escauriaza, L. and Seregin, G. A. and {\v{S}}ver{\'a}k, V.},
title = {{$L_{3,\infty}$}-solutions of the {N}avier--{S}tokes equations and backward uniqueness},
journal = {Russian Math. Surveys},
volume = {58},
number = {2},
year = {2003},
pages = {211--250},
doi = {10.1070/RM2003v058n02ABEH000609},
}
@misc{aim-critical-velocity-control-for-a-rigid-body-moving-in-a-bounded-viscous-fluid,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Critical velocity control for a rigid body moving in a bounded viscous fluid}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/749-critical-velocity-control-for-a-rigid-body-moving-in-a-bounded-viscous-fluid.md},
note = {Entry 749 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-01149, "Critical velocity control for a rigid body moving in a bounded viscous fluid", clopen.solutions.