Long-time self-intersection problem in two-dimensional fluid–structure interaction
Consider the deterministic two-dimensional viscous incompressible fluid–structure system in which a perfectly elastic shell is governed by a linearised beam equation and forms the boundary of the fluid domain. Starting from an initially non-self-intersecting shell configuration, determine whether the shell remains non-self-intersecting for every time ; equivalently, whether its parametrization remains injective for all .
References
Primary source
Additional references
- Strong transport noise prevents self-intersection in 2D fluid-structure interaction — arXiv — Dominic Breit, Prince Romeo Mensah
Progress summary
A noisy version of the fluid–structure model reportedly avoids self-intersection with high-probability control, but the original deterministic long-time question remains open.
The problem asks whether a two-dimensional fluid–structure evolution can avoid geometric self-intersection for all time. No deterministic resolution of that question was found.
Known results
- Breit and Mensah, 2023: a stochastic model has weak martingale solutions with continuation possible unless the displacement reaches the self-intersection threshold.
- Related stochastic fluid–structure results establish only local-in-time existence before loss of admissible geometry.
October 2026 transport-noise result
On October 7, 2026, Dominic Breit and Prince Romeo Mensah reported a probabilistic no-self-intersection result after adding strong transport noise. This controls self-intersection only in the modified stochastic model, not in the original deterministic problem; the claim has not been independently verified.
Current status (as of October 2026): The deterministic long-time problem remains open; a claimed stochastic result gives no-self-intersection control only after adding strong transport noise.
Solutions 0
No solutions have been posted yet.