Geometric Hydrodynamics in Open Problems — Problem 18 (Aging Problem)

Definitions and mathematical statement

Let $M$ denote the spatial domain appearing in the source, let $I\subseteq{\mathbb{R}}$ be a time interval, and let $u(t,\cdot)$ denote the velocity field of a solution of the Euler equations on $M$, defined for $t\in I$. The supplied record does not specify $M$, the version of the Euler equations, boundary conditions, or the solution class; these remain unresolved parameters of the question.

Aging Problem. For an arbitrary such solution $u$ and any two distinct times $t_1,t_2\in I$, can one determine, from the two velocity fields $u(t_1,\cdot)$ and $u(t_2,\cdot)$ alone, which field corresponds to the earlier time?

One possible precise interpretation of this question would ask for a single rule \[ A:\mathcal V\times\mathcal V\longrightarrow\{-1,1\}, \qquad A\left(u(t_1,\cdot),u(t_2,\cdot)\right) =\operatorname{sgn}(t_2-t_1) \] for every admissible solution $u$ and all distinct $t_1,t_2$ in its interval of existence, where $\mathcal V$ is the class of admissible instantaneous velocity fields. This is only a candidate interpretation: the source does not define $\mathcal V$, require a universal rule explicitly, or specify how coincident snapshots and exceptional solutions are to be treated.

The source also asks whether fluid "aging" can be quantified. Its stated motivation is that numerical experiments suggest that, starting from a smooth initial velocity $u_0=u(0,\cdot)$, the velocity becomes "wrinkled" and its derivatives generally grow. The record attributes the expression "regularity deterioration" for this phenomenon to V. Yudovich. These observations are supplied motivation, not a stated theorem or a specified monotonicity property. No derivative order, norm, aging functional, time direction, growth rate, or quantifier making "generally" precise is supplied. Consequently, the request to quantify aging remains an open-ended formulation requiring further hypotheses.

Short English statement

The source asks whether two velocity snapshots from an arbitrary Euler flow reveal their chronological order, and whether the suggested growth of spatial derivatives can quantify fluid "aging." The domain, solution class, exceptions, and intended quantitative criterion are unspecified. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, permanent ID $800819$, source problem code catalogue entry, title: Geometric Hydrodynamics in Open Problems --- Problem 18 (Aging Problem). The supplied record attributes the problem to Khesin, Misiolek, and Modin, with proposed year 2022, and labels its status open.
• Khesin, Misiolek, and Modin, Geometric Hydrodynamics in Open Problems (2022), Problem 18. Supplied source link: https://arxiv.org/abs/2205.01143. The background records an access date of 2026-10-16 and extraction method arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• The supplied statement attributes the term "regularity deterioration" to V. Yudovich and reports numerical experiments suggesting aging. No separate bibliographic reference for either attribution or the experiments is supplied.