Finite-Time Perfect Mixing Under Power Constraints

Definitions and mathematical statement

The supplied question is: Determine whether finite-time perfect mixing can occur for a two-dimensional incompressible flow subject to a finite power constraint.

Let $\Omega$ denote the two-dimensional spatial domain, whose geometry and boundary conditions are not specified in the supplied record. Write $u(t,x)$ for the velocity field; incompressibility means \[ \nabla\cdot u=0, \] with the sense of this equation dependent on the unspecified regularity class.

The following notation describes the missing ingredients without selecting them. Let $q$ denote the state being mixed, let $\mathcal M(q)\geq 0$ denote a specified measure of its departure from perfect mixing, and let $\mathcal A_T(q_0)$ denote the class of admissible evolutions $(u,q)$ on $[0,T]$ with initial state $q_0$, incorporating the intended evolution equation, incompressibility, boundary conditions, regularity, and finite power constraint. If perfect mixing is defined by $\mathcal M(q)=0$, then the question has the schematic form \[ \text{Can there exist }T\in(0,\infty) \text{ and }(u,q)\in\mathcal A_T(q_0) \text{ such that }\mathcal M(q_0)>0 \text{ and }\mathcal M(q(T))=0? \] This is a formulation template, not a fully specified conjecture: the record does not define $q$, $\mathcal M$, $\mathcal A_T(q_0)$, or the intended quantifier over initial states. The initial nonzero mixing measure expresses the nontrivial interpretation of reaching perfect mixing; the source does not explicitly state an initial-data hypothesis.

In particular, the record does not specify whether the transported state obeys pure advection or an equation with diffusion, what norm or functional defines perfect mixing, what quantity constitutes power, or whether its constraint is instantaneous or integrated in time. These choices cannot be supplied silently.

The accompanying supplied literature assessment describes a finite-power exponential lower-bound conjecture. Its precise inequality, constants, parameter dependence, admissible class, and quantifiers are absent. An exponential lower bound and exclusion of finite-time perfect mixing are distinct assertions; their equivalence is not supplied and is not asserted here. The source labels the problem open, but that label is supplied curation rather than an independently verified mathematical conclusion.

Short English statement

Can an initially imperfectly mixed state become perfectly mixed in finite time under a two-dimensional incompressible flow with finite power? The supplied record leaves the domain, evolution equation, power constraint, mixing criterion, and initial-data quantifiers unspecified, so a unique precise conjecture cannot yet be stated. Its accompanying exponential lower-bound formulation also requires clarification.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002292$, source code catalogue entry, title Finite-Time Perfect Mixing Under Power Constraints; collection: Oberwolfach Reports Open Problems. The supplied record gives proposed year 2013 and identifies no individual proposer.
• Supplied report citation: Geophysical Fluid Dynamics (2013), Oberwolfach Reports, pp. 521--577, DOI 10.4171/owr/2013/10. https://doi.org/10.4171/owr/2013/10. The record attributes the question to workshop open-problem discussions documented in this report; no specific problem page or passage is supplied.
• Supplied literature source: Lunasin--Lin--Novikov--Mazzucato--Doering, Optimal mixing and stirring. https://doi.org/10.1063/1.4752098.
• Supplied literature source: Zhu et al., Optimal flows under constraints. https://arxiv.org/abs/2401.05684.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. It labels the problem open, describes the finite-power exponential lower-bound problem as a conjecture, and states that later instantaneous optimization results under related constraints do not prove the global finite-power statement. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry marks the statement as exact and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized. This is a supplied source-verification assertion; the references have not been independently consulted for this draft.