Convergence of Dynamic Monge–Kantorovich Systems

Definitions and mathematical statement

The supplied record denotes by $\mu(t)$ the density variable and by $u(t)$ the potential variable of a dynamic Monge--Kantorovich system. It denotes by $(\mu^*,u^*)$ an optimal-transport density and potential satisfying the associated stationary Monge--Kantorovich equations. Neither the evolution system nor the stationary equations are supplied explicitly.

Conjectural convergence target. For a solution trajectory $t\mapsto(\mu(t),u(t))$ defined for all $t\geq 0$, the proposed assertion is \[ \left(\mu(t),u(t)\right)\longrightarrow\left(\mu^*,u^*\right) \qquad\text{as }t\to\infty. \] This formula records the supplied target only: a fully quantified convergence conjecture cannot be specified from this record. The spatial domain, transport data and cost, evolution equations, initial and boundary conditions, admissible solution class, and convergence topology are absent. The record also does not specify how the limiting potential is selected or normalized, whether convergence of potentials is understood modulo additive constants, or whether the assertion concerns every admissible initial state and every corresponding global solution. No uniqueness of the limiting pair or global existence is asserted here.

Short English statement

The conjecture asks whether the dynamic Monge--Kantorovich density and potential approach a stationary optimal-transport density and potential as time tends to infinity. The supplied record leaves the equations, admissible data, solution quantifiers, potential normalization, and meaning of convergence unspecified, so this remains a schematic statement requiring mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30003937$, source problem code catalogue entry, title: Convergence of Dynamic Monge--Kantorovich Systems. The record assigns proposed year 2018 and names no individual proposer. Its supplied status is open; its statement-status label is corrected_verified. These are registry labels, not independent mathematical verification.
• Supplied report citation: Reactive Flows in Deformable, Complex Media (2019), Oberwolfach Reports, pp. 2385--2473, DOI 10.4171/owr/2018/39. https://doi.org/10.4171/owr/2018/39. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports; the literature-source list identifies this reference as Oberwolfach Report 39/2018.
• Supplied literature reference: Towards a stationary Monge--Kantorovich dynamics. https://doi.org/10.1137/16M1098383.
• Supplied literature assessment, checked on 2026-08-21: the proposed Lyapunov functional is reported to have the optimal-transport density as its unique minimizer; numerical experiments are reported to support convergence; and global convergence of the continuous dynamics is reported to remain conjectural. This is supplied dated curation derived from the OWR record, not independently verified. Its time-sensitive status and cited mathematical claims require independent verification.