Dynamic Limits for Congested and Branched Transport

Definitions and mathematical statement

Let $\beta>0$ denote the exponent in the dynamic optimal-transport system referred to in the supplied record, and let $(\mu(t),u(t))$ denote its solution pair at time $t$. The record does not specify the evolution equations, the spatial domain, the meanings or function spaces of $\mu$ and $u$, the forcing data, initial conditions, boundary conditions, or the solution concept. Thus these symbols identify the source's intended objects but do not define a self-contained dynamical system.

The supplied conjecture asks whether the solution pair converges to an equilibrium as $t\to\infty$, schematically \[ (\mu(t),u(t))\longrightarrow(\mu_\infty,u_\infty), \qquad t\to\infty, \] where the limiting equilibrium is associated with congested transport for $0<\beta<1$, and with branched transport for $\beta>1$.

The convergence topology, the equilibrium equations, and the precise meaning of either transport association are unspecified. In particular, no transport functional, admissible class, or criterion relating an equilibrium to a transport solution is supplied. The record also does not specify which data or solutions the conjecture quantifies over, whether solutions exist for all positive times, or how nonuniqueness would affect the assertion. Although the parameter is introduced as $\beta>0$, the supplied regime descriptions give no conclusion for $\beta=1$. Consequently, this is an unresolved formulation of the reported conjecture; a precise quantified mathematical statement requires the missing definitions and hypotheses.

Short English statement

The supplied conjecture asks whether the dynamic optimal-transport solution converges to an equilibrium describing congested transport for $0<\beta<1$ and branched transport for $\beta>1$. The equations, admissible data, convergence notion, and transport characterization are missing, and the case $\beta=1$ is unspecified. These omissions prevent a self-contained precise formulation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003938$, title Dynamic Limits for Congested and Branched Transport, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2018.
• 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 record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no location within the cited page range is supplied.
• Supplied literature reference: Extended Dynamic-Monge-Kantorovich equations. https://doi.org/10.48550/arXiv.1811.12691.
• Supplied literature reference: Branching structures emerging from a continuous optimal transport model. https://doi.org/10.1016/j.jcp.2021.110700.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the congested and branched transport interpretations and long-time convergence for the extended dynamics are conjectural, with later work primarily variational or numerical rather than a general convergence theorem. This is supplied dated curation derived from the OWR record, not independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.