Convergence of Complex-Balanced Systems with Boundary Equilibria

Definitions and mathematical statement

For terminology, consider a reaction network with species concentrations $c=(c_1,\ldots,c_m)\in[0,\infty)^m$. A reaction $r$ has input and output complexes $y_r,y'_r\in\mathbb{Z}_{\geq0}^m$ and reaction flux $v_r(c)$. Its contribution to the reaction vector field is $(y'_r-y_r)v_r(c)$. A strictly positive concentration vector $c_*$ is complex balanced if, for every complex $y$, \[ \sum_{r:\,y_r=y}v_r(c_*) = \sum_{r:\,y'_r=y}v_r(c_*). \] These equations explain the complex-balance terminology; the supplied record does not specify the reaction kinetics or the precise class of networks under consideration. A boundary equilibrium, in concentration-vector notation, is a nonnegative equilibrium $c_b$ with $(c_b)_i=0$ for at least one species $i$. For the reaction--diffusion problem, the intended interpretation of boundary equilibrium, including whether equilibria must be spatially homogeneous, is not specified.

The requested problem is to establish convergence to equilibrium for renormalized solutions of general complex-balanced reaction--diffusion systems that admit boundary equilibria. Schematically, the proposed conclusion is \[ \text{for every admissible global renormalized solution }c, \quad \exists\,c_\infty\text{ an equilibrium compatible with the solution's conserved quantities}, \quad c(t)\longrightarrow c_\infty\quad(t\to\infty), \] or to establish an entropy--dissipation estimate sufficient to imply this conclusion. The terms "admissible" and "compatible" here mark hypotheses that must be supplied, rather than additional hypotheses asserted by this draft.

The source does not specify the spatial domain, its dimension or regularity, the diffusion operators or coefficients, boundary conditions, admissible initial data, the definition of renormalized solution, or the topology of convergence. It also does not identify the limiting equilibrium as positive or on the boundary, state uniqueness of that equilibrium, prescribe a convergence rate, or specify an entropy functional and the required dissipation estimate. Consequently, the displayed formulation records the intended quantifier structure but is not a fully determined theorem statement. These missing qualifications require mathematical review before a precise assertion can be fixed.

The supplied dated assessment says that entropy methods establish convergence for important complex-balanced classes and under additional trajectory conditions, while the general problem with boundary equilibria remains open. That assessment and the supplied open label are source qualifications, not independently verified mathematical or literature conclusions.

Short English statement

Determine whether renormalized solutions of general complex-balanced reaction--diffusion systems admitting boundary equilibria converge to equilibrium, or prove an entropy--dissipation estimate implying convergence. The supplied record leaves essential hypotheses, the solution concept, and the meaning of convergence unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003384$, exact source problem code catalogue entry, titled "Convergence of Complex-Balanced Systems with Boundary Equilibria," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2017 and names no individual proposer.
• Supplied report citation: Applications of Optimal Transportation in the Natural Sciences (2018), Oberwolfach Reports, pp. 339--416, DOI 10.4171/owr/2017/7. https://doi.org/10.4171/owr/2017/7. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Fellner and Tang, "Convergence to equilibrium." https://imsc.uni-graz.at/fellnerk/preprints/FTRenorm.pdf.
• Supplied literature source: "Oberwolfach Report 7/2017." https://ems.press/content/serial-article-files/46673.
• Literature assessment date, copied exactly: 2026-08-21. This is supplied dated curation derived from the supplied OWR curation, not independently verified. It labels the problem open and reports convergence results for important complex-balanced classes and under additional trajectory conditions, while describing the general case with boundary equilibria as open. Time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and explains that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This is supplied editorial metadata and does not establish independent mathematical verification of this draft.