Finite-Time Blowup for Three-Dimensional Zakharov–Kuznetsov

Definitions and mathematical statement

The supplied question asks whether the three-dimensional Zakharov--Kuznetsov equation in the $L^2$-critical regime has a solution that blows up in finite forward time. The record does not specify the differential equation, its nonlinear term and sign, the spatial domain, boundary conditions, admissible initial data, solution concept, or the meaning of blow-up. Consequently, it does not determine a unique mathematical proposition.

To describe the missing specifications, let $\Omega\subseteq{\mathbb{R}}^3$ denote a spatial domain, let $X(\Omega)$ be a normed space of functions with a specified scalar field, and let $\mathcal E[u]=0$ denote a specified evolution equation with specified boundary conditions and solution concept. If blow-up is intended to mean unboundedness in $X(\Omega)$, a conditional formulation of the question is whether \[ \begin{gathered} \exists u_0\in X(\Omega)\;\exists T_*\in(0,\infty)\;\exists u:[0,T_*)\longrightarrow X(\Omega),\\ u(0)=u_0,\qquad \mathcal E[u]=0,\qquad T_*=T_{\max}(u),\\ \limsup_{t\uparrow T_*}\|u(t)\|_{X(\Omega)}=\infty, \end{gathered} \] where $T_{\max}(u)$ is the maximal forward existence time in the chosen solution class. This is a formulation template, not a selection of the missing hypotheses or a claim that this norm criterion is the source's intended definition of blow-up.

The designation $L^2$-critical refers to invariance of the $L^2$ norm under the relevant equation's scaling. The equation and scaling must be specified to make that designation precise; it does not itself identify the norm in which blow-up is sought. The supplied expectation of finite-time blow-up is motivation, not an established existence assertion.

Short English statement

The question asks whether an $L^2$-critical three-dimensional Zakharov--Kuznetsov equation has a solution that blows up in finite time. The supplied record omits the equation and analytic setting, so a definitive mathematical statement requires further source clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003148$, title "Finite-Time Blowup for Three-Dimensional Zakharov--Kuznetsov," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2016.
• Supplied report citation: Nonlinear Evolution Problems (2017), Oberwolfach Reports, pp. 1517--1576, DOI 10.4171/owr/2016/27. https://doi.org/10.4171/owr/2016/27. The record describes these reports as documenting mathematical workshops and their open-problem discussions. The supplied citation does not locate the question more precisely within this page range.
• Supplied literature reference: Klein--Roudenko--Stoilov, "3D critical ZK." https://arxiv.org/abs/2608.15086.
• Supplied literature reference labeled "original OWR report." https://doi.org/10.4171/owr/2016/27.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that finite-time blow-up for the three-dimensional $L^2$-critical ZK equation remains unproved, while the August 2026 work provides formal analysis and numerical evidence rather than a rigorous existence theorem. This is supplied dated curation, not an independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired. This supplied label does not constitute independent verification of this draft or resolve its missing hypotheses.