Nonlocal Water Waves with Generalized Vorticity

Definitions and mathematical statement

Write $\Omega(t)$ for the fluid region, $\Gamma(t)$ for its moving free surface, and $u$ for its velocity. In classical local coordinates, vorticity can be represented by the antisymmetric tensor \[ \omega_{ij}=\partial_i u_j-\partial_j u_i. \] This notation does not select a spatial dimension or prescribe a weak interpretation of vorticity. The supplied record uses both "general vorticity" and "generalized vorticity" without defining either class.

Statement of the supplied research problem. Does the nonlocal interface formulation of the water-wave problem for constant vorticity admit an extension to general or generalized vorticity? If such an extension exists, determine whether it yields additional conservation laws. Here a conserved quantity $Q$ would satisfy \[ Q(t_2)=Q(t_1) \] for every admissible solution and every pair of times $t_1,t_2$ in its interval of existence. The record does not specify the admissible solutions, the quantities already known to be conserved, or the permitted class of additional conserved quantities. The possibility of new conservation laws is a question, not an asserted consequence.

Unresolved formulation requirements. The source record supplies neither the constant-vorticity nonlocal interface equations nor the governing water-wave equations and boundary conditions. It does not specify spatial dimension, fluid depth, horizontal geometry, regularity, surface tension, or the meaning of generalized vorticity. Consequently, a fully quantified extension or equivalence statement cannot be extracted from this record alone. The notation above describes the intended objects without supplying these missing hypotheses.

Short English statement

The problem asks whether the constant-vorticity nonlocal interface description of water waves extends to a broader vorticity class and, if so, whether it produces new conserved quantities. The intended equations, admissible vorticities, and boundary conditions require clarification before a precise theorem-level statement can be given.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004187$, title "Nonlocal Water Waves with Generalized Vorticity," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposed year is supplied as 2019; no individual proposer is identified.
• Supplied source citation: Mathematical Theory of Water Waves (2020), Oberwolfach Reports, pp. 1919--1979, DOI 10.4171/owr/2019/32. Supplied literature-source label: "original OWR report." https://doi.org/10.4171/owr/2019/32. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; it gives no precise location for this question within the report.
• Supplied literature source: "Mobbs, vorticity conservation laws." https://doi.org/10.1017/S002211208100222X. No further bibliographic details or specific supporting theorem are supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and describes a conjectural extension of conservation laws to generalized vorticity, stating that existing vorticity laws do not establish this particular water-wave extension. 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 supplied statement status is corrected_verified, with an editorial explanation that notation, typography, and sentence structure were repaired while mathematical content was preserved. This supplied label does not establish mathematical correctness or independent source verification of this draft.