Stable High-Order Regularization of Momentum Sheets

Definitions and mathematical statement

The source poses four tasks for a momentum-sheet application. The governing PDE, singular Green kernel, sheet variables, and optimization problem are not reproduced, so these tasks form a source-dependent numerical and geometric programme.

First, construct a high-order regularization $K_\varepsilon$ of the singular Green function $K$ without temporal instability. Here $\varepsilon$ denotes a regularization scale, not necessarily a time step. A precise order claim would require a specified consistency quantity and norm with an error bound such as $C\varepsilon^p$; the record supplies neither the target order, norm, nor admissible solutions. Temporal stability likewise needs a defined linearized or nonlinear evolution, time interval, and allowed parameter regime. No arbitrary smoothing kernel or stability criterion is chosen.

Second, preserve potential vorticity in the intended model or its discretization. The source does not define that quantity, its transport law, or whether preservation means an exact invariant, an advected field, or a controlled numerical error. A fluid-specific potential-vorticity formula cannot be imported without identifying the equations.

Third, integrate the normal motion of straight momentum sheets. The sheet geometry, normal velocity law, initial data, interactions, boundary conditions, and whether the requested integration is analytic or numerical are not supplied. The problem is not silently reduced to tangential reparameterization or to a different vortex-sheet model.

Fourth, relate symplecticity of the integrator directly to optimization behavior. In a symplectic phase-space formulation, a one-step map $\Psi_h$ is symplectic when \[ \Psi_h^*\Omega=\Omega, \] where $\Omega$ is the symplectic form and $h$ the time step. The phase space, objective function, optimization iteration, and desired convergence or performance criterion are absent. The source asks for a relationship, not an assertion that symplecticity alone implies descent or optimization convergence.

The supplied literature assessment checked 2026-08-21 reports these high-order stable-regularization and momentum-sheet structural questions as open, with no general resolution located. The exact source says all four requests are retained, and none is treated as solved or omitted here.

Short English statement

For the intended momentum-sheet model, seek stable high-order kernel regularization, preservation of potential vorticity, integration of normal sheet motion, and a precise link between symplectic integration and optimization. The equations and quantitative success criteria are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000443, catalogue entry, Stable High-Order Regularization of Momentum Sheets.
• Geometric Numerical Integration (2006), Oberwolfach Reports, pp. 805--882, DOI https://doi.org/10.4171/owr/2006/14.
• Supplied literature triage checked 2026-08-21 cites only that report; no governing equations, kernel formula, optimization functional, or separate theorem citation is supplied.