An Infinite Family of Approximate Conservation Laws for Point Vortex Dynamics
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I report an infinite family of approximate conservation laws for two-dimensional point vortex dynamics. For any smooth function f, the pairwise-weighted quantity Q_f = Σ Γ_i Γ_j f(r_ij) is approximately conserved during Kirchhoff evolution. Two members reduce to known exact invariants (energy and angular impulse); the remaining members are new. The optimal non-trivial member is f(r) = √r, with fractional variance 3×10⁻¹¹. In 3D vortex filament dynamics, Q_{1/r} achieves the best conservation. Both optimal choices are Green's functions of the Laplacian, and both equal the kinetic energy up to constants. The family exhibits a dichotomy between concentration-detecting and stretch-resistant members relevant to Navier-Stokes regularity. Higher-order (triplet) generalizations do not exist, confirming the pairwise structure is special. Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20 Updated 2026-03-20
Fields of Study & Subjects
Navier-Stokes equation solutions
Fluid Dynamics and Turbulent Flows
Nonlinear Waves and Solitons
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 16 March 2026
- Language
- EN
- Document type
- Preprint