An Infinite Family of Approximate Conservation Laws for Point Vortex Dynamics
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2026
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| author | Sanchez, Bryan |
| author_facet | Sanchez, Bryan |
| contents | 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 |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19136335 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | An Infinite Family of Approximate Conservation Laws for Point Vortex Dynamics Sanchez, Bryan point vortex dynamics approximate conservation laws Kirchhoff equations Green's function vortex filaments Navier-Stokes regularity fractional variance invariant families 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 |
| title | An Infinite Family of Approximate Conservation Laws for Point Vortex Dynamics |
| topic | point vortex dynamics approximate conservation laws Kirchhoff equations Green's function vortex filaments Navier-Stokes regularity fractional variance invariant families |
| url | https://doi.org/10.5281/zenodo.19136335 |