Scalarization and Superlevel Directional Control in Localized Two-Point Vortex Stretching
Year: 2026
Type: preprint
Abstract: We study how absolute majorants interact with the two symmetry components of the direct one-sided localization of Constantin's principal-value vortex-stretching representation. In the increment-weighted antisymmetric component, replacing the cross product by scalar amplitudes after a mean–increment factorization destroys an endpoint zero of the exact density. For compactly supported fields, this produces a separation-independent floor and long-range divergence whenever the relevant cutoff or compact set captures nonzero vorticity. The endpoint-preserving majorant vanishes in the same vacuum sector; its separation-weighted far-field integral admits an unrestricted L² bound, while compact incidence refines that bound by localizing the quadratic L² mass through a geometry-generated weight. In the mean-weighted symmetric component, crude scalarization is locally integrable but incurs a scale-compatible amplitude cost, and within the declared direct Hölder factorization, controlling both gradient factors solely in L² leaves that amplitude unavoidable. We then estimate the increment budget of this component at two separations at once: below a freeinternal scale h∗ the increment is paid by halo palinstrophy, above it by halo enstrophy, and the intermediate integration produces a logarithm. Our main result expresses the high–high contribution through the associated half-Hölder cross-product modulus on pairs whose two endpoints satisfy |ω| ≥ M, and pays the low-vorticity complement by the two-scale budget (1/8) M h∗² Q_{W,ρ} + 6 M E_{W,ρ} log(ρ/h∗), where Q_{W,ρ} and E_{W,ρ} are the palinstrophy and the enstrophy over the ρ-halo of the cutoff support. A common Hardy–Littlewood–Sobolev estimate controls the all-pairs and high–high sectors. Because h∗ is a parameter of the proof and not theobservation radius, the viscous share of the complement can be made as small as desired at the price of a finite deterministic logarithmic enstrophy coefficient; that coefficient vanishes exactly when M ρ²/(8ν) does not exceed the chosen budget parameter, so the one-scale absorption window is the condition for a logarithm-free budget and not an obstruction intrinsic to the complement. We also show that all-pairs finiteness is not automatic for smooth compactly supported solenoidal fields,whereas the high–high modulus is finite pointwise for fixed M > 0. No regularity criterion is asserted.
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212158099