The rectifiability of singular sets for geometric flows (I)--Yang-Mills flow

In integral geometry generalized with Aleksandrov's variational theory, Lutwak-Xi-Yang-Zhang [Comm. Pure Appl. Math. 77 (2024)] recently opened the door to researching the cone-chord measures and their log-Minkowski problem stemming from the chord integrals, named as the chord log-Minkowski problem. In the smooth category, the solvability of the chord log-Minkowski problem amounts to dealing with a nonlocal {M}onge-{A}mpère equation involving a Riesz potential. In this paper, to study the chord log-Minkowski problem, we first present some novel results on the boundary regularity of the Riesz potential. Based on these results, we obtain the regularity and existence for the chord log-Minkowski problem from the perspective of a nonlocal {M}onge-{A}mpère equation and a nonlocal Gauss curvature flow equation.

math.AP↗

arXiv2026

A proof of Onsager's Conjecture for the SQG equation

We construct solutions to the SQG equation that fail to conserve the Hamiltonian while having the maximal allowable regularity for this property to hold. This result solves the generalized Onsager conjecture on the threshold regularity for Hamiltonian conservation for SQG. We also prove nonuniqueness for arbitrary smooth data and an h-principle result.

math.AP↗

arXiv2026

A quantitative Borg-Levinson theorem for a large class of unbounded potentials

We prove a quantitative Borg-Levinson theorem for a broad class of unbounded potentials when the spatial dimension is five or greater. In the final section, we outline the modifications required to extend the main result to the anisotropic case.

math.AP↗