Hölder regularity for collapses of point vortices

The paper proves 1/(α+1)-Hölder rates at vortex collapse in the plane (Theorem 1.3) and a 1/2-Hölder rate in bounded domains under an interior-adherence hypothesis (Theorem 1.4), via a cluster decomposition and a quantitative “no-collapse-too-fast” criterion (Proposition 2.4) that implies a minimal-time barrier comparable to η^{α+1} for shrinking separations. These results and their proofs are internally consistent and carefully track the role of non-neutrality and boundary effects. The candidate’s plane-case argument reproduces the 1/(α+1) rate through a clean Dini-derivative bound for the minimal separation and is essentially correct. However, for bounded domains the candidate omits the crucial hypothesis that the collapsing vortex has an interior adherence point and incorrectly treats the smooth harmonic terms as uniformly Lipschitz with constants depending only on Ω and the intensities; the paper shows that ∇_xγ_Ω can blow up like 1/dist(x,∂Ω), and therefore the bounded “forcing” reduction requires staying away from the boundary. The paper explicitly builds this into the statement and proof of Theorem 1.4, whereas the model’s statement would be false without that interior-distance control. See Theorem 1.3 and its use of Proposition 2.4 for the plane-case rate, and Theorem 1.4 plus the decomposition/estimates for the bounded domain case .