2026/08/03 by Marc Magaña
Mathematics · #math.AP #msc:35Q35 #msc:35B65 #msc:35A01 #msc:35A02 #msc:45G05 #msc:76B47
arxiv created 2026/08/03 · arxiv updated 2026/08/04
We establish a unified local theory for the persistence of Sobolev regularity of vortex patch boundaries in a family of two-dimensional active scalar equations with radial convolution kernels K(|x-y|). The class includes the 2D Euler equation, the generalized SQG equation in the locally integrable range 0<β<1, and the quasi-geostrophic shallow water equation. Under natural assumptions on K (smoothness, integrability near the origin, monotonicity, and polynomial growth), we prove that if the initial boundary belongs to H3(\mathbb T) and satisfies the arc-chord condition, then the contour dynamics equation admits a unique local solution in C([0,T];H3(\mathbb T)). Under a stronger integrability condition on the kernel, we also obtain local existence of H2 solutions. The proof combines Sobolev energy estimates for the contour equation with quantitative control of the arc-chord quantity.