2024/11/18 by Takashi Sakajo, Sakajo, Takashi, Changjun Zou +1
Mathematics · #76B47 #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2411.11388
openalex publication_date 2024/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct a series of patch type solutions for incompressible Euler equation on \mathbb S2, which constitutes the regularization for steady or traveling point vortex systems. We first prove the existence of k-fold symmetric patch solutions, whose limit is the well-known von Kármán point vortex street on \mathbb S2; then we consider the general steady case, where besides a non-localized part induced by the sphere rotation, j positive and k negative patches are located near a nondegenerate critical point of the Kirchhoff--Routh function on \mathbb S2. Our construction is accomplished by Lyapunov--Schmidt reduction argument, where the traveling speed or vortex patch location are used to eliminate the degenerate direction of a linearized operator. We also show that the boundary of each vortex patch is a C1 close curve, which is a perturbation of a small ellipse in the spherical coordinates. As far as we know, this is the first attempt for a regularization of the point-vortex equilibria on \mathbb S2.