Rotating vortex patches for the planar Euler equations in a disk

2019/08/29 by Daomin Cao, Jie Wan, Cao, Daomin +5
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1908.11093

openalex publication_date 2019/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a family of rotating vortex patches with fixed angular velocity for the two-dimensional Euler equations in a disk. As the vorticity strength goes to infinity, the limit of these rotating vortex patches is a rotating point vortex whose motion is described by the Kirchhoff-Routh equation. The construction is performed by solving a variational problem for the vorticity which is based on an adaption of Arnold's variational principle. We also prove nonlinear orbital stability of the set of maximizers in the variational problem under L^p perturbation when p∈[3/2,+∞).

Related

• Existence of steady symmetric vortex patch in a disk
• On radial symmetry of rotating vortex patches in the disc
• Desingularization of nondegenerate rotating vortex patches
• On the trivial solutions for the rotating patch model
• On the rigidity of uniformly rotating vortex patch near the Rankine vortex