Global Regularity and Fast Small Scale Formation for Euler Patch Equation in a Smooth Domain

2018/06/19 by Alexander Kiselev, Chao Li, Kiselev, Alexander +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1806.07744

openalex publication_date 2018/06/19 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

It is well known that the Euler vortex patch in ℝ² will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. In this paper, we study Euler vortex patches in a general smooth bounded domain. We prove global in time regularity by providing an upper bound on the growth of curvature of the patch boundary. For a special symmetric scenario, we construct an example of double exponential curvature growth, showing that our upper bound is qualitatively sharp.

Cited by

• Unified theory for regularity persistence of vortex patch boundaries

Related

• Global regularity and fast small scale formation for Euler patch equation in a disk
• Existence and Regularity for Vortex Patch Solutions of the 2D Euler Equations
• Regularity of the velocity field for Euler vortex patch evolution
• The Slightly Supercritical Euler Equations: Smooth Solutions and Vortex\n Patches
• Contour dynamics and global regularity for periodic vortex patches and layers