Stability of the Rankine Vortex and Perimeter Growth in Vortex Patches

2025/11/12 by John Brownfield, Brownfield, John
Mathematics · Engineering · Physics and Astronomy · #Navier-Stokes equation solutions #Fluid Dynamics and Turbulent Flows #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2511.09772

Abstract

We prove that for ω: ℝ² → [0,1] sharing the same total vorticity and center of vorticity as the Rankine vortex, the L¹ deviation from the Rankine patch can be bounded by a function of the pseudo-energy deviation and the angular momentum of ω. In the case of m-fold symmetry, the dependence on the angular momentum can be dropped. Using this, we affirm the results of prior simulations by demonstrating linear in time perimeter growth for a simply connected perturbation of the Rankine vortex.

Related

• Stability in L¹ of circular vortex patches
• Growth of perimeter for vortex patches in a bulk
• On the rigidity of uniformly rotating vortex patch near the Rankine vortex
• On the trivial solutions for the rotating patch model
• Boundary effects on the emergence of quasi-periodic solutions for Euler equations