On the steady axisymmetric vortex rings for 3-D incompressible Euler flows

2020/09/28 by Daomin Cao, Cao, Daomin, Weicheng Zhan +1
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2009.13210

openalex publication_date 2020/09/28 · openalex created_date 2020/10/01 · openalex updated_date 2026/07/28

Abstract

In this paper, we study nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler flows. We construct a family of steady vortex rings (with and without swirl) which constitutes a desingularization of the classical circular vortex filament in ℝ³. The construction is based on a study of solutions to the similinear elliptic problem -(1)/(r)(∂)/(∂ r)((1)/(r)(∂ψ^ε)/(∂ r))-(1)/(r²)(∂²ψ^ε)/(∂ z²)=(1)/(ε²)(g(ψ^ε)+(f(ψ^ε))/(r²)), where f and g are two given functions of the Stokes stream function ψ^ε, and ε>0 is a small parameter.

Citations

• Asymptotic estimates for the plasma problem
• A non-linear eigenvalue problem: The shape at equilibrium of a confined plasma
• Desingularization of vortices for 2D steady Euler flows via the vorticity method

Related

• Desingularization of vortex rings in 3 dimensional Euler flows: with swirl
• Desingularization of vortex rings in 3 dimensional Euler flows
• Asymptotic behavior of global vortex rings
• Uniqueness and stability of steady vortex rings for 3D incompressible Euler equation
• Existence, uniqueness and stability of steady vortex rings of small cross-section