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 R. The construction is based on a study of solutions to the similinear elliptic problem -1/r partial derivative/partial derivative r (1/r partial derivative Psi/partial derivative z(2)) = 1/epsilon(2 )(g(Psi)) + f(Psi)/r(2), r > 0, z is an element of R with suitable boundary conditions, where and are two given functions of the Stokes stream function , and is a small parameter.

Publication:

SIAM JOURNAL ON MATHEMATICAL ANALYSIS

http://dx.doi.org/10.1137/24M1678076

Author:

DAOMIN CAO

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

dmcao@amt.ac.cn

WEICHENG ZHAN

School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R. China

zhanweicheng@amss.ac.cn