Existence, uniqueness and stability of steady vortex rings of small cross-section

This paper is concerned with steady vortex rings in an ideal fluid of uniform density, which are special global axi-symmetric solutions of the three-dimensional incompressible Euler equation. We systematically establish the existence, uniqueness and nonlinear orbital stability of steady vortex rings of small cross-section for which the potential vorticity is constant throughout the core. The latter two answer a long-standing question since the pioneering work of Fraenkel and Berger \cite{BF1} (Acta Math., 1974). To achieve our goal, we rescale the Stokes stream function of vortex ring by its cross-section radius, and expand it at the well-known Rankine vortex using Taylor's formula, where the estimates for coefficients are obtained by a decomposition according to Green's function and local Pohozaev identities. The main observations are: The stream function is even and has a translational invariance in $z$-direction; the zero point for the $r$-coefficient of linear term in its expansion determines the asymptotic location of vortex ring, which appears as the condition to eliminate the degenerate direction in Lyapunov-Schmidt reduction argument for existence; the non-vanishing condition of the second order $r$-coefficient at foresaid zero point is verified as one of the essential factors for uniqueness, while the negativity means that these vortex rings maximizes the functional composed of kinetic energy and impulse. By applying the Arnol'd's dual variational principle together with the uniqueness result, we are finally able to prove the nonlinear orbital stability of thin vortex rings. This result gives a large class of stable vortex rings supported on topological tori, which is different from Hill's spherical vortex discussed by Choi \cite{Choi20} (Comm. Pure Appl. Math., 2023).