Solutions to the two-dimensional steady incompressible Euler equations in an annulus

Abstract

This paper investigates the well-posedness of five classes of boundary value problems for the two-dimensional steady incompressible Euler equations in an annular domain. Three of these boundary conditions can be effectively addressed using the Grad-Shafranov method, and the well-posedness of solutions in the $C^{1,\al}$ space is established via variational techniques. We demonstrate that all five classes of boundary value problems are solvable through the vorticity transport method. Based on this approach, we further prove the well-posedness of $C^{2,\al}$ solutions under a perturbation framework.