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

TL;DR

This paper studies the existence and uniqueness of solutions to various boundary value problems for the steady incompressible Euler equations in an annular domain, using different mathematical methods to establish well-posedness.

Contribution

It introduces a comprehensive analysis of five classes of boundary value problems, applying the Grad-Shafranov and vorticity transport methods to prove well-posedness in different function spaces.

Findings

Three boundary conditions are effectively addressed with the Grad-Shafranov method.

All five classes of problems are solvable via the vorticity transport method.

Well-posedness of $C^{2,eta}$ solutions is established under perturbations.

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.