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

Wengang Yang

TL;DR

This work analyzes five boundary-value problems for the 2D steady incompressible Euler equations in an annulus, establishing well-posedness and connecting two principal methodologies. The Grad-Shafranov reduction converts the problem to a nonlinear elliptic equation for a stream function, enabling variational and fixed-point solutions under boundary data constraints. The vorticity-transport framework provides a unified route to all boundary conditions by propagating vorticity along characteristics and solving a subsequent div-curl system, with perturbative results yielding $C^{2,\alpha}$ regularity. Together, these results advance understanding of boundary-driven rotational Euler flows in annular domains and relate to magnetohydrostatic analogies via the Bernoulli/MHS correspondence.

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.

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

TL;DR

This work analyzes five boundary-value problems for the 2D steady incompressible Euler equations in an annulus, establishing well-posedness and connecting two principal methodologies. The Grad-Shafranov reduction converts the problem to a nonlinear elliptic equation for a stream function, enabling variational and fixed-point solutions under boundary data constraints. The vorticity-transport framework provides a unified route to all boundary conditions by propagating vorticity along characteristics and solving a subsequent div-curl system, with perturbative results yielding regularity. Together, these results advance understanding of boundary-driven rotational Euler flows in annular domains and relate to magnetohydrostatic analogies via the Bernoulli/MHS correspondence.

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 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 solutions under a perturbation framework.

Paper Structure

This paper contains 11 sections, 14 theorems, 178 equations, 1 table.

Table of Contents

  1. Introduction and main results
  2. Main results and overview
  3. Notation
  4. Paper Organization
  5. Grad-Shafranov approach
  6. variational method
  7. Iteration method
  8. The vorticity transport method
  9. Boundary condition
  10. Boundary condition
  11. Alternative approach for boundary condition

Key Result

Theorem 2.1

Let $f_0(\theta),f_1(\theta),b_0(\theta)\in C^{1,\alpha}(\mathbb{T}_{2\pi})$ and $f_0(\theta)\geq\delta_0>0$ for all $\theta\in\mathbb{T}_{2\pi}$. Then the steady incompressible Euler equation InE_Pol admits a solution $(\mathbf{u},p)\in (C^{1,\alpha}(\Omega))^3$ satisfying the boundary condition BC

Theorems & Definitions (27)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.1
  • proof
  • ...and 17 more