Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Helically symmetric solution of 3D Euler equations with vorticity and its free boundary

Lili Du, Feng Ji

TL;DR

This work studies steady incompressible Euler flow with helically symmetric free boundary and nontrivial vorticity, reducing the 3D problem to a 2D cross-section through an orthogonality condition to the symmetry lines and a stream function $\psi$. A variational formulation with the functional $\mathcal J_h(\psi,D)$ yields a semilinear free boundary problem $-\nabla\cdot(K\nabla\psi)=f(\psi)$ in $\{\psi>0\}$ with a Bernoulli-type boundary condition, and the authors prove key regularity results: the local minimizer $\psi$ is Lipschitz and non-degenerate, and the free boundary in a cross-section is $C^{1,\alpha}$, which lifts to a $C^{1,\alpha}$ surface for the 3D free boundary. The analysis combines an almost-minimizer reduction via coefficient freezing, a Weiss-type monotonicity formula, blow-up arguments, and De Silva’s flatness-implies-regularity theory. These results establish a rigorous regularity theory for helically symmetric Euler free boundary problems and illustrate a robust variational approach with variable coefficients. The methodology and findings have implications for understanding geometric and stability properties of helically symmetric flows in fluid mechanics.

Abstract

This paper investigates an incompressible steady free boundary problem of Euler equations with helical symmetry in $3$ dimensions and with nontrivial vorticity. The velocity field of the fluid arises from the spiral of its velocity within a cross-section, whose global existence, uniqueness and well-posedness with fixed boundary were established by a series of brilliant works. A perplexing issue, untouched in the literature, concerns the free boundary problem with (partial) unknown domain boundary in this helically symmetric configuration. We address this gap through the analysis of the optimal regularity property of the scalar stream function as a minimizer in a semilinear minimal problem, establishing the $C^{0,1}$-regularity of the minimizer, and the $C^{1,α}$-regularity of its free boundary. More specifically, the regularity results are obtained in arbitrary cross-sections through smooth helical transformation by virtue of variational method and the rule of "flatness implies $C^{1,α}$".

Helically symmetric solution of 3D Euler equations with vorticity and its free boundary

TL;DR

This work studies steady incompressible Euler flow with helically symmetric free boundary and nontrivial vorticity, reducing the 3D problem to a 2D cross-section through an orthogonality condition to the symmetry lines and a stream function . A variational formulation with the functional yields a semilinear free boundary problem in with a Bernoulli-type boundary condition, and the authors prove key regularity results: the local minimizer is Lipschitz and non-degenerate, and the free boundary in a cross-section is , which lifts to a surface for the 3D free boundary. The analysis combines an almost-minimizer reduction via coefficient freezing, a Weiss-type monotonicity formula, blow-up arguments, and De Silva’s flatness-implies-regularity theory. These results establish a rigorous regularity theory for helically symmetric Euler free boundary problems and illustrate a robust variational approach with variable coefficients. The methodology and findings have implications for understanding geometric and stability properties of helically symmetric flows in fluid mechanics.

Abstract

This paper investigates an incompressible steady free boundary problem of Euler equations with helical symmetry in dimensions and with nontrivial vorticity. The velocity field of the fluid arises from the spiral of its velocity within a cross-section, whose global existence, uniqueness and well-posedness with fixed boundary were established by a series of brilliant works. A perplexing issue, untouched in the literature, concerns the free boundary problem with (partial) unknown domain boundary in this helically symmetric configuration. We address this gap through the analysis of the optimal regularity property of the scalar stream function as a minimizer in a semilinear minimal problem, establishing the -regularity of the minimizer, and the -regularity of its free boundary. More specifically, the regularity results are obtained in arbitrary cross-sections through smooth helical transformation by virtue of variational method and the rule of "flatness implies ".

Paper Structure

This paper contains 14 sections, 7 theorems, 131 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. Preliminary about helical symmetry
  3. Physical setting of the free boundary problem
  4. Dimensionality reduction and the stream function
  5. Formulation of the free boundary
  6. Free boundary problem
  7. Recovering to three-dimensional solution
  8. Variational approach and main theorems
  9. Regularity of the stream function
  10. Lipschitz regularity
  11. Non-degeneracy
  12. Regularity of the free boundary in a cross-section
  13. The field of tangents of the symmetry lines
  14. Basic properties of minimizers

Key Result

Lemma 1.1

Suppose $f$ is a differentiable function and $\bm v=(v_1,v_2,v_3)$ is a differentiable vector field in the helical domain $D\subset\mathbb{R}^3$. Then the function $f$ is helical if and only if and the vector field $\bm v$ is helical if and only if

Figures (4)

  • Figure 1: Sketch of the far-wake model.
  • Figure 2: An example of helical pipe.
  • Figure 3: Helical transformation
  • Figure 4: Dimensionality reduction of $\Omega$.

Theorems & Definitions (21)

  • Definition 1.1
  • Lemma 1.1
  • Remark 1.3
  • Remark 1.4
  • Definition 1.2
  • Corollary 1.5
  • proof
  • Remark 1.6
  • Remark 1.7
  • Proposition 2.1
  • ...and 11 more