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Self-similar algebraic spiral vortex sheets of 2-D incompressible Euler equations

Feng Shao, Dongyi Wei, Zhifei Zhang

TL;DR

This work provides the first rigorous construction of self-similar algebraic spiral vortex sheets for the 2-D incompressible Euler equations. By recasting the vortex-sheet dynamics into a self-similar Birkhoff-Rott formulation and perturbing around Kaden’s spiral via a nonlinear operator 𝒩, the authors establish invertibility and apply a Banach fixed-point framework to obtain existence of m-fold spiral sheets for large m. A delicate decomposition of the principal-value Cauchy integral 𝓘_m into oscillatory series and weighted Hölder/W^{1,∞} estimates show contraction and yield quantitative control of the sheet geometry and the velocity field, including the near-origin spiral behavior and far-field decay. They also prove that the constructed sheets define weak solutions to the Euler equations in the whole plane, thereby connecting rigorous BR dynamics to physically relevant vortex-sheet evolution. The results illuminate the roll-up patterns observed after curvature singularities and provide a robust mathematical foundation for self-similar spiral vortex-sheet solutions in 2-D flows.

Abstract

This paper provides the first rigorous construction of the self-similar algebraic spiral vortex sheet solutions to the 2-D incompressible Euler equations. These solutions are believed to represent the typical roll-up pattern of vortex sheets after the formation of curvature singularities. The most challenging part of this paper is to handle the Cauchy integral for the algebraic spiral curve, which falls outside the classical theory of singular integral operators.

Self-similar algebraic spiral vortex sheets of 2-D incompressible Euler equations

TL;DR

This work provides the first rigorous construction of self-similar algebraic spiral vortex sheets for the 2-D incompressible Euler equations. By recasting the vortex-sheet dynamics into a self-similar Birkhoff-Rott formulation and perturbing around Kaden’s spiral via a nonlinear operator 𝒩, the authors establish invertibility and apply a Banach fixed-point framework to obtain existence of m-fold spiral sheets for large m. A delicate decomposition of the principal-value Cauchy integral 𝓘_m into oscillatory series and weighted Hölder/W^{1,∞} estimates show contraction and yield quantitative control of the sheet geometry and the velocity field, including the near-origin spiral behavior and far-field decay. They also prove that the constructed sheets define weak solutions to the Euler equations in the whole plane, thereby connecting rigorous BR dynamics to physically relevant vortex-sheet evolution. The results illuminate the roll-up patterns observed after curvature singularities and provide a robust mathematical foundation for self-similar spiral vortex-sheet solutions in 2-D flows.

Abstract

This paper provides the first rigorous construction of the self-similar algebraic spiral vortex sheet solutions to the 2-D incompressible Euler equations. These solutions are believed to represent the typical roll-up pattern of vortex sheets after the formation of curvature singularities. The most challenging part of this paper is to handle the Cauchy integral for the algebraic spiral curve, which falls outside the classical theory of singular integral operators.
Paper Structure (27 sections, 39 theorems, 332 equations, 2 figures)

This paper contains 27 sections, 39 theorems, 332 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Difficulties and key ideas
  4. Notations and conventions
  5. A road map of the proof
  6. Birkhoff-Rott equation
  7. Self-similar formulation
  8. Kaden's algebraic spiral solutions
  9. Existence results for self-similar Birkhoff-Rott
  10. Proof of Theorem \ref{['Thm.main']}
  11. Sketch of the proof: Proposition \ref{['Prop.N']}
  12. Sketch of the proof: Proposition \ref{['Prop.I']}
  13. Properties of the weighted Hölder spaces
  14. Invertibility of $\mathcal{N}$
  15. Regularity of nonlinear map $\mathcal{N}$
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\mu>1/2$, there exists $m_*\in\mathbb Z$ such that for all $m\in \mathbb Z\cap(m_*,+\infty)$, the following Birkhoff-Rott equation where $\xi_m:=\mathrm e^{\frac{2\pi\mathrm{i}}{m}}$ is the $m$-th unit root, with the initial data ($Z_m^*$ denoting the complex conjugate of $Z_m$) possesses a solution $Z_m(t,\Gamma)$ with the following properties. See Figure Fig.solution for the shape of the

Figures (2)

  • Figure 1: Computation of the periodic vortex sheet after singularity by means of a desingularization of the kernel \ref{['Eq.kernel-K2']}. The computation is shown at $t=3.9$, well after the initial singularity. Note the approximation to a double-branched spiral. This picture is MB.
  • Figure 2: Symmetric spiral vortex sheets: $m=4$.

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Proposition 2.2: Invertibility of ${\mathcal{N}}$
  • Proposition 2.3: Contraction of $\mathcal{I}_m$
  • proof : Proof of Theorem \ref{['Thm.main']}
  • Proposition 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 65 more