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Self-similar algebraic spiral solution of 2-D incompressible Euler equations

Feng Shao, Dongyi Wei, Zhifei Zhang

TL;DR

This work constructs new self-similar algebraic spiral solutions to the 2-D incompressible Euler equations for initial vorticities of the form $|y|^{-\frac{1}{\mu}}\mathring{\omega}(\theta)$ with $\mu>\frac{1}{2}$ and $\mathring{\omega}\in L^1(\mathbb{T})$ (with $m$-fold symmetry). It introduces a novel coordinate system $(\beta,\phi)$ and a weighted, $m$-uniform functional-analytic framework to reformulate the problem as $\mathcal{F}(\psi,\Omega)=0$, solved via the implicit function theorem, after a careful linear theory that handles high/low frequency contributions. The paper proves the existence of weak solutions when the initial vorticity is a Radon measure, thereby linking to vortex-sheet dynamics, and provides a robust construction that can accommodate measure-valued data within a Delort-type setting. The approach advances the understanding of self-similar spiral structures in 2-D Euler and suggests a pathway to analyzing inviscid limits and non-uniqueness questions for weak solutions. The results are significant for vortex-sheet dynamics and the long-standing problem of self-similar spiral formation in incompressible flows.

Abstract

In this paper, we prove the existence of self-similar algebraic spiral solutions for 2-D incompressible Euler equations for the initial vorticity of the form $|y|^{-\frac1μ}\ \mathringω(θ)$ with $μ>\frac12$ and $\mathringω\in L^1(\mathbb T)$ satisfying $m$-fold symmetry ($m\geq 2$) and a dominant condition. As an important application, we prove the existence of weak solution when $\mathringω$ is a Radon measure on $\mathbb T$ with $m$-fold symmetry, which is related to the vortex sheet solution.

Self-similar algebraic spiral solution of 2-D incompressible Euler equations

TL;DR

This work constructs new self-similar algebraic spiral solutions to the 2-D incompressible Euler equations for initial vorticities of the form with and (with -fold symmetry). It introduces a novel coordinate system and a weighted, -uniform functional-analytic framework to reformulate the problem as , solved via the implicit function theorem, after a careful linear theory that handles high/low frequency contributions. The paper proves the existence of weak solutions when the initial vorticity is a Radon measure, thereby linking to vortex-sheet dynamics, and provides a robust construction that can accommodate measure-valued data within a Delort-type setting. The approach advances the understanding of self-similar spiral structures in 2-D Euler and suggests a pathway to analyzing inviscid limits and non-uniqueness questions for weak solutions. The results are significant for vortex-sheet dynamics and the long-standing problem of self-similar spiral formation in incompressible flows.

Abstract

In this paper, we prove the existence of self-similar algebraic spiral solutions for 2-D incompressible Euler equations for the initial vorticity of the form with and satisfying -fold symmetry () and a dominant condition. As an important application, we prove the existence of weak solution when is a Radon measure on with -fold symmetry, which is related to the vortex sheet solution.
Paper Structure (28 sections, 43 theorems, 372 equations, 1 figure)

This paper contains 28 sections, 43 theorems, 372 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sketch of the proof
  3. New coordinates and reformulation
  4. New coordinates
  5. Formulation in the new coordinates
  6. Radially symmetric solutions
  7. Weighted functional spaces with $m$-fold symmetry
  8. Weighted Hölder type spaces
  9. Functional spaces for $\psi$ and $\Omega$
  10. Functional spaces with $m$-fold symmetry
  11. Nonlinear problem in the new coordinates
  12. $C^2$ regularity of nonlinear map
  13. Proof of Theorem \ref{['IFT']}
  14. Solvability of the linearized problem
  15. Uniqueness of the solution
  16. ...and 13 more sections

Key Result

Theorem 1.1

Given $\varepsilon>0$ and $\mu>\frac{2}{3}$, there exists an $N_0\in \mathbb{N}$ so that a weak solution of 2DEuler and initial_data exists for all initial data $\mathring{\omega}$ satisfying the following conditions:

Figures (1)

  • Figure 1: The algebraic spiral $\theta=\frac{3}{2} r^{-4/3}$ obtained by taking $c_0=1,\mu=3/4$.

Theorems & Definitions (90)

  • Theorem 1.1
  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Lemma 4.1: Equivalent norm on $C_\beta^\alpha$
  • proof
  • Lemma 4.2
  • proof
  • ...and 80 more