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Existence of nonsymmetric logarithmic spiral vortex sheet solutions to the 2D Euler equations

T. Cieślak, P. Kokocki, W. S. Ożański

TL;DR

This work constructs a new family of time-dependent vortex-sheet solutions to the 2D Euler equations, modeled as $M$ concentric logarithmic spirals with nonuniform angular distribution. By reducing the existence problem to a finite-dimensional discrete system and performing a careful asymptotic and matrix-analytic analysis, the authors establish the existence of nonsymmetric spirals for $M\\in\\{2,3,5,7,9\\}$ and $n\\in\\{1,2\\}$, with angles near the explicit $\\overline{\\Theta}$ and nonzero densities. The approach hinges on explicit determinant formulas, a limiting analysis as $a\\to\\infty$, nondegeneracy of the limit angles, and an implicit-function argument supported by a detailed examination of linear independence of key vectors. The results expand the catalogue of rigorous, nontrivial time-dependent vortex-sheet solutions and illuminate the role of angular nondegeneracy in enabling such nonsymmetric configurations, with numerical evidence suggesting the framework extends to all odd $M>9$.

Abstract

We consider solutions of the 2D incompressible Euler equation in the form of $M\geq 1$ cocentric logarithmic spirals. We prove the existence of a generic family of spirals that are nonsymmetric in the sense that the angles of the individual spirals are not uniformly distributed over the unit circle. Namely, we show that if $M=2$ or $M\geq 3 $ is an odd integer such that certain non-degeneracy conditions hold, then, for each $n \in \{ 1,2 \}$, there exists a logarithmic spiral with $M$ branches of relative angles arbitrarily close to $\barθ_{k} = knπ/M$ for $k=0,1,\ldots , M-1$, which include halves of the angles of the Alexander spirals. We show that the non-degeneracy conditions are satisfied if $M\in \{ 2, 3,5,7,9 \}$, and that the conditions hold for all odd $M>9$ given a certain gradient matrix is invertible, which appears to be true by numerical computations.

Existence of nonsymmetric logarithmic spiral vortex sheet solutions to the 2D Euler equations

TL;DR

This work constructs a new family of time-dependent vortex-sheet solutions to the 2D Euler equations, modeled as concentric logarithmic spirals with nonuniform angular distribution. By reducing the existence problem to a finite-dimensional discrete system and performing a careful asymptotic and matrix-analytic analysis, the authors establish the existence of nonsymmetric spirals for and , with angles near the explicit and nonzero densities. The approach hinges on explicit determinant formulas, a limiting analysis as , nondegeneracy of the limit angles, and an implicit-function argument supported by a detailed examination of linear independence of key vectors. The results expand the catalogue of rigorous, nontrivial time-dependent vortex-sheet solutions and illuminate the role of angular nondegeneracy in enabling such nonsymmetric configurations, with numerical evidence suggesting the framework extends to all odd .

Abstract

We consider solutions of the 2D incompressible Euler equation in the form of cocentric logarithmic spirals. We prove the existence of a generic family of spirals that are nonsymmetric in the sense that the angles of the individual spirals are not uniformly distributed over the unit circle. Namely, we show that if or is an odd integer such that certain non-degeneracy conditions hold, then, for each , there exists a logarithmic spiral with branches of relative angles arbitrarily close to for , which include halves of the angles of the Alexander spirals. We show that the non-degeneracy conditions are satisfied if , and that the conditions hold for all odd given a certain gradient matrix is invertible, which appears to be true by numerical computations.
Paper Structure (15 sections, 7 theorems, 161 equations, 1 figure, 2 tables)

This paper contains 15 sections, 7 theorems, 161 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm_main']}
  3. The determinants
  4. Limiting behaviour as $a\to \infty$
  5. Nondegeneracy of the point $\overline{\Theta}$
  6. Case $M=2$
  7. The case of odd $M\geq 3$
  8. The case of even $M\geq 4$
  9. Asymptotic expansions near $a=\infty$
  10. Linear independence of $E_1$, $E_2$
  11. Case $M=2$
  12. The case of odd $M\geq 3$
  13. Appendix
  14. Appendix - Invertibility of $\mathcal{A}$
  15. Appendix - Nontriviality of nonsymmetric spirals

Key Result

Theorem 1

Given $M\in \{ 2 , 3, 5 , 7 , 9 \}$ and $n\in \{1,2 \}$, let and $\varepsilon >0$. Then there exists $a_0 > 0$ such that for every $a\geq a_0$ there exists a unique choice of such that eq-disc holds and In particular (as discussed above), then there exist weak solutions of the 2D incompressible Euler equations in the form of logarithmic spirals spirals with angles halves.

Figures (1)

  • Figure :

Theorems & Definitions (16)

  • Theorem 1: The main result
  • Remark 2
  • Theorem 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Proposition 7
  • proof
  • Proposition 8
  • proof
  • ...and 6 more