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Time-zero limits of Kaden's spirals and 2D Euler

Bartosz Bieganowski, Tomasz Cieślak, Jakub Siemianowski

TL;DR

This work examines the time-zero limits of Kaden spirals in the 2D Euler setting and shows they satisfy an inhomogeneous Euler equation in a weak sense, depending on the spiral parameter $oldsymbol{ u}$. A steady-state weak-solution lemma for vortex sheets is developed to relate boundary behavior across a sheet to the presence of delta sources in the Euler equation: for $oldsymbol{ u}> rac{2}{3}$ a delta on a half-line $ ext{Sigma}$ appears, while for $oldsymbol{ u}= rac{2}{3}$ a delta at the origin can arise. The analysis demonstrates the necessity of both velocity matching and decay of spherical averages of $|v|^2$ near the origin for weak Euler solvability, and provides partial results and numeric evidence for the critical case $oldsymbol{ u}= rac{2}{3}$, where the velocity-matching condition appears not to hold. Collectively, the results illuminate the delicate structure of vortex-sheet limits and their implications for uniqueness questions in Delort-type theories and possible non-uniqueness scenarios driven by Kaden-like perturbations.

Abstract

The present note is devoted to the studies of the relation of the time-zero limits of Kaden's spirals and the 2D Euler equation. It is shown that the time-zero limits of Kaden's spirals satisfy inhomogeneous 2D Euler in a weak sense. As a corollary, the necessity of both, the decay of spherical averages around the origin of the spiral as well as the velocity matching condition, for the 2D Euler equation to hold in a weak sense, is shown. Finally, some preliminary results concerning the Kaden spirals are obtained.

Time-zero limits of Kaden's spirals and 2D Euler

TL;DR

This work examines the time-zero limits of Kaden spirals in the 2D Euler setting and shows they satisfy an inhomogeneous Euler equation in a weak sense, depending on the spiral parameter . A steady-state weak-solution lemma for vortex sheets is developed to relate boundary behavior across a sheet to the presence of delta sources in the Euler equation: for a delta on a half-line appears, while for a delta at the origin can arise. The analysis demonstrates the necessity of both velocity matching and decay of spherical averages of near the origin for weak Euler solvability, and provides partial results and numeric evidence for the critical case , where the velocity-matching condition appears not to hold. Collectively, the results illuminate the delicate structure of vortex-sheet limits and their implications for uniqueness questions in Delort-type theories and possible non-uniqueness scenarios driven by Kaden-like perturbations.

Abstract

The present note is devoted to the studies of the relation of the time-zero limits of Kaden's spirals and the 2D Euler equation. It is shown that the time-zero limits of Kaden's spirals satisfy inhomogeneous 2D Euler in a weak sense. As a corollary, the necessity of both, the decay of spherical averages around the origin of the spiral as well as the velocity matching condition, for the 2D Euler equation to hold in a weak sense, is shown. Finally, some preliminary results concerning the Kaden spirals are obtained.
Paper Structure (4 sections, 9 theorems, 78 equations)

This paper contains 4 sections, 9 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Main lemma
  3. Zero-time limits of Kaden's spirals
  4. Kaden's spirals partial results

Key Result

Theorem 1.1

Let $\mu=2/3$, moreover assume that $v$ is a divergence-free velocity corresponding to $\omega_0$ given in stanys via the Biot-Savart law B-S. Then $v$ satisfies in the sense of distributions where $\delta_{(0,0)}$ is the Dirac delta centered at the origin, while $Y=\int_{\partial B(0,1)}\frac{1}{2}|v(x)|^2x-\left(v(x)\cdot x\right)v(x) dS(x)$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['wlasnosci']}
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['tw_2/3']}
  • ...and 7 more