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Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in $\mathbb{R}^3$

Dengjun Guo, Lifeng Zhao

Abstract

We consider the three-dimensional incompressible Euler equation \begin{equation*}\left\{\begin{aligned} &\partial_t Ω+U \cdot \nabla Ω+Ω\cdot \nabla U=0 \\ &Ω(x,0)=Ω_0(x) \end{aligned}\right. \end{equation*} in the whole space $\mathbb{R}^3$. Under the assumption that the initial velocity is helical and without swirl, we prove the global well-posedness of weak solutions in $L^1_1 \bigcap L^{\infty}_1(\mathbb{R}^3)$. The vortex transport formula is also obtained in our article.

Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in $\mathbb{R}^3$

Abstract

We consider the three-dimensional incompressible Euler equation \begin{equation*}\left\{\begin{aligned} &\partial_t Ω+U \cdot \nabla Ω+Ω\cdot \nabla U=0 \\ &Ω(x,0)=Ω_0(x) \end{aligned}\right. \end{equation*} in the whole space . Under the assumption that the initial velocity is helical and without swirl, we prove the global well-posedness of weak solutions in . The vortex transport formula is also obtained in our article.
Paper Structure (17 sections, 32 theorems, 362 equations)

This paper contains 17 sections, 32 theorems, 362 equations.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries and main result.
  3. Function spaces
  4. Helical functions and vector fields.
  5. Biot-Savart law
  6. Reduction to the two-dimensional helical Euler equation.
  7. Key estimates for the Biot-Savart kernel.
  8. Estimates for the modified Biot-Savart kernel $H$ and $K$.
  9. Osgood property of $Hw$.
  10. Estimates for the three-dimensional modified Biot-Savart kernel.
  11. Uniqueness.
  12. Uniqueness of the three-dimensional helical Euler equation \ref{['eq 3euler z']}.
  13. Uniqueness of the two-dimensional helical Euler equation \ref{['eq 2euler']} in $L^1_1 \bigcap L^{\infty}_1(\mathbb{R}^2)$.
  14. Global existence of weak solutions and conservation laws.
  15. A formal argument
  16. ...and 2 more sections

Key Result

Theorem 1.1

The two-dimensional helical Euler equation eq 2euler is globally well-posed in $L^1_1\bigcap L^{\infty}_1(\mathbb{R}^2)$ in the sense that

Theorems & Definitions (68)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Definition 2.2: helical function
  • Definition 2.3: helical vector field
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 58 more