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On the Cauchy problem for 3D Navier-Stokes helical vortex filament

Francisco Gancedo, Antonio Hidalgo-Torné

Abstract

This paper studies the Cauchy problem for a helical vortex filament evolving by the 3D incompressible Navier-Stokes equations. We prove global-in-time well-posedness and smoothing of solutions with initial vorticity concentrated on a helix. We provide a local-in-time well-posedness result for vortex filaments periodic in one spatial direction, and show that solutions with helical initial data preserve this symmetry. We follow the approach of [4], where the analogue local-in-time result has been obtained for closed vortex filaments in $\mathbb{R}^3$. Next, we apply local energy weak solutions theory with a novel estimate for helical functions in non-helical domains to uniquely extend the solutions globally in time. This is the first global-in-time well-posedness result for a vortex filament without size restriction and without vanishing swirl assumptions.

On the Cauchy problem for 3D Navier-Stokes helical vortex filament

Abstract

This paper studies the Cauchy problem for a helical vortex filament evolving by the 3D incompressible Navier-Stokes equations. We prove global-in-time well-posedness and smoothing of solutions with initial vorticity concentrated on a helix. We provide a local-in-time well-posedness result for vortex filaments periodic in one spatial direction, and show that solutions with helical initial data preserve this symmetry. We follow the approach of [4], where the analogue local-in-time result has been obtained for closed vortex filaments in . Next, we apply local energy weak solutions theory with a novel estimate for helical functions in non-helical domains to uniquely extend the solutions globally in time. This is the first global-in-time well-posedness result for a vortex filament without size restriction and without vanishing swirl assumptions.
Paper Structure (22 sections, 43 theorems, 126 equations)

This paper contains 22 sections, 43 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. General notation
  3. Statement of results and outline of the proof
  4. Construction of fixed point scheme
  5. Geometrical setting
  6. Approximate solution
  7. Unknowns and fixed point scheme
  8. Norms
  9. Mapping from Fixed Point Theorem
  10. Fixed Point Estimates, Short Time Existence
  11. Preliminary estimates
  12. Estimates on $\eta^{c_1}$
  13. Estimates on $\omega^{c_2}$
  14. Short time existence
  15. Preservation of helical symmetry
  16. ...and 7 more sections

Key Result

Theorem 1.3

For any $\alpha\in \mathbb R$, and any $x_3$-periodic, smooth, and non-self-intersecting curve $\Gamma$ there exist $T>0$ and a mild solution to the Navier-Stokes equations in the sense of definition def:mildsolution with initial data vortexfilamentdata. The solution $\omega\in C^0((0,T],L^1\cap L^2

Theorems & Definitions (86)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Local-in-time existence $x_3$-periodic vortex filaments
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Proposition 1.7
  • Definition 1.8
  • Definition 1.9: Local energy weak solutions
  • Remark 1.10
  • ...and 76 more