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Quantitative enstrophy bounds for measure vorticities

Luigi De Rosa, Margherita Marcotullio

Abstract

We consider the two-dimensional incompressible Navier-Stokes equations with measure initial vorticity. By means of improved Nash inequalities, we establish quantitative estimates for the enstrophy depending on the absolute vorticity decay on balls. The bounds are optimal in several aspects and yield to a conjecturally sharp rate of the dissipation in the Delort's class.

Quantitative enstrophy bounds for measure vorticities

Abstract

We consider the two-dimensional incompressible Navier-Stokes equations with measure initial vorticity. By means of improved Nash inequalities, we establish quantitative estimates for the enstrophy depending on the absolute vorticity decay on balls. The bounds are optimal in several aspects and yield to a conjecturally sharp rate of the dissipation in the Delort's class.
Paper Structure (10 sections, 9 theorems, 164 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 164 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Main context and related literature
  4. Sharpness of the bounds
  5. Improved Nash inequalities
  6. Quantitative enstrophy bounds
  7. Failed sharp attempts in the Delort's class
  8. Construction by rescaling
  9. Construction with an integrable initial datum
  10. Conclusions

Key Result

Theorem 1.1

Let $\{u^\nu_0\}_\nu\subset L^2(\mathbb{T}^2)$ be such that $\{\omega^\nu_0\}_\nu\subset \mathcal{M} (\mathbb{T}^2)$ is bounded. Let $\{\omega^\nu\}_\nu$ be the corresponding solutions to NS-Vort and let $\mathbb{M}_\omega$ be defined as in vort on balls.

Figures (1)

  • Figure 1: Iterations $n=1$ (on the left) and $n=2$ (on the right) of the self-similar Cantor set.

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Conjecture 1.6
  • Proposition 2.1: ELL25*Proposition 3.2
  • Proposition 2.2
  • proof
  • Remark 2.3
  • ...and 11 more