Quantitative enstrophy bounds for measure vorticities

TL;DR

This paper derives optimal quantitative bounds on enstrophy for 2D Navier-Stokes equations with measure initial vorticity, using improved Nash inequalities to analyze vorticity decay and dissipation rates.

Contribution

It introduces new enstrophy bounds based on vorticity decay, providing sharper estimates and insights into dissipation in measure-valued vorticity scenarios.

Findings

Bounds are optimal in several aspects

Establishes a conjecturally sharp dissipation rate

Uses improved Nash inequalities for analysis

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.