Dissipation concentration in two-dimensional fluids

TL;DR

This paper investigates the nature of energy dissipation in two-dimensional incompressible fluids as viscosity vanishes, revealing conditions under which dissipation is absolutely continuous or trivial, and exploring implications for anomalous dissipation.

Contribution

It establishes new results on the absolute continuity and triviality of dissipation measures in the inviscid limit, especially for measure-valued initial vorticity, and introduces criteria for anomalous dissipation.

Findings

Dissipation is Lebesgue in time and absolutely continuous for almost every time.

Trivial dissipation occurs if initial vorticity has a singular part of fixed sign.

Dissipation is absolutely continuous with respect to a quadratic space-time vorticity measure for measure-valued initial vorticity.

Abstract

We study the dissipation measure arising in the inviscid limit of two-dimensional incompressible fluids. It is proved that the dissipation is Lebesgue in time and, for almost every time, it is absolutely continuous with respect to the defect measure of strong compactness of the solutions. When the initial vorticity is a measure, the dissipation is proved to be absolutely continuous with respect to a ''quadratic'' space-time vorticity measure. This results into the trivial measure if the initial vorticity has singular part of distinguished sign, or a spatially purely atomic measure if wild oscillations in time are ruled out. In fact, the dynamics at the Batchelor-Kraichnan dissipative scale is the only relevant one, in turn offering new criteria for anomalous dissipation. We provide kinematic examples highlighting the strengths and the limitations of our approach. Quantitative rates, dissipation life-span and steady fluids are also investigated.