Dissipation Concentration in Two-Dimensional Fluids

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.