No anomalous dissipation in two-dimensional incompressible fluids

TL;DR

This paper proves that in two-dimensional incompressible fluids, solutions with certain initial conditions do not exhibit anomalous energy dissipation as viscosity vanishes, contrasting with three-dimensional turbulence.

Contribution

It establishes the absence of anomalous dissipation for a class of solutions with measure-valued vorticity, using concentration-compactness and linking dissipation to the Kolmogorov scale.

Findings

No anomalous dissipation occurs in the studied class of solutions.

Energy dissipation is bounded by the vorticity measure at the Kolmogorov scale.

The result contrasts two-dimensional turbulence with three-dimensional cases.

Abstract

We prove that any sequence of vanishing viscosity Leray-Hopf solutions to the periodic two-dimensional incompressible Navier-Stokes equations does not display anomalous dissipation if the initial vorticity is a measure with positive singular part. A key step in the proof is the use of the Delort-Majda concentration-compactness argument to exclude formation of atoms in the vorticity measure, which in particular implies that the limiting velocity is an admissible weak solution to Euler. This is the first result proving absence of dissipation in a class of solutions in which the velocity fails to be strongly compact in $L^2$, putting two-dimensional turbulence in sharp contrast with respect to that in three dimensions. Moreover, our proof reveals that the amount of energy dissipation can be bounded by the vorticity measure of a disk of size $\sqrt \nu$, matching the two-dimensional Kolmogorov dissipative length scale which is expected to be sharp.