Comment on ‘Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence’

Abstract

The current claim by Grebenev et al (2019 J. Phys. A: Math. Theor. 52 335501), namely that the inviscid and unclosed 2D Lundgren–Monin–Novikov (LMN) equations on a zero-vorticity Lagrangian path admit conformal invariance, is based on a flawed analysis published earlier by Grebenev et al (2017 J. Phys. A: Math. Theor. 50 435502). All results and conclusions made before in the Eulerian picture were now extended by Grebenev et al to the Lagrangian picture. Although we have already commented on these errors in Frewer and Khujadze (2018 arXiv:1802.02490; 2021 J. Phys. A: Math. Theor. 54 438002) and consistently refuted their previous study, we deem it necessary to address and discuss these errors again in the new formulation and notation of Grebenev et al (2019 J. Phys. A: Math. Theor. 52 335501) as it will offer new insights into this issue.

An example from turbulence how not to use the invariant function method of Lie-group symmetries

“…These are the so-called non-negativity, normalization, coincidence, separation and conditional constraints[14,16]. In particular, the reviews[19][20][21][22] explicitly show what happens if one of these constraints are violated 5. Obviously, if an invariant transformation already violates from the outset one of the dynamical constraints as causality, as the two invariant transformations Eqs.…”