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Conformal invariance of the probability density function for passive scalar fields (including arbitrary virus) in 2D turbulence - the proof of Polyakov conjecture.


Task 1: Symmetry analysis of the governing equation for the $n$-point PDF $f_n$ for passive scalar fieldsbased on the direct method.Task 2: Conformal invariance of the probability measure.Task 3: The derivation of a Lie group of the symmetry transformations calculated.Task 4: The fine structure of a Lie algebra to classify the zero-isolines by the central charge. Then the effects of non-zero Gaussian white-in-time forcing and large-scale friction along flow surfaces will be included into the consideration. The method and calculations that worked out in (Grebenev, Wac\l{}awczyk and M Oberlack 2017, Wac\l{}awczyk, Grebenev and M Oberlack 2019, (Grebenev, Grishkov, Oberlack, Wac\l{}awczyk 2021) to find symmetries and calculate invariantsof LMN for 2D and 3D vorticity and velocity fields enable us to conclude that this work programme will be successfully completed.It will be published minimum two papers in higher-rank Journals. (AU)

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