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 Uspekhi Mat. Nauk, 2007, Volume 62, Issue 3(375), Pages 193–206 (Mi umn6118)

Conformal invariance in hydrodynamic turbulence

G. Falkovich

Weizmann Institute of Science

Abstract: This short survey is written by a physicist. It contains neither theorems nor precise definitions. Its main content is a description of the results of numerical solution of the equations of fluid mechanics in the regime of developed turbulence. Due to limitations of computers, the results are not very precise. Despite being neither exact nor rigorous, the findings may nevertheless be of interest for mathematicians. The main result is that the isolines of some scalar fields (vorticity, temperature) in two-dimensional turbulence belong to the class of conformally invariant curves called SLE (Scramm–Löwner evolution) curves. First, this enables one to predict and find a plethora of quantitative relations going far beyond what was known previously about turbulence. Second, it suggests relations between phenomena that seemed unrelated, like the Euler equation and critical percolation. Third, it shows that one is able to get exact analytic results in statistical hydrodynamics. In short, physicists have found something unexpected and hope that mathematicians can help to explain it.

DOI: https://doi.org/10.4213/rm6118

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English version:
Russian Mathematical Surveys, 2007, 62:3, 497–510

Bibliographic databases:

UDC: 517.54+517.957+517.958:531.32
MSC: Primary 60K35, 76F55; Secondary 35Q35, 60J65, 76F05, 76F25, 81T40, 82B27, 82C05

Citation: G. Falkovich, “Conformal invariance in hydrodynamic turbulence”, Uspekhi Mat. Nauk, 62:3(375) (2007), 193–206; Russian Math. Surveys, 62:3 (2007), 497–510

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn6118
• https://doi.org/10.4213/rm6118
• http://mi.mathnet.ru/eng/umn/v62/i3/p193

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Falkovich G., “Nodal lines in turbulence”, The European Physical Journal - Special Topics, 145:1 (2007), 211–216
2. Eyink G., Frisch U., Moreau R., Sobolevskiǐ A., “General introduction”, Proceedings of an international conference. Euler equations: 250 Years on (EE250), Phys. D, 237:14–17 (2008), xi–xv
3. V. N. Grebenev, M. Oberlack, A. N. Grishkov, “Infinite dimensional Lie algebra associated with conformal transformations of the two-point velocity correlation tensor from isotropic turbulence”, Z. Angew. Math. Phys, 2012
4. V. N. Grebenev, A. N. Grishkov, M. Oberlack, “The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence”, Advances in Mathematical Physics, 2013 (2013), 1
5. Gregory Falkovich, Alexander Zamolodchikov, “Operator product expansion and multi-point correlations in turbulent energy cascades”, J. Phys. A: Math. Theor, 48:18 (2015), 18FT02
6. Batista-Tomas A.R., Diaz O., Batista-Leyva A.J., Altshuler E., “Classification and dynamics of tropical clouds by their fractal dimension”, Q. J. R. Meteorol. Soc., 142:695, B (2016), 983–988
7. Grebenev V.N., Waclawczyk M., Oberlack M., “Conformal Invariance of the Lungren-Monin-Novikov Equations For Vorticity Fields in 2D Turbulence”, J. Phys. A-Math. Theor., 50:43 (2017), 435502
8. Waclawczyk M., Grebenev V.N., Oberlack M., “Lie Symmetry Analysis of the Lundgren-Monin-Novikov Equations For Multi-Point Probability Density Functions of Turbulent Flow”, J. Phys. A-Math. Theor., 50:17 (2017), 175501
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