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When will there be a theory of multifractal turbulence?

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TURW01 - Turbulence: where do we stand and where are we heading?

About twenty years after Kolmogorov (1941) developed a theory of fullydevelopped incompressed 3D turbulence, he thought that experimentaltechniques had made enough progress to test the theory, for examplethe power-law with exponent -5/3 predicted for the energyspectrum. The theory seemed close to working fine, with howevermoderately small-scale deviations from the predictedself-similarity. These took the form of intermittent bursts ofactivity, also seen by Batchelor and Townsend in their 1949experiments. Kolmogorov deemed that the 1941 theory was in need of revisiting. Heand his collaborators Obukhov and Yaglom developed a number of modelsintended to match the data more adequately. Mandelbrot suggested thatthe proper explanation of turbulence required that the energydissipation would be concentrated on a fractal with some non-integerdimension. Then, in the early 1980, Anselmet et al. performedstate-of-the-art measurements of small-scale intermittency forturbulence. Georgio Parisi and this author looked at the data of Anselmet et al. and found that they could not be explained with a single fractal dissipation, set of a prescribed dimension. They tried a multifractaldescription, which seemed to fit the data. It took a few years to realizethat the multifractal model is the turbulence counterpart of the probabilistic theory of large deviations in finances, due to Cramer 1938.Large deviations are able to capture tiny deviations from the law of large numbers. They  played a key role in the foundations of statisticalmechanics (Cramer’s rate function is basically the entropy).In the first part of the lecture we shall give some highlights of the Parisi’s and Frisch’s original 1983 multifractal approach.The theory of multifractal turbulence was probably one of the manyfields of activity of Giorgio Parisi, which convinced the NobelCommittee to grant him the 2021 Physics Nobel Prize.  Nevertheless, so far “multifractal turbulence” is just a fit toexperimental data (or later to numerical data) with little contact tothe basic hydrodynamical theory.  It would indeed be unreasonable todemand a full mathematical theory of such turbulence: we do not evenknow if the solution to the Euler/Navier-Stokes equations in 3D, withnice initial data, do remain so for a finite or infinite time. Hence,it will take some time before we can derive multifractality from thebasic hydrodynamical equations. In the mean time, it would be nice to derive multifractality from theBurgers’ equation.  The latter is not just a poor-man’s look-alike ofthe Euler/Navier-Stokes equation, but is also important in condensedmatter physics, cosmology and plays an important role in Parisi’s keycontributions. In the second part of my conference, I shall presentbriefly some results established with K. Khanin (Toronto), R. Panditand D. Roy (Bangalore) on the Burgers equation with Brownian initialvelocity or potential and their generalizationa to arbitrary Hurstexponents h between 0 and 1.  In 1992, Sinai proved rigorously that ifthe initial velocity is a Brownian motion function, then the Lagrangemap is a Devil’s staircase with fractal dimension 1/2 (She et alfirst obtained  this result first by numerical simulations). RecentlyG. Molchan (2017) extended this result to generalized Brownian motionwhose Hurst exponent is not 1/2.  Such results look like monofractalsolution, at least for the Lagrangian map. However, without knowingthe velocity structure functions (moments of velocity spatialincrements), we do not know if the solutions of such Burgers equationsare monofractal or multifractal. Another case of possible (large-scale)multifractal behavior arises if the initial potential has a Hurstexponent that changes very slowly as a function of space.

This talk is part of the Isaac Newton Institute Seminar Series series.

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