BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:When will there be a theory of multifractal turbulence? - Uriel Fr isch (Observatoire de la Côte d' Azur) DTSTART:20220105T130000Z DTEND:20220105T140000Z UID:TALK166393@talks.cam.ac.uk DESCRIPTION:About twenty years after Kolmogorov (1941) developed a theory of fullydevelopped incompressed 3D turbulence\, he thought that experiment altechniques 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 deviatio ns from the predictedself-similarity. These took the form of intermittent bursts ofactivity\, also seen by Batchelor and Townsend in their 1949exper iments.\nKolmogorov deemed that the 1941 theory was in need of revisiting. Heand his collaborators Obukhov and Yaglom developed a number of modelsin tended to match the data more adequately. Mandelbrot suggested thatthe pro per explanation of turbulence required that the energydissipation would be concentrated on a fractal with some non-integerdimension. Then\, in the e arly 1980\, Anselmet et al. performedstate-of-the-art measurements of smal l-scale intermittency forturbulence.\nGeorgio Parisi and this author looke d at the data of Anselmet \;et al. and found that they could not be ex plained with a single fractal \;dissipation\, set of a prescribed dime nsion. They tried a multifractaldescription\, which seemed to fit the data . It took a few years to realizethat the multifractal model is the turbule nce counterpart of \;the probabilistic theory of large deviations in f inances\, due to Cramer 1938.Large deviations are able to capture tiny dev iations from the law of large \;numbers. They  \;played a key role in the foundations of statisticalmechanics (Cramer's rate function is bas ically the entropy).In the first part of the lecture we shall give some hi ghlights of the \;Parisi's and Frisch's original 1983 multifractal app roach.The theory of multifractal turbulence was probably one of the manyfi elds of activity of Giorgio Parisi\, which convinced the NobelCommittee to grant him the 2021 Physics Nobel Prize. \;\nNevertheless\, so far "mu ltifractal turbulence" is just a fit toexperimental data (or later to nume rical data) with little contact tothe basic hydrodynamical theory.  \; It would indeed be unreasonable todemand a full mathematical theory of suc h turbulence: we do not evenknow if the solution to the Euler/Navier-Stoke s equations in 3D\, withnice initial data\, do remain so for a finite or i nfinite time. Hence\,it will take some time before we can derive multifrac tality from thebasic hydrodynamical equations.\nIn the mean time\, it woul d be nice to derive multifractality from theBurgers' equation.  \;The latter is not just a poor-man's look-alike ofthe Euler/Navier-Stokes equat ion\, but is also important in condensedmatter physics\, cosmology and pla ys an important role in Parisi's keycontributions. In the second part of m y conference\, I shall presentbriefly some results established with K. Kha nin (Toronto)\, R. Panditand D. Roy (Bangalore) on the Burgers equation wi th Brownian initialvelocity or potential and their generalizationa to arbi trary Hurstexponents h between 0 and 1.  \;In 1992\, Sinai proved rigo rously that ifthe initial velocity is a Brownian motion function\, then th e Lagrangemap is a Devil's staircase with fractal dimension 1/2 (She et al first obtained  \;this result first by numerical simulations). Recentl yG. Molchan (2017) extended this result to generalized Brownian motionwhos e Hurst exponent is not 1/2.  \;Such results look like monofractalsolu tion\, at least for the Lagrangian map. However\, without knowingthe veloc ity structure functions (moments of velocity spatialincrements)\, we do no t know if the solutions of such Burgers equationsare monofractal or multif ractal. 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. LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR