Besov spaces and the multifractal hypothesis |
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Authors: | Gregory L. Eyink |
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Affiliation: | (1) Department of Mathematics, University of Arizona, 85721 Tucson, Arizona |
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Abstract: | Parisi and Frisch proposed some time ago an explanation for multiscaling of turbulent velocity structure functions in terms of a multifractal hypothesis, i.e., they conjecture that the velocity field has local Hölder exponents in a range [hmin,hmax], with exponents <h occurring on a setS(h) with a fractal dimensionD(h). Heuristic reasoning led them to an expression for the scaling exponentzp ofpth order as the Legendre transform of the codimensiond-D(h). We show here that a part of the multifractal hypothesis is correct under even weaker assumptions: namely, if the velocity field hasLp-mean Hölder indexs, i.e., if it lies in the Besov spaceBps,, then local Hölder regularity is satisfied. Ifs<d/p, then the hypothesis is true in a generalized sense of Hölder space with negative exponents and we discuss the proper definition of local Hölder classes of negative index. Finally, if a certain box-counting dimension exists, then the Legendre transform of its codimension gives the scaling exponentzp, and, more generally, the maximal Besov index of order,p, assp=zp/p. Our method of proof is derived from a recent paper of S. Jaffard using compactly-supported, orthonormal wavelet bases and gives an extension of his results. We discuss implications of the theorems for ensemble-average scaling and fluid turbulence. |
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Keywords: | Multifractals Besov spaces turbulence |
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