Besov spaces and the multifractal hypothesis
Besov spaces and the multifractal hypothesis
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Besov 空间和多重分形假设
DOI：
10.1007/bf02183353
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发表时间：
1995
影响因子：
1.6
通讯作者：
G. Eyink
中科院分区：
文献类型：
--
作者：
G. Eyink
Parisi和Frisch不久前提出了一种用“多重分形假设”解释湍流速度结构函数“多重标度”的方法，即他们猜想速度场具有在[hmin，hmax]范围内的局部Hölder指数，指数<；h出现在具有分形维D(H)的集合(H)上。启发式推理使他们得到了p阶标度指数Zp的表达式，即余维-D(H)的勒让德变换。我们证明了在更弱的假设下，多重分形假设的一部分是正确的：即，如果速度场有Lp-平均Hölder指数，即如果它位于Besov空间Bps，∞，则满足局部Hölder正则性。如果<；d/p，则该假设在广义负指数Hölder空间意义下成立，并讨论了负指数的局部Hölder类的适当定义。最后，如果存在某一“盒数维”，则其余维的勒让德变换给出了标度指数Zp，以及更一般的最大阶Besov指数p，asp=Zp/p。我们的证明方法是从S.Jaffard最近的一篇文章中用紧支撑的标准小波基得到的，并推广了他的结果。我们讨论了系综平均标度定理和流体湍流定理的含义。
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.