Geometry of surfaces with Caputo fractional derivatives and applications to incompressible two-dimensional flows

Geometric structures of surfaces are formulated based on Caputo fractional derivatives. The Gauss frame of a surface with fractional order is introduced. Then, the non-locality of the fractional derivative characterizes the asymmetric second fundamental form. The mean and Gaussian curvatures of the surface are defined in the case of fractional order. Based on the fractional curvatures, incompressible two-dimensional flows are discussed. The stream functions are obtained from a fractional continuity equation. The asymmetric second fundamental form of stream-function surface is related to the path dependence of flux. Moreover, the fractional curvatures are calculated for the stream-function surfaces of uniform and corner flows. The uniform flow with fractional order is characterized by the non-vanishing mean curvature. The non-locality of corner flow is expressed by the mean and Gaussian curvatures with fractional order. In particular, the fractional order within the stream-function of corner flow determines the change of sign of Gaussian curvature. Therefore, the non-local property of incompressible flows can be investigated by the fractional curvatures.

基于卡普托分数阶导数构建了曲面的几何结构。引入了分数阶曲面的高斯标架。然后，分数阶导数的非局部性刻画了非对称第二基本形式。在分数阶情形下定义了曲面的平均曲率和高斯曲率。基于分数阶曲率，讨论了不可压缩二维流。从分数阶连续性方程得到了流函数。流函数曲面的非对称第二基本形式与通量的路径依赖性相关。此外，计算了均匀流和角流的流函数曲面的分数阶曲率。分数阶均匀流的特征是平均曲率不为零。角流的非局部性由分数阶平均曲率和高斯曲率表示。特别地，角流流函数内的分数阶决定了高斯曲率符号的变化。因此，可通过分数阶曲率研究不可压缩流的非局部性质。