An insight on the fractal power law flow: from a Hausdorff vector calculus perspective

TL;DR

This paper introduces a novel Hausdorff vector calculus framework to analyze fractal power-law flows, deriving fundamental theorems and proposing a new approach to model anomalous diffusion and fractal flow equations.

Contribution

It develops the first Hausdorff vector calculus with key theorems, providing a new mathematical tool for fractal flow and anomalous diffusion modeling.

Findings

Derived Gauss-Ostrogradsky-like theorem in Hausdorff calculus

Established Stokes-like and Green-like theorems for fractal flows

Proposed a conjecture related to fractal flow equations and Millennium Prize Problem

Abstract

In the article we suggest the Hausdorff vector calculus based on the Chen Hausdorff calculus for the first time. The Gauss-Ostrogradsky-like, Stokes-like, and Green-like theorems, and Green-like identities are obtained in the framework of the Hausdorff vector calculus. The formula is proposed as a mathematical tool to describe the real world problems for the fractal power-law flow equations with the anomalous diffusion equation. A conjecture for the fractal power-law flow equations analogous to the Smales 15th Problem (one of the Millennium Prize Problems for the Navier--Stokes equations) is also addressed.