Energy Dissipation and Regularity in Quaternionic Fluid Dynamics using Sobolev-Besov Spaces

TL;DR

This paper introduces a novel mathematical framework combining quaternionic analysis with Sobolev-Besov spaces to study energy dissipation and regularity in complex fluid flows, especially turbulence and bifurcation phenomena.

Contribution

It presents two new theorems that analyze energy dissipation and solution regularity in quaternionic fluid systems within Sobolev-Besov spaces, advancing the mathematical understanding of such systems.

Findings

Energy dissipation is dominated by high-frequency components in Besov spaces.

Conditions for regularity and existence of solutions are established.

The framework addresses frequency-specific dissipation and rotational symmetries.

Abstract

This study investigates the dynamics of incompressible fluid flows through quaternionic variables integrated within Sobolev-Besov spaces. Traditional mathematical models for fluid dynamics often employ Sobolev spaces to analyze the regularity of the solution to the Navier-Stokes equations. However, with the unique ability of Besov spaces to provide localized frequency analysis and handle high-frequency behaviors, these spaces offer a refined approach to address complex fluid phenomena such as turbulence and bifurcation. Quaternionic analysis further enhances this approach by representing three-dimensional rotations directly within the mathematical framework. The author presents two new theorems to advance the study of regularity and energy dissipation in fluid systems. The first theorem demonstrates that energy dissipation in quaternionic fluid systems is dominated by the higher-frequency component in Besov spaces, with contributions decaying at a rate proportional to the frequency of the quaternionic component. The second theorem provides conditions for regularity and existence of solutions in quaternionic fluid systems with external forces. By integrating these hypercomplex structures with Sobolev-Besov spaces, our work offers a new mathematically rigorous framework capable of addressing frequency-specific dissipation patterns and rotational symmetries in turbulent flows. The findings contribute to fundamental questions in fluid dynamics, particularly by improving our understanding of high Reynolds number flows, energy cascade behaviors, and quaternionic bifurcation. This framework therefore paves the way for future research on regularity in complex fluid dynamics.