This study presents a rigorous mathematical framework for the analysis of the Navier-Stokes equations within the context of Sobolev and Besov functional spaces, with a particular emphasis on the regularity of solutions, hypercomplex bifurcations, and turbulence in fluid dynamics. By employing advanced mathematical tools such as interpolation theory, Littlewood-Paley decomposition, and energy cascade models, we provide a comprehensive analysis of the intricate behaviors exhibited by fluid systems. The research establishes higher-order Sobolev regularity for solutions to the Navier-Stokes equations, demonstrating enhanced smoothness under appropriate conditions on initial data and external forces. Additionally, the characterization of Besov spaces through the Littlewood-Paley decomposition captures multifractal and irregular behaviors in turbulent flows, offering critical insights into energy dissipation mechanisms. A quaternionic formulation of the Navier-Stokes equations is introduced, providing a novel approach to modeling rotational symmetries and bifurcation phenomena in three-dimensional fluid dynamics. The study further confirms the regularity and uniqueness of solutions in Besov spaces, contributing to the ongoing exploration of the Millennium Prize Problem. Overall, this work advances the mathematical understanding of fluid dynamics and establishes a robust foundation for future research in this challenging field.
Santos et al. (2025) studied this question.