Treatment for regularity of the Navier-Stokes equations based on Banach and Sobolev functional spaces coupled to anisotropic viscosity for analysis of vorticity transport

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The mathematical analysis employed in this study constitutes an essential foundation for a broader investigation into the regularity of the Navier-Stokes Equations. Within this context, this work represents a significant advance with the Smagorinsky model integrated into the LES methodology. Using the Banach and Sobolev functional spaces, we developed a new theorem that points out a path towards the creation of an anisotropic viscosity model, formulated in the present work. Initially, our effort focuses on providing a comprehensive mathematical analysis, with the aim of promoting a deeper understanding of the challenge inherent in the regularity of the Navier-Stokes equations.