A Geometric Approach to the Navier-Stokes Equations

Abstract: Introduction: the Navier-Stokes equations are essential in fluid dynamics, describing the motion of fluids like liquids and gases. Solving these equations, especially in complex flows and high-Reynolds-number regimes, is a significant challenge. Numerical simulations provide some insights, but often under restrictive assumptions that limit applicability. Recent geometric and algebraic methods have emerged, focusing on the equations' structure, yet questions about the uniqueness and stability of weak solutions persist. Objective: this paper aims to reformulate the Navier-Stokes equations in covariant form and develop new equations that facilitate the search for potential solutions, emphasizing symmetries. Geometric Approach: a covariant formulation of the Navier-Stokes equations is presented, applying a Fourier transform on a bounded manifold and seeking smoothness and viable solutions through convergence of manifold elements. Transformations: the study examines transformations between manifolds, investigating symmetries and interpretations related to homeomorphisms, isometries, and diffeomorphisms, including inertial frames of reference. Discussion and Conclusions: this study introduces a geometric reformulation of the Navier-Stokes equations, proposing new equations to enhance convergence and smoothness of solutions. It presents a novel class of solutions and transformations, with significant interdisciplinary connections. Further simulations, experimental validation, and ongoing development are essential to broaden the applicability of these equations and their solutions.