Classical Physics

    

Global Existence and Smoothness of Solutions to the Navier Stokes Equations: An Energy Pertubation Approach

Authors: Felipe G. G. de Carvalho

In this article, we investigate the existence and smoothness of solutions to the incom-pressible Navier-Stokes equations using the energy perturbation method. By assuming thatexternal forces are either null or sufficiently small and that the initial conditions are smo-othly compact, we demonstrate, through an energy inequality, that the energy associatedwith perturbations decays exponentially over time. This energy dissipation implies thatsmall perturbations vanish, ensuring the global stability and regularity of solutions. Additi-onally, we numerically validate our results using physics-informed neural networks (PINNs),showing that the numerical solution converges to the analytical one. These findings pro-vide strong evidence toward a potential proof of the global existence and smoothness ofNavier-Stokes solutions, contributing to the discussion of millennium prize problems.

Comments: 8 Pages.

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Submission history

[v1] 2025-04-01 20:54:02

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