New Concept of Relativity of Time Applied to Navier-Stokes Equations

This paper is dedicated to study the three-dimensional periodic Navier-Stokes equations in Gevrey-Sobolev spaces. We start by proving the local in time well-posedness for arbitrary large in $H_{a,s}^{r}(\mathbb{T}^3)$ initial data. We also prove the global in time well-posedness provided that the initial data satisfies a smallness condition. Finally, we propose the new theory \textit{Hierarchical Relativity of Time}, which posits that every object perceives time according to its position within the universal hierarchy. Applying this theory to the field of fluid mechanics has yielded significant insights. It allows us to dissect the complex Navier-Stokes problem into two distinct regimes:\\ Viscous Fine Regime: In this regime, the effects of viscosity are highly significant. Each elementary action occurs within an infinitesimally short time interval, which is approximately proportional to the size of the individual particles within the fluid.\\ Continuum Coarse Regime: In contrast, the second regime is characterized by a continuum on a much larger scale. Here, all particle activity below a certain level can be effectively suppressed. In this regime, each elementary action requires a fixed time interval of constant length, denoted as $h>0$. This length of time is directly proportional to the size of the elementary continuum.