The Navier-Stokes Existence and Smoothness Poser in $\mathbb{R}^n$

TL;DR

This paper presents a method for deriving smooth solutions to the incompressible Navier-Stokes equations in rf, providing insights into the potential blowup phenomena and the longstanding open problem of existence and smoothness.

Contribution

It introduces a novel approach to construct smooth solutions for the Navier-Stokes system in rf, advancing understanding of the smoothness and blowup issues.

Findings

Existence of smooth solutions for given initial data in rf.

Periodic solutions illustrate potential blowup time phenomena.

Method provides a new perspective on Navier-Stokes regularity problem.

Abstract

In this paper we describe a method to derive solutions of the incompressible Navier- Stokes system of equations for non-stationary initial value problems in $\mathbb{R}^n$. We show that for a given smooth solenoidal initial velocity vector field there exist smooth spatially periodic solutions of pressure and velocity in $\mathbb{R}^n$. An illustrative example in $\mathbb{R}^3$ provides important insights into the ostensible phenomenon of the blowup time.