Global in time solution to the incompressible Navier-Stokes equations on $\Real^n$. An Elementary Approach

TL;DR

This paper proves a global-in-time existence theorem for classical solutions to the incompressible Navier-Stokes equations in any dimension greater than or equal to two, using an elementary approach based on boundedness of the solution's time derivative.

Contribution

It introduces a straightforward method to establish global solutions for Navier-Stokes equations in $ eal^n$, expanding the understanding of solution longevity under regular initial conditions.

Findings

Global existence of solutions in $ eal^n$ for $n geq 2$

Boundedness of the time derivative of the $L^$ norm

Elementary proof technique for Navier-Stokes solutions

Abstract

In this paper we prove a theorem of global time-extension for the local classical solution of Navier-Stokes's evolution problem in $\Real^n$ with $n\geqslant2$ for incompressible fluids subjected to external forces and regular initial conditions. This will be achieved by expressing the boundedness of the time derivative of the $L^\infty$ solution norm.