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

In this paper we prove a theorem of global time-extension for the local classical solution of Navier-Stokes’s evolution problem in ${\text{Real}}^n$ with $n⩾2$ 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 $L^{\mathrm{\infty }}$ solution norm.