Remarks on smoothness and finite-time blowup solutions of the incompressible Navier-Stokes equation

TL;DR

This paper investigates the smoothness and finite-time blowup of solutions to the incompressible Navier-Stokes equations, showing solutions are smooth only for finite times and identifying blowup times for various cases.

Contribution

It introduces a novel perspective on the solution smoothness and demonstrates finite-time blowup for different cases of the Navier-Stokes equations.

Findings

Solutions are smooth only for finite times.

Finite-time blowup occurs in various cases.

Existence of blowup times for solutions.

Abstract

This article examines the smoothness of the solution to the Navier-Stokes equation from a novel perspective. Here, the existence of the smoother solution relative to x and to the time t was shown only for a finite time. Moreover, for each considered case of the problem, a blowup time for its solutions can be demonstrated.