A condition for the finite time blow up of the incompressible Navier–Stokes equations in the whole space

Abstract

This paper is interested in the existence of singularities for solutions of the Navier–Stokes equations in the whole space. We demonstrate the existence of initial data that leads to the unboundedness of the corresponding strong solution within a finite time. Our approach relies on lower and upper bounds of rates of decay for solutions to the Navier–Stokes equations. This result provides valuable insights into significant open problems in both physics and mathematics.