An important and still unresolved problem in fluid dynamics is the question of global regularity in three dimensional Euclidian space, utilizing Navier-Stokes equation for incompressible fluid:

In this post and related paper, we share specific example of the fluid velocity vector field for fluid occupying all of space, as well as the vector field, for which we prove that the Navier-Stokes equation for incompressible fluid does not have a solution at any position in space at . The velocity vector field

;

where

is smooth, divergence-free, continuously differentiable . Interestingly, vector field has bounded energy over whole space equal to :

Vector field has zero velocity at coordinate origin, and velocity converges to zero for . The vector field

is smooth, continuously differentiable , converging to zero for .

Applying and in the Navier-Stokes equation for incompressible fluid results with three mutually different solutions for pressure , one of which includes zero division with zero term at , which is indeterminate for all positions .

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