# Incompressible Navier-Stokes Equations: Example of no solution at R3 and t=0

An important and still unresolved problem in fluid dynamics is the question of global regularity in three dimensional Euclidian $R^{3}$ space, utilizing Navier-Stokes equation for incompressible fluid:

$\large \frac{\partial \vec{u}}{\partial t} +(\vec{u}\cdot \nabla )\vec{u}=-\frac{\nabla p}{\rho } +\nu \Delta \vec{u}+\vec{f}$

In this post and related paper, we share specific example of the fluid velocity vector field $\vec{u}^{0}(\vec{x})=\vec{u}(\vec{x})$ for fluid occupying all of $R^{3}$ space, as well as the $\vec{f}(\vec{x},t)$ vector field, for which we prove that the Navier-Stokes equation for incompressible fluid does not have a solution at any position in space $\vec{x} \in R^{3}$ at $t=0$. The velocity vector field

$\large u_{i} (\vec{x})=2\frac{x_{h(i-1)} -x_{h(i+1)} }{\left(1+\sum _{j=1}^{3}x_{j}^{2} \right)^{2} }$ ; $i=\{ 1,2,3\}$

where

$h(l)=\left\{\begin{array}{ccc} {l} & {;1\le l\le 3} & {} \\ {1} & {;l=4} & {} \\ {3} & {;l=0} & {} \end{array}\right.$

is smooth, divergence-free, continuously differentiable $u(\vec{x})\in C^{\infty }$. Interestingly, vector field has bounded energy over whole $R^3$ space equal to $\pi^2$:

$\large \int _{R^{3} }\left|\vec{u}\right|^{2} dx=\pi ^{2}$

Vector field has zero velocity at coordinate origin, and velocity converges to zero for $\left|\vec{x}\right|\to \infty$. The vector field

$\large \vec{f}(\vec{x},t)=(0,0,\frac{1}{1+t^{2} (\sum _{j=1}^{3}x_{j} )^{2} )}$

is smooth, continuously differentiable $f(\vec{x},t)\in C^{\infty }$, converging to zero for $\left|\vec{x}\right|\to \infty$.

Applying $\vec{u}(\vec{x})$ and $\vec{f}(\vec{x}, t)$ in the Navier-Stokes equation for incompressible fluid results with three mutually different solutions for pressure $p(\vec{x}, t)$, one of which includes zero division with zero $\frac{0}{0}$ term at $t=0$, which is indeterminate for all positions $\vec{x} \in R^{3}$.