Incompressible Navier-Stokes Equations: Example of no solution at $R^3$ and $t=0$

We provide an example of a smooth, divergence-free $\nabla \cdot \vec{u}(\vec{x})=0$velocityvectorfield$\vec{u}(\vec{x})$ for incompressible fluid occupying all of $R^3$ space, and smooth vector field $\vec{f}(\vec{x}, t)$ for which the Navier-Stokes equation for incompressible fluid does not have a solution for any position in space $\vec{x}\in R^3$ at $t=0$. The velocity vector field $u_{i} (\vec{x})=2\frac{x_{h(i-1)} -x_{h(i+1)} }{\left(1+\sum {j=1}^{3}x{j}^{2} \right)^{2} }$;$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 $u(\vec{x})\in C^{\mathrm{\infty }}$, has bounded energy ${\int }_{R^3}{\mid \vec{u}\mid }^{2}dx={\pi }^2$, zero velocity at coordinate origin, and velocity converges to zero for $\left|\vec{x}\right|\to \infty$\mathrm{.}Thevectorfield$\vec{f}(\vec{x},t)=(0,0,\frac{1}{1+t^{2} (\sum {j=1}^{3}x{j} )^{2} )} $issmooth,continuouslydifferentiable$f(\vec{x},t)\in C^{\infty }$,convergingtozerofor$\left|\vec{x}\right|\to \infty$\mathrm{.}Applying$\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 $\vec{x}\in R^3$.