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We provide an example of a smooth, divergence-free $\nabla \cdot\vec{u}(\vec{x})=0$ velocity vector field $\vec{u}(\vec{x})$ for incompressiblefluid 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 havea solution for any position in space $\vec{x}\in R^{3} $ at $t=0$. The velocityvector 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 $u(\vec{x})\in C^{\infty }$, has bounded energy$\int _{R^{3} }\left|\vec{u}\right|^{2} dx=\pi ^{2}$, zero velocity atcoordinate origin, and velocity converges to zero for $\left|\vec{x}\right|\to\infty$. The vector field $\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 theNavier-Stokes equation for incompressible fluid results with three mutuallydifferent solutions for pressure $p(\vec{x}, t)$, one of which includes zerodivision with zero $\frac{0}{0}$ term at $t=0$, which is indeterminate for allpositions $\vec{x} \in R^{3}$.

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