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

We provide an example of a smooth, divergence-free \(\nabla \cdot \vec{u}(\vec{x})=0\) velocity vector field \(\vec{u}(\vec{x})\) 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 \(u(\vec{x})\in C^{\infty }\), has bounded energy \(\int _{R^{3} }\left|\vec{u}\right|^{2} dx=\pi ^{2}\), zero velocity at coordinate 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 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 \(\vec{x} \in R^{3}\).