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\}


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}.

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