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

TL;DR

This paper constructs a smooth, divergence-free velocity field and a force for incompressible Navier-Stokes equations in three-dimensional space, demonstrating that no solution exists at time zero due to an indeterminate pressure term.

Contribution

It provides a specific example of smooth initial data and forcing that lead to the non-existence of solutions at t=0 for the Navier-Stokes equations in R^3.

Findings

Existence of a smooth divergence-free velocity field with bounded energy.

Construction of a smooth force field that converges to zero at infinity.

Demonstration of an indeterminate pressure term at t=0, implying no solution exists.

Abstract

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 $\vec{x} \in R^{3}$.