  We introduce a new modified Navier-Stokes model in $3$ dimensions by
modifying the convection term in the ordinary Navier-Stokes equations. This is
done by replacing the convective term $(\textbf{u}\cdot \nabla) \textbf{u}$ by
$(\textbf{v}\cdot \nabla)\textbf{u}$ with
$\textbf{v}=c\textbf{u}/\sqrt{c^2+|\textbf{u}|^2}$ where $c$ is the speed of
light. Thus we have that $|\textbf{v}|\leq c$ and for $|\textbf{u}|\ll c$ we
have $\textbf{v} \approx \textbf{u}$. Thus the solutions to this system should
yield a good approximation to the solutions of the ordinary Navier-Stokes
equations under physically reasonable conditions. The modification of the
convective term is a natural progression of the work done in \cite{JaraczLee}.
The property that $|\textbf{v}|\leq c$ embodies the notion that in relativity
matter can't travel faster than the speed of light, giving the model its name.
We prove that there exists a strong solution $\textbf{u} \in L^2(0, T;
\textbf{H}^2) \cap L^{\infty}(0, T; \textbf{V})$ with $\textbf{u}' \in L^2(0,
T; \textbf{L}^2)$ to our system of equations on either a smooth bounded domain
$U\subset \textbf{R}^3$ or the flat $3$-torus $\mathbb{T}$ for any initial
velocity $\textbf{u}_0 \in \textbf{V}$ and any forcing function $\textbf{f}\in
L^2(0, T; \textbf{L}^2)$. No assumption on the smallness of the data is
necessary. Here $\textbf{V}$ is the space of weakly divergence free vector
fields with components in $H^1$ which vanish on the boundary. We also prove the
uniqueness of this strong solution. Though our modification is somewhat ad-hoc,
it suggests that though more complicated, equations incorporating aspects of
special and general relativity might have better existence and uniqueness
properties than the ordinary Navier-Stokes equations.
