The Global Existence of Solutions to a Quasi-Relativistic Incompressible Navier-Stokes Model

Abstract

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 $(\text{u}\cdot \nabla )\text{u}$ by $(\text{v}\cdot \nabla )\text{u}$ with $\text{v}=c\text{u}/\sqrt{c^2+|\text{u}{|}^2}$ where $c$ is the speed of light. Thus we have that $|\text{v}|\le c$ and for $|\text{u}|\ll c$ we have $\text{v}\approx \text{u}$. 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 $|\text{v}|\le c$ 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 $\text{u}\in L^2(0,T;{\text{H}}^2)\cap L^{\infty }(0,T;\text{V})$ with ${\text{u}}^{\prime }\in L^2(0,T;{\text{L}}^2)$ to our system of equations on either a smooth bounded domain $U\subset {\text{R}}^3$ or the flat $3$-torus $\mathbb{T}$ for any initial velocity ${\text{u}}_0\in \text{V}$ and any forcing function $\text{f}\in L^2(0,T;{\text{L}}^2)$. No assumption on the smallness of the data is necessary. Here $\text{V}$ 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.