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

TL;DR

This paper introduces a modified Navier-Stokes model inspired by relativity, proving the existence and uniqueness of strong solutions without small data assumptions, potentially improving understanding of fluid dynamics under relativistic conditions.

Contribution

The paper establishes the global existence and uniqueness of strong solutions for a new relativistic-inspired Navier-Stokes model in three dimensions.

Findings

Existence of strong solutions on bounded domains and tori.

No smallness condition required on initial data.

Model aligns with relativistic speed constraints.

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 $(\textbf{u}\cdot \nabla) \textbf{u}$ 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.