Global solutions to the homogeneous and inhomogeneous Navier-Stokes equations

TL;DR

This paper introduces a new approach to proving the existence and uniqueness of solutions for the 3D-Navier-Stokes equations, unifying the treatment for various domain types and fluid conditions.

Contribution

It constructs a maximal separable Hilbert space where Leray-Hopf solutions become strong solutions, simplifying the proof across different scenarios.

Findings

Established a unified proof method for Navier-Stokes solutions

Constructed a new Hilbert space ${\bf{SD}}^2[\R^3]$

Demonstrated solutions are weak in space but strong in time.

Abstract

In this paper we take a new approach to a proof of existence and uniqueness of solutions for the 3D-Navier-Stokes equations, which leads to essentially the same proof for both bounded and unbounded domains and for homogeneous or inhomogeneous incompressible fluids. Our approach is to construct the largest separable Hilbert space ${\bf{SD}}^2[\R^3]$, for which the Leray-Hopf (type) solutions in $L^2[{\mathbb R}^3]$ are strong solutions in ${\bf{SD}}^2[\R^3]$. We say Leray-Hopf type because our solutions are weak in the spatial sense but not in time.