Global (in Time) Solutions to the 3D-Navier-Stokes Equations on R^3

TL;DR

This paper proves the existence and uniqueness of global in time solutions to the 3D Navier-Stokes equations on unbounded domain R^3, addressing a fundamental open problem in fluid mechanics.

Contribution

It introduces a new approach based on physical insight to establish conditions for global solutions in an unbounded domain, extending previous bounded domain results.

Findings

Existence of global solutions on R^3

Uniqueness of solutions under certain conditions

Extension of previous bounded domain results

Abstract

A well-known unsolved problem (in the classical theory of fluid mechanics) is to identify a set of initial velocities, which may depend on the viscosity, the body forces and possibly the boundary of the fluid that will allow global in time solutions to the three-dimensional Navier-Stokes equations. (These equations describe the time evolution of the fluid velocity and pressure of an incompressible viscous homogeneous Newtonian fluid in terms of a given initial velocity and given external body forces.) A related problem is to provide conditions under which we can be assured that the numerical approximation of these equations, used in a variety of fields from weather prediction to submarine design, have only one solution. In earlier papers, we solved this problem for a bounded domain. In this paper, we use an approach based on additional physical insight, that allows us to prove that there exists unique global in time solutions to the Navier-Stokes equations on R^3.