Abstract
There is currently no proof guaranteeing that, given a smooth initial condition, the three-dimensional Navier-Stokes equations have a unique solution that exists for all positive times. This paper reviews the key rigorous results concerning the existence and uniqueness of solutions for this model. In particular, the link between the regularity of solutions and their uniqueness is highlighted. This article is part of the theme issue 'Stokes at 200 (Part 1)'.
Highlights
The 3D Navier–Stokes equations (NSE) that model the flow of a viscous, incompressible fluid, are
It may be that an eventual proof/disproof of the regularity of solutions of the NSE will involve such particular features of the equations that it says little about other models
Spaces in which the natural norm have this scaling-invariance property are termed ‘critical’, and much of the recent progress in the analysis of the NSE has been in extending arguments similar to those
Summary
The 3D Navier–Stokes equations (NSE) that model the flow of a viscous, incompressible fluid, are. This is one of the seven Clay Millennium Prize Problems, the solution of which (either positive or negative) will be awarded with a prize of one million dollars. This paper will show that much of our current understanding of the problem is based on simple bounds that can be derived for smooth solutions. These employ only techniques that are likely to have been available to Stokes, turning these into rigorous proofs required. This paper is not intended as an in-depth review of the mathematical state-of-the-art for this model, and experts are quite likely to find that their favourite recent result is not here; the book by Lemarié-Rieusset [20] would be a good place to start for something in this direction
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