Conjectures
Navier-Stokes Existence and Smoothness
nav-YAY stokes
Also Known As Navier-Stokes Problem
Analysis
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One of the seven Clay Mathematics Institute Millennium Prize Problems, asking for a proof settling the most basic open question about the Navier-Stokes equations that govern the flow of fluids such as water and air: whether smooth, physically reasonable solutions always exist. The equations themselves were formulated by Claude-Louis Navier in 1821 and 1822 and completed by George Gabriel Stokes in 1845, but whether their three dimensional solutions always exist and stay smooth, or can instead break down in finite time, was not known when the Clay Mathematics Institute named the problem in 2000. The official problem statement was written by Charles Fefferman of Princeton University, who allows a solver to prove existence and smoothness on ordinary three dimensional space or on the three dimensional torus, or instead to exhibit a case where a solution breaks down.
Facts
StatementA correct solution must prove either that smooth, globally defined solutions to the three dimensional Navier-Stokes equations always exist given smooth initial data, on ordinary space or on the three dimensional torus, or must exhibit a case in which such a solution breaks down in finite time. 1 Proposed Year Prize StatusOne of the seven Clay Mathematics Institute Millennium Prize Problems, named in 2000; a correct proof carries a one million dollar award. 1 Progress Toward ResolutionNo general proof or counterexample is known. In the official problem statement's own words, standard methods from partial differential equations appear inadequate and some deep, new idea is probably needed; even whether the solutions exist at all was, as of the statement's writing, unknown, leaving understanding at what its author called a very primitive level. 1 Cross-Tradition Connections
Associated With
Navier and Stokes formulated the underlying equations in the nineteenth century; the Millennium Prize question about existence and smoothness of their solutions was posed by the Clay Mathematics Institute in 2000, not by Navier or Stokes themselves.
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1. Clay Mathematics Institute
Clay Mathematics Institutehttps://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdfQuote, https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf
Since we don't even know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas.
1. Clay Mathematics Institute
Clay Mathematics InstituteMillennium Prize Problems page, announcement paragraphQuote, Millennium Prize Problems page, announcement paragraph
The prizes were announced at a meeting in Paris, held on May 24, 2000 at the Collège de France.
MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsAssociated With: Claude-Louis Navier, Biography section, paragraph on major contributionsQuote, Associated With: Claude-Louis Navier, Biography section, paragraph on major contributions
He gave the well known Navier-Stokes equations for an incompressible fluid in 1821 while in 1822 he gave equations for viscous fluids.
MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsAssociated With: George Gabriel Stokes, Biography section, paragraph after the 1842-1843 publicationsQuote, Associated With: George Gabriel Stokes, Biography section, paragraph after the 1842-1843 publications
Stokes then continued his investigations, looking at the situation where he took into account internal friction in fluids in motion.
Navier-Stokes Existence and Smoothness (Wikipedia)
Wikimedia FoundationIn Branch: Analysis, The Navier-Stokes equations sectionQuote, In Branch: Analysis, The Navier-Stokes equations section
The Navier-Stokes equations are nonlinear, meaning that the terms in the equations do not have a simple linear relationship
Navier-Stokes Existence and Smoothness (Wikipedia)
Wikimedia FoundationIn Branch: Applied and Computational Mathematics, IntroductionQuote, In Branch: Applied and Computational Mathematics, Introduction
Solutions to the Navier-Stokes equations are used in many practical applications.
Open Questions (1 open question)
Do smooth, globally defined solutions to the three dimensional Navier-Stokes equations always exist, or can a solution break down in finite time?
As the Clay Mathematics Institute's own official problem statement puts it, standard methods from partial differential equations appear inadequate, and whether solutions even exist in general was, at the time of writing, unknown.
What would resolve this A proof that smooth solutions always exist on ordinary three dimensional space or the three dimensional torus given smooth initial data, or a single explicit case where a solution breaks down in finite time.
Partial differential equations and fluid dynamicsClay Mathematics Institute
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