Navier-Stokes Existence and Smoothness Problems

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That solutions to the motion of fluids in three dimensions always exist (existence);
and that if they do exist, then they do not contain any singularity, infinity or discontinuity
(smoothness).

Also stated as: Show that the Navier–Stokes equations on Euclidean 3-space have a unique,
smooth, finite energy solution for all time greater than or equal to zero, given smooth, divergencefree,
initial conditions which decay rapidly at large distances. Or show that there is no such solution.

Discipline: Analysis.

Originators: French mathematician Claude-Louis Navier and English mathematician George Gabriel
Stokes in 1822.

Incentive: US$1million, one of the seven Millennium Prize Problems - the most important
open problems in mathematics, according to the USA-based Clay Mathematics Institute.

Usefulness: The Navier-Stokes equations are nonlinear partial differential equations in almost
every real situation. They describe the physics of weather, ocean currents, water flow in a pipe, air
flow around a wing, and the motion of stars in a galaxy as well as help with design for aircraft, cars
and power stations, the study of blood flow, and pollution analysis.

Explorations: One approach - constructing a weak solution and showing that any weak solution
is smooth - has had partial success. It is believed, though not known with certainty, that the Navier-
Stokes equations describe turbulence properly. However, the equations are supercritical - energy
can interact much more forcefully at fine scales than it can at coarse scales. There is no good large data
global theory for any supercritical equation, without additional constraints.

Almost all the equations are written for Newtonian fluids, which continue to flow regardless of forces
acting on them. Models for other kinds of fluid flows, such as blood, do not yet exist.

State of play: Since we don’t even know whether solutions exist, our understanding is primitive. Some
exact solutions of degenerate cases and non-linear equations do exist. Solutions may lie in related
models, such as the Euler equations.

Published in IMAges 7 - November 2009