The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, one of the pillars of fluid mechanics (such as with turbulence). These equations describe the motion of a fluid (that is, a liquid or a gas) in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.
Contents
- The NavierStokes equations
- Two settings unbounded and periodic space
- Hypotheses and growth conditions
- The Millennium Prize conjectures in the whole space
- Hypotheses
- The periodic Millennium Prize theorems
- Partial results
- Attempt at solution
- References
Even much more basic properties of the solutions to Navier–Stokes have never been proven. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proved that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass. This is called the Navier–Stokes existence and smoothness problem.
Since understanding the Navier–Stokes equations is considered to be the first step to understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute in May 2000 made this problem one of its seven Millennium Prize problems in mathematics. It offered a US$1,000,000 prize to the first person providing a solution for a specific statement of the problem:
Prove or give a counter-example of the following statement:
In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.
The Navier–Stokes equations
In mathematics, the Navier–Stokes equations are a system of nonlinear partial differential equations for abstract vector fields of any size. In physics and engineering, they are a system of equations that models the motion of liquids or non-rarefied gases (in which the mean free path is short enough so that it can be thought of as a continuum mean instead of a collection of particles) using continuum mechanics. The equations are a statement of Newton's second law, with the forces modeled according to those in a viscous Newtonian fluid—as the sum of contributions by pressure, viscous stress and an external body force. Since the setting of the problem proposed by the Clay Mathematics Institute is in three dimensions, for an incompressible and homogeneous fluid, only that case is considered below.
Let
where
then for each
The unknowns are the velocity
Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of solenoidal ("divergence-free") functions. For this flow of a homogeneous medium, density and viscosity are constants.
The pressure p can be eliminated by taking an operator rot (alternative notation curl) of both sides of the Navier–Stokes equations. In this case the Navier–Stokes equations reduce to the vorticity-transport equations.
Two settings: unbounded and periodic space
There are two different settings for the one-million-dollar-prize Navier–Stokes existence and smoothness problem. The original problem is in the whole space
Hypotheses and growth conditions
The initial condition
The external force
For physically reasonable conditions, the type of solutions expected are smooth functions that do not grow large as
-
v ( x , t ) ∈ [ C ∞ ( R 3 × [ 0 , ∞ ) ) ] 3 , p ( x , t ) ∈ C ∞ ( R 3 × [ 0 , ∞ ) ) - There exists a constant
E ∈ ( 0 , ∞ ) such that∫ R 3 | v ( x , t ) | 2 d x < E for allt ≥ 0 .
Condition 1 implies that the functions are smooth and globally defined and condition 2 means that the kinetic energy of the solution is globally bounded.
The Millennium Prize conjectures in the whole space
(A) Existence and smoothness of the Navier–Stokes solutions in
Let
(B) Breakdown of the Navier–Stokes solutions in
There exists an initial condition
Hypotheses
The functions sought now are periodic in the space variables of period 1. More precisely, let
Then
Notice that this is considering the coordinates mod 1. This allows working not on the whole space
Now the hypotheses can be stated properly. The initial condition
Just as in the previous case, condition 3 implies that the functions are smooth and globally defined and condition 4 means that the kinetic energy of the solution is globally bounded.
The periodic Millennium Prize theorems
(C) Existence and smoothness of the Navier–Stokes solutions in
Let
(D) Breakdown of the Navier–Stokes solutions in
There exists an initial condition
Partial results
- The Navier–Stokes problem in two dimensions has already been solved positively since the 1960s: there exist smooth and globally defined solutions.
- If the initial velocity
v ( x , t ) is sufficiently small then the statement is true: there are smooth and globally defined solutions to the Navier–Stokes equations. - Given an initial velocity
v 0 ( x ) there exists a finite time T, depending onv 0 ( x ) such that the Navier–Stokes equations onR 3 × ( 0 , T ) have smooth solutionsv ( x , t ) andp ( x , t ) . It is not known if the solutions exist beyond that "blowup time" T. - Jean Leray in 1934 proved the existence of so-called weak solutions to the Navier–Stokes equations, satisfying the equations in mean value, not pointwise.
- Terence Tao in 2016 published a finite time blowup result for an averaged version of the 3-dimensional Navier–Stokes equation. He writes that the result formalizes a "supercriticality barrier" for the global regularity problem for the true Navier–Stokes equations, and claims that the method of proof in fact hints at a possible route to establishing blowup for the true equations.
Attempt at solution
In 2013, Mukhtarbay Otelbaev of the Eurasian National University in Astana, Kazakhstan, proposed a solution. As an attempt to solve an important open problem, the proof was immediately inspected by others in the field, who found at least one serious flaw. Otelbaev is attempting to fix the proof, but other mathematicians are skeptical.