Navier-Stokes existence and smoothness
Encyclopedia
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.
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 kinetic energy
. This is called the Navier–Stokes existence and smoothness problem.
Since understanding the Navier–Stokes equations is considered to be the first step for understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute
offered a US$
1,000,000 prize in May 2000, not to whomever constructs a theory of turbulence, but (more modestly) to the first person providing a hint on the phenomenon of turbulence. In that spirit of ideas, the Clay Institute set a concrete mathematical problem:
s 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 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
be a 3-dimensional vector, the velocity of the fluid, and let 
be the pressure of the fluid.More precisely, 
is the pressure divided by the fluid density
, and the density is constant for this incompressible and homogeneous fluid. The Navier–Stokes equations are:
where
is the kinematic viscosity, 
the external force, 
is the gradient
operator and
is the Laplacian operator, which is also denoted by 
. Note that this is a vector equation, i.e. it has three scalar equations. Writing down the coordinates of the velocity and the external force
then for each
there is the corresponding scalar Navier–Stokes equation:
The unknowns are the velocity
and the pressure 
. Since in three dimensions, there are three equations and four unknowns (three scalar velocities and the pressure), then a supplementary equation is needed. This extra equation is the continuity equation
describing the incompressibility of the fluid:
Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of "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. In two dimensions (2D), these equations are well-known [6, p. 321].
, which needs extra conditions on the growth behavior of the initial condition and the solutions. In order to rule out the problems at infinity, the Navier–Stokes equations can be set in a periodic framework, which implies that they are no longer working on the whole space 
but in the 3-dimensional torus 
. Each case will be treated separately.
is assumed to be a smooth and divergence-free function (see smooth function
) such that, for every multi-index
(see multi-index notation
) and any
, there exists a constant 
(i.e. this "constant" depends on 
and K) such that
The external force
is assumed to be a smooth function as well, and satisfies a very analogous inequality (now the multi-index includes time derivatives as well):
For physically reasonable conditions, the type of solutions expected are smooth functions that do not grow large as
. More precisely, the following assumptions are made:
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.
Let
. For any initial condition 
satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector 
and a pressure 
satisfying conditions 1 and 2 above.
(B) Breakdown of the Navier–Stokes solutions in
There exists an initial condition
and an external force 
such that there exists no solutions 
and 
satisfying conditions 1 and 2 above.
be the unitary vector in the j- direction:
Then
is periodic in the space variables if for any 
, then:
Notice that this is considering the coordinates mod 1
. This allows working not on the whole space
but on the quotient space
, which turns out to be the 3-dimensional torus:
Now the hypotheses can be stated properly. The initial condition
is assumed to be a smooth and divergence-free function and the external force 
is assumed to be a smooth function as well. The type of solutions that are physically relevant are those who satisfy these conditions:
3.
4. There exists a constant
such that 
for all 
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.
Let
. For any initial condition 
satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector 
and a pressure 
satisfying conditions 3 and 4 above.
(D) Breakdown of the Navier–Stokes solutions in
There exists an initial condition
and an external force 
such that there exists no solutions 
and 
satisfying conditions 3 and 4 above.
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
(such as with turbulence
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
). 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
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
, which remains one of the greatest unsolved problems in physics
Unsolved problems in physics
This is a list of some of the major unsolved problems in physics. Some of these problems are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result...
, despite its immense importance in science and engineering.
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 kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
. This is called the Navier–Stokes existence and smoothness problem.
Since understanding the Navier–Stokes equations is considered to be the first step for understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute
Clay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...
offered a US$
United States dollar
The United States dollar , also referred to as the American dollar, is the official currency of the United States of America. It is divided into 100 smaller units called cents or pennies....
1,000,000 prize in May 2000, not to whomever constructs a theory of turbulence, but (more modestly) to the first person providing a hint on the phenomenon of turbulence. In that spirit of ideas, the Clay Institute set a concrete mathematical problem:
The Navier–Stokes equations
In mathematics, the Navier–Stokes equations are a system of nonlinear partial differential equationPartial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s 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 using continuum mechanics
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
. The equations are a statement of Newton's second law, with the forces modeled according to those in a viscous
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
Newtonian fluid
Newtonian fluid
A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...
—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
be a 3-dimensional vector, the velocity of the fluid, and let be the pressure of the fluid.More precisely, is the pressure divided by the fluid densityDensity
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
, and the density is constant for this incompressible and homogeneous fluid. The Navier–Stokes equations are:
where
is the kinematic viscosity, the external force, is the gradientGradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
operator and
is the Laplacian operator, which is also denoted by . Note that this is a vector equation, i.e. it has three scalar equations. Writing down the coordinates of the velocity and the external force
then for each
there is the corresponding scalar Navier–Stokes equation:
The unknowns are the velocity
and the pressure . Since in three dimensions, there are three equations and four unknowns (three scalar velocities and the pressure), then a supplementary equation is needed. This extra equation is the continuity equationContinuity equation
A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
describing the incompressibility of the fluid:
Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of "divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
-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. In two dimensions (2D), these equations are well-known [6, p. 321].
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, which needs extra conditions on the growth behavior of the initial condition and the solutions. In order to rule out the problems at infinity, the Navier–Stokes equations can be set in a periodic framework, which implies that they are no longer working on the whole space but in the 3-dimensional torus . Each case will be treated separately.Hypotheses and growth conditions
The initial condition is assumed to be a smooth and divergence-free function (see smooth functionSmooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
) such that, for every multi-index
(see multi-index notationMulti-index notation
The mathematical notation of multi-indices simplifies formulae used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices....
) and any
, there exists a constant (i.e. this "constant" depends on and K) such that-
for all 
The external force
is assumed to be a smooth function as well, and satisfies a very analogous inequality (now the multi-index includes time derivatives as well):-
for all 
For physically reasonable conditions, the type of solutions expected are smooth functions that do not grow large as
. More precisely, the following assumptions are made:-
- There exists a constant
such that 
for all 
Condition 1 implies that the functions are smooth and globally defined and condition 2 means that the kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
of the solution is globally bounded.
The million-dollar-prize conjectures in the whole space
(A) Existence and smoothness of the Navier–Stokes solutions inLet
. For any initial condition satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector and a pressure satisfying conditions 1 and 2 above.(B) Breakdown of the Navier–Stokes solutions in
There exists an initial condition
and an external force such that there exists no solutions and satisfying conditions 1 and 2 above.Hypotheses
The functions sought now are periodic in the space variables of period 1. More precisely, let be the unitary vector in the j- direction:
Then
is periodic in the space variables if for any , then:
Notice that this is considering the coordinates mod 1
Fractional part
All real numbers can be written in the form n + r where n is an integer and the remaining fractional part r is a nonnegative real number less than one...
. This allows working not on the whole space
but on the quotient spaceQuotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
, which turns out to be the 3-dimensional torus:
Now the hypotheses can be stated properly. The initial condition
is assumed to be a smooth and divergence-free function and the external force is assumed to be a smooth function as well. The type of solutions that are physically relevant are those who satisfy these conditions:3.
4. There exists a constant
such that for all 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
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
of the solution is globally bounded.
The periodic million-dollar-prize theorems
(C) Existence and smoothness of the Navier–Stokes solutions inLet
. For any initial condition satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector and a pressure satisfying conditions 3 and 4 above.(D) Breakdown of the Navier–Stokes solutions in
There exists an initial condition
and an external force such that there exists no solutions and satisfying conditions 3 and 4 above.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
is sufficiently small then the statement is true: there are smooth and globally defined solutions to the Navier–Stokes equations. - Given an initial velocity
there exists a finite time T, depending on 
such that the Navier–Stokes equations on 
have smooth solutions 
and 
. It is not known if the solutions exist beyond that "blowup time" T. - The mathematician Jean LerayJean LerayJean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology....
in 1934 proved the existence of so called weak solutionWeak solutionIn mathematics, a weak solution to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for...
s to the Navier–Stokes equations, satisfying the equations in mean value, not pointwise.
External links
- The Clay Mathematics Institute's Navier–Stokes equation prize
- Why global regularity for Navier–Stokes is hard — Possible routes to resolution are scrutinized by Terence TaoTerence TaoTerence Chi-Shen Tao FRS is an Australian mathematician working primarily on harmonic analysis, partial differential equations, combinatorics, analytic number theory and representation theory...
. - Fuzzy Fluid Mechanics
- Navier–Stokes existence and smoothness (Millennium Prize Problem) A lecture on the problem by Luis CaffarelliLuis CaffarelliLuis A. Caffarelli is an Argentinian mathematician and leader in the field of partial differential equations and their applications....
.