The Navier–Stokes existence and smoothness problem concerns the
mathematical properties of solutions to the
Navier–Stokes equations, a system of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s that describe the motion of a
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
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
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
, which remains one of the greatest
unsolved problems in physics, despite its immense importance in science and engineering.
Even more basic (and seemingly intuitive) properties of the solutions to Navier–Stokes have never been proven. For the three-dimensional system of equations, and given some
initial condition
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). Fo ...
s, mathematicians have neither proved that
smooth solutions always exist, nor found any counter-examples. 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
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
, the
Clay Mathematics Institute
The Clay Mathematics Institute (CMI) is a private, non-profit foundation dedicated to increasing and disseminating mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address is now in Denver, Colorado. CMI's sc ...
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:
The Navier–Stokes equations
In mathematics, the Navier–Stokes equations are a system of
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s for abstract
vector fields of any size. In physics and engineering, they are a system of equations that model the motion of liquids or non-
rarefied gases (in which the
mean free path
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as ...
is short enough so that it can be thought of as a continuum mean instead of a collection of particles) using
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such mo ...
. The equations are a statement of
Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in mo ...
, with the forces modeled according to those in a
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the in ...
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 field, the velocity of the fluid, and let
be the pressure of the fluid.
[More precisely, is the pressure divided by the fluid ]density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
, and the density is constant for this incompressible and homogeneous fluid. The Navier–Stokes equations are:
:
where
is the
kinematic viscosity,
the external volumetric force,
is the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
operator and
is the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
operator, which is also denoted by
or
. 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
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
for
incompressible fluids that describes the
conservation of mass of the fluid:
:
Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of
solenoidal ("
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
-free") functions. For this flow of a homogeneous medium, density and viscosity are constants.
Since only its gradient appears, the pressure ''p'' can be eliminated by taking the
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was ...
of both sides of the Navier–Stokes equations. In this case the Navier–Stokes equations reduce to the
vorticity-transport equations.
Now, we are going to look at Nonlinearity
The Navier-Stokes equations are nonlinear because the terms in the equations do not have a simple linear relationship with each other. This means that the equations cannot be solved using traditional linear techniques, and more advanced methods must be used instead. Nonlinearity is important in the Navier-Stokes equations because it allows the equations to describe a wide range of fluid dynamics phenomena, including the formation of shock waves and other complex flow patterns. However, the nonlinearity of the Navier-Stokes equations also makes them more difficult to solve, as traditional linear methods may not work.
One way to understand the nonlinearity of the Navier-Stokes equations is to consider the term (v · ∇)v in the equations. This term represents the acceleration of the fluid, and it is a product of the velocity vector v and the gradient operator ∇. Because the gradient operator is a linear operator, the term (v · ∇)v is nonlinear in the velocity vector v. This means that the acceleration of the fluid depends on the magnitude and direction of the velocity, as well as the spatial distribution of the velocity within the fluid.
The nonlinear nature of the Navier-Stokes equations can be seen in the term
, which represents the acceleration of the fluid due to its own velocity. This term is nonlinear because it involves the product of two velocity vectors, and the resulting acceleration is therefore dependent on the magnitude and direction of both vectors.
Another source of nonlinearity in the Navier-Stokes equations is the pressure term
. The pressure in a fluid depends on the density and the gradient of the pressure, and this term is therefore nonlinear in the pressure.
One example of the nonlinear nature of the Navier-Stokes equations can be seen in the case of a fluid flowing around a circular obstacle. In this case, the velocity of the fluid near the obstacle will be higher than the velocity of the fluid farther away from the obstacle. This results in a pressure gradient, with higher pressure near the obstacle and lower pressure farther away.
To see this more explicitly, consider the case of a circular obstacle of radius
placed in a uniform flow with velocity
and density
. Let
be the velocity of the fluid at position
and time
, and let
be the pressure at the same position and time.
The Navier-Stokes equations in this case are:
:
:
where
is the kinematic viscosity of the fluid.
Assuming that the flow is steady (meaning that the velocity and pressure do not vary with time), we can set the time derivative terms equal to zero:
:
:
We can now consider the flow near the circular obstacle. In this region, the velocity of the fluid will be higher than the uniform flow velocity
due to the presence of the obstacle. This results in a nonlinear term
in the Navier-Stokes equations that is proportional to the velocity of the fluid.
At the same time, the presence of the obstacle will also result in a pressure gradient, with higher pressure near the obstacle and lower pressure farther away. This can be seen by considering the continuity equation, which states that the mass flow rate through any surface must be constant. Since the velocity is higher near the obstacle, the mass flow rate through a surface near the obstacle will be higher than the mass flow rate through a surface farther away from the obstacle. This can be compensated for by a pressure gradient, with higher pressure near the obstacle and lower pressure farther away.
As a result of these nonlinear effects, the Navier-Stokes equations in this case become difficult to solve, and approximations or numerical methods must be used to find the velocity and pressure fields in the flow.
Consider the case of a two-dimensional fluid flow in a rectangular domain, with a velocity field
and a pressure field
. We can use a finite element method to solve the Navier-Stokes equation for the velocity field:
To do this, we divide the domain into a series of smaller elements, and represent the velocity field as:
where
is the number of elements, and
are the shape functions associated with each element. Substituting this expression into the Navier-Stokes equation and applying the finite element method, we can derive a system of ordinary differential equations:
where
is the domain, and the integrals are over the domain. This system of ordinary differential equations can be solved using techniques such as the finite element method or spectral methods.
Such method can be applied as:
we can use a variety of techniques, such as the finite element method or spectral methods.
One common approach is to use a finite difference method, which involves approximating the derivative terms in the equation using finite differences. To do this, we can divide the time interval