The question of whether the Navier–Stokes equations always have smooth solutions in three-dimensional Euclidean space, given some initial conditions and possibly an external force, has been a longstanding unsolved problem in mathematics since the early 20th century. The equations are a system of partial differential equations that describe the motion of a fluid in space. Although computational solutions to the Navier–Stokes equations are used in a large number of practical applications, there is no complete analytical understanding of the solutions. 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.
In 2000, the Clay Mathematics Institute named four instances of the Navier–Stokes existence and smoothness problem, two of which dealing with the Navier–Stokes equation with no external force and the other two dealing with the Navier–Stokes equation with a smooth external force, as one of seven Millennium Prize Problems because of the Navier–Stokes equations' critical importance to multiple fields. The institute offered a Millennium Prize to the first person providing a solution for one of the four specific statements of the problem as given by Charles Fefferman.
On 8 September 2026, artificial intelligence company OpenAI claimed to have found a solution to the Navier–Stokes equations with a smooth external force that develops a singularity in finite time in three-dimensional Euclidean space, along with a formalization in the Lean proof assistant, using a swarm of around 10,000 AI agents running an internal frontier model. The solution to the Navier–Stokes equation resembles a spinning top that tightens to a singularity with diverging velocities.
The announcement was accompanied by a priority dispute with Levent Alpöge (employed at rival AI company Anthropic) and Tristan Buckmaster, who had derived a set of closely related results on the Euler equations used in the work. The method used to generate the solution to the Navier–Stokes equation built upon a method developed by Diego Córdoba and Luis Martínez-Zoroa in 2023 to find blowup phenomena in related fluid equations. OpenAI stated in its announcement that it would not claim the $1 million Clay Millennium Prize for the solution it posted. As of October 2026, the Clay Mathematics Institute currently considers the problem "active".
The Navier–Stokes existence and smoothness problem with a real-analytic external force or no external force remains unsolved.
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 model 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 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, and the density is constant for this incompressible and homogeneous fluid. The Navier–Stokes equations are:
where is the kinematic viscosity, is the external specific force (which may be understood as the component of local acceleration of the fluid that is due to an external force), is the gradient operator and is the Laplacian 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 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-free") functions. For this flow of a homogeneous medium, density and viscosity are constants.
Since only its gradient appears, the pressure can be eliminated by taking the curl of both sides of the Navier–Stokes equations. In this case the Navier–Stokes equations reduce to the vorticity-transport equations.
The Navier–Stokes equations are nonlinear, meaning that 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. This nonlinearity allows the equations to describe a wide range of fluid dynamics phenomena, including the formation of shock waves and other complex flow patterns.
One way to understand the nonlinearity of the Navier–Stokes equations is to consider the term in the equations. This term represents the acceleration of the fluid, and it is a product of the velocity vector and the gradient operator . Because the gradient operator is a linear operator, the term is nonlinear in the velocity vector . 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.
(In the case of compressible flow, 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.)
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.
Here, we will use the finite difference method. To do this, we can divide the time interval into a series of smaller time steps, and approximate the derivative at each time step using a finite difference formula:
where is the size of the time step, and and are the values of and at time step .
Using this approximation, we can iterate through the time steps and compute the value of at each time step. For example, starting at time step and using the approximation above, we can compute the value of at time step :
This process can be repeated until we reach the final time step .
There are many other approaches to solving ordinary differential equations, each with its own advantages and disadvantages. The choice of approach depends on the specific equation being solved, and the desired accuracy and efficiency of the solution.
The Navier–Stokes equations are a set of partial differential equations that describe the motion of fluids. They are given by:
where is the velocity field of the fluid, is the pressure, is the density, is the kinematic viscosity, and is an external force. The first equation is known as the momentum equation, and the second equation is known as the continuity equation.
These equations are typically accompanied by boundary conditions, which describe the behavior of the fluid at the edges of the domain. For example, in the case of a fluid flowing through a pipe, the boundary conditions might specify that the velocity and pressure are fixed at the walls of the pipe.
The Navier–Stokes equations are nonlinear and highly coupled, making them difficult to solve in general. In particular, the difficulty of solving these equations lies in the term , which represents the nonlinear advection of the velocity field by itself. This term makes the Navier–Stokes equations highly sensitive to initial conditions, and it is the main reason why the Navier–Stokes existence and smoothness problems are so challenging.
In addition to the mathematical challenges of solving the Navier–Stokes equations, there are also many practical challenges in applying these equations to real-world situations. For example, the Navier–Stokes equations are often used to model fluid flows that are turbulent, which means that the fluid is highly chaotic and unpredictable. Turbulence is a difficult phenomenon to model and understand, and it adds another layer of complexity to the problem of solving the Navier–Stokes equations. To solve the Navier–Stokes equations, we need to find a velocity field and a pressure field that satisfy the equations and the given boundary conditions. This can be done using a variety of numerical techniques, such as finite element methods, spectral methods, or finite difference methods.
For example, consider the case of a two-dimensional fluid flow in a rectangular domain, with velocity and pressure fields and a pressure field , respectively. The Navier–Stokes equations can be written as:
where is the density, is the kinematic viscosity, and is an external force. The boundary conditions might specify that the velocity is fixed at the walls of the domain, or that the pressure is fixed at certain points.
To solve these equations numerically, we can divide the domain into a series of smaller elements, and solve the equations locally within each element. For example, using a finite element method, we might represent the velocity and pressure fields as: