Navier–Stokes existence and smoothness

Summarize Timeline Top Qs Fact Check

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 ${\displaystyle \mathbf {v} ({\boldsymbol {x}},t)}$ be a 3-dimensional vector field, the velocity of the fluid, and let ${\displaystyle p({\boldsymbol {x}},t)}$ be the pressure of the fluid.^{[note 1]} The Navier–Stokes equations are:

${\displaystyle {\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} =-{\frac {1}{\rho }}\nabla p+\nu \Delta \mathbf {v} +\mathbf {f} ({\boldsymbol {x}},t)}$

where ${\displaystyle \nu >0}$ is the kinematic viscosity, ${\displaystyle \mathbf {f} ({\boldsymbol {x}},t)}$ the external force per unit of mass, ${\displaystyle \nabla }$ is the gradient operator and ${\displaystyle \displaystyle \Delta }$ is the Laplacian operator, which is also denoted by ${\displaystyle \nabla \cdot \nabla }$ or ${\displaystyle \nabla ^{2}}$. 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

${\displaystyle \mathbf {v} ({\boldsymbol {x}},t)={\big (}\,v_{1}({\boldsymbol {x}},t),\,v_{2}({\boldsymbol {x}},t),\,v_{3}({\boldsymbol {x}},t)\,{\big )}\,,\qquad \mathbf {f} ({\boldsymbol {x}},t)={\big (}\,f_{1}({\boldsymbol {x}},t),\,f_{2}({\boldsymbol {x}},t),\,f_{3}({\boldsymbol {x}},t)\,{\big )}}$

then for each ${\displaystyle i=1,2,3}$ there is the corresponding scalar Navier–Stokes equation:

${\displaystyle {\frac {\partial v_{i}}{\partial t}}+\sum _{j=1}^{3}{\frac {\partial v_{i}}{\partial x_{j}}}v_{j}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial x_{i}}}+\nu \sum _{j=1}^{3}{\frac {\partial ^{2}v_{i}}{\partial x_{j}^{2}}}+f_{i}({\boldsymbol {x}},t).}$

The unknowns are the velocity ${\displaystyle \mathbf {v} ({\boldsymbol {x}},t)}$ and the pressure ${\displaystyle p({\boldsymbol {x}},t)}$. 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:

${\displaystyle \nabla \cdot \mathbf {v} =0.}$

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 p 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 ${\displaystyle (\mathbf {v} \cdot \nabla )\mathbf {v} }$ 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.

(In the case of compressible flow, another source of nonlinearity in the Navier–Stokes equations is the pressure term ${\displaystyle -{\frac {1}{\rho }}\nabla p}$. 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 ${\displaystyle R}$ placed in a uniform flow with velocity ${\displaystyle \mathbf {v_{0}} }$ and density ${\displaystyle \rho }$. Let ${\displaystyle \mathbf {v} (\mathbf {x} ,t)}$ be the velocity of the fluid at position ${\displaystyle \mathbf {x} }$ and time ${\displaystyle t}$, and let ${\displaystyle p(\mathbf {x} ,t)}$ be the pressure at the same position and time.

The Navier–Stokes equations in this case are:

${\displaystyle {\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} =-{\frac {1}{\rho }}\nabla p+\nu \Delta \mathbf {v} }$

${\displaystyle \nabla \cdot \mathbf {v} =0}$

where ${\displaystyle \nu }$ 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:

${\displaystyle (\mathbf {v} \cdot \nabla )\mathbf {v} =-{\frac {1}{\rho }}\nabla p+\nu \Delta \mathbf {v} }$

${\displaystyle \nabla \cdot \mathbf {v} =0}$

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 ${\displaystyle \mathbf {v_{0}} }$ due to the presence of the obstacle. This results in a nonlinear term ${\displaystyle (\mathbf {v} \cdot \nabla )\mathbf {v} }$ 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 ${\displaystyle \mathbf {v} (x,t)}$ and a pressure field ${\displaystyle p(x,t)}$. We can use a finite element method to solve the Navier–Stokes equation for the velocity field:

${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}+\nu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right)+f_{x}(x,y,t)}$

To do this, we divide the domain into a series of smaller elements, and represent the velocity field as:

${\displaystyle u(x,y,t)=\sum _{i=1}^{N}U_{i}(t)\phi _{i}(x,y)}$

where ${\displaystyle N}$ is the number of elements, and ${\displaystyle \phi _{i}(x,y)}$ 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:

${\displaystyle {\frac {dU_{i}}{dt}}=-{\frac {1}{\rho }}\sum _{j=1}^{N}\left({\frac {\partial p}{\partial x}}\right)j\int {\Omega }\phi _{j}{\frac {\partial \phi _{i}}{\partial x}}d\Omega +\nu \sum _{j=1}^{N}\int _{\Omega }\left({\frac {\partial ^{2}u}{\partial x^{2}}}\right)\phi _{j}{\frac {\partial ^{2}\phi _{i}}{\partial x^{2}}}d\Omega +\int {\Omega }f_{x}\phi _{i}d\Omega }$

where ${\displaystyle \Omega }$ 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 ${\displaystyle [t_{0},t_{f}]}$ into a series of smaller time steps, and approximate the derivative at each time step using a finite difference formula:

${\displaystyle {\frac {U_{i+1}-U_{i}}{\Delta t}}\approx -{\frac {1}{\rho }}\sum _{j=1}^{N}\left({\frac {\partial p}{\partial x}}\right)j\int {\Omega }\phi _{j}{\frac {\partial \phi _{i}}{\partial x}}d\Omega +\nu \sum _{j=1}^{N}\int _{\Omega }\left({\frac {\partial ^{2}u}{\partial x^{2}}}\right)j\phi _{j}{\frac {\partial ^{2}\phi _{i}}{\partial x^{2}}}d\Omega +\int {\Omega }f_{x}\phi _{i}d\Omega }$

where ${\displaystyle \Delta t=t_{i+1}-t_{i}}$ is the size of the time step, and ${\displaystyle U_{i}}$ and ${\displaystyle t_{i}}$ are the values of ${\displaystyle U_{i}}$ and ${\displaystyle t}$ at time step ${\displaystyle i}$.

Using this approximation, we can iterate through the time steps and compute the value of ${\displaystyle U_{i}}$ at each time step. For example, starting at time step ${\displaystyle i}$ and using the approximation above, we can compute the value of ${\displaystyle U_{i}}$ at time step ${\displaystyle i+1}$:

${\displaystyle U_{i+1}=U_{i}+\Delta t\cdot \left(-{\frac {1}{\rho }}\sum _{j=1}^{N}\left({\frac {\partial p}{\partial x}}\right)j\int {\Omega }\phi _{j}{\frac {\partial \phi _{i}}{\partial x}}d\Omega +\nu \sum _{j=1}^{N}\int _{\Omega }\left({\frac {\partial ^{2}u}{\partial x^{2}}}\right)_{j}\phi _{j}{\frac {\partial ^{2}\phi _{i}}{\partial x^{2}}}d\Omega +\int _{\Omega }f_{x}\phi _{i}d\Omega \right)}$

This process can be repeated until we reach the final time step ${\displaystyle t_{f}}$.

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:

${\displaystyle {\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} =-{\frac {1}{\rho }}\nabla p+\nu \nabla ^{2}\mathbf {v} +\mathbf {f} }$

${\displaystyle \nabla \cdot \mathbf {v} =0}$

where ${\displaystyle \mathbf {v} (x,t)}$ is the velocity field of the fluid, ${\displaystyle p(x,t)}$ is the pressure, ${\displaystyle \rho }$ is the density, ${\displaystyle \nu }$ is the kinematic viscosity, and ${\displaystyle \mathbf {f} (x,t)}$ 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 ${\displaystyle (\mathbf {v} \cdot \nabla )\mathbf {v} }$, 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 Millennium Prize's Navier–Stokes existence and smoothness problem is 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 ${\displaystyle \mathbf {v} (x,t)}$ and a pressure field ${\displaystyle p(x,t)}$ 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 ${\displaystyle \mathbf {v} (x,t)}$ and a pressure field ${\displaystyle p(x,t)}$,respectively. The Navier–Stokes equations can be written as:

${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}+\nu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right)+f_{x}(x,y,t)}$

${\displaystyle {\frac {\partial v}{\partial t}}+u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial y}}+\nu \left({\frac {\partial ^{2}v}{\partial x^{2}}}+{\frac {\partial ^{2}v}{\partial y^{2}}}\right)+f_{y}(x,y,t)}$

${\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0}$

where ${\displaystyle \rho }$ is the density, ${\displaystyle \nu }$ is the kinematic viscosity, and ${\displaystyle \mathbf {f} (x,y,t)=(f_{x}(x,y,t),f_{y}(x,y,t))}$ 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:

${\displaystyle u(x,y,t)=\sum _{i=1}^{N}U_{i}(t)\phi _{i}(x,y)}$

${\displaystyle v(x,y,t)=\sum _{i=1}^{N}V_{i}(t)\phi _{i}(x,y)}$

${\displaystyle p(x,y,t)=\sum _{i=1}^{N}P_{i}(t)\phi _{i}(x,y)}$

where ${\displaystyle N}$ is the number of elements, and ${\displaystyle \phi _{i}(x,y)}$ are the shape functions associated with each element. Substituting these expressions into the Navier–Stokes equations and applying the finite element method, we can derive a system of ordinary differential equations.

Summarize Timeline Top Qs Fact Check

There are two different settings for the Navier–Stokes existence and smoothness problem. The original problem is in Euclidean space ${\displaystyle \mathbb {R} ^{3}}$, 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 Euclidean space ${\displaystyle \mathbb {R} ^{3}}$ but in the 3-dimensional torus ${\displaystyle \mathbb {T} ^{3}=\mathbb {R} ^{3}/\mathbb {Z} ^{3}}$. Each setting will be treated separately.

In Euclidean space

For the case of the Navier–Stokes existence and smoothness problem on Euclidean space ${\displaystyle \mathbb {R} ^{3}}$, the initial condition ${\displaystyle \mathbf {v} _{0}(x)}$ is assumed to be a smooth and divergence-free function (see smooth function) such that, for every multi-index ${\displaystyle \alpha }$ (see multi-index notation) and any ${\displaystyle K>0}$, there exists a constant ${\displaystyle c=c(\alpha ,K)>0}$ such that

${\displaystyle \vert \partial ^{\alpha }\mathbf {v_{0}} (x)\vert \leq {\frac {c}{(1+\vert x\vert )^{K}}}\qquad }$ for all ${\displaystyle \qquad x\in \mathbb {R} ^{3}.}$

The external force ${\displaystyle \mathbf {f} (x,t)}$ is assumed to be a smooth function as well, and satisfies a very analogous inequality (now the multi-index includes time derivatives as well):

${\displaystyle \vert \partial ^{\alpha }\mathbf {f} (x,t)\vert \leq {\frac {c}{(1+\vert x\vert +t)^{K}}}\qquad }$ for all ${\displaystyle \qquad (x,t)\in \mathbb {R} ^{3}\times [0,\infty ).}$

For physically reasonable conditions, the type of solutions expected are smooth functions that do not grow large as ${\displaystyle \vert x\vert \to \infty }$. More precisely, the following assumptions are made:

1. ${\displaystyle \mathbf {v} (x,t)\in C^{\infty }(\mathbb {R} ^{3}\times [0,\infty )),\qquad p(x,t)\in C^{\infty }(\mathbb {R} ^{3}\times [0,\infty ))}$
2. There exists a constant ${\displaystyle E\in (0,\infty )}$ such that ${\displaystyle \int _{\mathbb {R} ^{3}}\vert \mathbf {v} (x,t)\vert ^{2}\,dx<E}$ for all ${\displaystyle t\geq 0\,.}$

Condition 1 implies that the functions are smooth and globally defined (i.e. infinitely differentiable) and condition 2 means that the kinetic energy of the solution is globally bounded.

There are multiple different versions of the Navier–Stokes existence and smoothness problem on Euclidean space, depending on what kind of external force is used in the statement of the problems. Examples include a smooth external force, a real analytic external force,^[8] or zero external force.

Existence and smoothness of the Navier–Stokes solutions in ${\displaystyle \mathbb {R} ^{3}}$

For any initial condition ${\displaystyle \mathbf {v} _{0}(x)}$ and external force ${\displaystyle \mathbf {f} (x,t)}$ with certain characteristics, such as smoothness, analyticity, or zero, satisfying the above hypotheses on ${\displaystyle \mathbb {R} ^{3}}$ there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector ${\displaystyle \mathbf {v} (x,t)}$ and a pressure ${\displaystyle p(x,t)}$ satisfying conditions 1 and 2 above.

Breakdown of the Navier–Stokes solutions in ${\displaystyle \mathbb {R} ^{3}}$

There exists an initial condition ${\displaystyle \mathbf {v} _{0}(x)}$ and an external force ${\displaystyle \mathbf {f} (x,t)}$ with certain characteristics, such as smoothness, analyticity, or zero, on ${\displaystyle \mathbb {R} ^{3}}$ such that there exists no solutions ${\displaystyle \mathbf {v} (x,t)}$ and ${\displaystyle p(x,t)}$ satisfying conditions 1 and 2 above.

On the torus

For the case of Navier–Stokes existence and smoothness problem on the torus ${\displaystyle \mathbb {T} ^{3}}$, the functions sought now are periodic in the space variables of period 1. More precisely, let ${\displaystyle e_{i}}$ be the unitary vector in the i- direction:

${\displaystyle e_{1}=(1,0,0)\,,\qquad e_{2}=(0,1,0)\,,\qquad e_{3}=(0,0,1)}$

Then ${\displaystyle \mathbf {v} (x,t)}$ is periodic in the space variables if for any ${\displaystyle i=1,2,3}$, then:

${\displaystyle \mathbf {v} (x+e_{i},t)=\mathbf {v} (x,t){\text{ for all }}(x,t)\in \mathbb {R} ^{3}\times [0,\infty ).}$

Notice that this is considering the coordinates mod 1. This allows working not on the whole space ${\displaystyle \mathbb {R} ^{3}}$ but on the quotient space ${\displaystyle \mathbb {R} ^{3}/\mathbb {Z} ^{3}}$, which turns out to be the 3-dimensional torus:

${\displaystyle \mathbb {T} ^{3}=\{(\theta _{1},\theta _{2},\theta _{3}):0\leq \theta _{i}<1\,,\quad i=1,2,3\}.}$

Now the hypotheses can be stated properly. The initial condition ${\displaystyle \mathbf {v} _{0}(x)}$ is assumed to be a smooth and divergence-free function and the external force ${\displaystyle \mathbf {f} (x,t)}$ is assumed to be a smooth function as well. The type of solutions that are physically relevant are those who satisfy these conditions:

3. ${\displaystyle \mathbf {v} (x,t)\in C^{\infty }(\mathbb {T} ^{3}\times [0,\infty )),\qquad p(x,t)\in C^{\infty }(\mathbb {T} ^{3}\times [0,\infty ))}$
4. There exists a constant ${\displaystyle E\in (0,\infty )}$ such that ${\displaystyle \int _{\mathbb {T} ^{3}}\vert \mathbf {v} (x,t)\vert ^{2}\,dx<E}$ for all ${\displaystyle t\geq 0\,.}$

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.

Similarly to the problems on Euclidean space, there are multiple different versions of the Navier–Stokes existence and smoothness problem on the torus, depending on what kind of external force is used in the statement of the problems. Examples include a smooth external force, a real analytic external force, or zero external force.

Existence and smoothness of the Navier–Stokes solutions in ${\displaystyle \mathbb {T} ^{3}}$

For any initial condition ${\displaystyle \mathbf {v} _{0}(x)}$ and external force ${\displaystyle \mathbf {f} (x,t)}$ with certain characteristics, such as smoothness, analyticity, or zero, satisfying the above hypotheses on ${\displaystyle \mathbb {T} ^{3}}$ there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector ${\displaystyle \mathbf {v} (x,t)}$ and a pressure ${\displaystyle p(x,t)}$ satisfying conditions 3 and 4 above.

Breakdown of the Navier–Stokes solutions in ${\displaystyle \mathbb {T} ^{3}}$

There exists an initial condition ${\displaystyle \mathbf {v} _{0}(x)}$ and an external force ${\displaystyle \mathbf {f} (x,t)}$ with certain characteristics, such as smoothness, analyticity, or zero, on ${\displaystyle \mathbb {T} ^{3}}$ such that there exists no solutions ${\displaystyle \mathbf {v} (x,t)}$ and ${\displaystyle p(x,t)}$ satisfying conditions 3 and 4 above.

Summarize Timeline Fact Check

The Millennium Prize problems were chosen by the Clay Mathematics Institute as the most important unsolved problems in mathematics. One of them concerns the Navier-Stokes equations and requires the prizewinner to have proven one of four statements. The first statement (A), which is known as the "smoothness" problem, states that there should always exist smooth and globally defined solutions to the Navier–Stokes equations in three-dimensional space. The second statement (C), known as the "breakdown" problem, states that there should be at least one set of initial conditions and external forces for which there are no smooth solutions to the Navier–Stokes equations. The other two statements are the equivalent statements on the torus, see statements (B) and (D).

There are four different statements for the Millennium Prize's Navier–Stokes existence and smoothness problem, two dealing with proving that smooth solutions always exist in the Navier-Stokes equations, and two dealing with finding a breakdown in the Navier-Stokes equations. There are different requirements for proving smooth solutions always exist for the Navier-Stokes equations and proving breakdown of the Navier-Stokes equations: A proof that smooth solutions always exist for the Navier-Stokes equations, given in statements (A) and (B), requires zero external force, but proving breakdown of the Navier-Stokes equations, given in statements (C) and (D), does require a smooth external force. Thus, statements (A) is not the negation of statement (C), and likewise statement (B) is not the negation of statement (D). Instead, we only have the following implications: ${\displaystyle \neg (A)\implies (C)}$, ${\displaystyle \neg (C)\implies (A)}$, ${\displaystyle \neg (B)\implies (D)}$, ${\displaystyle \neg (D)\implies (B)}$, and none of the reverse implications. Solving the Millennium Prize's Navier–Stokes existence and smoothness problem only requires solving one of statements (A), (B), (C), (D), which means that the Millennium Prize's Navier–Stokes existence and smoothness problem may be considered solved while one or more of the four statements (A), (B), (C), (D) may still remain unsolved.

In addition, there are other statements of the Navier–Stokes existence and smoothness problem unrelated to the Millennium Prize, such as the Navier–Stokes existence and smoothness problem with a real analytic external force.^[8]

Summarize Timeline Top Qs Fact Check

In 1934, Jean Leray proved that there are smooth and globally defined solutions to the Navier–Stokes equations under the assumption that the initial velocity ${\displaystyle \mathbf {v} _{0}(x)}$ is sufficiently small.^[1] He also proved the existence of so-called weak solutions to the Navier–Stokes equations, which may not satisfy the equations pointwise but do satisfy them in mean value.^[9]

In the 1960s, the finite difference method was proven to be convergent for the Navier–Stokes equations and the equations were numerically solved. It was also proven that there are smooth and globally defined solutions to the Navier–Stokes equations in two dimensions.^[10]

It is known that given an initial velocity ${\displaystyle \mathbf {v} _{0}(x)}$ there exists a finite "blowup time" T, depending on ${\displaystyle \mathbf {v} _{0}(x)}$, such that the Navier–Stokes equations on ${\displaystyle \mathbb {R} ^{3}\times (0,T)}$ have smooth solutions ${\displaystyle \mathbf {v} (x,t)}$ and ${\displaystyle p(x,t)}$. These solutions may, however, not hold for values of ${\displaystyle t}$ beyond the blowup time.^[1]

2016 result by Terence Tao

In 2016, Terence Tao published a paper titled "Finite time blowup for an averaged three-dimensional Navier–Stokes equation", in which he formalizes the idea of a "supercriticality barrier" for the global regularity problem for the true Navier–Stokes equations, and said that his method of proof hints at a possible route to establishing blowup for the true equations.^[11]

Tao proved that for ${\displaystyle \nu >0}$, through rescaling of time normed to ${\displaystyle \nu =1}$ without loss of generality, and ${\displaystyle \mathbf {f} =\mathbf {0} }$, the existence of a smooth divergence-free (df) velocity ${\displaystyle \mathbf {v} }$ with given initial velocity ${\displaystyle \mathbf {v} (-,0)}$, which solves the Navier–Stokes equations, is equivalent to the existence of a "mild" solution ${\displaystyle \mathbf {v} \colon [0,\infty )\rightarrow H_{\mathrm {df} }^{10}(\mathbb {R} ^{3})}$. Let ${\displaystyle \Omega \subset \mathbb {R} ^{3}}$ be an open subset, then ${\displaystyle H^{k}(\Omega ):=W^{k,2}(\Omega )\subset L^{2}(\Omega )}$ is the Sobolev space of square-integrable functions, for which up to ${\displaystyle k}$ partial differentiations are also square-integrable. Through the restriction of the ${\displaystyle L^{2}}$-scalar product:^[12]

${\displaystyle \langle -,-\rangle \colon H^{k}(\Omega )\times H^{k}(\Omega )\rightarrow \mathbb {R} ,\langle u,v\rangle$ :=\int _{\Omega }u(\mathbf {x} )\cdot v(\mathbf {x} )\mathrm {d} \mathbf {x} }

it becomes a Hilbert space. There is a canonical inclusion ${\displaystyle C_{\mathrm {c} }^{\infty }(\Omega )\subset W^{k,p}(\Omega )}$ of smooth functions with compact support. Tao hence weakened the search for solutions with compact support to the larger space of the inclusion ${\displaystyle C_{\mathrm {c,df} }^{\infty }(\mathbb {R} ^{3})\subset H_{\mathrm {df} }^{10}(\mathbb {R} ^{3})}$. Let:

${\displaystyle B\colon H_{\mathrm {df} }^{10}(\mathbb {R} ^{3})\times H_{\mathrm {df} }^{10}(\mathbb {R} ^{3})\rightarrow L^{2}(\mathbb {R} ^{3})\subset H_{\mathrm {df} }^{10}(\mathbb {R} ^{3})^{*},(u,v)\mapsto -{\frac {1}{2}}L((u\cdot \nabla )v+(v\cdot \nabla )u)}$

be the Euler bilinear form with ${\displaystyle \langle B(u,u),u\rangle =0}$ for all ${\displaystyle u\in H_{\mathrm {df} }^{10}(\mathbb {R} ^{3})}$.^[13] ${\displaystyle L}$ is the Leray projection onto divergence-free vector fields, which in coordinates is given by ${\displaystyle Lu_{i}=u_{i}-\Delta ^{-1}\partial _{i}\partial _{j}u_{j}}$ (using the Einstein summation convention)^[13] with the inverted Laplace operator ${\displaystyle \Delta ^{-1}}$ being defined using the Fourier transform, hence ${\displaystyle {\widehat {Lu_{i}}}(\xi )={\widehat {u_{i}}}(\xi )-\xi ^{-2}\xi _{i}\xi _{j}{\widehat {u}}_{j}}$. With the normalized pressure ${\displaystyle p:=-\Delta ^{-1}\partial _{i}\partial _{j}(u_{i}u_{j})}$, one has ${\displaystyle B(u,u)=-(u\cdot \nabla )u-\nabla p}$ and the pressure is removed from the Navier–Stokes equations:^[14]

${\displaystyle {\frac {\partial u}{\partial t}}=\Delta u+B(u,u).}$

Tao now considered this equation with an averaged Euler bilinear form ${\displaystyle {\widetilde {B}}\colon H_{\mathrm {df} }^{10}(\mathbb {R} ^{3})\times H_{\mathrm {df} }^{10}(\mathbb {R} ^{3})\rightarrow H_{\mathrm {df} }^{10}(\mathbb {R} ^{3})^{*}}$ with weaker properties, leading to fulfilling more inequalities, and still ${\displaystyle \langle {\widetilde {B}}(u,u),u\rangle =0}$ for all ${\displaystyle u\in H_{\mathrm {df} }^{10}(\mathbb {R} ^{3})}$.^[14] In this case there is indeed an initial velocity ${\displaystyle \mathbf {v} (-,0)}$, for which no "mild" solution ${\displaystyle \mathbf {v} \colon [0,\infty )\rightarrow H_{\mathrm {df} }^{10}(\mathbb {R} ^{3})}$ exists.²⁰¹⁶

This shows that a successful method to solve Navier–Stokes existence and smoothness must either rely on a finer structure than the simplifications above or on a property of the Euler bilinear form ${\displaystyle B}$, which its averaged variant ${\displaystyle {\widetilde {B}}}$ does not have.

2026 claims of counter-example

Main article: Navier–Stokes priority controversy

On 8 September 2026, OpenAI stated that it had solved the Navier–Stokes Millennium Prize problem using an internal frontier large language model, demonstrating that there were initial conditions leading to finite time blowup.^[15] The company released a paper as well as a Lean formalization.^[2]

OpenAI stated its research on Millennium Prize problems had begun on 1 September, following later-disproved rumors that rival AI company Anthropic had solved two Millennium Prize problems. OpenAI researchers used a model they had begun training on 28 August, as a swarm of around 10,000 AI agents, partially orchestrated by the OpenAI Codex tool. The agents initially evaluated all six open Millennium Prize problems before being redirected to focus solely on Navier–Stokes after showing promising results. The swarm produced ~4.9 million messages and ~300 billion output tokens, with ~2.7 million messages and ~130 billion output tokens on Navier–Stokes specifically. The claimed solution was reached on 5 September after 88 hours of work.^[2] The New York Times estimated that the effort likely cost millions of dollars in computation and electricity costs.^[15]

Dispute with Alpöge and Buckmaster

Shortly before the announcement, Tristan Buckmaster (a New York University professor) and Levent Alpöge (a researcher at Anthropic) had reported advances in Euler equations, a related field.^[4] A priority dispute started immediately.^[16] Buckmaster said that OpenAI adopted their research methods to solve the problem and attempted to exclude Alpöge from a joint publication proposal. OpenAI denied this and stated that its model had independently solved the Euler problem with different methods.^[15][4]

Also on 8 September 2026, OpenAI researcher Sébastien Bubeck wrote that OpenAI had begun work on Navier–Stokes partially following vague tweets by Alpöge, and that he had attempted to coordinate the publication of "concurrent discoveries" with Alpöge.^[17]

Buckmaster suggested that his work, which made extensive use of Codex, could have become training data for the internal model that OpenAI began training on 28 August and used for the solution. OpenAI acknowledged but downplayed this, announcing:^[16]

While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models⁠. However, our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced).

Construction outline

The following is a rough outline of the construction given in the OpenAI paper. It does not give a direct expression for its solution, but rather constructs it implicitly. As of September 18, 2026, the counterexample to the proof has not yet been independently verified.^{[when?][citation needed]}. The construction is a smooth solution, which resembles a spinning top that gets increasingly thinner as well as shorter and creates a velocity singularity for ${\displaystyle t=1}$ in a neighborhood around the origin, meaning:

${\displaystyle \lim _{t\rightarrow 1^{-}}\|\mathbf {v} \|_{L^{\infty }(\mathbb {R} ^{3})}=\lim _{t\rightarrow 1^{-}}\max\{|\mathbf {v} (t,\mathbf {x} )|\,|\mathbf {x} \in \mathbb {R} ^{3}\}=\infty .}$

Nonetheless, for ${\displaystyle t=1}$ it is still required to fulfill ${\displaystyle \|\mathbf {v} \|_{L^{2}(\mathbb {R} ^{3})}<\infty }$ for finite kinetic energy. The basic description uses cylindrical coordinates ${\displaystyle (r,\theta ,z)}$ with the solution being independent of the azimuthal angle ${\displaystyle \theta }$. Let ${\displaystyle \ell _{r}}$ be the characteristic radius and ${\displaystyle \ell _{z}}$ be the characteristic height, then ${\displaystyle \ell _{r}\sim \tau ^{1/2}}$ and ${\displaystyle \ell _{z}\sim \tau ^{1/2-h}}$^[18] with the time ${\displaystyle \tau =1-t}$ to the singularity and a parameter ${\displaystyle 0<h<1/100}$, with ${\displaystyle \sim }$ meaning that the expressions can be estimated against each other with just constants. The spinning top gets thinner more rapidly than shorter with ${\displaystyle \ell _{r}/\ell _{z}\sim \tau ^{h}}$. Let ${\displaystyle |u_{r}^{(0)}|}$, ${\displaystyle |u_{\theta }^{(0)}|}$ and ${\displaystyle |u_{z}^{(0)}|}$ be the average velocities, then ${\displaystyle |u_{\theta }^{(0)}|,|u_{z}^{(0)}|\sim \tau ^{-1/2-h}}$ and ${\displaystyle |u_{r}^{(0)}|\in {\mathcal {O}}(\tau ^{-1/2})}$^[18] with Landau notation. For the characteristic volume ${\displaystyle V}$, the characteristic velocity square ${\displaystyle |u|^{2}}$ and the total kinetic energy ${\displaystyle T}$, one has:^[18]

${\displaystyle V\sim \ell _{r}^{2}\ell _{z}\sim \tau ^{3/2-h};}$

${\displaystyle |u|^{2}=|u_{r}^{(0)}|^{2}+|u_{\theta }^{(0)}|^{2}+|u_{z}^{(0)}|^{2}\sim \tau ^{-1-2h};}$

${\displaystyle T\sim |u|^{2}V\sim \tau ^{1/2-3h}.}$

For ${\displaystyle t\rightarrow 1^{-}}$, hence ${\displaystyle \tau \rightarrow 0^{+}}$, the velocities diverge, although the kinetic energy ${\displaystyle T}$ does not, with ${\displaystyle T\rightarrow 0^{+}}$. The further description then transforms the time ${\displaystyle t}$ and the cylinder coordinates ${\displaystyle (r,z)}$ implicitly onto new coordinates ${\displaystyle \eta \in (-1,+1)}$ and ${\displaystyle X,q>0}$, the velocities ${\displaystyle (u_{r}^{(0)},u_{\theta }^{(0)},u_{z}^{(0)})}$ onto new functions ${\displaystyle (V_{0},E,U)}$ and the pressure ${\displaystyle p}$ onto a new function ${\displaystyle \Pi }$ by:^[19]

${\displaystyle A:={\frac {1}{2}}+h,D:={\frac {1}{2}}-h;}$

${\displaystyle \tau =q(1-\eta ^{2}),z=q^{D}\eta ,X={\frac {r^{2}}{2q}};}$

${\displaystyle ru_{r}^{(0)}=V_{0},ru_{\theta }^{(0)}=q^{-A}E,ru_{z}^{(0)}=q^{-A}U,p=q^{-2A}\Pi .}$

These notations are then used throughout the paper, in particular its central theorem 4.6. Translating additional conditions is not straightforward due to implicit coordinate definitions, although some already fit together. For example, the incompressibility becomes ${\displaystyle \nabla \cdot u^{(0)}=\partial _{r}(ru_{r}^{(0)})+\partial _{z}(ru_{z}^{(0)})=0}$, in which ${\displaystyle V_{0}}$ and ${\displaystyle q^{-A}U}$ can be directly substituted. The derivations are also not given in detail, only the results for incompressibility and regularity on the axis as well as radial pressure balance and vanishing at infinity:^[19]

${\displaystyle V_{0}(0,\eta )=0;}$

${\displaystyle \Pi (X,\eta )=\int _{X}^{\infty }{\frac {E(x,\eta )}{2x}}\mathrm {d} x.}$

The construction uses ${\displaystyle \nu =1}$ without loss of generality, like the 2016 result by Terence Tao, and expresses the forcefield-free Navier–Stokes equations as ${\displaystyle {\mathcal {R}}(u,p)=0}$ using the residual:^[20]

${\displaystyle {\mathcal {R}}(u,p):={\frac {\partial u}{\partial t}}+(u\cdot \nabla )u-\Delta u+\nabla p.}$

It is now required to show that the residual can be extended smoothly through the singular time ${\displaystyle t=1}$, although the velocities, which appear in the former three terms, diverge on their own. The proof uses the usual strategy established for partial differential equations to show this by inequalities for the partial derivatives (with respect to both space and time).

For the construction, the not-yet-determined force field ${\displaystyle \mathbf {f} }$ then allows a reverse approach: Instead of finding a solution of the Navier–Stokes equations with diverging velocities, one can use smooth functions ${\displaystyle u}$ and ${\displaystyle p}$ on ${\displaystyle \mathbb {R} ^{3}\times [0,1)}$ and set ${\displaystyle \mathbf {f} ={\mathcal {R}}(u,p)}$ on ${\displaystyle \mathbb {R} ^{3}\times [0,1)}$ to make them into solutions automatically.^[21] Special functions ${\displaystyle u}$ and ${\displaystyle p}$ are constructed, which both have compact support in a compact subset ${\displaystyle {\mathcal {K}}\subset \mathbb {R} ^{3}}$. For every multi-index ${\displaystyle \alpha \in \mathbb {N} ^{3}}$ and every natural number ${\displaystyle j\in \mathbb {N} }$, one then has that ${\displaystyle \partial _{x}^{\alpha }\partial _{t}^{j}f}$ for ${\displaystyle t\rightarrow 1^{-}}$ converges uniformly on ${\displaystyle \mathbb {R} ^{3}}$ and there exist smooth functions ${\displaystyle F_{j}\in C_{\mathrm {c} }^{\infty }(\mathbb {R} ^{3},\mathbb {R} )}$ with compact support in ${\displaystyle {\mathcal {K}}}$ and:^[22]

${\displaystyle \lim _{t\rightarrow 1^{+}}\partial _{x}^{\alpha }\partial _{t}^{j}f(t,x)=\partial _{x}^{\alpha }F_{j}(x);}$

${\displaystyle \partial _{x}^{\alpha }F_{j}(0)=0.}$

This can be used to smoothly extend ${\displaystyle \mathbf {f} }$ onto ${\displaystyle \mathbb {R} ^{3}\times [0,\infty )}$ with compact support in ${\displaystyle {\mathcal {K}}\times [0,2]}$.^[23]

Inability to be extended to the unforced case

On September 17, 2026, Peter Constantin, Mihaela Ignatova, and Vlad Vicol released a preprint claiming that OpenAI's method does not extend to the Navier-Stokes existence and smoothness problem with a real analytic external force instead of merely a smooth force.^[8] This implies that OpenAI's method does not extend to statements (A) and (B) of the Millennium Prize's Navier-Stokes existence and smoothness problem,^[24] since a zero external force is real analytic.

The Navier–Stokes problem features in The Mathematician's Shiva (2014), a book telling the fictional story of Rachela Karnokovitch, a prestigious mathematician who takes the proof to her grave in protest of academia.^[25][26] The movie Gifted (2017) referenced the Millennium Prize problems, including the Navier–Stokes problem.^[27]

• List of unsolved problems in mathematics
• List of unsolved problems in physics