Navier–Stokes existence and smoothness

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Flow visualization of a turbulent jet, made by laser-induced fluorescence. The jet exhibits a wide range of length scales, an important characteristic of turbulent flows.

 

The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid 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 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. 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 ${\displaystyle \mathbf {v} ({\boldsymbol {x}},t)}$ be a 3-dimensional vector field, the velocity of the fluid, and let ${\displaystyle p({\boldsymbol {x}},t)}$ be the pressure of the fluid. 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 volumetric force, ${\displaystyle \nabla }$ 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. 

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 ${\displaystyle \mathbb {R} ^{3}}$, 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 the whole space ${\displaystyle \mathbb {R} ^{3}}$ but in the 3-dimensional torus ${\displaystyle \mathbb {T} ^{3}=\mathbb {R} ^{3}/\mathbb {Z} ^{3}}$. Each case will be treated separately. 

Statement of the problem in the whole space

Hypotheses and growth conditions

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

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 ${\displaystyle \mathbb {R} ^{3}}$

Let ${\displaystyle \mathbf {f} (x,t)\equiv 0}$. For any initial condition ${\displaystyle \mathbf {v} _{0}(x)}$ satisfying the above hypotheses 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.

(B) 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 ${\displaystyle \mathbf {f} (x,t)}$ such that there exists no solutions ${\displaystyle \mathbf {v} (x,t)}$ and ${\displaystyle p(x,t)}$ satisfying conditions 1 and 2 above.

Statement of the periodic problem

Hypotheses

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:

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

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

The periodic Millennium Prize theorems

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

Let ${\displaystyle \mathbf {f} (x,t)\equiv 0}$. For any initial condition ${\displaystyle \mathbf {v} _{0}(x)}$ satisfying the above hypotheses 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.

(D) 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 ${\displaystyle \mathbf {f} (x,t)}$ such that there exists no solutions ${\displaystyle \mathbf {v} (x,t)}$ and ${\displaystyle p(x,t)}$ satisfying conditions 3 and 4 above. 

Partial results

1. The Navier–Stokes problem in two dimensions has already been solved positively since the 1960s: there exist smooth and globally defined solutions.
2. If the initial velocity ${\displaystyle \mathbf {v} (x,t)}$ is sufficiently small then the statement is true: there are smooth and globally defined solutions to the Navier–Stokes equations.
3. Given an initial velocity ${\displaystyle \mathbf {v} _{0}(x)}$ there exists a finite 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 ${\displaystyle p(x,t)}$. It is not known if the solutions exist beyond that "blowup time" T.
4. 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.
5. 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.
6. Tristan Buckmaster and Vlad Vicol showed in 2018 that very weak solutions (weaker than Leray's above) are not unique in the class of finite energy solutions 

In popular culture

Unsolved problems have been used to indicate a rare mathematical talent in fiction. The Navier-Stokes problem features in The Mathematician's Shiva (2014), a book about a prestigious, deceased, fictional mathematician named Rachela Karnokovitch taking the proof to her grave in protest of academia. The movie Gifted (2017) referenced the Millennium Prize problems and dealt with the potential for a 7-year-old girl and her deceased mathematician mother for solving the Navier–Stokes problem.

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