Navier–Stokes existence and smoothness

The Navier–Stokes equations are one of the pillars of fluid mechanics. 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 in May 2000 a US$1,000,000 prize, not to whoever 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:

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

Main article: 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 using continuum mechanics. The equations are a statement of the second law of Newton, 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, we will consider only that case.

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

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

where $\nu >0$ is the kinematic viscosity, $\mathbf {f} ({\boldsymbol {x}},t)$ the external force, $\nabla$ is the gradient operator and $\displaystyle \Delta$ is the Laplacian operator, which is also denoted by $\nabla \cdot \nabla$. Note that this is a vector equation, i.e. it has three scalar equations. If we write down the coordinates of the velocity and the external force

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

then for each $i=1,2,3$ we have the corresponding scalar Navier–Stokes equation:

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

The unknowns are the velocity $\mathbf {v} ({\boldsymbol {x}},t)$ and the pressure $p({\boldsymbol {x}},t)$. Since in three dimensions we have three equations and four unknowns (three scalar velocities and the pressure), we need a supplementary equation. This extra equation is the continuity equation describing the incompressibility of the fluid:

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

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.

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 $\mathbb {R} ^{3}$, 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 we are no longer working on the whole space $\mathbb {R} ^{3}$ but in the 3-dimensional torus $\mathbb {T} ^{3}=\mathbb {R} ^{3}/\mathbb {Z} ^{3}$. We will treat each case separately.

Statement of the problem in the whole space

Hypotheses and growth conditions

The initial condition $\mathbf {v} _{0}(x)$ is assumed to be a smooth and divergence-free function (see smooth function) such that, for every multi-index $\alpha$ (see multi-index notation) and any $K>0$, there exists a constant $C=C(\alpha ,K)>0$ (i.e. this "constant" depends on $\alpha$ and K) such that

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

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

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

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

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

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 million-dollar-prize conjectures in the whole space

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

Let $\mathbf {f} (x,t)\equiv 0$. For any initial condition $\mathbf {v} _{0}(x)$ satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector $\mathbf {v} (x,t)$ and a pressure $p(x,t)$ satisfying conditions 1 and 2 above.

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

There exists an initial condition $\mathbf {v} _{0}(x)$ and an external force $\mathbf {f} (x,t)$ such that there exists no solutions $\mathbf {v} (x,t)$ and $p(x,t)$ satisfying conditions 1 and 2 above.

Statement of the periodic problem

Hypotheses

The functions we seek now are periodic in the space variables of period 1. More precisely, let $e_{i}$ be the unitary vector in the j- direction:

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

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

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

Notice that we are considering the coordinates mod 1. This allows us to work not on the whole space $\mathbb {R} ^{3}$ but on the quotient space $\mathbb {R} ^{3}/\mathbb {Z} ^{3}$, which turns out to be the 3-dimensional torus

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

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

3. $\mathbf {v} (x,t)\in \left[C^{\infty }(\mathbb {T} ^{3}\times [0,\infty ))\right]^{3}\,,\qquad p(x,t)\in C^{\infty }(\mathbb {T} ^{3}\times [0,\infty ))$

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

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 million-dollar-prize theorems

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

Let $\mathbf {f} (x,t)\equiv 0$. For any initial condition $\mathbf {v} _{0}(x)$ satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector $\mathbf {v} (x,t)$ and a pressure $p(x,t)$ satisfying conditions 3 and 4 above.

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

There exists an initial condition $\mathbf {v} _{0}(x)$ and an external force $\mathbf {f} (x,t)$ such that there exists no solutions $\mathbf {v} (x,t)$ and $p(x,t)$ 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 $\mathbf {v} (x,t)$ 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 $\mathbf {v} _{0}(x)$ there exists a finite time T, depending on $\mathbf {v} _{0}(x)$ such that the Navier–Stokes equations on $\mathbb {R} ^{3}\times (0,T)$ have smooth solutions $\mathbf {v} (x,t)$ and $p(x,t)$. It is not known if the solutions exist beyond that "blowup time" T.
4. The mathematician 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.

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 Tao.