Navier-Stokes existence and smoothness

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 (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 conditions, 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, 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:

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

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

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

\(\mathbf{v}(\boldsymbol{x},t)=\big(\,v_1(\boldsymbol{x},t),\,v_2(\boldsymbol{x},t),\,v_3(\boldsymbol{x},t)\,\big)\,,\qquad \mathbf{f}(\boldsymbol{x},t)=\big(\,f_1(\boldsymbol{x},t),\,f_2(\boldsymbol{x},t),\,f_3(\boldsymbol{x},t)\,\big)\)

then for each \(i=1,2,3\) there is the corresponding scalar Navier-Stokes equation:

\(\frac{\partial v_i}{\partial t} +\sum_{j=1}^{3}\frac{\partial v_i}{\partial x_j}v_j= -\frac{1}{\rho}\frac{\partial p}{\partial x_i} + \nu\sum_{j=1}^{3}\frac{\partial^2 v_i}{\partial x_j^2} +f_i(\boldsymbol{x},t).\)

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

\(\nabla\cdot \mathbf{v} = 0.\)

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.

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

\(\nabla\cdot \mathbf{v} = 0\)

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

\(\nabla\cdot \mathbf{v} = 0\)

Condensed: the full section is in Wikipedia.

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 they are no longer working on the whole space \(\mathbb{R}^3\) but in the 3-dimensional torus \(\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3\). Each case will be treated separately.

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 \(C=C(\alpha,K)>0\) such that

\(\vert \partial^\alpha \mathbf{v_0}(x)\vert\le \frac{C}{(1+\vert x\vert)^K}\qquad\) 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):

\(\vert \partial^\alpha \mathbf{f}(x,t)\vert\le \frac{C}{(1+\vert x\vert + t)^K}\qquad\) 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. \(\mathbf{v}(x,t)\in C^\infty(\mathbb{R}^3\times[0,\infty)),\qquad p(x,t)\in C^\infty(\mathbb{R}^3\times[0,\infty))\)
2. There exists a constant \(E\in (0,\infty)\) such that \(\int_{\mathbb{R}^3} \vert \mathbf{v}(x,t)\vert^2 \, dx

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.

(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.

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 (B), 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 points (C) and (D) below.

The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluids. They are given by:

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

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

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

\(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0\)

Condensed: the full section is in Wikipedia.

The functions sought now are periodic in the space variables of period 1. More precisely, let \(e_i\) be the unitary vector in the i- 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 \(i=1,2,3\), then:

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

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

\(\mathbb{T}^3=\{(\theta_1,\theta_2,\theta_3): 0\le \theta_i<1\,,\quad i=1,2,3\}.\)

Now the hypotheses can be stated 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:

1. \(\mathbf{v}(x,t)\in C^\infty(\mathbb{T}^3\times[0,\infty)),\qquad p(x,t)\in C^\infty(\mathbb{T}^3\times[0,\infty))\)
2. There exists a constant \(E\in (0,\infty)\) such that \(\int_{\mathbb{T}^3} \vert \mathbf{v}(x,t)\vert^2 \, dx

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.

(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.

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 \(\mathbf{v}_0(x)\) is sufficiently small. 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.

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 2 dimensions.

It is known that given an initial velocity \(\mathbf{v}_0(x)\) there exists a finite "blowup 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 \(p(x,t)\). These solutions may, however, not hold for values of \(t\) beyond the blowup time.

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 claims that his method of proof hints at a possible route to establishing blowup for the true equations.