Contents

This is part 1 of a series of posts on the Navier Stokes Millennium Prize Problem in which I endeavour to express the problem statement in my own words (IMOW) with some help from ChatGPT.

Here is the list of all the posts in this series:

See references for links to the original problem statement.

Introduction

The problem is concerned with solutions for the Navier Stokes equations in $\mathbb{R}_3$ for a viscous incompressible fluid.

  • incompressible: $\rho$, the fluid density is constant
  • viscous: $\mu$, the fluid viscosity is non-zero and thus $\nu = \mu / \rho > 0$

For such a fluid the Navier-Stokes equations are given by

\(\frac{\partial u_i}{\partial t} + \sum_{j=1}^n u_j \frac{\partial u_i}{\partial x_j} = \nu \Delta u_i - \frac{\partial p}{\partial x_i} + f_i(x, t) \quad (x \in \mathbb{R}^n, t \geq 0) \quad (1)\)

\(\text{div} \, u = \sum_{i=1}^n \frac{\partial u_i}{\partial x_i} = 0 \quad (x \in \mathbb{R}^n, t \geq 0) \quad (2)\)

with initial conditions

\(u(x, 0) = u^\circ(x) \quad (x \in \mathbb{R}^n) \quad (3)\)

  • Equation 1: Newton’s 2nd law for a fluid element subject to the following forces
    • External force per unit mass $f = \sum_i f_i(x, t)\mathbf{e}_i$ e.g. gravity where $f = -g\mathbf{e}_z$
    • Pressure forces
    • Friction forces, due to viscosity. Viscosity is the measure of a fluid’s resistance to flow. It describes the internal friction of a moving fluid, where a higher viscosity implies a greater resistance to fluid deformation.
  • Equation 2: conservation of mass for an incompressible fluid.
  • Equation 3: $u^\circ(x)$ is a given $C^\infty$ divergence-free vector field on $\mathbb{R}^n$

In this context, $C^\infty$ refers to a differentiable function whose derivatives of all orders exist and are continuous. $C^\infty$ is also often referred to as a smooth function.

The Euler equations are the NS-equations $(1), (2), (3)$ for an inviscid fluid i.e. $\mu = 0 \implies \nu = 0$.

NOTE: The pressure gradient term in 1 is usually written as $\frac{1}{\rho}\frac{\partial p}{\partial x_i}$. This allows for a potentially non-constant density. However the form used above would be consistent with a constant density $\rho$ if $p$ is interpreted as pressure per unit density. I am a bit confused as I have not seen pressure written this way before. However it does not seem to be a typo as there is no mention of the equation in the errata.

Physical reasonableness

For a solution to be accepted it must be physically reasonable. In particular, the magnitude of the velocity must be bounded as $\lvert x \rvert \rightarrow \infty$.

The initial velocity $u^\circ(x)$ should satisfy the following inequality

\(\left\vert\partial_x^{\alpha} u^{\circ}\right\vert \leq C_{\alpha K} (1 + \left\vert x \right\vert)^{-K} \quad \text{on} \ \mathbb{R}^n \quad \text{for any} \ \alpha \ \text{and} \ K \quad (4)\)

What this means:

  • The magnitude of the spatial partial derivative of the $u^\circ$ to any order $\alpha$, $\left\vert\frac{\partial^{\alpha} u^{o}}{\partial x}\right\vert$ should be bounded as $\lvert x \rvert \rightarrow \infty$.
  • The inequality should hold for any rate of decrease $K$ of the right hand side.
  • The upper bound should decrease as $\lvert x \rvert$ gets larger.

A similar requirement exists for the spatial and time partial derivatives of the external force $f$

\(\left\vert\partial_x^{\alpha} \partial_t^m f(x, t) \right\vert \leq C_{\alpha m K} (1 + \left\vert x \right\vert + t)^{-K} \quad \text{on} \ \mathbb{R}^n \times [0, \infty) \quad \text{for any} \ \alpha,\ m,\ K \quad \text{(5)}\)

$x$ is a vector in $\mathbb{R}^3$, hence the magnitude $\left\vert x \right\vert$ is used in the two inequalities. $t$ is a non-negative scalar so its raw value is used in the second inequality.

Moreover a solution is accepted as physically reasonable only if it satisfies

\(p, u \in C^\infty(\mathbb{R}^n \times [0, \infty))\quad (6)\)

What this means:

  • $p, u$ infinitesimally smooth meaning they are differentiable an infinite number of times over the domain which is $\mathbb{R}^n \times [0, \infty)$
  • The condition ensures that there are no discontinuities which would lead to numerical problems or physical impossibilities in calculation.

Moreover the kinetic energy must be bounded for all time

\(\int_{\mathbb{R}^n} |u(x, t)|^2 \, dx < C\quad \text{for all} \ t\geq 0 \quad (7)\)

Existence (Statement A of the paper)

Let $u^\circ(x)$ be an initial velocity vector that satisfies (4).

The task is to prove that for any such $u^\circ(x)$, there exists a velocity function $u(x, t)$ and a pressure function $p(x, t)$ which solve the Navier Stokes equations (1), (2), (3) and satisfy (6), (7)

For the purposes of this task it is assumed that there are no external forces acting on the fluid, $f(x, t) = 0$.

This is in order to focus on intrinsic fluid behavior derived from its pressure and velocity in response to initial conditions, without the overcomplications of additional external forces.

Breakdown (Statement C of the paper)

There exist

  • a smooth and divergence-free initial velocity vector field $u^\circ(x)$ on $\mathbb{R}^3$ that satisfies (4).
  • a smooth externally applied force $f(x, t)$ on $\mathbb{R}^3 \times [0, \infty)$ that satisfies (5)

for which you cannot find any velocity and pressure functions $u(x, t)$ and $p(x, t)$ which solve the Navier Stokes equations (1), (2), (3) and satisfy (6), (7).

Periodic solutions (Statements B, D of the paper)

There are a couple of variations of these two problems that cover existence (statement B) and breakdown (statement D) with periodic solutions.

The physical reasonableness conditions for the initial velocity $u^\circ$ and the external force $f$ (Equations 4, 5), become spatial periodicity conditions

\(u^\circ\left(x + e_j\right) = u^\circ(x), \quad f\left(x + e_j, t\right) = f(x, t) \quad \text{for} \ 1 \leq j \leq n \quad (8)\)

where $e_j$ represents the unit vector in the direction of the $j$-th dimension. In other words, $e_j$ is a vector in $\mathbb{R}^n$ that has a magnitude (or length) of 1 and points in direction $j$.

The equations simply express that the initial velocity $u^\circ$ and the external force $f$ must be the same when you move a unit step $e_j$ in any spatial direction. This means that both $u^\circ$ and $f$ are repeating after each unit step in space, hence they are periodic in all spatial dimensions.

There is also a condition similar to (5) but which only requires that $f$ be bounded in time

\(\left\vert\partial_x^{\alpha} \partial_t^m f(x, t) \right\vert \leq C_{\alpha m K} (1 + \left\vert t \right\vert)^{-K} \quad \text{on} \ \mathbb{R}^n \times [0, \infty) \quad \text{for any} \ \alpha,\ m,\ K \quad (9)\)

As for the physical reasonable conditions for the solutions $p, u$, Equation 6 continues to apply (although for some reason the problem statement lists it again and subsequently refers to it as Equation 11).

However instead of the bounded kinetic energy requirement for the velocity solution $u(x, t)$, (Equation 7) there is a periodicity condition

\(u(x, t) = u\left(x + e_j, t\right) \quad \text{on} \ \mathbb{R}^3 \times [0, \infty) \quad \text{for} \ 1 \leq j \leq n \quad (10)\)

Due to the inherent boundedness in space brought about by periodicity, Equation 4 becomes unnecessary and unlike Equation 3, there is no spatial component (i.e. $\lvert x \rvert$) in Equation 9. A periodic function repeats its values at regular intervals, which means that it does not become arbitrarily large in any spatial direction, thereby automatically bounding the magnitude of any spatial derivative. This spatial boundedness negates the need for concern over extreme spatial cases. However, time, being unbounded and continuously evolving, needs to be considered as part of Equation 9.

Not entirely sure though why $\lvert t \rvert$ rather than $t$ in Equation 9 given that the Navier Stokes equations apply only for $t \geq 0$. Here is what ChatGPT had to say:

The reason for using $|t|$ instead of $t$ in the equation is related primarily to mathematical formality in expressing absolute bounds. Even though in the context of the Navier-Stokes equations, $t$ is non-negative, absolute value is often used to represent strict positivity in mathematics and hence, to denote that a quantity is always non-negative. This notation helps to emphasize the non-negativity of $t$, thereby highlighting the direction of time as forward moving. Even though $t$ yields the same number, using $|t|$ can hence help enhance clarity in mathematical inequalities.

Existence and breakdown for periodic solutions

The existence and breakdown statements now concern solutions on $\mathbb{R}^3/\mathbb{Z}^3$ instead of $\mathbb{R}^3$. In this case, $\mathbb{R}^3/\mathbb{Z}^3$ instead of $\mathbb{R}^3$ is used to denote a torus, or a physically equivalent situation where the fluid is considered in a periodic box. This implies that the fluid in question is in a closed loop or in a periodic condition, repeating itself after every unit step, and allowing the study of the periodic solutions.

The statements are virtually identical except for the equations to be satisfied by the initial conditions, external forces and solutions

  • Initial condition $u^\circ(x)$ must satisfy (8) instead of (4).
  • External force (non-zero for breakdown only) must satisfy (9) instead of (5).
  • Solutions $u(x, t), p(x, t)$ must satisfy (10) instead of (7).

References