Francais | English | Espanõl

From Wikipedia, the free encyclopedia

Millennium Prize Problems
P versus NP
The Hodge conjecture
The Poincaré conjecture
The Riemann hypothesis
Yang–Mills existence and mass gap
Navier-Stokes existence and smoothness
The Birch and Swinnerton-Dyer conjecture

A $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute to whoever proves the following statement about the Navier-Stokes equations. These equations describe the flow of nearly all practical fluids, but can be extremely complicated and difficult to solve.

Contents 1 Problem description 1.1 Existence and smoothness of Navier-Stokes solutions on <math>\mathbb{R}^3</math> 1.2 Existence and smoothness of Navier-Stokes solutions on <math>\mathbb{R}^3/\mathbb{Z}^3</math> 1.3 Breakdown of Navier-Stokes solutions on <math>\mathbb{R}^3</math> 1.4 Breakdown of Navier-Stokes solutions on <math>\mathbb{R}^3/\mathbb{Z}^3</math> 2 Background 3 External links

[edit] Problem description

Let <math>u(x, t) = (u_i(x, t))_{1 \le i \le 3} \mathcal{2} \mathbb{R}^3</math> be the unknown velocity vector field, defined for positions <math>x \mathcal{2} \mathbb{R}^3</math> and times <math>t \ge 0</math> and let <math>p(x, t) \mathcal{2} \mathbb{R}</math> be the unknown pressure, defined likewise.

Let <math>f(x, t) = (f_i(x, t))_{1 \le i \le 3} \mathcal{2} \mathbb{R}^3</math> be a known external force, again defined for positions <math>x \mathcal{2} \mathbb{R}^3</math> and times <math>t \ge 0</math>.

Also let <math>u^\circ(x)</math> be the known initial velocity vector field on <math>\mathbb{R}^3</math>, which is divergence-free on C^∞.

Finally, let <math>\nu > 0</math> be a known constant (the viscosity).

Then the Navier-Stokes equations for incompressible viscous fluids filling <math>\mathbb{R}^3</math> are given by <math>\forall i \mathcal{2} {1, 2, 3}:</math>

<math>\frac{\partial u_i}{\partial t} + \sum_{j=1}^3 u_j \frac{\partial u_i}{\partial x_j} = \nu \Delta u_i - \frac{\partial p}{\partial x_i} + f_i(x, t)</math>	<math>(x \mathcal{2} \mathbb{R}^3, t \ge 0)</math>	(1)
<math>\operatorname{div}\ u = \sum_{i=1}^3 \frac{\partial u_i}{\partial x_i} = 0</math>	<math>(x \mathcal{2} \mathbb{R}^3, t \ge 0)</math>	(2)

and the initial condition:

<math>u(x,0) = u^\circ(x)</math>

<math>(x \mathcal{2} \mathbb{R}^3)</math>

(3)

The problem then is to prove one of the following four statements:

[edit] Existence and smoothness of Navier-Stokes solutions on <math>\mathbb{R}^3</math>

Assume in addition that:

• There are no external forces, i.e.:

<math>( \forall x \mathcal{2} \mathbb{R}^3 )( \forall t \ge 0 )\ f(x, t) = 0</math>

• <math>u^\circ</math> is bounded, i.e.:

<math>( \forall \alpha \mathcal{2} \mathbb{R} )( \forall K \mathcal{2} \mathbb{R} )( \exists C \mathcal{2} \mathbb{R} )( \forall x \mathcal{2} \mathbb{R}^3 )( \forall t \ge 0 )\ | \partial_x^\alpha u^\circ(x) | \le C(1 + |x|)^{-K}</math>

Then there exists <math>p \mathcal{2} C^\infty(\mathbb{R}^3 \times [0,\infty))</math> and <math>u \mathcal{2} (C^\infty(\mathbb{R}^3 \times [0,\infty)))^3</math> that satisfy (1), (2) and (3) as well as having bounded energy, i.e.:

<math>( \exists C \mathcal{2} \mathbb{R} )( \forall t \ge 0 )\ \int_{\mathbb{R}^3} |u(x, t)|^2 dx < C</math>

[edit] Existence and smoothness of Navier-Stokes solutions on <math>\mathbb{R}^3/\mathbb{Z}^3</math>

Assume in addition that:

• There are no external forces, i.e.:

<math>( \forall x \mathcal{2} \mathbb{R}^3 )( \forall t \ge 0 )\ f(x, t) = 0</math>

• <math>u^\circ</math> is periodic, i.e.:

<math>(\forall j \mathcal{2} {1, 2, 3})( \forall x \mathcal{2} \mathbb{R}^3 )\ u^\circ(x + e_j) = u^\circ(x)</math>, where <math>e_j</math> is the j^th unit vector in <math>\mathbb{R}^3</math>.

Then there exists <math>p \mathcal{2} C^\infty(\mathbb{R}^3 \times [0,\infty))</math> and <math>u \mathcal{2} (C^\infty(\mathbb{R}^3 \times [0,\infty)))^3</math> that satisfy (1), (2) and (3) and have a periodic u, i.e.:

<math>( \forall x \mathcal{2} \mathbb{R}^3 )( \forall t \ge 0 )\ u(x, t) = u(x + e_j, t)</math>

[edit] Breakdown of Navier-Stokes solutions on <math>\mathbb{R}^3</math>

There exists an <math>f \mathcal{2} (C^\infty(\mathbb{R}^3))^3</math> and a divergence-free <math>u^\circ \mathcal{2} (C^\infty(\mathbb{R}^3))^3</math> for which there are no <math>p \mathcal{2} C^\infty(\mathbb{R}^3 \times [0,\infty))</math> and <math>u \mathcal{2} (C^\infty(\mathbb{R}^3 \times [0,\infty)))^3</math> satisfying (1), (2), (3) and also having bounded energy, i.e.:

<math>( \exists C \mathcal{2} \mathbb{R} )( \forall t \ge 0 )\ \int_{\mathbb{R}^3} |u(x, t)|^2 dx < C</math>

[edit] Breakdown of Navier-Stokes solutions on <math>\mathbb{R}^3/\mathbb{Z}^3</math>

There exists an <math>f \mathcal{2} (C^\infty(\mathbb{R}^3))^3</math> and a divergence-free <math>u^\circ \mathcal{2} (C^\infty(\mathbb{R}^3))^3</math> for which there are no <math>p \mathcal{2} C^\infty(\mathbb{R}^3 \times [0,\infty))</math> and <math>u \mathcal{2} (C^\infty(\mathbb{R}^3 \times [0,\infty)))^3</math> satisfying (1), (2), (3) and also having a periodic u, i.e.:

<math>( \forall x \mathcal{2} \mathbb{R}^3 )( \forall t \ge 0 )\ u(x, t) = u(x + e_j, t)</math>

[edit] Background

The analogous problem for <math>\mathbb{R}^2</math> has already been solved positively (it is known that there are smooth solutions on <math>\mathbb{R}^2</math>).

[edit] External links

• The Clay Mathematics Institute's Navier-Stokes equation prize
  • Official statement of the problem
• QEDen Millennium Prize Problems Wiki

---

This article contains public-domain material taken from QEDen.

General subfields within physics v • d • e</div>

Classical mechanics | Electromagnetism | Thermodynamics | General relativity | Quantum mechanics

Particle physics | Condensed matter physics | Atomic, molecular, and optical physics