Navier-Stokes existence and smoothness

From Wikipedia, the free encyclopedia

This article or section is in need of attention from an expert on the subject.
Please help recruit one or improve this article yourself. See the talk page for details.
Please consider using {{Expert-subject}} to associate this request with a WikiProject

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

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

1 Problem description
2 Background
3 External links

[edit] Problem description

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

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

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

Finally, let $ν > 0$ be a known constant (the viscosity).

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

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

and the initial condition:

$u(x,0) = u^\circ(x)$

$(x \mathcal{2} \mathbb{R}^3)$

(3)

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

[edit] Existence and smoothness of Navier-Stokes solutions on $\mathbb{R}^3$

Assume in addition that:

There are no external forces, i.e.:

$( \forall x \mathcal{2} \mathbb{R}^3 )( \forall t \ge 0 )\ f(x, t) = 0$

$u^\circ$ is bounded, i.e.:

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

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

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

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

Assume in addition that:

There are no external forces, i.e.:

$( \forall x \mathcal{2} \mathbb{R}^3 )( \forall t \ge 0 )\ f(x, t) = 0$

$u^\circ$ is periodic, i.e.:

$(\forall j \mathcal{2} {1, 2, 3})( \forall x \mathcal{2} \mathbb{R}^3 )\ u^\circ(x + e_j) = u^\circ(x)$ , where

e j

is the j^th unit vector in $\mathbb{R}^3$ .

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

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

[edit] Breakdown of Navier-Stokes solutions on $\mathbb{R}^3$

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