Navier-Stokes existence and smoothness

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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

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:

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.:

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

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.:

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

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.: