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Navier-Stokes_existence_and_smoothness

Navier-Stokes existence and smoothness

The Navier-Stokes equations, which describe the flow of ordinary fluids such as water, can be extremely complicated and difficult to solve. Numerical solutions to these equations are frequently used in many fields, such as aircraft design and weather forecasting. However, the theory behind these solutions is unsatisfactory. For the three dimensional system of equations, mathematicians have proved neither that smooth solutions always exist (existence), nor that if they do exist they do not contain any infinities or singularities (smoothness). Showing that solutions exist, and that they are smooth, is therefore called the Navier-Stokes existence and smoothness problem. A $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute to whomever proves any of the following statements about the Navier-Stokes equations.

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

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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 R³, 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 R³ 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)

Where $Δ$ is the Laplacian in the space variables.

$\operatorname{div}\ u = \sum_{j=1}^3 \frac{\partial u_j}{\partial x_j} = 0$

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

(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:

(A) 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.:

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

(B) 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 R³.

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.:

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

(C) 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.:

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

(D) Breakdown of Navier-Stokes solutions on $\mathbb{R}^3/\mathbb{Z}^3$

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

Background

The analogous problem for R² has already been solved positively (it is known that there are smooth solutions on R²). From the Clay math official problem description:

In two dimensions, the analogues of assertions (A) and (B) have been known for a long time (Ladyzhenskaya^[1]), both for the Navier-Stokes equations and the more difficult Euler equations. This gives no hint about the three–dimensional case, since the main difficulties are absent in two dimensions.

References

^ O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows (2nd edition), Gordon and Breach, 1969.

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