Navier-Stokes existence and smoothness
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Problem description
Let [u(x, t) = (u_i(x, t))_1<=i<=3 \mathcal \mathbb^3] be the unknown velocity vector field, defined for positions [x \mathcal \mathbb^3] and times [t \ge 0] and let [p(x, t) \mathcal \mathbb] be the unknown pressure, defined likewise.
Suppose they satisfy the Navier-Stokes equations for incompressible fluids filling [\mathbb^3] which are given by [\forall i \mathcal :]
| [\frac u_i + \sum_^3 u_j \frac = \nu \Delta u_i - \frac + f_i(x, t)] | [(x \mathcal \mathbb^3, t \ge 0)] | (1) |
| [\operatorname\ u = \sum_^3 \frac = 0] | [(x \mathcal \mathbb^3, t \ge 0)] | (2) |
and the initial conditions:
| [u(x,0) = u^\circ(x)] | [(x \mathcal \mathbb^3)] | (3) |
Where the equations are to be solved for an unknown velocity vector (and unknown pressure p(x,t)) given by:
[u(x,t) = u_i(x,t)_ \mathcal \mathbb^n]
[f_i(x, t)] represent an external force (i.e. gravity) A $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute to whoever proves the following four statements about the Navier-Stokes equations:
- Existence and smoothness of Navier-Stokes Solutions on [\mathbb^3]
- Existence and smoothness of Navier-Stokes Solutions on [\mathbb^3]/[\mathbb^3]
- Breakdown of Navier-Stokes Solutions on [\mathbb^3]
- Breakdown of Navier-Stokes Solutions on [\mathbb^3]/[\mathbb^3]
Background
External links
- [The Clay Mathematics Institute's Navier-Stokes equation prize]
- *[Official statement of the problem]
- [QEDen] Millennium Prize Problems Wiki
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