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NavierStokes existence and smoothness
Additional recommended knowledge
Problem descriptionLet be the unknown velocity vector field, defined for positions and times and let be the unknown pressure, defined likewise. Let be a known external force, again defined for positions and times . Also let be the known initial velocity vector field on R^{3}, which is divergencefree on C^{∞}. Finally, let ν > 0 be a known constant (the viscosity). Then the NavierStokes equations for incompressible viscous fluids filling R^{3} are given by
Where Δ is the Laplacian in the space variables.
And the initial condition:
(A) Existence and smoothness of NavierStokes solutions onAssume in addition that:
Then there exists and that satisfy (1), (2) and (3) as well as having bounded energy, i.e.: (B) Existence and smoothness of NavierStokes solutions onAssume in addition that:
Then there exists and that satisfy (1), (2) and (3) and have a periodic u, i.e.: (C) Breakdown of NavierStokes solutions onThere exists an and a divergencefree for which there are no and satisfying (1), (2), (3) and also having bounded energy, i.e.: (D) Breakdown of NavierStokes solutions onThere exists an and a divergencefree for which there are no and satisfying (1), (2), (3) and also having a periodic u, i.e.: BackgroundThe analogous problem for R^{2} has already been solved positively (it is known that there are smooth solutions on R^{2}). 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 NavierStokes 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
This article contains publicdomain material taken from QEDen.


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "NavierStokes_existence_and_smoothness". A list of authors is available in Wikipedia. 