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Navier-Stokes 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 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
Where Δ is the Laplacian in the space variables.
And the initial condition:
(A) Existence and smoothness of Navier-Stokes 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 Navier-Stokes 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 Navier-Stokes solutions onThere exists an and a divergence-free for which there are no and satisfying (1), (2), (3) and also having bounded energy, i.e.: (D) Breakdown of Navier-Stokes solutions onThere exists an and a divergence-free for which there are no and satisfying (1), (2), (3) and also having a periodic u, i.e.: BackgroundThe 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
This article contains public-domain material taken from QEDen.
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Navier-Stokes_existence_and_smoothness". A list of authors is available in Wikipedia. |