# Then we divide by c2 f x y g t 1 1 g t f x y c2

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Unformatted text preview: = ν (∆v)(x, y, z, t) − ∇p(x(t), y (t), z (t), t). Here the Laplace operator is applied componentwise and ν is a constant, called the viscosity of the ﬂuid. The equations used by Navier and Stokes to model an incompressible viscous ﬂuid (with ρ constant) are then ∇ · v = 0, ∂v 1 + (v · ∇)v = ν ∆v − ∇p. ∂t ρ It is remarkable that these equations, so easily expressed, are so diﬃcult to solve. Indeed, the Navier-Stokes equations form the basis for one of the seven Millenium Prize Problems, singled out by the Clay Mathematics Institute as central problems for mathematics at the turn of the century. If you can show that under reasonable initial conditions, the Navier-Stokes equations possess a unique well-behaved solution, you may be able to win one million dollars. To ﬁnd more details on the prize oﬀered for a solution, you can consult the web address: http://www.claymath.org/millennium/ Exercise: 5.4.1. Show that if the tension and density of a membrane are given by variab...
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## This document was uploaded on 01/12/2014.

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